TEL AVIV UNIVERSITY The Iby and Aladar Fleischman Faculty of Engineering

Analysis of Optimum

Detector of TCM in Phase

Noise Channels

A thesis submitted toward the degree of

Master of Science in Electrical and Electronic Engineering

by

Oded Bialer

This research was carried out in the School of Electrical Engineering

under the supervision of Dr. Dan Raphaeli

September 2008

TEL AVIV UNIVERSITY The Iby and Aladar Fleischman Faculty of Engineering

Analysis of Optimum

Detector of TCM in Phase

Noise Channels

A thesis submitted toward the degree of

Master of Science in Electrical and Electronic Engineering

by

Oded Bialer

This research was carried out in the School of Electrical Engineering

under the supervision of Dr. Dan Raphaeli

September 2008

Acknowledgments

I would like to express my sincere thanks to my mentor, Dr. Dan Raphaeli, whose knowledge

and wisdom greatly contributed to this research. Working with him was a unique experience,

which enriched me both personally and professionally.

I would also like to thank my dearest wife Orly, who supported and encouraged me all the

way.

I

Abstract

As communication extends to higher carrier frequencies, the phase noise problem becomes

more severe and conventional phase tracking methods become inadequate. Join phase symbol

MAP sequence estimation (JMAP) is a practical tracking method achieving near optimal

performance. Conventionally the phase noise is modeled as a Weiner process resulting in a

relatively simple JMAP decoder. However, in many practical cases the Weiner model does

not model accurately the phase noise of the RF synthesizer. In such cases, implementing a

JMAP decoder matched to the actual phase noise is highly complex and hence impractical. A

practical solution that usually gives good performance is to use a JMAP decoder designed

based on the assumption that the phase noise is a Weiner process while the actual phase noise

is different i.e. decoder mismatch.

An analytical tool for performance evaluation of coded JMAP is not available. In prior

researches, analytical expressions that approximate the BER for uncoded JMAP were

developed by methods which are not suitable for coded JMAP. Furthermore, the phase noise

was limited to the Weiner model and the JMAP decoder was matched.

In this thesis, we developed a novel closed form analytical expression which approximates the

bit error rate of the JMAP decoder for Trellis Code Modulation with MPSK modulation, any

arbitrary phase noise model (i.e. not limited to Weiner process) and either a matched or

mismatched decoder.

The analytical expression showed to be tight and efficient for MPSK constellations with M>2

and code rates equal or greater then 0.5.

The developed tool will enable designers to take into consideration the influences of the code

characteristics, decoder implementation and RF synthesizer phase noise on the receiver

performance. It further enables joint optimization of the synthesizer, the code and the decoder

for achieving lowest error rate or other design targets.

II

III

Table of Contents

1 Introduction.................................................................................................................... 1

2 Background.................................................................................................................... 5

2.1 Phase Noise............................................................................................................ 5

2.2 System Model....................................................................................................... 11

2.3 Joint Coded Phase MAP (JMAP).......................................................................... 12

3 Literature Survey.......................................................................................................... 16

3.1 JMAP Implementation.......................................................................................... 16

3.2 JMAP Performance Analysis ................................................................................ 19

3.2.1 Jackson and Omura [6,17,18,19] ................................................................... 19

3.2.2 Ungerboeck [7] ............................................................................................. 23

3.2.3 Robinson & Meer [20] .................................................................................. 27

4 Analysis of JMAP Error Rate ....................................................................................... 30

4.1 Introduction.......................................................................................................... 30

4.2 Pairwise Error Analysis ........................................................................................ 32

4.2.1 Pairwise Error Probability Approximation .................................................... 38

4.2.2 Matched Decoder Pairwise Error Probability ................................................ 45

4.2.3 Mismatched Decoder Pairwise Error Probability........................................... 52

4.3 Trellis Code Error Rate Analysis .......................................................................... 53

5 Analysis of the Phase Estimation in the JMAP ............................................................. 62

6 Results and Discussion................................................................................................. 66

7 Summary...................................................................................................................... 83

Appendix A ......................................................................................................................... 85

Bibliography........................................................................................................................ 87

IV

List of Symbols

n - Discrete time index

nθ - Phase noise at time n

θ - Sequence of phase noise values

θ - JMAP estimation of the phase noise sequence

nθ - JMAP estimation of the phase noise at time n

0ˆnθ - The JMAP phase noise estimation attaining the minimum accumulated metric

for the correct code sequence

1ˆnθ - The JMAP phase noise estimation attaining the minimum accumulated metric

for the incorrect code sequence

( )zΘ - The Z transform of nθ

nξ - Complex thermal noise

2

ξσ - Variance of real and imaginary components of nξ

m

na - A Trellis coded MPSK symbol, m is the codeword index and n is the time index

ma - The m-th code word symbol sequence

ma - JMAP m-th codeword hypothesis

m

na - The symbol at time n of the JMAP m-th codeword hypothesis

0a - The JMAP correct codeword hypothesis

1a - The JMAP incorrect codeword hypothesis

m

nφ - The phase of m

na

mφ - The phase of ma

mφ - The phase of ma

m

nφ - The phase of m

na

0φ - The phase of 0a

1φ - The phase of 1a

nv - White Gaussian noise which is the input to phase noise ARMA filter

2

vσ - The variance of nv

nu - Thermal noise in the equivalent model

2

uσ - The variance of un which equal to 2

ξσ

V

nx - Channel output at base band system model (un-approximated model)

x - Sequence of channel outputs (sequence of xn)

ny - Equivalent channel model output at time n

y - Sequence of equivalent channel model outputs

( )zY - The Z transform of yn

nη - Total noise in equivalent channel model (sum of thermal and phase noises)

( )ωηηjeS - Spectrum of

nη

( )ωjeA - Used for the representation of ( )ωηηjeS such that ( ) ( )2

2 ωωηη σ j

u

j eAeS =

( )zH - A filter which in the matched decoder case whitens the noise spectrum ( )ωηηjeS ,

in the mismatch decoder H(z) whitens the noise spectrum which the JMAP

decoder metric is matched to

G(z) - Phase noise ARMA filter, the phase noise is obtained by filtering vn with G(z)

Sθθ(z) - Phase noise spectrum

β -Weighting factor used in the accumulated metric of JMAP decoder matched to

Weiner phase noise

λ - The pole of the whitening filter for JMAP matched to Weiner model (inverse

monotonic function of β)

( )maΓ - Likelihood of the m-th

codeword, i.e. the JMAP decoder minimum metric for

the m-th

codeword

( )ma~Γ - Approximation of ( )maΓ

( )kmk aaaP /2 → - The probability that the decoder will choose the codeword am given that

the transmitted codeword was ak (pairwise error probability)

( )eP2 - The probability that the decoder will decide on an error sequence with phase

difference e from the phase of the transmitted codeword

ne - The difference at time n, between the phase of the correct and incorrect code

sequences

e - Sequence of en

( )zE - The Z transform of en

ne~ - The result of filtering en with H(z)

m

nφ~

- The result of filtering m

nφ with H(z)

ny~ - The result of filtering yn with H(z)

nη~ - The result of filtering nη with H(z)

( )zS - The JMAP phase estimation filter

VI

List of Figures

Figure 1: Band-Pass Communication System ................................................................6

Figure 2: Base band digital system ................................................................................6

Figure 3: Phase noise affect on 8PSK modulated symbols .............................................7

Figure 4: Typical oscillator phase noise power spectrum density...................................8

Figure 5: Frequency Synthesizer scheme.......................................................................9

Figure 6: Synthesizer output phase noise.....................................................................10

Figure 7: Trellis code ..................................................................................................14

Figure 8: Phase symbol JMAP trellis single code branch.............................................15

Figure 9: PSP-PLL illustration ....................................................................................18

Figure 10 Illustration of Omura & Jacksons union bound approach.............................20

Figure 11: Pairwise error between 2 code sequences ...................................................31

Figure 12: Equivalent channel system .........................................................................33

Figure 13: Equivalent communication model ..............................................................35

Figure 14: Generelized pairwise error expression block diagram.................................37

Figure 15: Arc approximation of a chord.....................................................................40

Figure 16: Approximation insight................................................................................41

Figure 17: Matched JMAP decoder pairwise error expression block diagram ..............46

Figure 18: Equivalent model - matched channel ..........................................................46

Figure 19: Filtered error event.....................................................................................49

Figure 20: Illustration of independent error event ........................................................58

Figure 21: Union bound error search algorithm ...........................................................61

Figure 22: Illustration of JMAP phase noise estimated ................................................64

Figure 23: Decision Directed 1st Order PLL ................................................................65

Figure 24: Phase noise spectrums................................................................................67

Figure 25: Pairwise error probability, Weiner phase noise ...........................................69

Figure 26: Pairwise error probability, non Weiner phase noise ....................................70

Figure 27: Code A trellis .............................................................................................72

Figure 28: Code A symbol mapping ............................................................................72

Figure 29: Code B encoder..........................................................................................73

Figure 30: Code B symbol mapping ............................................................................73

Figure 31: Code C 4D encoder ....................................................................................74

Figure 32: Code C 2D symbol mapping ......................................................................74

Figure 33: Code A performance - Weiner phase noise & matched JMAP decoder .......77

Figure 34: Code B performance - Weiner phase noise & matched JMAP decoder .......78

Figure 35: Code C performance - Weiner phase noise & matched JMAP decoder .......79

Figure 36: Code A performance - non Weiner phase noise & mismatch JMAP............80

Figure 37: Code B performance - non Weiner phase noise & mismatch JMAP............81

Figure 38: Code C performance - non Weiner phase noise & mismatch JMAP............82

1 Introduction

Phase noise is encountered in communication systems using a carrier frequency. The phase of

the local oscillators used for up and down conversion of the base-band to pass-band and pass-

band to base-band signal has inherently phase noise i.e. a random fluctuating in the phase. As

the carrier frequency increases the phase noise intensifies. The effect of phase noise on the

complex base band signal is a random rotation of the complex symbol. Thus a modulated

symbol affected by phase noise will appear at the receiver rotated (in the 2D constellation).

The effect increases the probability of an error in the symbol detection.

There are many applications where the phase noise is a limiting factor in performance or

modem cost. For example: satellite communication in high frequencies (e.g. Ku, Ka bands),

60 GHz short range links, microwave links (e.g. 23 GHz), coherent optical communications.

Noncoherent methods are less sensitive to phase noise however, they suffer from a

performance degradation compared to coherent methods. Examples are differential PSK

(DPSK), differential QPSK and multiple-symbol noncoherent detection [1], [2] and [3].

Coherent methods have simpler detectors but require phase estimation or tracking.

The conventional method to track the channel phase is the Phase Locked Loop (PLL). It is

easy to implement but fails to track the channel when the channel has relatively high phase

noise coupled with low signal to noise ration (SNR). PLL is not suitable for coded systems in

high phase noise. The decoder requires a delay in decisions making. Hence, updating the PLL

with relatively reliable decisions of the decoder requires a delay in the loop which imposes

narrowing the PLL loop filter, causing the PLL to fail in tracking of large phase fluctuations.

On the other hand, updating the PLL with the tentative zero delayed decisions (i.e. not from

the decoder output) will increase the probability of errors in the decision making. A possible

solution is Per Surviving Processing (PSP) PLL [4]. In a PSP PLL receiver a PLL is kept for

each Trellis code state. Each PLL is updated every symbol interval according to the best

2

survivor entering the state at that time. The PSP-PLL performs better then conventional PLL

with a significant increase in complexity, but still is degraded in respect to the optimal

receiver.

Optimum receiver for TCM over the phase noise channel in terms of bit error probability is

the Maximum Apostriori Probability (MAP) estimator of the transmitted information symbol

given the received sequence, i.e. choosing the transmitted symbol which maximizes the

probability that it was transmitted given the received sequence.

Though the optimal receiver above can be implemented by the BCJR algorithm, joint phase

and symbol MAP sequence estimation (JMAP) achieves practically the same performance but

with lower complexity. In this case the detector chooses the joint phase and symbol sequence

that assigns the maximum value to the joint apostriori probability i.e. assign the maximum to

the probability of the transmitted symbol sequence and channel phase noise sequence given

the received sequence. The phase state is a continuous variable, thus it is approximated by a

discrete variable, and then the JMAP receiver can be implemented by the well known Viterbi

Algorithm. The combined effect of the code and the phase noise as a joint trellis. The JMAP

outperforms the PSP PLL at the expense of increased complexity. Today's technology makes

the JMAP practical for commercial applications.

Traditionally, the phase noise is modeled as a discrete Weiner process, i.e. independent

Gaussian increments. However, in many practical cases the Weiner model does not model

accurately the phase noise of the RF synthesizer [5,9] which produces the local oscillator

frequency used for up and down conversion of base-band to pass-band and pass-band to base

band.

In such cases, implementing a JMAP receiver matched to the actual phase noise is highly

complex and hence impractical. A practical solution that usually gives good performance is to

use a JMAP receiver designed based on the assumption that the phase noise is a Weiner

process while the actual phase noise is different i.e. channel mismatch.

3

There is a lack of analytical tools to evaluate the bit error rate (BER) of coded JMAP (i.e.

joint MAP of phase and coded symbol sequences) for either the matched or mismatched case.

Such tools are vital since they enable the analytical comparison of the matched and

mismatched JMAP detector performance for various codes and phase noise models. It also

enables to compares their performance to other phase tracking methods. Furthermore, if the

BER analytical expression takes in account the phase noise model, JMAP receiver

implementation (which varies in the mismatch case) and specific code characteristics then it

can be used as a powerful tool to jointly optimize them (RF synthesizer, JMAP and code) in

order to achieve lowest error rate.

Unlike the coherent trellis decoding, in the joint code phase trellis there are several paths with

correct code symbols, each with different phase states. Every error code sequence has also

several paths, each with different combination of phase states. The number of correct and

incorrect paths grows as the number of phase states increases. Hence finding the error rate of

such joint decoder is not trivial.

The straight forward approach taken by Jackson and Omura [6,17] was to apply a union

bound on the pairwise error probabilities to all error paths. The result is a very loose upper

bound. Ungerboeck [7] developed an analytical expression which approximated the pairwise

error probability between two symbol sequences. However, the expression is not general in

the sense that obtaining the pairwise error probability for each different error event requires a

specific analytical calculation and hence cannot serve as a practical tool for coded error rate

analysis where there are many different error events.

In the work of both Ungerboeck and Omura the phase noise was assumed as a Weiner process

and the JMAP was matched to the channel phase noise.

In this research, a different approach is taken which yields a tight approximation to the

performance of the JMAP for Trellis Code Modulation (TCM) with MPSK modulation for

both matched and mismatched decoders. The outline of the derivation is as follows.

4

First, the JMAP metric for any arbitrary phase noise model is derived and then a first order

linear approximation is applied to the nonlinear term of the metric. The approximation is

usually sufficiently accurate for TCM codes with rate equal or greater then 1/2 and MPSK

constellations with M>2. Next, a communication system is introduced, which its JMAP

decoder metric is equal to the approximated JMAP metric and hence serves as an equivalent

model for the purpose of error rate analysis. From the equivalent model a closed-form

approximation to the pairwise error probability is derived. The resulting pairwise error

probability expression can be used for any arbitrary channel phase noise and for either a

matched or mismatched JMAP decoder. Finally, the union bound is applied on the pairwise

errors. The result is a novel generalized analytical expression which approximates the bit error

rate of a coded JMAP decoder suited for any arbitrary phase noise spectrum and for either

matched or mismatched decoder. The resulting expression is simple to use, affords important

insight to the problem and enables to jointly optimize the code, synthesizer components

(modifying the phase noise spectrum) and JMAP decoder metric.

5

2 Background

In this chapter we provide an overview on relevant background metrical and define the system

model that is used throughout this book.

2.1 Phase Noise

A band-pass communication system is presented in Figure 1 and described briefly in the

following (for detailed description see [8 chapter 4]). The transmitted bits are modulated in

base band and then up converted to the carrier frequency. Denote by a(t) (t is the time index)

the complex modulated signal in base band. The base band signal is up converted to the

carrier frequency by multiplying it with the local oscillator outputs ( )( )tt mc θω +cos and

( )( )tt mc θω +sin . The phase noise unit is radian. Let p(t) be the transmitted pass band signal

thus ( ) ( ) ( )( ) ( ) ( )( )tttatttatp mcmc θωθω +++= sinImcosRe , where cc fπω 2= , fc is the

carrier frequency, Re is the real operation and θm(t) is the transmitter local oscillator phase

noise. The transmitted signal passes threw a channel followed by additive noise. At the

receiver, the received signal, denoted by r(t) is down converted to base band by multiplying it

with the receivers local oscillator outputs ( )( )tt dc θω +cos and ( )( )tt dc θω +− sin followed by

low pass filtering to yield the base band real and imaginary components. The received base

band signal is denoted by x(t) and the receivers local oscillator phase noise by θd(t). The

resulting base band signal is then demodulated to yield the detected bits. Let θ(t)= θm(t)+ θd(t)

be the total phase noise in the system (from transmitter and receiver local oscillator) and ξ (t)

be the additive noise after applying down conversion. Assuming that the channel has no

dispersion i.e. a single delta function (all pass channel) then the received complex base band

signal is ( ) ( ) ( )( ) ( )tetatxttj c ξθω += +

.

6

Base Band

Modulator

Local

Oscillator

( )( )tt tc θω +cos

( )( )tt tc θω +sin

Local

Oscillator

LPF

LPF

( )( )tt dc θω +cos

( )( )tt dc θω +− sin

Channel

Noise

Base Band

Demodulator

( ) taRe

( ) taIm

( )tp ( )tr

( ) txRe

( ) txIm

Figure 1: Band-Pass Communication System

The equivalent digital base band system is presented in Figure 2. The transmitted symbol is

m

na , where n is the symbol index and in case that the symbol is coded then m represents the

codeword index. The transmitted symbol is rotated by the phase noise θn (the sum of

transmitter and receiver phase noise) followed by additive white complex Gaussian noise nξ .

nx

njeθ

nξ

m

na

Figure 2: Base band digital system

The phase noise θn is a random process causing the demodulated symbol to suffer from

random rotational fluctuations. Figure 3 presents the received complex symbols (xn) of 8PSK

modulated symbols that experience phase noise. It is shown that the phase noise rotates the

symbols towards each other and hence increasing the symbol detection error rate.

7

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

Real

Imag

Received symbols

Transmitted symbols

Figure 3: Phase noise affect on 8PSK modulated symbols

Denote by ( )fLθ the phase noise power spectrum density. Figure 4 presents a typical

spectrum of an oscillator output phase noise. Essentially the phase noise is assembled from

several noise sources. In the spectrum each noise source is dominant in different frequency

regions as can be seen in Figure 4. The noise sources are as follows:

a. Thermal noise - white Gaussian noise (f 0 region in Figure 4)

b. Fliker noise - noise from transistors with spectrum proportional to f -1

.

c. Integrated white noise - white Gaussian noise which changes the frequency of the

oscillator causes phase noise which is integrating white noise (since phase is the

integral of frequency). The spectrum is proportional to f -2

.

d. Integrated Fliker noise - Fliker noise which changes the frequency of the oscillator

causes phase noise which is integrating Fliker noise. The spectrum is proportional

to f -3

.

8

The most dominant phase noise source is integrated white Gaussian noise with spectrum

proportional to f -2

.

f

f -3

f -2

f -1

f 0

f -3

f -2

f -1

f 0

( )fLθ

fc

Figure 4: Typical oscillator phase noise power spectrum density

Consider a general cyclic wave such as sin or cosine with phase noise e.g. ( )( )ttfc θπ +2sin

where the carrier frequency is fc=1/Tc, Tc is the cycle time and θ(t) is the phase noise. The

phase noise origin is random changes in the cycle time of the wave. Denote by τ(t) the change

in the cycle time measured in seconds. The relationship between the noise causing the change

in cycle time τ(t) to the change in the wave phase θ(t) (measured in radians) is given

by ( ) ( )π

τθ 2

cT

tt = . Since Tc=1/fc it follows that the phase noise is ( ) ( ) cftt πτθ 2= . From this

relationship it is clear that the phase noise increases as the carrier frequency grows.

Multiplying the carrier frequency by N causes the phase noise spectrum given in Figure 4 to

increase by 20log(N). Thus, the phase noise problem becomes more sever in high carrier

communication systems.

In practice, the local oscillator frequency is produced by a frequency synthesizer. A frequency

synthesizer generates a high frequency signal from an input low frequency oscillator. The

9

synthesizer is implemented with a phase lock loop as presented in Figure 5 [9]. The reference

signal is the low frequency oscillator and the synthesizer output frequency (at the VCO

output) is R/N times the reference frequency. The phase noise at the synthesizer output

consists of two noise sources. The reference oscillator phase noise and the VCO phase noise.

Denote by BL the PLL closed loop low pass filter bandwidth. The synthesizer phase noise

spectrum is presented in Figure 6 [ref].

It is shown that in the range of LB± around the synthesizer output frequency (fc) the

synthesizer output phase noise spectrum is affected by the reference oscillator phase noise

after being increased by 20log(R/N) dB while in the rest of the spectrum range the phase noise

is the VCO phase noise. More specific, the phase noise at the synthesizer output consists of

the reference phase noise (increased by 20log(R/N) dB) filtered by the PLL closed loop low

pass filter and from the VCO phase noise filtered with a high pass filter which is the

complementary of the PLL closed loop filter (1-LPF).

Figure 5: Frequency Synthesizer scheme

10

( )fLθ

Figure 6: Synthesizer output phase noise

Since the oscillator phase noise presented in Figure 4 is mostly dominated by the f -2

region, it

is common to model the phase noise power spectrum as proportional to f -2

. In the digital base

band equivalent model such spectrum is obtained by modeling the phase noise as a Weiner

process. The Weiner process is independent Gaussian increments i.e. nnn v+= −1θθ , where n

is the time index and vn is zero mean white Gaussian noise. However, from Figure 6 we see

that even if each, the reference and VCO phase noise sources have a power spectrum which is

proportional to f -2

, the phase noise at the synthesizer output is different. If the loop bandwidth

(BL) is small or the reference phase noise is negligible relatively to the VCO phase noise then

the phase noise spectrum can be approximated as being proportional to f -2

. However, if this is

not the case, then the specific phase noise spectrum should be considered and modeling the

phase noise as a Weiner process is inappropriate.

Furthermore, Figure 6 represents the phase noise of an ideal synthesizer, in practical cases the

phase noise at the synthesizer output may have other dominant noise components such as

spurs in the spectrum.

11

2.2 System Model

The discrete time model of a communication channel in this research is presented in Figure 2.

The received symbol (xn) is given byn

jm

nnneax ξθ += , where n is the time index,

nθ is the

phase noise and nξ is a complex white Gaussian thermal noise with zero mean and

( )( ) ( )( ) 222ImRe ξσξξ == nn EE . Denote by m

na a Trellis coded MPSK symbol, where m is

the codeword index. Let m

nφ be the phase of m

na .

It is assumed that all code sequences are transmitted with equal probability and m

nφ , nξ and

nθ are independent. The phase noise nθ is Wide Sense Stationary Gaussian process, generated

by the following model. Let vn be a white Gaussian noise process with zero mean and

variance 2

vσ then nθ is generated by the autoregressive moving average model (ARMA)

∑∑−

=−−

=

+=1

01

K

k

knkmn

M

m

mn vpq θθ , where qm , pk are the model coefficients. The phase noise

spectrum is therefore ( ) ( ) 22zGzS vσθθ = where ( )

∑

∑−

=

−

=

−

=1

0

0

K

k

k

k

M

m

m

m

zp

zq

zG .

Throughout this work the time index is omitted when referring to a vector of time elements.

For example, we denote by θ the phase noise sequence and by am the m-th code word symbol

sequence.

Throughout this work the notation x is used to represent the estimation of the variable x. Also

the notation X(z) represent the Z transform of the time series xn.

12

2.3 Joint Coded Phase MAP (JMAP)

The joint code-phase maximum apostriori probability (MAP) estimator is defined as

( )xaPm

m|ˆ,ˆmax ˆ,

θθ

i.e. the phase and code symbol sequences which assign the maximum

value to the joint apostriori probability, where x is the received symbol sequence. Using

Bayes’ rule and assuming that all sequences am are a priori equally probable, we have

( ) ( ) ( )θθθθθ

ˆˆ,ˆ|max|ˆ,ˆmax ˆ,ˆ,PaxPxaP

m

m

m

m∝ (1)

Since nξ is white Gaussian noise then

( )[ ] ∑ −=−n

jm

nn

u

m neaxaxPθ

σθ

ˆ

2ˆ

2

1ˆ,ˆ|ln (2)

The summation in (2) is over the codeword length. It follows from (1) and (2) that

( ) ( )[ ]∑ −−=n

jm

nn

u

m

m

mPeaxxaP n θ

σθ θ

θθˆlnˆ

2

1min|ˆ,ˆmax

2ˆ

2ˆ,ˆ, (3)

The sequences θ,ˆ ma that minimize the metric in (3) are determined by applying the Viterbi

algorithm [8 chapter 8-2-2]. Each state in the trellis consists of a code state and quantized

value of the phase noise state variables according to the ARMA model (see section 2.2) i.e.

MnnnKnnn vvv −−−−−−− ,..,,,,.., 21)1(21 θθθ . The branch metric is given by

( )[ ]MnnnKnnnn

jm

nn

u

vvvPeax n

−−−−−−−−− ˆ,..,ˆ,ˆˆ,..,ˆ,ˆ/ˆlnˆ2

121),1(21

2ˆ

2θθθθ

σθ

and the accumulated metric

of the pair of sequences θ,ˆ ma is given by

( )[ ]∑ −−n

jm

nn

u

Peax n θσ

θ ˆlnˆ2

1 2ˆ

2 (4)

Unless K+M is very small, implementing a JMAP Viterbi decoder for a ARMA phase noise

model is impractical. However it is important to derive the generalized JMAP metric in the

context of a JMAP decoder unmatched to the channel phase noise as will be discussed later on

in this work.

13

In the following of this section, the phase noise is modeled as a Weiner process. In this case

we can write

( )[ ] ( )[ ] ( )∑ ∑ −

−

−∝−=−

n n v

nnnnPP

2

2

11

2

ˆˆˆ/ˆlnˆln

σθθ

θθθ (5)

Substituting (5) into (4) results in the following Viterbi algorithm branch metric

( )21

2ˆ ˆˆˆ −−+− nn

jm

nnneax θθβ θ

and the accumulated metric is given by

( )∑ −−+−n

nn

jm

nnneax

2

1

2ˆ ˆˆˆ θθβ θ (6)

where 2

2

u

v

σσ

β = .

The continuous phase nθ is quantized to a finite set in the range 0:2π. Each Trellis state

consists of a pair of code state and a quantized phase value (phase state) [11]. If the number of

code and phase states is Mc and Mp respectively then the total number of states is Mc*Mp.

The Viterbi algorithm for simultaneous phase and symbol decoding consists simply of an

algorithm which determines survivor phase-symbol sequences terminating at each possible

phase-symbol pair. One of these surviving sequences is ultimately decoded as the maximum

likelihood phase-symbol sequence. Figure 7 illustrates a standard coherent Trellis

containing only code states and Figure 8 presents the corresponding phase-symbol JMAP

Trellis where there are four different phase states. It is shown that the branch connecting code

state s0 and s1 in Figure 7 results in 16 branches connecting pairs of phase and code

states in Figure 8.

The complexity of the phase-symbol JMAP decoder is determined by the number of

calculations of the metric in (6) and the number of standard Viterbi algorithm add compare

select operations per each received symbol xn. Denote by Mb the number of different code

branches exiting each code state in the original Trellis code (i.e. the Trellis before adding

14

phase states). In a brute force implementation, the number of calculations of the metric in (6)

and the number of add compare select operations are Mb*Mp*Mc.

Efficient implementation can significantly reduce complexity. For all possible phase

transaction, the phase differences term ( )21

ˆˆ−− nn θθ in (6) can be pre-computed and stored in

memory as a look up table.

If the number of phase states is a product of the number of code states then the number of

different squared Euclidean distances 2ˆ

ˆ njm

nn eaxθ− in (6) required to compute per each

received symbol is reduced to Mp. These Mp distances are ( ) 22 kMj

npex

π− where

k=0,1,Mp-1. Hence, for each new received symbol, only Mp squared Euclidean distances are

computed and stored in memory as a look up table. Per each received symbol, it is required to

compute (6) for all Mb*Mp*Mc branches. For each branch, the appropriate squared Euclidean

distance and the phase square difference are simply read from the look up table. The

complexity per each received symbol is reduced to Mp calculations of the squared Euclidean

distances and Mb*Mp*Mc add compare select operations (reading from lookup table is

considered to have relatively negligible complexity).

Figure 7: Trellis code

15

code state

phase state

0,0S

2,0S

,0S

2

3,0S

2

3,1S

0,1S

2,1S

,1S

Figure 8: Phase symbol JMAP trellis single code branch

16

3 Literature Survey

In section 3.1 a brief overview will be given on published papers which discuss JMAP

implementation as well as various suboptimal decoder implementations. In section 3.2 we

describe prior published researches of JMAP performance analysis.

3.1 JMAP Implementation

In [12] the transmitted signal was an un-modulated carrier and the phase noise was modeled

as a Weiner process. The MAP phase sequence estimator was implemented by the Viterbi

algorithm where the trellis contains only phase states (no code/symbol states) and the

accumulated metric is as given in (6). The authors compared the mean square error (MSE),

obtained from simulations, of the MAP phase estimation to the PLL phase estimation. For

relatively low phase noise variance they have showed that the MAP MSE is better by 2 dB.

More dramatic performance gain may be achieved for larger phase noise variances. In severe

phase noise conditions and low SNR there is a significant probability that the PLL which is

updated with tentative decisions will lose lock resulting in an extreme increase in the mean

square error, a phenomena that does not occur in the MAP phase estimation.

In [11] the joint phase and symbol MAP sequence estimator was developed. The transmitted

symbols where uncoded and linear modulation was assumed i.e. M-array phase shift keying

(M-PSK) and M-array quadrature amplitude shift keying (M-QASK).

The phase noise was modeled as a random walk, each phase increment is an independent

identically distributed random variable which has a general symmetric probability density

function. For simulation results the phase increment was chosen to have zero mean Gaussian

distribution i.e. Weiner process. The JMAP was implemented with the Viterbi algorithm as

described in chapter 2.3.

17

The authors studied the effect of the Viterbi algorithm back tracking length on the

performance and claimed that infinity back tracking length offers 6 dB improvement over

zero length and back tracking length of 10 offers an improvement of 4-5 dB compared to zero

length.

The only parameter required in order to proceed with the Viterbi algorithm is 22

uv σσβ =

which is the ratio of phase variance to additive noise variance used in the JMAP metric in (6).

The Viterbi algorithm sensitivity to the imperfect knowledge of β was tested in simulation and

showed that performance is not appreciable degraded by an error of 6 dB in β.

Simulation results of the JMAP decoder where compared to other suboptimal phase tracking

methods for BPSK, 8PSK and 16-QASK modulations. The JMAP decoder outperformed the

other methods, its supremacy is due to the fact that it precludes the occurrence of error bursts

since even in case of decision errors the JMAP does not lose lock. Thus the JMAP decoder

can be used even at high error probabilities on the order of 10-2

.

In [13], [14] and [15] the joint data and phase MAP receiver was driven for continuous phase

modulation (CPM) and its Viterbi algorithm implementation is described. Various suboptimal

methods with reduced complexity are suggested and their simulation results are compared to

those of the optimal JMAP receiver.

In [16] an interesting suboptimal solution for simultaneously data decoding and phase

tracking was introduced. Instead of minimizing the JMAP metric with respect to both the

sequence of symbols and phase states, it was suggested to first minimize the JMAP metric

with respect to the current phase value at each step in the sequence and then minimize the

metric with respect to the data symbol sequence by the use of dynamic programming. The

phase maximization is decoupled to a step by step one by dropping all the possible values of

phase states but one for each symbol state. In practice, the Viterbi algorithm is implemented

18

with a Trellis that consist of only symbol states (no phase states). A different PLL is kept per

each symbol state. Each PLL is updated every symbol interval according to the best survivor

entering the state at that time i.e. the survivor with minimal accumulated metric. In [16] the

symbols were assumed to be uncoded. The same method was extended to joint TCM

decoding and phase tracking as part of the Per Survivor Processing (PSP) method presented in

[4] where a PLL is kept per each code state. Figure 9 illustrates the implementation of PSP-

PLL for four code states and binary transmission. For example, consider state 1 after the first

trellis interval. Since the state survivor is the branch exiting state 2 then after the first trellis

interval, the PLL of state 1 is replaced with the PLL of state 2 from the previous trellis

interval after being updated according to the coded symbol of the survivor branch (connecting

state 2 from previous interval to state 1 of current interval).

The conventional approach of separating the PLL from the decoder, i.e. using a single PLL

that is updated from the decoders delayed decisions requires a delay in the loop which

requires to narrow the loop filter enabling the PLL to track only slow varying phase noise.

Hence, PSP-PLL is a good solution which enables to update the PLL with the decoder output

with zero delay in the loop. Unlike JMAP, the PSP-PLL suffers from error propagation in

case that the decoder chooses an incorrect code sequence. Thus, JMAP decoder outperforms

the PSP-PLL decoder but at the expense of increased complexity.

Figure 9: PSP-PLL illustration

19

3.2 JMAP Performance Analysis

In section 3.2.1 and 3.2.2 we describe two prior researches that develop an analytical

expression for the BER of a JMAP decoder when the transmitted symbols are uncoded. We

will show that the analytical expression given by Jackson and Omura in section 3.2.1 results

in a very loose upper bound and that the result of Ungerboeck in section 3.2.2 is only

applicable to the uncoded case. In section 3.2.3 we discuss the research of Robinson and Meer

who addressed the case that the transmitted signal is an un-modulated carrier and the receiver

performs MAP phase noise estimation. They have developed an analytical expression that

lower bounds the mean square error of the phase estimation. The lower bound showed to be

tight only fort high SNR.

3.2.1 Jackson and Omura [6,17,18,19]

In [6] and [17] Jackson and Omura developed an analytical expression for the bit error rate

(BER) of uncoded phase-symbol JMAP with continuous phase modulation (CPM). With

minor modifications, their method is extended in this section to support also coded JMAP

with MPAK and QAM modulations. The continuous phase is quantized into M values in the

range 0:2π i.e. phase states. The authors used the assumption that the random phase varies

slowly and hence the phase state jumps during a symbol interval are limited to adjacent phase

state i.e.

−∈− −

MMnn

ππθθ

2,0,

2ˆˆ1 . Let PJ be the probability of a phase state change between

two consecutive symbols then Jnnnn PM

PM

P =

+=−=

−=− −−

πθθ

πθθ

2ˆˆ2ˆˆ11 and

( ) Jnn PP 210ˆˆ1 −==− −θθ .

The bit error probability is obtained by applying the union bound on pairwise error

probabilities.

20

Error events are considered between pairs of code and phase sequences. The error event

begins when code states diverge and terminates when both pair of code and phase states

remerge for the first time. At the time of code divergence, the error path may have already

diverged in the phase states. Paths that diverge and remerge only in the phase states are not

considered as error events. An illustration is given in Figure 10 where for each code states

there are four phase states. The blue path is the correct code and phase pair and the green

paths are potential error code phase pairs. The pairwise error between the correct path to each

potential is calculated and added to the union bound. This process is repeated for each

different correct code phase pair.

Figure 10: Illustration of Omura & Jacksons union bound approach

Let Yk,t

=ak,θ

t and Y

m,i=a

m,θ

i represent the correct and incorrect pair of code and phase

sequences respectively, where k and m represent the code sequence indexes and t and i

represent the phase sequence indexes. Denote by ( )tkY ,Ψ and ( )imY ,Ψ the JMAP accumulated

metric (in the Viterbi algorithm) for Yk,t

and Ym,i

respectively. A pairwise error is considered

when at the time in which Yk,t

and Ym,i

remerge to the same pair of code-phase states, the

accumulated metric of the incorrect code-phase pair (Ym,i

) is lower then the correct pair (Yk,t

)

21

i.e the pairwise error probability is ( ) ( ) ( ) tkimtktkimtk YYYYYYP ,,,,,,

2 /Pr/ Ψ>Ψ=→ . By

substituting the general accumulated metric given in (4) for the correct and incorrect sequence

in ( )tkimtk YYYP ,,,

2 /→ we arrive at

( )=→ tkimtk YYYP ,,,

2 /

( )[ ] ( )[ ]

−−>−− ∑∑n

ijm

nn

un

tjk

nn

u

PeaxPeaxin

tn θ

σθ

σθθ ˆlnˆ

2

1ˆlnˆ2

1Pr

2ˆ

2

2ˆ

2 (7)

Since ak and θ

t are the correct sequence then n

jk

nn

tneax ξθ += . By applying simple algebra (7)

becomes

( )=→ tkimtk YYYP ,,,

2 /

( ) ( )[ ] ( )[ ]( )

−+−>

− ∑∑

n

it

u

jk

n

jm

n

n

jk

n

jm

nn PPeaeaeaeatn

in

tn

in θθσξ θθθθ ˆlnˆlnˆˆ

2

1ˆˆRePr

22ˆˆ*ˆˆ

(8)

where X* is the conjugate of X and ReX is the real part of X. Since

nξ is zero mean white

complex Gaussian noise with ( )( ) ( )( ) 222ImRe unn EE σξξ == then

( )∑

−

n

jk

n

jm

nn

tn

in eaea

*ˆˆˆˆRe

θθξ is also a zero mean Gaussian random variable with variance of

2ˆˆ2 ˆˆtn

in jk

n

jm

nu eaeaθθσ − . The term on the right hand of the inequality in (8) is a deterministic

value and hence we arrive at the following pairwise error probability

( )( )[ ] ( )[ ]( )

−

−+−

=→∑

2ˆˆ2

2

22ˆˆ

,,,

2

ˆˆ

ˆlnˆlnˆˆ2

1

/tn

in

tn

in

jk

n

jm

nu

n

it

u

jk

n

jm

n

tkimtk

eaea

PPeaea

QYYYPθθ

θθ

σ

θθσ

where Q(x) is the Gaussian cumulative distribution function.

Let Ω(s) be the set of codewords which are initiated from state s. Let Ωe(s,k) be the set of all

error codewords per transmitted codeword with index k which belongs to the set Ω(s). The bit

error probability is upper bounded by

22

( ) ( ) ( ) ( )( )( )

∑ ∑ ∑Ω∈ Ω∈

→→≤s skt ksmi

imtk

b

tkimtktk

bite

YYNYYYPYsP, ,,

,,,,,

2

, /PrPr (9)

where Pr(s) is the probability of the encoder being at code state s, ( )tkY ,Pr is the probability

that the correct code phase sequence pair is ak,θ

t and ( )imtk

b YYN ,, → is the number of bit

errors that result when path, imY , , is chosen instead of the correct path tkY , . In (9), the outer

summation is on all code states (s) and the two inner summations are on all possible pairs of

correct code phase sequences (t,k) and all possible pairs of code phase error sequence (i,m).

Both correct and error sequences belong to the set Ω(s).

The authors make an inaccurate assumption, that the estimated phase sequence for the correct

code sequence is always equal to the channel phase noise sequence. This assumption results in

an upper bound on the pairwise error probability since in many cases, choosing for the correct

code sequence a phase noise sequence which differs from the channel phase noise will yield a

lower (better) accumulated metric.

As realized from the MAP metric in (3) the JMAP sequence estimator chooses only one phase

sequence per each code sequence hypothesis. Jackson and Omura consider all possible pairs

of code phase sequence and add them to the union bound instead of one pair per each

codeword hypothesis. Since these events are statistically dependent, the overhead in the error

event summation causes the union bound to increase dramatically. When the phase noise

variance is not negligible, rapid and large phase fluctuations occur and hence the limitation

taken by the authors on the phase transaction between consecutive states is incorrect.

Excluding the phase transaction limitation increases even more significantly the number of

code-phase error pairs which are summed in the union bound. Furthermore, as the phase noise

increases there are more pairwise errors with significant probability. Hence the conclusion is

that the resulting union bound is very loose especially in high phase noise values.

23

In the coded case, the number of pairwise error to be included in the summations of (9) is

extremely high and hence it is impractical to be implemented. For the uncoded case, the

authors developed a transfer function technique which results in a close form expression

presented in [6], [18] and [19]. Since the transfer function derivation is suited only for the

uncoded case, in the coded case all possible pairwise error should be summed.

In summery, the suggested solution is suited only for very low phase noise. For un-negligible

phase noise variations the resulting upper bound is very loose and uninformative. In any case

it is impractical to be implemented for coded JMAP.

3.2.2 Ungerboeck [7]

Ungerboeck developed and analytical expression for the uncoded phase-symbol JMAP

sequence estimation. The expression is suited only for phase modulation and phase noise

modeled as a Weiner process. The estimated phase values are considered continuous rather

then quantized phase states. In this case, the accumulated metric used for the Viterbi

algorithm is given in (6). The notations in this section are according to the definitions in

chapter 2.2 (system model). Thus the phase modulated symbol is knj

n eaφ= where k

nφ is the

phase of the n-th symbol from the k-th codeword. By substituting the above into (3) the

following metric for the likelihood of the q-th

codeword is given as

( ) ( ) ( ) ( )∑ −++ −+−+=Γ

n

nn

j

n

jq nqnn

kn eea

2

1

2ˆˆ

ˆˆˆminˆ θθξβ θφθφ

θ (10)

Let the correct and incorrect code sequences index be k and m respectively. The code

sequence pairwise error probability is given by

( ) ( ) ( ) kmkmk aaPaaaP ˆˆ/2 Γ<Γ=→ (11)

Next, an approximation is applied to the norm term in (10).

24

We can write

( ) ( ) ( ) ( )( ) ( ) =−+=−+ +−++++2ˆˆ2ˆˆ

nknn

qnn

knn

qnn

kn jj

n

jj

n

jeeeee

θφθφθφθφθφ ξβξβ

( ) ( ) 2ˆˆ1 n

knn

qnn

kn jj

n eeθφθφθφξβ −−++− −+ (12)

where ( )n

knj

neθφξ +−

is the rotated Gaussian noise which can be written as ( )

nn

j

n juue nkn +=+−

1

θφξ ,

where u1n , un are independent white Gaussian noise processes with zero mean and variance

2

uσ . Assuming that

( )2

ˆˆ πθφθφ ≤+−+ n

k

nn

q

n (13)

and then use the first order Taylor approximation to yield

( ) ( )n

k

nn

q

n

jje n

knn

qn θφθφθφθφ −−++≈−−+ ˆˆ1

ˆˆ (14)

The meaning of the left hand term in (13) is the phase distance between the rotated

transmitted symbol ( )n

k

n θφ + and the JMAP estimated rotated symbol ( )n

q

n θφ ˆˆ + . For the

incorrect code sequence ( )m

nφ , the JMAP will tend to choose an estimated phase sequence

which will stretch the estimated code-phase sequence ( )n

m

n θφ ˆˆ + towards the received code-

phase sequence ( )n

k

n θφ + in order to minimize the metric in (10), this justifies the assumption

of small phase difference in (13).

Substituting (14) and (12) into (10) and omitting terms which are insignificant for the

minimization inequality in (11) results in the following approximated likelihood metric of the

q-th

codeword

( ) ( ) ( ) ( )21ˆ

ˆˆˆminˆ~

ˆ −−+−−+=Γ≈Γ ∑ nn

n

nnnn

qq ueaa θθθθβθ

(15)

25

where k

n

q

nne φφ −= ˆ is the symbol phase error. The minimization in (15) can be solved by

using the following power series property

( ) ( ) 0

12 −∞

−∞=

=∑ zFzFfn

n (16)

F(z) is the Z transform of the sequence fn and the notation 0 means the coefficient belonging

to the zero'st power in the Z transform power series. By applying (16) to (15) we arrive at

( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )( ) ( )( ) 0

1111 1ˆ1ˆˆˆminˆ~

zzzzzSzzSza z

q −Θ−Θ++Θ+Θ=Γ −−−−Θ β (17)

where ( ) ( ) ( ) ( )zUzzEzS −Θ−= and ( ) ( ) ( ) ( )zzUzzE ΘΘ ˆ,,, are the Z transforms of

nnnn ue θθ ˆ,,, respectively. Setting the derivation of (17) to zero yields

( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] 0111ˆ~ −−− −Θ−−Θ−=Γ zUzzEzWzUzzEaq (18)

where

( ) ( )( )( )( ) ( ) ( )1

1

1

11

11 −−

−

=−−−−

= zHzHzz

zzzW βλ

λλβλ , β

ββλ +−+=

421

2

and ( )1

1

1

1−

−

−−

=z

zzH

λ.

Substituting the approximated likelihood metric in (18) into (11) for both the correct and

incorrect code sequence (for the correct code sequence E(z)=0) and applying simple algebra

yields

( )

=>+=→

22 /ˆˆ

r

u

kmk TQTrrPaaaP

σθ (19)

where ( )22

ur rr += θσ and the notation ( )2urr +θ means the expectation of ( )2urr +θ , Q(x) is

the Gaussian cumulative distribution function, ( ) ( ) ( ) 0

1

2

1zEzWzET

−= and rθ, ru are

independent Gaussian random variables given by

( ) ( ) ( ) ( ) ( ) ( )0

1

1

0

1

1

1

−=Θ= −

−−

zzVzWzEzzWzErθ and ( ) ( ) ( ) 01

zUzWzEru

−= .

26

In the definition of θr , V(z) is the Z transform of vn. Recall from the definitions in the system

model that nnn v+= −1θθ i.e. vn is the white Gaussian noise source in the Weiner process

phase noise model.

Ungerboeck treated the uncoded case where he had assumed that there are only two dominant

error events for which he had calculated the pairwise error probability given in (19). The first

error sequence is a "single error" (SE) at time n=0 i.e. ( )M

zEπ2

±= where M is the

modulation constellation size (i.e. MPSK modulation). For this case (19) results in

( )

+=

λσλπ

1

22

u

SEM

QP . The second error event is a "single slip" (SS) which is a constant

phase error staring from time n=0 i.e. ( ) ( )zM

zzM

zE−

±=+++±=1

12...1

2 2 ππ.

In this case (19) results in ( )

−=

22 1

2

λσλπ

u

SSM

QP .

When using phase-differential coding scheme with Gray-encoding, a single error causes two

bit errors in consecutive words, whereas a single slip causes one bit-error. The bit error

probability of uncoded MPSK is then approximated by

( ) ( )

−+

+=+≈

222 1

22

1

2424

λσλπ

λσλπ

uu

SSSEBERM

QM

QPPP

For the analysis of coded phase-symbol JMAP sequence estimation, per each specific code

there may be many dominant error events for which (19) is calculated for each separately.

Since there is no simple closed-form analytical expression for solving T and ( )[ ]2

urrE +θ in

(19), the approximated pairwise error probability given in (19) is not suited for the analysis of

coded JMAP. Furthermore, Ungerboeck's analysis limited the phase noise to be a Weiner

process and the JMAP decoder to be matched to the channel phase noise.

27

3.2.3 Robinson & Meer [20]

In [20], the transmitted signal was an un-modulated carrier and the MAP sequence estimator

was used to estimate the phase of the signal. The MAP phase sequence estimator was

implemented as described in [12]. The authors developed a closed form analytical expression

which approximates the mean square error (MSE) of the MAP phase sequence estimation

which is summarized in the following.

The MAP phase estimator chooses either the correct phase sequence or an error sequence.

These two events are mutually exclusive and exhaustive, therefore the MSE at any point in

time on the trellis is defined as

( ) ( ) ( )∑+≡i

ii

C PMSEPCMSEMSEγ

γγ// (20)

where iγ is the i-th phase estimation error sequence i.e. θθγ −= ii ˆ ( iθ is the i-th phase

estimated sequence and θ is the correct phase noise sequence), ( )CMSE / is the MSE given

that the estimator chose the correct path and PC is the probability that the estimator chose the

correct path. The average MSE at a point in time given that the estimator has chosen the error

path iγ is denoted by ( )iMSE γ/ . The ( )iMSE γ/ is averaged over the length of the error

event i.e. ( ) ( )21

0

ˆ1/ ∑

−

=

−=L

n

n

i

n

i

LMSE θθγ where L is the error path length. The probability that the

phase estimation error is iγ is denoted by ( )iP γ .

The term ( )CMSE / is unequal to zero due to phase quantization noise caused by the

approximation of the phase noise by one of a set of M discrete levels in the interval -π,+π.

This quantization error is assumed to be uniform distributed over each quantization interval,

therefore ( )2

2

3/

MCMSE

π= . The assumption of uniformity over the quantization interval is

reasonable if M is sufficiently large. The received signal is defined as

( ) ( ) ( )( ) ( )tttftAtr c ηθπ ++= 2cos where A(t) is the fading amplitude, fc is the carrier

28

frequency, θ(t) is the phase process to be estimated and ( )tη is zero mean, white Gaussian

noise with two sided power spectrum N0/2 w/Hz.

The probability that the MAP estimators chooses an error phase sequence is given by

( )( )( ) ( )( )( )

( )

+−+

=∑ ∫−

=

+

0

1

0

12

2

2cosˆ2cos2

N

dtttfttfT

E

QP

L

n

Tn

nT

c

i

c

i

θπθπγ (21)

where T is the Trellis interval time, E is the energy of the signal in the T second interval and L

is the phase error sequence length in Trellis intervals, hence the error sequence length in time

is L*T. The nominator in (21) is the square Euclidian distance between the correct and

incorrect sequences.

The MSE in (20) was lower bounded by considering only the error event with the largest

probability instead of summing all error events in the term ( ) ( )∑i

ii PMSEγ

γγ/ in (20).

Let m be the index of the phase error sequence with maximum probability then we arrive at

( ) ( ) ( )mm

C PMSEPCMSEMSE γγ// +>

The error event with largest probability ( )mγ was chosen as the sequence which contains a

single phase error to an adjacent phase state. In this case the resulting approximated MSE is

given by

−

+≈MN

EQ

MMMSE

πππ 2cos1

3

22

3 0

2

2

2

2

In the case of a Rayleigh fading signal, the predicted MAP estimator MSE is obtained by

considering the amplitude of the signal as a Rayleigh random variable which is independent in

time and constant over each T second interval. It is assumed that the MAP estimator has

perfect knowledge of the Rayleigh fading amplitude. The Q function is approximated by

( )

−=

2exp

2

1 2xxQ .

29

The resulting predicted MSE for this case is given by

( ) ( ) ( )( )

−+−+≈

MNEMMMSE fading π

ππ2cos122

13/28

3 0

2

2

2

2

where E is the (ensemble) average signal energy received in an interval T.

The MSE analytical expressions for the case with and without fading showed to give a good

approximation to the simulation results at high SNR values and large quantization levels. The

approximation fails at low SNR due to the incorrect assumption that there is only one

dominant error event. When the number of phase quantization levels is small, the assumption

that the quantization error is uniformly distributed in the quantization interval is inaccurate.

The authors had compared MSE simulation results of the MAP phase estimator to the PLL

MSE and reported that the PLL MSE is higher by 2-6 dB when there is no fading and 10 dB

when fading is considered. This comparison is extremely conservative since the PLL MSE

was measured when it did not loss lock. However, when the PLL losses lock the MSE is

increased dramatically. As the phase noise increases and the SNR decreases the probability of

the PLL to loss lock increases. On the other hand the MAP does not suffer from the losing

lock phenomena. The PLL mean time to loss lock was not considered in the analysis.

30

4 Analysis of JMAP Error Rate

4.1 Introduction

The key figure of merit for performance evaluation of coded systems is the bit error rate

(BER). For trellis codes with modest complexity, a usually tight upper bound on the BER is

obtained by applying the union bound on the code sequence pairwise error probability.

A pairwise error occurs when an incorrect code sequence achieves a lower accumulated

metric then the correct code sequence, where for each code sequence hypothesis, the phase

sequence attaining minimum accumulated metric is taken.

An example of an error event is shown in Figure 11. The green paths in Figure 11 are paths

with the same incorrect code sequence, each has a different phase sequence. The blue paths

are paths with the correct code sequence and different phase sequences. The black and red

paths are the paths which achieve the minimum metric out of all green and blue paths

respectively. The JMAP decoder will make a decision error between the two code sequences

if at the point where the codewords end, the black path achieves a lower accumulated metric

than the red path. It is also evident from this figure that the merging point of two sequences is

not necessarily happening at the point of code state merge, since the phase state has to merge

as well. Chapter 4.3 elaborates on the definition of an error event.

31

error paths

correct paths

best metric among

correct paths

____

__

code state 0

code state 1

code state 2

best metric among

incorrect paths__

Figure 11: Pairwise error between 2 code sequences

We will assume, for simplifying the analysis, that the quantization level used by the JMAP is

high, and then the estimated phase sequence θ can be approximated by continuous values.

This approximation is good when the phase quantization is much smaller than the minimum

phase between transmitted symbols, or than the phase noise estimation error. Denote the

correct and incorrect code sequences as a0 and a

1, respectively. Define the likelihood of the

m-th codeword by

( ) ( )[ ]∑ −−=Γn

jm

nn

u

m Peaxa n θσ

θθ

ˆlnˆ2

1minˆ

2ˆ

2ˆ (22)

which is the JMAP decoder minimum accumulated metric (given in (4)) for the m-th

codeword. The code sequences pairwise error probability is defined as

( ) ( ) ( ) 01010

2ˆˆ|ˆˆ aaPaaaP Γ<Γ=→ (23)

32

4.2 Pairwise Error Analysis

In this subsection we shall apply suitable approximations to develop the pairwise error

probability into a convenient closed form expression. Substituting (22) into (23) yields

( )=→ 010

2 |ˆˆ aaaP

( )[ ] ( )[ ]

−−>

−− ∑∑ θσ

θσ

θθ

θθ

ˆlnˆ2

1minˆlnˆ

2

1min

2ˆ1

2ˆ

2ˆ0

2ˆ PeaxPeaxPn

j

nn

un

j

nn

u

nn (24)

Solving (24) is highly complex and hence requires an appropriate approximation.

Ungerboek [7] based his derivations on the condition that the phase difference between the

two JMAP estimations is very small (<π/8). Here we relax the condition and require only

2ˆˆˆˆ 1100 πθφθφ ≤−−+ nnnn (25)

where 0ˆnθ , 1ˆ

nθ are the phase noise estimations attaining minimum accumulated metric for the

correct and incorrect code sequence respectively (attaining the minimum in (22)) and 0ˆnφ , 1ˆ

nφ

are the phase of the code symbols 0ˆna , 1ˆ

na respectively.

As detailed later in Section 4.2.1, we can approximate the pairwise error probability by

( )≈→ 010

2 |ˆˆ aaaP

( ) ( )[ ] ( ) ( )[ ]

−−−<

−−−+− ∑∑n

nnn

un

nnnnn

u

PuPuP θθθσ

θθθφφσ θθ

ˆlnˆ2

1minˆlnˆˆ

2

1min

2

2ˆ

201

2ˆ (26)

where un is the thermal noise component which is a zero mean Gaussian random variable with

variance 22

ξσσ =u (recall from the system model in section 2.2, that nξ is the thermal noise).

By comparing (26) to (24) it can be seen that (26) can be obtained using a candidate

sequences likelihood of the form

( ) ( ) ( )[ ]∑ −−−+−≈Γn

nnnn

m

n

u

m Pua θθθφφσθ

ˆlnˆˆ2

1minˆ

20

2ˆ (27)

33

In other words, a receiver that computes the approximated likelihood metric given in (27) for

each candidate code sequence and then chooses the sequence with minimum metric has the

pairwise error probability given in (26).

We like to show that (27) is also the likelihood metric of the optimal receiver for the

equivalent channel model presented in Figure 12. In this channel a sequence of PAM symbols

m

nφ (each phase is treated as amplitude rather then as phase, i.e. becomes PAM sequence) is

corrupted by additive colored Gaussian noise nη where

nnn u+=θη , yielding the channel

output nn

m

nn

m

nn uy ++=+= θφηφ .

Figure 12: Equivalent channel system

Using Bayes’ rule and assuming that all sequences mφ are apriori equally probable, the joint

phase symbol MAP sequence estimation is ( ) ( ) ( )θθφθφθθ

ˆˆ,ˆ|max|ˆ,ˆmax ˆ,ˆ,PyPyP m

m

m

m∝ .

Since the sequence un is white Gaussian then

( ) ( )( )

( )θπσ

θθφ σ

θφ

θθˆ

2

1maxˆˆ,ˆ|max

2

2

2

ˆˆ

2ˆ,ˆ,

PePyPn

y

u

m

m

m

u

nmnn

∏−−

−

= . By substituting

n

m

nny ηφ += , applying log and eliminating insignificant terms we arrive at

34

( ) ( ) ( ) ( )

+−−+−∝ ∑ θθθφφσ

θθφθθ

ˆˆˆ2

1minˆˆ,ˆ|max

20

2ˆ,ˆ,PuPyP

n

nnnn

m

n

um

m

m. Thus, the JMAP

estimator for the m-th cowdeowrd hypothesis is

( ) ( )

+−−+−∑ θθθφφσθ

ˆˆˆ2

1min

20

2ˆ Pun

nnnn

m

n

u

which is equal to (27). Therefore, the pairwise

error probability of the JMAP decoder for the equivalent channel is given in (26) and thus for

the purpose of the approximate error rate analysis of this research we can consider this

channel model followed by a joint phase symbol MAP sequence estimator. Note that the

equivalent channel is not suited for actual receiver implementation since it is based on the

approximations of Section 4.2.1 which holds only when the JMAP decoder is implemented

with the un-approximated metric given in (22).

Apparently, the estimated transmitted symbol phase sequence mφ obtained from the JMAP

criterion

( ) yP m

m /,maxmaxarg θφθ (28)

may be different then the sequence obtained from the symbol sequence MAP criterion

( )yP m

m /maxarg φ

However, in Appendix A we prove that in the case of the equivalent channel model the two

criteria are identical.

Since it is assumed that all code sequences are transmitted with equal probability then it can

be shown that ( ) ( )m

m

m

m yPyP φφ /maxarg/maxarg = . Hence we conclude that in the

equivalent model, the joint phase symbol MAP estimator (28) yields the same code sequence

estimation as the Maximum Likelihood criterion

( )m

m yP φ/maxarg (29)

and therefore both decoders have the same pairwise error rate given in (26). It is known [8

chapter 10-1] that the Maximum Likelihood decoder (29) for the equivalent channel model

can be implemented with a noise whitening filter, denoted by H(z), followed by a detector

35

which calculates the minimum Euclidean distances between the whitening filter output and all

possible transmitted phase sequences after being filtered with H(z) and chooses the

transmitted sequence with minimum distance, as presented in Figure 13. Next we derive the

pairwise error probability of this detector (which is actually the solution of (26)) and use it as

an approximation for (24).

Figure 13: Equivalent communication model

The spectrum of nη is given by

( ) ( ) 22 ωω

ηη σ j

u

j eAeS = , where

( ) ( )22

1 ωω β jj eGeA += (30)

and 2

2

u

v

σσ

β = . Thus, the whitening filter H(z) is chosen such that

( )( ) 2

2 1

zAzH = (31)

Denote by m

nφ~

, ny~ and nη~ the results of filtering m

nφ , ny and nη , respectively, with the

whitening filter H(z). As before, let 10 , nn φφ be the phase of the correct and incorrect code

sequences, respectively, so that nnny ηφ ~~~ 0 += and nη~ is a zero mean Gaussian random

36

variable with spectrum

( ) ( ) ( )2

~~ωω

ηηω

ηηjjj eHeSeS = (32)

Let 01

nnne φφ −= . From the equivalent channel MAP detector (Figure 13) the pairwise error

probability in (23) is approximated by

( ) ( ) ( ) ( )

−<−≈=→ ∑∑n

nn

n

nn yyPePaaaP2021

2

010

2

~~~~|ˆˆ φφ (33)

The filtered error sequence ne~ is the code phase error (e) after being filtered with H(z) i.e.

01 ~~~nnne φφ −= . Substituting nnny ηφ ~~~ 0 += and ne~ into (33) and applying simple algebra results

in

( )

=

>≈Λ

∑∑∑ 2

2

2

2

2

~

~~~2σ

η n

n

n

n

n

nn

e

QeePeP (34)

Where 2

Λσ is the variance of the zero mean Gaussian variable ∑=Λn

nne η~~2 . It can be shown

that ( )∑∑ −=Λn m

mn nmRee ηησ ~~2 ~~4 , where ηη ~~R is the autocorrelation of nη

~ given by

( ) ( ) ( ) ( ) 2221~~

1~~ zHzAZzSZnR uσηηηη

−− == and Z-1

is the inverse Z transform. Further

defining ∑ −=k

knkn fer ~ where ( ) ( ) zHzAZfn

1−= then

( )∑∑

∑∑ ∑∑ ∑∑∑

Λ

−−−−

=−

===

k q

qk

k q n

qnknuqk

n q

qnq

k

knku

n

nu

kqRee

ffeefefer

2~~

2222

~~4

~~4~~44

σ

σσσ

ηη

i.e.

∑=Λn

nu r 222 4σσ (35)

37

From the definition of ne~ it follows that

∑ ∑ ∑

= −

n n k

knkn hee

2

2~ (36)

where hn=Z-1

H(z).

By substituting (35) and (36) into (34) we arrive at

( )

≈∑

∑ ∑ −

n

nu

n k

knk

r

he

QeP22

22

24σ

(37)

A convenient physical interpretation of this expression is as follows. Each of the nominator

and denominator in (37) are obtained by filtering the code phase error sequence en with a

different filter and calculating the energy of the filters outputs as described in Figure 14.

Figure 14: Generelized pairwise error expression block diagram

Next we will express the pairwise error of (37) in the spectrum domain rather then the time

domain. Recall that rn is the result of the convolution between ne~ and fn. Then from the

definition of ne~ and fn we can write ( ) ( ) ( ) zAzHzEZrn

21−= . By using Parseval’s theorem

we obtain

( ) ( ) ( ) ωπ

πωωω

deAeEeHrjjj

n

n ∫∑ =2

0

242

2

1 (38)

and

38

( ) ( ) ωπ

πωω

deHeEejj

n

n ∫∑ =2

0

22

2

1~ (39)

Substituting (38), (39) and (30) into (37) results in

( )( ) ( )

( ) ( ) ( )

+

≈

∫

∫

ωσσπ

ωπ

πωωω

πωω

deGeHeE

deHeE

QePj

vu

jj

jj

2

0

222

42

22

0

2

2

2

4

2

1

(40)

The careful reader could notice that throughout our derivation we did not rely on the fact that

the phase noise probability function ( )θP used for the JMAP decoder metric is perfectly

matched to the actual channel phase noise spectrum. In fact, use of some approximated and

simplified phase noise model can be made for the JMAP decoder implementation which is

unmatched to the actual channel phase noise model. It is important to discuss such mismatch

since a decoder matched to the actual model might be impractical to be implemented, and a

simple model can lead to a practical implementation. A JMAP decoder based on an

unmatched phase noise model is equivalent to substituting the whitening filter H(z) (in Figure

13) chosen according to (31) with a filter that whitens the corresponding simplified noise

spectrum rather then the true channel noise spectrum, i.e. mismatched decoder. The JMAP

pairwise error probability for the mismatched case is obtained by substituting into (37) and

(40) the H(z) filter which whitens the simplified phase noise spectrum. In chapters 4.2.2 and

4.2.3 we further analyze the pairwise error probability of the matched and mismatched JMAP

decoders respectively.

4.2.1 Pairwise Error Probability Approximation

In chapter 4.2 (24) it was shown that the code sequences pairwise error probability is

( )

( )[ ] ( )[ ]

−−>

−−

=→

∑∑ θσ

θσ

θθ

θθ

ˆlnˆ2

1minˆlnˆ

2

1min

|ˆˆ

2ˆ1

2ˆ

2ˆ0

2ˆ

010

2

PeaxPeaxP

aaaP

n

j

nn

un

j

nn

u

nn

39

In the following, we show that this pairwise error probability can be approximate by

( )≈→ 010

2 |ˆˆ aaaP

( ) ( )[ ] ( ) ( )[ ]

−−−<

−−−+− ∑∑n

nnn

un

nnnnn

u

PuPuP θθθσ

θθθφφσ θθ

ˆlnˆ2

1minˆlnˆˆ

2

1min

2

2ˆ

201

2ˆ

where the notations are as given in chapter 4.2.

We start by writing the un-approximated code sequence pairwise error probability (24) as

( ) ( )[ ] ( )[ ]

−>−−−=→ ∑ 10

2ˆ1

2

2ˆ0

2

010

2ˆlnˆlnˆ

2

1ˆ

2

1|ˆˆ

10

θθσσ

θθPPeaxeaxPaaaP nn j

nn

un

j

nn

u

(41)

where 0θ and 1θ are the phase sequences attaining minimum for the correct and incorrect

code sequences respectively in (24) i.e.

( )[ ]θσ

θ θθ

ˆlnˆ2

1minargˆ

2ˆ

2ˆ Peaxn

jm

nn

u

m n −−= ∑ 1,0∈m (42)

Assuming MPSK modulation we substitute n

j

nnneax ξθ += 0 ,

00 nj

n eaφ= and

11 nj

n eaφ= into (41)

and then the term on the left hand side of the argument (in (41)) becomes

( ) ( ) ( ) ( ) ( ) ( )( ) ∑ +−+−++++ −+−−−n

jj

n

jjjj

u

nnnnnnnnnnnn eeeeee0011110000 ˆˆˆˆ2ˆˆ2ˆˆ

2Re2

2

1 θφθφθφθφθφθφ ξσ

(43)

We can rewrite the noise component in (43) as follows

( ) ( )( ) ( ) ( )00110011 ˆˆˆˆˆˆˆˆ2Re2 nnnnnnnn jj

n

jj

n eeueeθφθφθφθφξ +−+−+−+− −=− (44)

where ( ) ( )( )( ) ( )

−

−=

+−+−

+−+−

0011

0011

ˆˆˆˆ

ˆˆˆˆ

Rennnn

nnnn

jj

jj

nn

ee

eeu

θφθφ

θφθφ

ξ .

Let ( ) ( )000 ˆˆ

nnnn jj

n eeAθφθφ ++ −= ,

( ) ( )110 ˆˆnnnn jj

n eeBθφθφ ++ −= and

( ) ( )0011 ˆˆˆˆnnnn jj

n eeCθφθφ +−+− −= so that (43)

can be written as ∑ +−n

nnnn

u

CuBA 22

1 22

2σ.

We apply the following approximations,

( ) ( ) ( ) nnnnn

jj

n AeeA nnnn ′=−−+≈−= ++ 000ˆˆ ˆˆ000

θφθφθφθφ

40

( ) ( ) ( ) nnnnn

jj

n BeeB nnnn ′=−−+≈−= ++ 110ˆˆ ˆˆ110

θφθφθφθφ

( ) ( ) ( ) nnnnn

jj

n CeeC nnnn ′=−−+≈−= +−+− 1100ˆˆˆˆ ˆˆˆˆ0011

θφθφθφθφ

These approximations nn AA ′≈ , nn BB ′≈ and nn CC ′≈ essentially converts the length of a

chord connecting two points on the unit circle to the length of the arc connecting them.

Denote by ρ the phase difference between the two points, then the chord length is given by

2sin2 ρ and the arc length by ρ . Hence the approximation is

≈

2sin2

ρρ which is

presented in Figure 15.

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

Phase [radians]

Length

chord: length=2*sin(phase/2)

arc: length=phase

Figure 15: Arc approximation of a chord

Note that Cn is the distance between the JMAP estimations of the correct and incorrect

sequences, An and Bn are the distances between the transmitted symbol after being rotated by

the phase noise and the JMAP estimation for the correct and incorrect symbols respectively.

In Figure 16 we present an example of a received symbol and the JMAP estimations for the

correct and incorrect sequences. The values of An, Bn, Cn, A’n, B’n and C’n are also marked in

this figure.

41

An’

Bn’

Cn’

nx( )nnj

eθφ +0

An

Cn

Bn

( )11 ˆˆnnj

eθφ +

0nj

eφ

1nj

eφ

( )00 ˆˆnnj

eθφ +

Figure 16: Approximation insight

It is assumed that in error events the rotated transmitted symbol ( )nnj

eθφ +0

will be located along

the arc connecting the JMAP estimations ( )00 ˆˆ

nnje

θφ + and

( )11 ˆˆnnj

eθφ +

, since then the difference

between the likelihood metrics (22) of the correct and incorrect code sequences is small. In

this case Cn>An and Cn>Bn. Thus, the approximations nn AA ′≈ and nn BB ′≈ are at least as

accurate as nn CC ′≈ .

Each JMAP code and phase estimation ( 00 ˆˆnn θφ + and 11 ˆˆ

nn θφ + ) tends to stretch towards the

rotated transmitted symbol nn θφ +0 and hence shorten the distances An, Bn and Cn (see Figure

16). This effect depends on the bandwidth of JMAP phase estimation filter. In case that the

JMAP is matched to the phase noise being a Weiner process the bandwidth is controlled by

the parameter β in the accumulated metric (6). As beta increases, the bandwidth increases and

the estimated phase tracks faster the received phase, leading to smaller phase error, and vice

versa. Therefore, the approximation becomes better for higher levels of phase noise. We

consider the case of low phase noise level, where the phase noise and phase noise estimation

terms can be approximately disregards, as an upper bound to the approximation error. In this

case, An=0, Bn=Cn and the resulting pairwise error is

42

( )

=→∑

2

2

010

24

|ˆˆu

n

nC

QaaaPσ

(45)

For 2

π<′nC , applying the approximation nn CC ′≈ into (45) will give an error which is upper

bounded by 0.45 dB. Therefore, we consider the approximation to be accurate as long as the

phase difference between the two JMAP estimations ( )nC′ is smaller then π/2 i.e. the

condition given in (25) and rewritten here again 2

ˆˆˆˆ 1100 πθφθφ ≤−−+ nnnn .

Substituting the approximations nn AA ′≈ , nn BB ′≈ and nn CC ′≈ into (41) and rearranging it

results in the following approximated pairwise error

( )≈→ 010

2 |ˆˆ aaaP

( ) ( )[ ] ( ) ( )[ ]

−−−<−−−+− ∑∑

n

nnn

un

nnnnn

u

PuPuP0

20

2

12

101

2ˆlnˆ

2

1ˆlnˆˆ2

1θθθ

σθθθφφ

σ (46)

Comparing (46) to (24) we see that the likelihood of the m-th codeword given in (22) was

approximated by

( ) ( ) ( )[ ]∑ −−−+−≈Γn

m

nn

m

nn

m

n

u

m Pua θθθφφσ

ˆlnˆˆ2

1ˆ

20

2 (47)

where mθ is computed according to (42).

Denote by mθ ′ the phase noise estimation per each codeword hypothesis mφ obtained by

minimizing the approximated expression in (47) i.e.

( ) ( )[ ]∑ −−−+−=′n

nnnn

m

n

u

m Pu θθθφφσ

θθ

ˆlnˆˆ2

1minarg

20

2ˆ (48)

Finally, we argue that 0ˆnθ and 1ˆ

nθ (the phase noise estimations obtained by minimizing the

original likelihood expression in (22)) can be replaced with 0

nθ ′ and 1

nθ ′ respectively in (46)

43

without detrimentally affecting the approximation and hence obtain the approximated

pairwise error probability given in (26) and rewritten here again

( )≈→ 010

2 |ˆˆ aaaP

( ) ( )[ ] ( ) ( )[ ]

−−−<

−−−+− ∑∑n

nnn

un

nnnnn

u

PuPuPnn

θθθσ

θθθφφσ θθ

ˆlnˆ2

1minˆlnˆˆ

2

1min

2

2ˆ

201

2ˆ

Let us understand the effect of each approximation leading to (26). We can write the pairwise

error probability in (41) as

( ) ( ) ( )[ ] ( )[ ]

−+−>=→ ∑∑ 1022

22

010

2ˆlnˆln

2

11|ˆˆ θθ

σσPPABCuPaaaP

n

nn

un

nn

u

(49)

We assume that nn AB ′>′ (the distance between the transmitted symbols rotated by phase noise

and the JMAP estimation for the error sequence is larger then the distance from the JMAP

estimation for the correct code sequence) and now show that 2222

nnnn ABAB −≥′−′ . From the

definition of nB′ and

nA′ it follows that

22

22

2sin2

2sin2

′−

′=− nn

nn

ABAB . Applying

trigonometry identity we can write

′−′

′+′=

′−

′

2sin2

2sin2

2sin2

2sin2

22

nnnnnn ABABAB and hence

′−′

′+′=−

2sin2

2sin2

22 nnnnnn

ABABAB . For x>0 it is a fact that

≥2

sin2x

x then

′+′≥′+′

2sin2 nn

nn

ABAB and since we assume that nn AB ′>′ then

′−′≥′−′

2sin2 nn

nn

ABAB , using these last two inequalities we can write

( )( ) 2222

2sin2

2sin2 nnnnnn

nnnnnn ABABAB

ABABAB ′−′=′−′′+′≤

′−′

′+′=− i.e.

2222

nnnn ABAB −≥′−′ . Thus, the approximation causes an increase in the right hand side of the

inequality in (49). This increase reduces the pairwise error probability. However, the arc

44

connecting two points on the unit circle is always larger then the cord connecting them

therefore nn CC >′ and thus the approximation also increases the noise term on the left hand

side of (49). This effect increases the pairwise error probability.

As part of the approximation the phase minimization is done on the approximated metric (27)

rather than (22). In the approximated metric the term ( )20

2ˆˆ

2

1nnnn

m

n

u

u−−+− θθφφσ

is typically

larger than the original term 2ˆ

2ˆ

2

1njm

nn

u

eaxθ

σ− (since the arc connecting two points on the

unit circle is larger then the cord connecting). This increase in the distance term will cause the

phase estimation to stretch more (in respect to the phase estimation done on the original

metric (22)) towards the received sequence in order to minimize the metric in (27) and hence

its effect is widening the bandwidth of the phase estimation filter (equivalently to widening

the loop filter in a PLL). Because of the assumption thatnn AB ′>′ then the phase estimation

bandwidth extension for the error path is larger then for the correct path. This bias also

increases the pairwise error probability since it helps the JMAP estimation for the error

sequence to reduce the inherent phase difference between the error sequence and the

transmitted sequences.

We have seen three effects during the approximations:

1. The signals Euclidean distances replaced with arc lengths (A'n, B'n).

2. The same effect as 1 but on the noise (C'n).

3. The phase minimization effect.

While the first effect reduces the error rate the two latter effects increase it. We consider the

third effect to have a minor influence relatively to the first and second effects. In the

approximation worst case, where the phase noise is disregarded and the pairwise error

probability is given in (45), replacing Cn with C’n will result in a lower bound on the pairwise

error probability (since C’n> C’n). Simulation results confirm that the total effect (from effects

45

1-3) gives a lower than actual pairwise error probability. The bound gets tighter as the

distance in (25) is smaller.

Note that the condition (25) involves the estimated phase sequence. One may evaluate this

condition using the estimated phase sequence derived in Section 5. The condition should be

evaluated for each relevant error sequence, possibly ignoring the noise for simplicity.

If the constellation size is greater then two and the code rate is not smaller then 0.5, the

symbol phase difference between two code sequences ( )10

nn φφ − is relatively small ensuring

that the condition in (25) holds. Hence the conclusion is that for Trellis codes which use

MPSK modulation where M>2 and have a rate equal or greater then 0.5, the condition in (25)

holds and hence (37) and (40) give a tight approximation on the pairwise error probability.

4.2.2 Matched Decoder Pairwise Error Probability

In case that the JMAP decoder is matched to the channel phase noise then we substitute

2)(zH given in (31) into (40) and arrive at

( )( ) ( )

≈∫

2

2

0

2

24

2

1

u

jj deHeE

QePσ

ωπ

πωω

(50)

By applying Parseval’s theorem to the nominator in the argument of (50) we obtain

( )

≈∑ ∑ −

2

2

24 u

n k

knk he

QePσ

(51)

A block diagram describing the calculation of the pairwise error probability in (51) is

presented in Figure 17.

46

Figure 17: Matched JMAP decoder pairwise error expression block diagram

In (51) we arrived at a simple close form expression for the approximated pairwise error

probability of JMAP decoder matched to the channel phase noise. This result is useful not

only for evaluating the performance of matched JMAP decoder but also serves as a reference

for the mismatch decoder case. The mismatch decoder pairwise error probability is lower

bounded by the matched decoder pairwise probability. Furthermore, comparing them is useful

to evaluate the performance loss due to channel mismatch.

We understand from (51) that the pairwise error rate is determined by filtering the phase error

sequence en with H(z) and calculating the energy of the filter output. This flow is as obtained

from Figure 14 by choosing H(z)=1/A*(z) (note that the phase of H(z) and A(z) can be chosen

arbitrary) which yields nn er ~= (A*(z) is the conjugate of A(z)). In this case the equivalent

model in Figure 13 reduces to the model presented in Figure 18 where the transmitted coded

phase values pass the filter H(z) followed AWGN (δn) with zero mean and variance 2

uσ .

( )2,0~ un N σδ

Figure 18: Equivalent model - matched channel

47

The pairwise error probability obtained for the AWGN channel in the absence of phase noise

[8 Equation 5-2-6] is given by

( )

=→∑

2

2

010

24

|ˆˆu

n

nd

QaaaPσ

, (52)

where 10

nnn aad −= is the Euclidean distance between the two complex code symbols and 2

uσ

is the thermal noise variance of each real and imaginary components. Comparing (51) with

(52) shows that the Euclidean distance between the two code symbol sequences in the

coherent case (52) was replaced in (51) with the distance between their corresponding phases

after having passed the filter H(z).

4.2.2.1 Matched JMAP Decoder - Weiner phase noise model

We now use the common assumption that the phase noise is modeled as a discrete Weiner

process i.e. independent Gaussian increments, nnn v+= −1φφ . In this case the metric for the

matched JMAP decoder given in (22) becomes [7]

( ) ( )∑ −−+−=Γn

nn

jm

nn

m neaxa2

1

2ˆ

ˆˆˆˆminˆ θθβ θ

θ, (53)

and the phase noise generating filter (ARMA model filter) is given by ( )11

1−−

=z

zG .

Substituting the latter into (31) yields ( ) ( )( )( )( )zz

zzzH

λλλ

−−−−

= −

−

11

111

12

, where

βββ

λ +−+=42

12

. We choose the noise whitening filter to be

( ) ( )( )1

1

1

1−

−

−−

=z

zzH

λλ (54)

The pairwise error probability is obtained by applying (54) into (51). Note that when the

phase noise is modeled as a Weiner process the resulting filter H(z) is the transfer function of

a single pole HPF with pole at λ. Hence the performance of codes on such phase noise

channel are dominated by either short error sequences or slow varying phase error sequences,

48

as in both cases the residual energy after passing through the HPF will be relatively small (the

residual energy determines the pairwise error probability in (51)). Codes which have large

minimum distance and thus perform well in the coherent AWGN channel may not perform

well in phase noise channels. This happens when there exist long error events with constant or

close to constant phase, as demonstrated in Example 1.

The HPF cutoff frequency is given by -ln(λ)*(Fs/2π) where Fs is the sampling frequency.

Thus, the cutoff frequency is a monotonically increasing function of β, the phase to thermal

noise weighting factor. As the phase noise increases relative to the thermal noise, β grows and

λ decreases, causing stronger attenuation of low frequencies by H(z). Conversely, when the

thermal noise is dominant, λ approaches one and H(z) tends to have an all pass response. In

this case, the system converts to the coherent case and the pairwise error of (51) converts to

(52) with the exception that the square Euclidian distance∑n

nd 2 in (52) converts to the phase

square distance ∑n

ne2 in (51). For small phase values, the phase distance is approximately

equal to the Euclidian distance.

Example 1:

An example which illustrates the effect that the JMAP decoder has on long error events with

constant or close to constant phase is shown in Figure 19 for β=0.6. The blue line is the phase

error between the correct and incorrect code sequences ( )01 φφ −=e . The Red line is the

result of filtering the error sequence e with the HPF H(z). We can observe that even though

the original distance (blue line) which determines the pairwise error probability in the

coherent decoding case was relatively large, the filtered distance (red line) which determined

the error rate of the JMAP decoder is small.

49

5 10 15 20 25 30 35 40 45 50 55 60-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time Index (n)

Phase [

radia

n]

Figure 19: Filtered error event Blue – phase error sequence (e)

Red – filtered error sequence ( e~ )

4.2.2.2 Extension of Ungerboeck’s Derivation

As described in chapter 3.2.2, Ungerboeck has developed an analytical expression that

approximates the pairwise error probability (19) of a JMAP decoder matched to Weiner phase

noise spectrum with MPSK modulation. The resulting expression is not general in the sense

that obtaining the pairwise error probability for each different error event requires a specific

analytical calculation and hence cannot serve as a practical tool for coded error rate analysis

where there are many different error events. In this section we will further develop

Ungerboeck's derivation and yield a simple closed form expression suited for any arbitrary

error event. The resulting error rate expression consolidates with the expression given in

chapter 4.2.2.1.

Consider the probability that the detector will chose the code sequence a1 given that the

transmitted code sequence is a0. From Ungerboeck's derivation (see chapter 3.2.2) this

50

probability is given in (19) and rewritten here again ( )

=→

2

010

2 /

r

TQaaaP

σ

where ( )22

ur rr += θσ , ( ) ( ) ( ) 0

1

2

1zEzWzET −= , ( ) ( ) ( )

0

1

1

1

1

−= −

−

zzVzWzErθ and

( ) ( ) ( ) 01 zUzWzEru

−= . The notation x means the expectation of x and the notation 0

means the coefficient belonging to the zeroth

power in the Z transform power series. In 3.2.2

we have defined ( ) ( ) ( )1−= zHzHzW β where ( ) ( )( )1

1

1

1−

−

−−

=z

zzH

λλ , β

ββλ +−+=

421

2

and 2

2

u

v

σσ

β = . The notations E(z), U(z) and V(z) are the Z transform of the code phase error

sequence (en), the thermal noise (un) and the phase noise white Gaussian noise source (vn)

respectively.

We will now further develop the argument in the pairwise error probability given in (19).

Let us define ( ) ( ) ( )1

1

1

1−

−

−=

zzWzEzC and cn as the inverse Z transform of C(z). We can

write ( ) ( ) ( ) ( ) ( ) ∑ −−− ==

−=

n

nnvczVzCz

zVzWzEr 0

0

1

1

1

1θ and the variance of θr is given

by ∑∑∑∑ −−−− ==n m

mnmn

n m

mmnn vvccvcvcr 2

θ . Since vn is zero mean white Gaussian noise with

variance 2

vσ then ∑=n

nv cr 222 σθ . Using the power series property ( ) ( ) ∑∞

−∞=

−=n

n zFzFf 0

12 we

can write ( ) ( ) 012222 −== ∑ zCzCcr v

n

nv σσθ and by substituting ( ) ( ) ( )1

1

1

1−

−

−=

zzWzEzC we

arrive at

( ) ( ) ( ) ( )( ) ( )

0

1

1122

1

1

1

1

−−= −

−−

zzzEzWzWzEr vσθ (55)

By defining ( ) ( ) ( )zWzEzB 1−= we can write ( ) ( ) ( ) ( ) ( ) 00

1 zUzBzUzWzEru == − .

51

Following similar steps as for 2

θr results in

( ) ( ) ( ) ( ) 0

112222 −−== ∑ zEzWzWzEbr u

n

nuu σσ (56)

Since θr and ur are independent we can write

( ) 2222

uur rrrr +=+= θθσ (57)

Substituting (55) and (56) into (57) and after collecting terms we arrive at

( ) ( ) ( ) ( ) ( ) 0112 −−= zEzWzWzEzJrσ (58)

where ( ) ( )( )( )( )

−−−−

= −

−

1

12

11

11

zz

ZZzJ u λλ

λσ

.

From the definitions ( ) ( )( )

( )( )z

z

z

zzW

λλβλ

−−

−−

= −

−

1

1

1

11

1

and 2

2

u

v

σσ

β = we can write ( )( )zW

zJ v

2σ= and

by substituting it into (58) we obtain ( ) ( ) ( ) 0122 −= zEzWzEvr σσ . From the definition

( ) ( ) ( ) 0

1

2

1zEzWzET −= it follows that Tvr

22 2σσ = and hence we can write the pairwise error

probability given in (19) as

( )

=

=

=→

22

2

2

2010

222

/ˆˆvvr

TQ

T

TQ

TQaaaP

σσσ (59)

Let us further develop the expression for T. Recall that from the definition of W(z) we can

write ( ) ( ) ( ) ( ) ( ) ( ) ( ) 011

0

1

22

1zEzHzHzEzEzWzET −−− ==

β and by using the power series

property ( ) ( ) ∑∞

−∞=

−=n

n zFzFf 0

12 we obtain

( ) ( ) ( ) ( ) ∑ ∑

== −

−−

n k

knk hezEzHzHzET

2

0

11

22

ββ (60)

52

By substituting (60) into (59) we arrive at

( )

≈→∑ ∑ −

2

2

010

24

|ˆˆu

n k

knk he

QaaaPσ

which is the exact pairwise error probability obtained in chapter 4.2.2.1 by substituting (54)

into (51).

We have succeeded to extend Ungerboeck's derivation and yield a simple closed form

expression for the pairwise error probability of the JMAP decoder matched to Weiner phase

noise model. Unlike the pairwise error probability obtained by Ungerboeck in (19), the

resulting expression is simply applicable for any error event and hence it enables to achieve

an approximation on the code error rate by applying the union bound on the pariwise errors

probabilities.

Although the approach leading to the pairwise error analytical expression in (51) and (54) is

different then Ungerboeck's approach (which is extended in this section), the results

consolidate since both methods are based on the same linear approximation described in

chapter 4.2.1. However, our approach is more powerful then Ungerboeck's since it is suited

for any arbitrary phase noise and for both matched and mismatched decoder while

Ungerboeck's approach is limited to the phase noise modeled as Weiner process and only

matched JMAP decoder.

4.2.3 Mismatched Decoder Pairwise Error Probability

Traditionally, the phase noise is treated as a Weiner process, justified by the physical

mechanism of a typical oscillator phase noise. However, in practice, the phase noise spectrum

may significantly differ from the Weiner spectrum as it is the combination of many different

noise sources. For example, the phase noise of the RF synthesizer used to generate the local

53

oscillator is contributed from the reference, the VCO, the divider and the loop filter. On the

other hand, implementing a JMAP decoder with an accumulated metric suited for a specific

synthesizer phase noise spectrum is also impractical. Hence an approximated JMAP decoder

metric is required, resulting in a decoder mismatch. A good approximation for the JMAP

decoder is to consider the phase noise as Weiner process and hence remain with the relatively

simple metric of (6) in the Viterbi decoder while the actual phase noise in the channel is

different. In this case the optimal weighting factor β in (6) is no longer equal to 22

uv σσ and

should be optimized specifically per each phase noise spectrum and SNR value.

By substituting (54) into (40) we obtain the following approximated pairwise error probability

for mismatched JMAP decoder based on the Weiner phase noise model

( )( )

( ) ( )

−−

+

−−

≈

∫

∫

−

−

−

−

ωλ

σσπ

ωλπ

π

ω

ωωω

π

ω

ωω

de

eeGeE

de

eeE

QeP

j

jj

vu

j

j

jj

2

0

42

222

22

0

22

2

1

1

2

4

1

1

2

1

(61)

and the equivalent pairwise error probability expression in the time domain is obtained by

substituting (54) into (37).

Equation (61) is a convenient closed form expression which will enable designers to take into

consideration the influence of each of the error (E(ejω

)), receiver metric (β) and synthesizer

phase noise (G(ejω

)) on the receiver performance. It further enables joint optimization of the

synthesizer, the code and the decoder for achieving lowest error rate.

4.3 Trellis Code Error Rate Analysis

We now derive an upper bound on the BER of matched and mismatched JMAP decoders for a

given Trellis code. For this analysis it is important to know if the code is Rotationally

Invariant (RI). Since there is no single definition of RI let us define the terms that will be used

in this research.

54

A RI code is a code in which there is at least one angle ρ for which for each valid code

sequence there is another valid code sequence which differs from it by a phase shift ρ.

A partial RI code is a code in which there is at least one angle ρ for which for some valid

code sequences, but not all, there is another valid code sequence which differs from it by a

phase shift ρ.

Obviously, the JMAP decoder cannot decide between codewords with a constant phase shift.

A further assumption is that proper encoding of the input (usually differential encoding) is

used to prevent the decoder to output wrong bits if it selects a codeword with a constant phase

shift from the transmitted one.

A non RI code is a code that no two valid codewords differ by a constant phase shift.

The analysis of partial RI codes is complicated by the fact that the probability of having error

events which include a long section of constant phase shift error should be weighted by the

probability of transmitting such codeword sections which have a constant phase shift pair.

The probability of having such input sequence to the encoder that generate those sections is

exponentially decreasing with the length of the section, assuming independent and identically

distributed input bits to the encoder. It is conclusive that for sufficiently long error event

considered in the analysis, partial RI codes can be treated as non RI codes.

In RI codes for each code state there are corresponding twin state(s) which is its rotated

version(s). Let us have a pair of two state sequences on the trellis, which their outputs differ

by a constant phase shift. Let sn and tn be the state of these sequences at time n. We shall call

such pair sn and tn as twin states. In other words, if state sn is a twin of tn then all codewords

stemming from state sn are equal to those stemming from state tn except for a constant phase

shift.

A formal definition of the twin states is given below:

55

Let ( )xiµ denote the phase of the output symbol of the branch exiting the state x as a result of

the encoder input index i. If state sn and tn are twin states, then for each branch from state sn to

any state sn+1 there is a related branch from tn to tn+1 such that the encoder input is i and

j=π(i), respectively, and the phase of their output symbols are ( )ni sµ and ( ) ( ) ρµµ += ninj st ,

respectively, where, ρ is a constant phase shift, π(i) is a permutation, and states sn+1, tn+1 are

themselves twin states with the same phase offset ρ and the same permutation.

The JMAP decoder will randomly decide between sequences having constant phase shift

between them. As mentioned, proper differential encoding of the input bits such that both

sequences are decoded to the same information bits is essential, otherwise endless bursts of

errors would occur. Such codes can be called noncoherently catastrophic [3].

Let us consider the error probability per node (code state) which is denoted by Pe(s), where s

is the state index. The probability of a node error at state s is upper bound by summing all

pairwise error probabilities which are initiated from state s and multiply each one with the

probability of the transmitted (correct) sequence. Let Ω(s) be the set of codewords which are

initiated from state s. Let Ωe(s,i) be the set of all independent error events per transmitted

codeword ( )sa i Ω∈ . The node error rate is given by

( ) ( ) ( ) ( )( )( )

∑ ∑Ω∈ Ω∈

→≤sa isa

imii

ei em

aaaPassP,

2 /PrPr (62)

where Pr(s) is the probability of being at state s and ( )imi aaaP /2 → is the probability that the

decoder chooses the code sequence am given that the transmitted sequence was a

i (pairwise

error probability).

Let us denote by ( )mi

b aaN → the number of error bits caused by choosing ma instead of ia .

Multiplying each error event with the corresponding number of error bits Nb and summing the

node error probabilities (62) of all code states results in the following upper bound on the bit

error rate

56

( ) ( ) ( ) ( )( )( )

∑ ∑ ∑Ω∈ Ω∈

→→≤s sa isa

mi

b

imii

biti em

aaNaaaPasP,

2 /PrPr (63)

In order to define the independent error events to be included in (63) we return to the

equivalent linear model presented in Figure 13 (chapter 4.2). In the equivalent model, the

detector which follows the whitening filter H(z) is implemented by the Viterbi algorithm with

an equivalent trellis [8 chapter 10-1].

Consider the impulse response of the filter H(z) to have length L (approximately reaches zero

after L symbols). Then in the equivalent trellis, each trellis state represents a different

combination of current code state (denoted by sn) and previous L-1 symbols phases (which

determine a specific state of the whitening filter memory) i.e. the trellis state is given by

121ˆ,..,ˆ,ˆ, +−−− Lnnnns φφφ . The branch metric for the equivalent trellis decoder is given by

21

0

ˆ~

−∑

−

=−

L

k

knkn hy φ where ny~ is the received symbol after being filtered with H(z) and the

sequence hn is the impulse response of H(z). The accumulated metric of the equivalent trellis

is given by ∑ ∑

−

−

=−

n

L

k

knkn hy

21

0

ˆ~ φ . Since the JMAP decoder is approximated by the MAP

receiver in the equivalent communication system of Figure 13, we will use the equivalent

trellis decoder to define an independent error event.

For non RI codes, an independent error event occurs when a state sequences in the equivalent

trellis is identical to the correct sequence till some time, then diverges and remerges later to

the same equivalent trellis state and at the remerging state has a lower accumulated metric

then the correct sequence.

Equivalently, an independent error event can be defined by using the original code trellis

which has only code states. Consider two code state sequences (the transmitted and error

sequences from the code trellis) which are identical till some time, then diverge to different

code states, and remerge later to the same code states and staying same for at least L

57

consecutive symbols. An independent error event occurs if after the remerging of L symbols

the error sequence results in a lower metric then the transmitted sequence. Thus, due to the

effect of the filtering, error events will be independent only if they are separated sufficiently

from one another to let the filter settle. Two error events which are too close together should

be considered as one error event. The above is illustrated in Figure 20. In the upper part of the

Figure there are two code state sequences, the all zero state sequence which is considered the

correct code sequence and an error state sequence. The lower part of the Figure shows the

corresponding filtered phase error sequence (i.e. the phase error sequence e filtered with

H(z)). It is shown that although the code state sequence diverges and remerges twice, the error

event does not terminate before the filtered phase error sequence decays to zero only then, the

next code state divergence initiated a new independent error event.

For RI codes there is another types of independent error events to be considered. Let tn and sn

be twin code states at time n with a phase shift of ρ. In the equivalent trellis an independent

error event also terminates when the correct and incorrect sequences reach the states

121ˆ,..,ˆ,ˆ, +−−− Lnnnns φφφ and ρφρφρφ +++ +−−− 121

ˆ,..,ˆ,ˆ, Lnnnnt respectively and the incorrect

sequence has a lower accumulated metric then the correct sequence. At this point, the JMAP

decoder, which has no absolute phase reference, is unable to distinct between twin states and

hence will continue decoding the rotated sequence, this will not cause further errors since

proper differential decoding is assumes. From the perspective of the original code state trellis

this error event is described as when the error code sequence diverges from the correct code

sequence and then later on, reaches a sequence of L consecutive twin code states where at the

end of the sequence the accumulated metric of the incorrect code sequence is lower then the

correct sequence.

58

Code

States

Filtered

Phase

Error

Sequence

Begin

of error

event

Code

states

remerge but

error event

continues

Code

states diverges

& remerges

again, still

considered

the same error

event

End

of error

event

Begin of

new error

event

Figure 20: Illustration of independent error event

Summing all error events in (63) is not practical since the number of all error events is

infinite.

The common practice is to truncate the length of the error events at a point where the

probability is insignificant, assuming that further enlarging the event would reduce its

probability further. This is not generally true for our case since error event probability is

dependent on the symbols that follow the truncated event. Our solution is based on the

derivation of the worst case extension. Let [ ]TK

p eee ,..,, 10=ε an incomplete error event of

length K. Let [ ]TKK

f eee ∞++= ,..,, 21ε be an extension of pε that terminates the error event such

that its pairwise error probability is maximized.

The sequence fε is called the worst case extension. Since worst case is needed, for simplicity

the extension is using continuous values rather than the constellation grid. When the JMAP

decoder is matched to the channel phase noise the sequence fε can be obtained in a closed

form by minimizing the argument ∑ ∑

−

n k

knk he

2

in (51). Using matrix notation we can write

59

=

=

∑ ∑ − f

pT

f

p

f

p

T

T

f

p

n k

knkBB

BBAAhe

εε

εε

εε

εε

2221

1211

2

(64)

where

=

0

1

20

10

000

.00

.0

..

h

h

hh

hh

A and B11, B12, B21, B22 are the appropriate sub-matrixes of

TAA such that

( ) ( ) ( ) ( ) fTffTppTfpTp

n k

knk BBBBhe εεεεεεεε 22122111

2

+++=

∑ ∑ − (65)

Comparing the derivative of (65) (with respect to fε ) to zero yields the following worst case

extension

( ) ( ) ( )( ) 1

22221221

−++−= TTTpTf BBBBεε (66)

Substituting [ ]fpe εε= into (51) results in the worst case extension pairwise error

probability. For the channel mismatch case, we use (66) as a useful approximation to the

worst case extension. Using the worst case extension of the matched JMAP decoder for the

mismatched decoder is a reasonable approximation since we assume that the channel miss

match is not large enough to result in a significant difference between the worst extensions of

both cases.

Consider the trellis formed by pairing transmitted and candidate code states. The search for

the error events to be included in (63) is carried out by performing a standard recursive tree

search on this trellis [10 and 8 chapter 8-2-7]. The tree is extended one branch at a time i.e.

the transmitted and candidate code sequences are extended by one trellis interval and the

symbol phase error is stored. Then, a stopping condition is checked. If the condition passed

then the pairwise error probability is calculated and added to the union bound in (63) and the

search traces back to the previous branch and proceeds to the next leaf, otherwise the tree is

extended by another branch. The stopping condition is composed of meeting either one of the

60

following two criterions. The first criterion is that the transmitted and candidate states reached

the same code state or a twin code state (in case of an RI codes) and that the filtered error

sequence has decayed to zero (the phase error (e) filtered with H(z)). For the second

condition, the incomplete error event is terminated by the worst case extension given in (66).

The second stopping criterion is passed if the resulting pairwise error probability is negligible.

A block diagram of the union bound error search algorithm is presented in Figure 21.

Although the union bound in (63) requires the summation of an infinity number of error

events, the described technique resulted in a significant reduction in complexity for a given

accuracy which enables the use of the union bound as a practical tool for upper bounding the

code error rate. Codes which have the property of being Uniform Error Probability (UEP) [23]

reduce the search time dramatically since they require the finding only of error events

unmerging from a single reference sequence.

61

Figure 21: Union bound error search algorithm

62

5 Analysis of the Phase Estimation in the JMAP

The JMAP phase noise estimation mθ per each codeword hypothesis mφ is given by

( )[ ]θσ

θ θθ

ˆlnˆ2

1minargˆ

2ˆ

2ˆ Peaxn

jm

nn

u

m n −−= ∑ . In Chapter 4.2.1 (48) we approximated mθ by

mθ ′ where ( ) ( )[ ]∑ −−−+−=′n

nnnn

m

n

u

m Pu θθθφφσ

θθ

ˆlnˆˆ2

1minarg

20

2ˆ . Here again, the notation

0

nφ is the phase of the transmitted symbols. In this chapter we will explicitly find mθ ′ per each

hypothesis m

nφ . In the approximated linear channel model presented in Figure 12 the received

sequence is nnnn uy ++= θφ 0 . The underlying assumption in the JMAP operation is that each

hypothesis tested is correct, so the phase estimation performed at each hypothesis (in the

linear channel approximation) is MAP estimation of nθ given the sequence nnn u+=θψ .

The MAP phase sequence estimation of nθ given the observation sequence nψ is equivalent

to filtering nψ with the uncausal Weiner filter [21,[25]. Denote by S(z) the uncausal Weiner

filter which is given by ( ) ( )( )zP

zPzS

ψψ

ψθ= where the notation Pxx(z) is the power spectrum density

of the series xn. Since un is white Gaussian noise then ( ) ( ) ( )22 ω

θθψθ σ j

v eGzPzP == ,

( ) ( ) ( ) ( ) 22

2

u

j

vuu eGzPzPzP σσ ωθθψψ +=+= and thus we arrive at

( ) ( )( ) ( )

( )( ) 2

22

22

u

j

v

j

v

uu eG

eG

zPzP

zPzS

σσ

σω

ω

θθ

θθ

+=

+= (67)

In conclusion, the JMAP phase noise estimation per each code sequence hypothesis is

approximated by first subtracting the code sequence estimation m

nφ from the received

sequence yn and then filtering the remaining sequence ( )m

nny φ− with the Weiner filter S(z) i.e.

63

( ) ( ) ( ) ( )[ ] ( ) ( )[ ]zuzzEzSzzYzSz mm +Θ+−=Φ−=Θ′ )()(ˆ (68)

where ( )zmΘ′ is the Z transform of mθ ′ . The approximated JMAP phase noise estimation in

(68) is suited for both matched and mismatched decoders. For the JMAP mismatch decoder,

the phase noise spectrum ( ) ( )22 ω

θθ σ j

v eGzP = used in (67) is the phase noise spectrum that the

decoder is matched to rather then the actual channel phase noise spectrum.

In case that the phase noise is modeled by a Weiner process and the decoder is matched, the

resulting Weiner filter (67) is

( ) ( )( )zzzS

λλβλ

−−= − 11 1

(69)

which is an uncausal low pass filter.

Figure 22 illustrates the JMAP phase estimation. Let the blue line illustrate the transmitted

symbols phase sequence 0φ , and let the all zero symbol phase sequence 0,...,0,01 =φ be an

incorrect hypothesis. The red line is the JMAP estimation for the incorrect hypothesis, i.e.

11 ˆˆ θφ + where 1θ was computed according to (68) and (69) neglecting thermal and phase

noises ( ) ( )( )0==Θ zuz . Neglecting noises, the JMAP phase estimation for the correct

hypothesis is zero and hence 000 ˆˆˆ φθφ =+ which is the blue line. Thus, this example

demonstrates that both JMAP estimations for the correct and incorrect hypothesizes (blue and

red lines respectively) tend towards each other and reduce the distance 1100 ˆˆˆˆnnnn θφθφ −−+ , the

same distance that is used in the approximation condition (25).

64

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time Index (n)

Phase [

radia

n]

Figure 22: Illustration of JMAP phase noise estimated Blue - Transmitted phase sequence ( 0φ )

Red - JMAP estimation for the incorrect code sequence hypothesis ( 11 ˆˆ θφ + )

Finally, we wish to compare the JMAP phase noise estimation to the PLL. The linear model

of the first order decision directed PLL is presented in Figure 23 [22 chapter 6.4] where the

received symbol is yn and the estimated transmitted symbol phase is m

nφ . Denote by PLL

nθ the

phase noise estimation of the PLL which is given by

n

PLL

n

PLL

n K∆+= −1ˆˆ θθ (70)

where n∆ is the phase error given by ( )PLL

n

m

nnn y 1ˆˆ−+−=∆ θφ . Substituting n∆ into (70) results

in

( ) ( )m

nn

PLL

n

PLL

n yKK φθθ ˆˆ1ˆ1 −+−= − (71)

When the phase noise is modeled as a Weiner process, in the steady state, the phase noise

estimation minimum mean square error ( ) 2ˆPLL

nnE θθ − is obtained with the loop gain factor

K=1-λ [7]. Substituting this factor into (71) and applying the Z transform results in

( ) ( ) ( ) ( )[ ]zzYzLz mΦ−=Θ ˆˆ (72)

65

where ( )

−−

= −11

1

zzL

λλ

. Thus, when the phase noise is modeled as Weiner process, the phase

noise estimation of the decision directed first order PLL given in (72) is similar to the JMAP

phase noise sequence estimation given in (68) except that the Weiner filter S(z) in the JMAP

estimator (given in (69)) is replaced by the filter L(z) in the PLL. The filter L(z) is the causal

part of the filter S(z) (disregarding gain difference) i.e. the low pass filter used for the phase

noise estimation in the PLL is the casual part of the uncausal Weiner filter used in the JMAP

phase noise sequence estimator.

In conclusion, the JMAP phase estimation outperforms the PLL due to two reasons. The first

is that the JMAP can be considered as performing phase estimation for all codeword

hypotheses regardless of the detected symbols (which may be incorrect). The PLL phase

estimation is performed on a single hypothesis which is composed from tentative symbol

decisions of the detector. Any erroneous symbol detection is dragged into future phase

estimation. Furthermore, decision error may cause the PLL to loose lock, a phenomena that

cannot occur in the JMAP. The second reason is that the JMAP phase estimation is done with

an un-causal filter which utilizes information from both past and future to estimate the current

symbol phase while the PLL phase estimation is done with the causal part of the filter which

uses only past information.

n∆

m

nany

m

nφPLL

nθ

PLL

n 1ˆ−θ

Figure 23: Decision Directed 1st Order PLL

66

6 Results and Discussion

Simulations results are presented for two examples of phase noise spectrums. The first is the

spectrum of a Weiner process and the second is an arbitrary spectrum which might resemble

the phase noise of a RF synthesizer. The ARMA model coefficients (chapter 2.2) of the non

Wiener phase noise are given in the table bellow. Both spectrums are shown in Figure 24 for

σv2=1 [rad

2].

ARMA model coefficients of tested non Weiner phase noise (notations of chapter 2.2)

Nominator coefficients

p0 p1 p2 p3 p4 p5 p6 p7

5.8236 -11.1540 1.4022 8.6627 -4.2187 -1.3055 0.7977 0.0786

Denominator coefficients

q0 q1 q2 q3 q4 q5 q6 q7

1.0000 -3.2259 3.0350 0.5405 -2.3695 0.9777 0.1801 -0.1377

67

10-3

10-2

10-1

101

102

103

104

105

106

107

Normalized frequency , (Fsamp=1)

|G(z

)|2 [

dB

]

Weiner spectrum

non Weiner example

Figure 24: Phase noise spectrums

In order to evaluate the accuracy of the pairwise error probability in (37) and (51) a pairwise

decoder was simulated by forcing the JMAP to select between only two possible code

sequences. The phase was quantized to 32 values (phase states) for negligible degradation due

to phase quantization. To create a valid independent error event, both sequences were

preceded and extended by a sufficient number of common symbols. Figure 25 compares

pairwise error computer simulations with analytical results, for the case that the phase noise

was modeled as a Weiner process and the JMAP decoder was matched (the analytical

expression given in (51)). The phase error sequence was e=[0,0,..0,π/4, π/2, π/4, π/2, π/4,0,..0]

and the variances used for the phase noise in the channel are σv2=1e

-2 [rad

2] and σv

2=3e

-2

[rad2]. The analytical pairwise error probability plots for the high phase noise variance

(σv2=3e

-2) show good matching to simulation results. For the low phase noise variance

(σv2=1e

-2) the analytical results are slightly below simulation results. This is expected, since

68

as mentioned in chapter 4.2.1 the approximated pairwise error probability gives a lower than

actual pairwise error probability and the gap gets smaller as the phase noise increases.

In Figure 26 we show pairwise error results for the case that the phase noise has the non

Weiner spectrum presented in Figure 24 and the JMAP decoder assumed that the phase noise

is a Weiner process i.e. mismatched decoder. The phase error sequence was as used before,

and the variances used for the channel phase noise are σv2=1e

-4 [rad

2] and σv

2=3e

-4 [rad

2],

chosen such that the degradation in code performance is similar to the Wiener phase noise

example. Obviously, the loss of the mismatched decoder relative to the matched decoder

strongly depends on the β used in the mismatched decoder metric (6). Since we only wish to

compare analysis to simulation, we can choose an arbitrary β factor for the receiver (used in

the JMAP metric (6)). Note that using the expressions developed in this paper, an

optimization on β can be performed per each code, SNR and phase noise variance. For the

mismatch case the optimal choice of β which gives lowest error rate depends on σu2, σv

2 and

the error events. As the phase noise (σv2) increases in respect to the thermal noise (σu

2) the

phase estimation filter bandwidth should be wider in order to better track the phase noise, this

is at the expense of increasing the thermal noise. Widening the bandwidth of the phase

estimation filter is done by increasing β. However, the increase of β should be restricted

according to the phase variations of the error event. Widening the phase estimator bandwidth

will increase the error rate more dramatically for slow varying (low frequency error events)

phase error events then for high varying error events (high frequency error events).

The solid and dotted lines in Figure 26 are the analytical results of the mismatched and

matched decoders, respectively. The β factor was chosen as 205.0 uσβ = . In Figure 26 the

mismatch decoder loss in respect to the matched decoder (the gap between the solid and

dotted line) was 1 dB in lower phase noise (σv2=1e

-4 [rad

2]) and 0.5 dB in higher phase noise

(σv2=3e

-4 [rad

2]). The larger loss for the lower phase noise case can be explained by the

improper choice of β for the mismatch decoder, apparently more suitable for the high phase

69

noise case. The optimal choice of β for the tested error event and channel phase noise variance

of σv2=1e

-4, is 201.0 uσβ = .

We can see that also in this case the analytical results of the mismatched decoder are slightly

lower then simulation results.

2 4 6 8 10 12 14 16

10-6

10-5

10-4

10-3

10-2

SNR [dB]

Pairw

ise E

rror

Pro

babili

ty

Figure 25: Pairwise error probability, Weiner phase noise Circle blue - simulation results of matched JMAP decoder, σv

2 =1e

-2 rad

2

Circle red - simulation results of matched JMAP decoder, σv2 =3e

-2 rad

2

Solid black - analytical results of coherent decoding without phase noise

Solid blue - matched JMAP decoder analysis, σv2 =1e-2 rad2

Solid red - matched JMAP decoder analysis, σv2 =3e

-2 rad

2

70

2 4 6 8 10 12 14

10-7

10-6

10-5

10-4

10-3

10-2

SNR [dB]

Pairw

ise E

rror

Pro

babili

ty

Figure 26: Pairwise error probability, non Weiner phase noise Solid black - analytical results of coherent decoding without phase noise

Circle blue - simulation results of mismatch JMAP decoder, σv2 =1e-4 rad2

Circle red - simulation results of mismatch JMAP decoder, σv2 =3e

-4 rad

2

Solid blue - analysis results of mismatch JMAP decoder, σv2 =1e

-4 rad

2

Dotted blue - analysis results of matched JMAP decoder, σv2 =1e

-4 rad

2

Solid red - analysis results of mismatch JMAP decoder, σv2 =3e-4 rad2

Dotted red - analysis results of matched JMAP decoder analysis, σv2 =3e

-4 rad

2

71

In the following we present simulations results of a full JMAP decoder for three different

Trellis codes denoted by Code A,B and C. Code A [8 Fig. 8-3-7] is a 4 state 2D TCM code,

rate 2/3 with 8PSK modulation. The trellis of code A is presented in Figure 27 and the symbol

mapping in Figure 28.

Code B [8 page 493] is a 4 state 2D TCM code, rate 1/2 with 4PSK modulation. The encoder

and symbol mapping of code B are presented in Figure 29 and Figure 30 respectively. Code

C [23 Fig.4] is a 64 state 4D TCM code, rate 2/3 with 8PSK modulation designed for multiple

symbol noncoherent detection, a different approach for solving the same phase noise problem.

The 4D encoder of code C is presented in Figure 31 and the 2D symbol mapping in Figure 32.

Code A and C are RI and Code B is partial RI. The encoder of Code C already incorporates

differential encoding and for Codes A and B (originally designed for coherent decoding) an

appropriate differential encoding in Z4 and Z2 was added (the modulo 4 and 2 groups)

respectively. For Code B, although it is not a rotational invariant code, differential encoder

prevents from long bit error bursts caused by error events which have long portions of

constant phase shift. While implementing the Viterbi algorithm for the JMAP decoder, we

realized that it is important to perform the differential decoding on the survivor rather then at

the output of the Viterbi decoder. This is since constant phase shift sequences, if exist, are

equally likely and the decoder randomly switches between them. If the differential decoding

is done on the Viterbi decoder output (i.e. the symbol at the final branch of the back tracking),

then a switch between constant phase sequences (a switch occurs when the state with the best

metric from which the back tracking is done is the state of the rotated sequence) will produce

an error. However, applying differential decoding along a surviving sequence which has a

constant phase shift from the transmitted sequence will not cause an error in the detected bits.

If M is the number of phase states in the 2D TCM JMAP Trellis decoder then the number of

phase states in the 4D TCM JMAP Trellis decoder should be M2 since each branch transaction

results in two 2D symbols each with its separate phase estimations. In this case the resulting

72

overall number of states would be too large and therefore impractical to be implemented.

However, our simulation results showed that negligible degradation in performance is

obtained by using only M phase states which are the phase estimation for the second 2D

symbol (from the 4D symbol) in the branch transaction. The phase estimation of the first 2D

symbol is obtained by linear interpolation. For the BER analysis of the 4D TCM BER it was

assumed that the JMAP decoder was implemented optimally with M2 phase states.

Figure 27: Code A trellis

Figure 28: Code A symbol mapping

73

Figure 29: Code B encoder

Figure 30: Code B symbol mapping

74

D

+

+

D

x

4

+

Modulo 2

Modulo 8

x x x01 3

Modulo 4

+

+

D

x

4

+

Modulo 2

Modulo 8

x x x12 2

Modulo 4

a0

a1

D D

Figure 31: Code C 4D encoder

Figure 32: Code C 2D symbol mapping

75

In Figure 33, Figure 34 and Figure 35 we present simulations and analytical results of Code

A, B and C, respectively, for the case that the channel phase noise is modeled as a Weiner

process and the JMAP decoder was matched to the channel phase noise. In Figure 36, Figure

37 and Figure 38 we present the results of the Code A,B and C respectively for the case that

the channel phase noise spectrum is the non Weiner phase noise presented in Figure 24. The

computer simulated JMAP decoder assumed the phase noise is a Weiner process i.e. channel

mismatch. The BER of the simulation results are compared in the Figures to analytical results

of a matched (dotted line plots) and mismatched (solid line plots) JMAP decoders.

For both Wiener and non Wiener phase noise spectrums two phase noise variances were

chosen to test the analytical results accuracy for lower and higher phase noise values. The

blue plots are for the lower phase noise values of σv2=1e

-2 [rad

2]

and σv

2=1e

-4 [rad

2] for the

Wiener and non Wiener phase noise spectrums respectively. The red plots are for the higher

phase noise values of σv2=3e

-2 [rad

2]

and σv

2=3e

-4 [rad

2] for the Wiener and non Wiener phase

noise spectrums respectively. For the mismatch decoder metric we used 205.0 uσβ = which

showed to give overall good performance. For both matched and mismatched simulated

JMAP trellis decoder the number of phase states was 32.

It is evident that the BER analytical results for when the phase noise has a Weiner spectrum

and the JMAP decoder is matched (solid lines in Figures 32-34) and for the non Weiner

spectrum and mismatch decoder (solid lines in Figures 35-37), tightly upper bound the

simulation results for Code A and C but for Code B give slightly lower BER then simulation

results. This is explained by the fact that the code rate and constellation size of Code B are

lower then Code A and C and some of its high probability error events violate the

approximation condition in (25).

Applying the union bound on the pairwise error probabilities results in an upper bound on the

BER. However, since the approximated pairwise error probability is lower then the true error

76

probability, applying union bound to the approximated pairwise error probabilities results in

an approximation to the BER which is no longer an upper bound. For Codes A and C, due to

the relatively high code rate and constellation size the analytical pairwise error probability

tightly approximates the simulation pairwise error probability and hence the union bound

results (solid lines in Figures 32, 34, 35 and 37) should be slightly below the result of the

union bound on the accurate pairwise error probabilities and still gives a higher error rate then

simulation result. As the phase noise increases the approximation is more accurate and hence

the union bound on the approximated pairwise errors probability increases even more towards

the union bound on the accurate pairwise error probabilities. Thus, for code A and C, the gap

between the analytical (solid lines in Figures 32, 34, 35 and 37) and simulation results slightly

increases for high phase noise variance. For code B, due to the low code rate and constellation

size the approximation is less accurate and hence the union bound results (solid lines in

Figures 33 and 36) in a lower BER then simulation results. Again, as the phase noise variance

increases the approximation is more accurate which increases the BER result and the gap

from simulation results is smaller.

Although Code A and C have the same code rate and constellation size, the gap between the

analytical results and simulation results for Code A (solid lines in Figures 32 and 35) is larger

then for Code C (solid lines in Figures 34 and 37). The reason is that Code A has a significant

larger number of error events with high probability then Code C has, which increases the

union bound result. We also like to note that the performance of Code C which was designed

to cope with phase noise (originally for noncoherent decoding) is about 2 dB better then Code

A which was designed for coherent decoding.

The dotted lines in Figures 35-37 are the analytical results of the JMAP decoder matched to

the non Weiner phase noise spectrum. Implementing such a receiver is impractical, however it

is interesting for evaluating the loss of the mismatched decoder from the matched decoder.

77

In figures 35-37 it is seen that in high phase noise variance the mismatch decoder

performance of codes A and C have a relatively large loss of 2.5 dB in respect to the matched

decoder while the loss of code B is only 0.75 dB. This is due to the lower code rate and

constellation size of Code B which as a result has filtered error events (error events after being

filtered with H(z)) with larger distances then Code A and C and hence is less affected by the

residual untracked phase noise (due to the mismatch).

4 6 8 10 12 14 16

10-6

10-5

10-4

10-3

10-2

Eb/No [dB]

BE

R

Figure 33: Code A performance - Weiner phase noise & matched JMAP decoder Circle black - simulation results of coherent decoding no phase noise

Circle blue - simulation results of matched JMAP decoder, σv2 =1e

-2 rad

2

Circle red - simulation results of matched JMAP decoder, σv2 =3e

-2 rad

2

Solid blue - matched JMAP decoder analysis, σv2 =1e-2 rad2

Solid red - matched JMAP decoder analysis, σv2 =3e

-2 rad

2

78

4 5 6 7 8 9 10 11 12 13

10-6

10-5

10-4

10-3

10-2

Eb/No [dB]

BE

R

Figure 34: Code B performance - Weiner phase noise & matched JMAP decoder Circle black - simulation results of coherent decoding no phase noise

Circle blue - simulation results of matched JMAP decoder, σv2 =1e

-2 rad

2

Circle red - simulation results of matched JMAP decoder, σv2 =3e

-2 rad

2

Solid blue - matched JMAP decoder analysis, σv2 =1e-2 rad2

Solid red - matched JMAP decoder analysis, σv2 =3e

-2 rad

2

79

4 5 6 7 8 9 10 11 12 1310

-6

10-5

10-4

10-3

10-2

Eb/No [dB]

BE

R

Figure 35: Code C performance - Weiner phase noise & matched JMAP decoder Circle black - simulation results of coherent decoding no phase noise

Circle blue - simulation results of matched JMAP decoder, σv2 =1e

-2 rad

2

Circle red - simulation results of matched JMAP decoder, σv2 =3e

-2 rad

2

Solid blue - matched JMAP decoder analysis, σv2 =1e

-2 rad

2

Solid red - matched JMAP decoder analysis, σv2 =3e-2 rad2

80

4 6 8 10 12 14 16

10-6

10-5

10-4

10-3

10-2

Eb/No [dB]

BE

R

Figure 36: Code A performance - non Weiner phase noise & mismatch JMAP Circle black - simulation results of coherent decoding no phase noise

Circle blue - simulation results of mismatch JMAP decoder, σv2 =1e-4 rad2

Circle red - simulation results of mismatch JMAP decoder, σv2 =3e-4 rad2

Solid blue - analysis results of mismatch JMAP decoder, σv2 =1e

-4 rad

2

Dotted blue - analysis results of matched JMAP decoder, σv2 =1e

-4 rad

2

Solid red - analysis results of mismatch JMAP decoder, σv2 =3e-4 rad2

Dotted red - analysis results of matched JMAP decoder analysis, σv2 =3e-4 rad2

81

4 5 6 7 8 9 10 11

10-6

10-5

10-4

10-3

10-2

Eb/No [dB]

BE

R

Figure 37: Code B performance - non Weiner phase noise & mismatch JMAP Circle black - simulation results of coherent decoding no phase noise

Circle blue - simulation results of mismatch JMAP decoder, σv2 =1e

-4 rad

2

Circle red - simulation results of mismatch JMAP decoder, σv2 =3e

-4 rad

2

Solid blue - analysis results of mismatch JMAP decoder, σv2 =1e

-4 rad

2

Dotted blue - analysis results of matched JMAP decoder, σv2 =1e-4 rad2

Solid red - analysis results of mismatch JMAP decoder, σv2 =3e

-4 rad

2

Dotted red - analysis results of matched JMAP decoder analysis, σv2 =3e

-4 rad

2

82

4 5 6 7 8 9 10 11 12

10-6

10-5

10-4

10-3

10-2

Eb/No [dB]

BE

R

Figure 38: Code C performance - non Weiner phase noise & mismatch JMAP Circle black - simulation results of coherent decoding no phase noise Circle blue - simulation results of mismatch JMAP decoder, σv

2 =1e

-4 rad

2

Circle red - simulation results of mismatch JMAP decoder, σv2 =3e

-4 rad

2

Solid blue - analysis results of mismatch JMAP decoder, σv2 =1e-4 rad2

Dotted blue - analysis results of matched JMAP decoder, σv2 =1e

-4 rad

2

Solid red - analysis results of mismatch JMAP decoder, σv2 =3e

-4 rad

2

Dotted red - analysis results of matched JMAP decoder analysis, σv2 =3e

-4 rad

2

83

7 Summary

In this research, we developed a novel closed-form analytical expression which approximates

the bit error rate of the JMAP decoder for Trellis Code Modulation with MPSK modulation,

for any arbitrary phase noise model (i.e. not limited to Weiner process) and either a matched

or mismatched decoder. The closed form expressions are easy to use, and provide tools for

system optimization.

We have obtained an approximation on the pairwise error probability and then applied the

union bound on the pairwise errors. However, for our use a new form of error events had to be

defined, together with a novel method to search for error events to be included in the union

bound that resulted in a significant reduction in complexity for a given accuracy. This method

enables the use of the union bound as a practical tool for approximating the code error rate.

We have also analyzed the JMAP phase estimation and compared it to the PLL. We showed

that the matched JMAP phase estimation per each codeword hypothesis can be approximated

by applying the uncasual Weiner filter on the received sequence phase. The JMAP

outperforms the PLL since the PLL phase estimation is done by using only the causal part of

the Weiner filter. Furthermore, the PLL is updated with tentative decisions that may cause it

to lose lock while the JMAP can be considered as performing phase estimation for all

codeword hypotheses regardless of the detected symbols (which may be incorrect).

We have tested the accuracy of the analytical expressions for the following two phase noise

spectrums: the spectrum of a Weiner process and an arbitrary phase noise spectrum which

might resemble the phase noise of a RF synthesizer. For the former we compared simulation

and analytical results of a matched decoder and for the latter we compared mismatched

decoder simulation results to both matched and mismatched analytical results. Although,

implementing a matched decoder to the phase noise with non-Weiner spectrum is impractical,

84

the analytical result of such a decoder serves as lower bound on performance as well as a tool

for evaluating the loss between the mismatch and matched decoders.

We presented the simulation and analytical results for three different TCM codes and various

phase and thermal noise variances. In conclusion, the analytical expression tightly

approximates the simulation results (usually better than 1dB) when the code rate is equal or

greater then 1/2 and the MPSK constellation size is larger then two i.e. M>2.

The developed tool will enable designers to take into consideration the influence of the code

characteristics, decoder implementation (which varies in the mismatch case e.g. β in the case

of a JMAP decoder matched to Weiner phase noise model) and RF synthesizer phase noise on

the receiver performance. It further enables joint optimization of the synthesizer, the code and

the decoder for the achievement of lowest error rate as well as provides important insights

which is necessary for code design.

85

Appendix A

The Figure bellow presents the equivalent channel model introduced in chapter 4 (Figure 12).

Recall from the system model definition ( 2.2) that φ m, y, u and θ are vector representation of

m

nφ , yn, un and θn respectively.

The estimated transmitted symbol sequence obtained from the joint phase and symbol

sequence MAP estimation is given by

( ) yP m

m /,maxmaxarg θφθ (a.1)

The estimated transmitted symbol sequence obtained from the symbol sequence MAP decoder

is obtained by

( )yP m

m /maxarg φ (a.2)

In the following, we prove that in the described linear model where u and θ are additive

Gaussian noise both criterion are equivalent i.e. both criterion estimate the same code

sequence φ m and thus ( ) ( )yPyP m

m

m

m /maxarg/,maxmaxarg φθφθ = .

Using Bayes’ theorem we can write ( ) ( ) ( ) yPyPyP mmm /,/max/,max φφθθφ θθ = .

Since ( )yP m /φ is independent of θ then

( ) ( )[ ] ( )yPyPyPmmm

/*,/max/,max φφθθφ θθ = (a.3)

86

We use the following equality ( ) ( ) ( )xPyPyP mm //,/ θφθφθ =−= where

myx φ−= . Since x and θ are mutual Gaussian vectors of length m then it can be shown [24]

that ( )xx CNx // ,~/ θθηθ , where x/θη and xC /θ are the mean and covariance of θ/x. Since

( )xP /θ is Gaussian then ( ) ( ) x

mCxP /21/max θθ πθ = .

The conditional covariance is given by θθθθθ xxxxx CCCCC 1

/

−−= [24].

It can be shown that θθC , xCθ , 1−xxC and θxC are independent from θ, y and mφ and hence

xC /θ is also independent from them. We define ( ) x

mCK /21 θπ= so that

( ) ( ) KxPyP m == /max,/max θφθ θθ (a.4)

By substituting (a.4) into (a.3) and the result into (a.1) we arrive at the following joint phase

symbol MAP estimator:

( ) ( ) yPKyP m

m

m

m /*maxarg/,maxmaxarg φθφθ = (a.5)

Since K is not a function of mφ it has no effect on the maximization in (a.5) and hence (a.5) is

equal to (a.2) which means both estimators (a.1) and (a.2) yield the same estimated code

sequence.

87

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תקציר

ככל שתדר האפנון גדל בעיית רעש הפאזה מחריפה . בעיית רעש הפאזהבתקשורת בתדר אפנון גבוה קיימת

שערוך משותף של סדרת הקוד והפאזה .ים מתאימם לעקיבה אחר רעש הפאזה אינים הקונבנציונאליפתרונותוה

JMAP)( ,וםאופטימקרובים לה לעקיבה אחר רעש הפאזה בעל ביצועים ישים פתרון והינ) הפרש זניח(.

הינו יחסית JMAP מקלט ה ,עבור מודל זה. Weinerרעש הפאזה ממודל כתהליך אקראי מסוג , באופן טיפוסי

RF synthesizer -ל איננו מתאר נכון את רעש הפאזה של ה" המודל הנבמקרים מרובים, לעומת זאת. פשוט

רון פרקטי שבדרך כלל נותן ביצועים טובים תפ. פרקטימותאם לרעש הפאזה בערוץ איננו הJMAPמימוש מקלט ו

שונה הוא שרעש הפאזה בערוץ למורותזאת וWeiner שתוכנן עבור רעש פאזה מסוג JMAP במקלט שימושהינו

.כלומר מקלט לא מתואם לערוץ

, מיםבמחקרים קוד . עבור מערכות מקודדותJMAP -לא קיים כלי אנליטי לאנליזת ביצועים של מקלט הכיום

הגישות במחקרים . עבור המקרה הלא מקודדJMAP -ם להסתברות השגיאה של מקלט הי אנליטיביטוייםפותחו

והמקלט היה Weinerמודל רעש הפאזה הוגבל למודל , יתר על כן .הללו לא ניתנים להרחבה למקרה המקודד

.מתואם לערוץ

עבור JMAPשל מקלט ה ) (BERיאה לביט גרות הש קירוב להסתבהינוביטוי אנליטי סגור אשר פותח , בתזה הזו

Trellis Code Modulationת ושידור בקונסטלציי MPSK .לא (רעש פאזה כלשהוא ל הביטוי האנליטי מתאים

.כזה שאינו מתואם מקלט מתואם או עבורו) Weinerמוגבל למודל של

עם הינה MPSK - ה שקונסטלציית במקרההראו קירוב הדוק , שהושוו לתוצאות סימולציההביטויים האנליטיים

M>2הקוד גדול או שווה לחציבקצש ו .

מימוש המקלט ורעש , את השפעת תכונות הקודלהביןשפותח בעבודה זו יאפשר למתכננים האנליטי הייחודי הכלי

-ל, יאפשר הכלי לעשות אופטימיזציה משותפת לקוד,בנוסף. על ביצועי המקלטRF synthesizer -הפאזה של ה

RF synthesizerהשגת הסתברות שגיאה מינימאלית או לצורך אופטימיזציות אחרות ולמקלט לשם .

אביבאוניברסיטת תל ש איבי ואלדר פליישמ"הפקולטה להנדסה ע

סליינרש זנדמ"בית הספר לתארי מתקדמי ע

גלאי אנליזת ביצועי של

בערו TCM אופטימאלי של אזהע רעש פ

ואלקטרוניקה חשמלבהנדסה " מוסמ אוניברסיטה"חיבור זה הוגש כעבודת גמר לקראת התואר

ידיעל

עודד ביאלר

ר ד רפאלי" להנדסת חשמל ואלקטרוניקה בהנחית דס"בביההעבודה נעשתה ט"תשסתשרי

אביבאוניברסיטת תל ש איבי ואלדר פליישמ"עהפקולטה להנדסה

סליינרש זנדמ"בית הספר לתארי מתקדמי ע

גלאי אנליזת ביצועי של

בערו TCM אופטימאלי של ע רעש פאזה

ואלקטרוניקה חשמלבהנדסה " מוסמ אוניברסיטה"חיבור זה הוגש כעבודת גמר לקראת התואר

ידיעל

עודד ביאלר

ר ד רפאלי" להנדסת חשמל ואלקטרוניקה בהנחית דס"בביההעבודה נעשתה ט"תשסתשרי