PULSE SHAPED 8-PSK BANDWIDTH EFFICIENCY AND … · narrowing bandwidth and increasing band...

PULSE SHAPED 8-PSK

BANDWIDTH EFFICIENCY

AND SPECTRAL SPIKE ELIMINATION

JIANPING TAO, B.EBG.

NMSU-ECE-98-003 MAY 1998

Presented to Dr. James LeBlanc

in partial fulfillment of the requirements

for the Degree

Master of Science in Electrical Engineering

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New Mexico State University

Las Cruces, New Mexico

May 1998

https://ntrs.nasa.gov/search.jsp?R=19980055189 2018-07-09T23:06:39+00:00Z

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ACKNOWLEDGMENTS

I would like to thank Dr. James P. LeBlanc, my advisor, for his help and

direction in the development and preparation of this report. I am truly indebted to Dr.

Sheila B. Horan who brought me to the communication field, for her advice, help and

direction in my research. I also wish to thank Dr. Phillip De Leon for his direction on

the sampling theorem and advice in my report. I thank Dr. Tonghui Wang for his

valuable advice in my report. I am grateful to Dr. Marvin K. Simon at the Jet

Propulsion Laboratory, for providing me his report on discrete spectrum.

I would like to thank the National Aeronautics and Space Administration

(NASA) for their support under Grant NASA# NAG 5-1491 and for giving me the

chance to work on this project. Also a special thank you to the people in the Center

for Space Telemetry and Telecommunications Systems at New Mexico State

University for their assistance.

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ABSTRACT

PULSE SHAPED 8-PSK

BANDWIDTH EFFICIENCY

AND SPECTRAL SPIKE ELIMINATION

BY

JIANPING TAO, B.EBG.

Master of Science in Electrical Engineering


Las Cruces, New Mexico, 1998

Dr. James P. LeBlanc, Chair

The most bandwidth-efficient communication methods are imperative to cope

with the congested frequency bands. Pulse shaping methods have excellent effects on

narrowing bandwidth and increasing band utilization. The position of the baseband

filters for the pulse shaping is crucial. Post-modulation pulse shaping (a low pass

filter is located after the modulator) can change signals from constant envelope to

non-constant envelope, and non-constant envelope signals through non-linear device

(a SSPA or TWT) can further spread the power spectra [6]. Pre-modulation pulse

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shaping (a filter is located before the modulator) will have constant envelope. These

two pulse shaping methods have different effects on narrowing the bandwidth and

producing bit errors. This report studied the effect of various pre-modulation pulse

shaping filters with respect to bandwidth, spectral spikes and bit error rate. A pre-

modulation pulse shaped 8-ary Phase Shift Keying (8PSK) modulation was used

throughout the simulations.

In addition to traditional pulse shaping filters, such as Bessel, Butterworth

and Square Root Raised Cosine (SRRC), other kinds of filters or pulse waveforms

were also studied in the pre-modulation pulse shaping method. Simulations were

conducted by using the Signal Processing Worksystem (SPW) software package on

HP workstations which simulated the power spectral density of pulse shaped 8-PSK

signals, end to end system performance and bit error rates (BERs) as a function of

Eb/No using pulse shaping in an AWGN channel. These results are compared with

the post-modulation pulse shaped 8-PSK results.

The simulations indicate traditional pulse shaping filters used in pre-

modulation pulse shaping may produce narrower bandwidth, but with worse BER

than those in post-modulation pulse shaping. Theory and simulations show pre-

modulation pulse shaping could also produce discrete line power spectra (spikes) at

regular frequency intervals. These spikes may cause interference with adjacent

channel and reduce power efficiency. Some particular pulses (filters), such as

trapezoid and pulses with different transits (such as weighted raised cosine transit)

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were found to reduce bandwidth and not generate spectral spikes. Although a solid

state power amplifier (SSPA) was simulated in the non-linear (saturation) region,

output power spectra did not spread due to the constant envelope 8-PSK signals.

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TABLE OF CONTENTS

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LIST OF TABLES .................................................................................................................. viii

LIST OF FIGURES ................................................................................................................ ix

LIST OF ABBREVIATIONS .............................................................................................. xm

Chapter

1 INTRODUCTION ................................................................................................................

2 THEORY ................................................................................................................................ 5

2.1 8-PSK Pulse Shaping Methods ......................................................................... 5

2.2 Filters Used for 8-PSK Pulse Shaping ............................................................... 8

2.3 Power Spectral Components and Pulse Shaping Waveforms ...................... 15

2.3.1 Baseband Pulse Shaped 8-PSK Vector-Matrix Notation ..................... 15

2.3.2 Power Spectral Components of Pulse Shaped 8-PSK ........................ 18

2.3.3 Pulse Shaping Waveforms ................................................................. 23

2.4 Solid State Power Amplifier (SSPA) .......................................................... 32

2.5 Additive White Gaussian Noise Channel ........................................................... 33

2.6 Receiver And Error Rate Estimation System ........................................................ 34

3 PULSE SHAPED 8-PSK SIMULATIONS ................................................................. 37

3.1 Simulations on Power Spectral Density ............................................................... 37

3.1.1 PSD Simulation Results of Using Traditional Filters ......................... 39

3.1.2 PSD Comparison Between Two Pulse Shaping Methods ............... 47

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3.1.3 Simulation Results of Particular 8-PSK Waveforms ......................... 49

3.1.4 Analysis Results of Particular 8-PSK Waveforms ............................ 53

3.2 Bit Error Rates Simulations ................................................................................. 55

4 CONCLUSIONS AND RECOMMENDATIONS ....................................................... 61

REFERENCE ........................................................................................................ 62

vii

LIST OF TABLES

= =

Table 2-1

Table 3-1

Table 3-2

Table 3-3

Table 3-4

Table 3-5

Table 3-6

Gray Codes and Phase ¢ ................................................................................... 6

Traditional Filters Used in the Simulations ................................................. 38

Utilization Ratio for Post-Modulation Pulse Shaping 8-PSK .............. 48

Utilization Ratio for Pre-Modulation Pulse Shaping 8-PSK ................ 48

PSD Line Components for Trapeziod Shaping .................................. 53

PSD Line Components for Weighted Raised Cosine Transit

Shaping ............................................................................................... 53

PSD Line Components for Raised Cosine Shaping ........................... 53

Table 3-7 PSD Line Components for Cosine Transit Shaping .......................... 54

Table 3-8 Utilization Ratio of Pre-Modulation Pulse Shaping with Different

Waveforms ........................................................................................... 54

Table 3-9 ISI Loss Measurements at 10 -3 BER (Compared to Ideal 8-PSK) ....... 59

Table 3-10 ISI Loss Measurements at 10 _ BER (Compared to Ideal BPSK) ....... 59

VII1

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LIST OF FIGURES

Figure 2-1

Figure 2-2

Figure 2-3

Figure 2-4

Figure 2-5

Figure 2-6

Figure 2-7

Figure 2-8

Figure 2-9

8-PSK Signal Constellation ......................................................................... 6

Producing Phase Signals Block Diagram ........................................... 7

Modulation Schemes For Unfiltered And Filtered Phase Signals ....... 8

Amplitude Response of Various Orders of Butterworth Filters ......... 9

An Impulse Response of 5th-Order of Butterworth Filters .............. 9

Amplitude Response of Various Orders of Bessel Filters .................. 10

An Impulse Response of 3rd-Order of Bessel Filters ........................ 1I

Frequency Response: Raised Cosine Characteristics with Different

rolloff factor a .................................................................................. 12

Impulse Responses of Raised Cosine Filters for Different a ............. 12

Figure 2-10 The Waveforms of Phase _ and Filtered Phase (p' .......................... 13

Figure 2-11 The Waveforms of Real Part and Imaginary Part of 8-PSK .............. 14

Figure 2-12 The Waveforms of Real Part and Imaginary Part of Pre-Modulation

Filtered 8-PSK .................................................................................. 15

Figure 2-13 Pulses with Different Overlapping Index K from [1 ] ....................... 24

Figure 2-14 T-s Duration Cosine Pulses Used as Phase Pulse and 8-PSK

Signaling ........................................................................................... 25

Figure 2-15 2T-s Duration Cosine Pulses Used as Phase Pulse and 8-PSK

Signaling ........................................................................................... 26

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2 S

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Figure 2-16

Figure 2-17

Figure 2-18

Figure 2-19

T-s Duration Raised Cosine Pulses Used as Phase Pulse and 8-PSK

Signaling ........................................................................................... 27

2T-s Duration Raised Cosine Pulses Used as Phase Pulse and

8-PSK Signaling ............................................................................... 28

Weighted Raised Cosine Pulses Used as Phase Pulse and 8-PSK

Signaling ........................................................................................... 29

2T-s Duration Trapezoid Pulses Used as Phase Pulse and 8-PSK

Signaling ........................................................................................... 30

Figure 2-20 Weighted Raised Cosine Transit Pulses Used as Phase Pulse and

8-PSK Signaling ............................................................................... 31

Figure 2-21 The Output Characteristic of 10 Watts SSPA ................................... 32

Figure 2-22 The Structure of SSPA ...................................................................... 33

Figure 2-23 Power Spectral Density of White Gaussian Noise ............................. 33

Figure 2-24 Block Diagram for the Receiver Error Rate Estimator ..................... 34

Figure 2-25 Block Diagram of Synchronizer ....................................................... 35

Figure 2-26

Figure 3-1

Figure 3-2

Figure 3-3

Figure 3-4

Simple Error Rate Estimator Block Diagram ................................... 36

Block Diagram of Spectrum analysis ................................................ 37

Unfiltered 8-PSK Power spectral Density .......................................... 40

Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth

Filter (BT=I) ...................................................................................... 41

Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth

X

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=_

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Filter (BT=2) ......................................................................................41

Figure 3-5 Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth

Filter (BT=2.8) ................................................................................... 42

Figure 3-6 Power spectra of 8-PSK Pulse Shaped with 5th-Order Butterworth

Filter (BT=3) ....................................................................................... 43

Figure 3-7 Power spectra of 8-PSK Pulse Shaped with 5th-Order Bessel

Filter (BT=I) ...................................................................................... 43


Filter (BT= 1.2) ................................................................................... 44


Filter (BT=2) ...................................................................................... 45


Filter (BT=3) ...................................................................................... 45

Figure 3-11 Power spectra of 8-PSK Pulse Shaped with SRRC Filter (a =1) ..... 46

Figure 3-12 Utilization Ratio of Using 5th-Order Butterworth Filters in

Pre-Modulation Pulse Shaping ........................................................ 48

Figure 3-13 Utilization Ratio of Using 3rd-Order Bessel Filters in

Pre-Modulation Pulse Shaping ........................................................ 49

Figure 3-14 PSD Comparison of 8-PSK Between Two Pulse Shaping Methods

(5th-Order Butterworth Filter, BT =1 ) ............................................ 49

Figure 3-15 PSD of 8-PSK Used 2T-s Duration Trapezoid Pulse Shaping ......... 5o

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t --

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Figure 3-16 PSD of 8-PSK Used 3T-s Duration Trapezoid Pulse Shaping ......... 51

Figure 3-17 PSD of 8-PSK Used 2T-s Duration Cosine Transit Signaling Pulse

Shaping ............................................................................................. 51

Figure 3-18 PSD of 8-PSK Used 2T-s Raised Cosine Transit Signaling Pulse

Shaping ............................................................................................. 52

Figure 3-19 PSD of 8-PSK Used 2T-s Duration Weighted Raised Cosine

Transit Signaling Pulse Shaping ........................................................ 52

Figure 3-20 Utilization Ratio of Different Pulses in Pre-Modulation Pulse

Shaping .............................................................................................. 54

Figure

Figure

Figure

Figure

Figure

Figure

3-21 BER simulation System Block Diagram .......................................... 55

3-22 BER Plot for Pre-Modulation Pulse Shaping Filters with BT--1 ...... 56

3-23

3-24

3-25

3-25

BER Plot for Pre-Modulation Pulse Shaping Filters with BT=2 ...... 57

BER Plot for Pre-Modulation Pulse Shaping Filters with BT--3 ...... 57

BER Plot for Pre-Modulation Pulse Shaping With 3T Trapezoid ... 58

Plot for ISI Loss as Utilization Ratios for Pre-Modulation Pulse

Shaping .............................................................................................. 6O

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k..,LIST OF ABBREVIATIONS

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- ==-

•r rmmB

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(X

P

AWGN

dB

BER

BPSK

BW

BT

CCSDS

Eb/No

ESA

fs

fc

FFT

Hz

HP

IIR

ISI

JPL

LSB

MSB

MSK

NASA

NMSU

NRZ-L

QPSK

Rolloff Factor for SRRC filters

Band Utilization Ratio

Additive White Gaussian Noise

Decibels

Bit Error Rate

Binary Phase Shift Keying

Bandwidth

Bandwidth-Time (symbol) product

Consultative Committee for Space Data Systems

Energy (bit) to Noise ratio

European Space Agency

Sampling Frequency

Carrier Frequency

Fast Fourier Transform

Hertz

Hewlett Packard

Infinite Impulse Response (Filter)

Intersymbol Interference

Jet Propulsion Laboratory

Least Significant Bit

Most Significant Bit

Minimum Shift Keying

National Aeronautics and Space Administration


Non-Return-to-Zero Logic

Quaternary Phase Shift Keying

xiii

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.__

i

W

= --

I

W

-- I

PA

PCM

PM

PSD

PSK

QPSK

Rb

Rs

RC

sec

SFCG

SNR

SPW

SSPA

SRRC

Tb

TWT

Power Amplifier

Pulse Coded Modulation

Phase Modulation

Power Spectral Density

Phase Shift Keying

Quartenary Phase Shift Keying

Bit Rate

Symbol or Baud Rate

Raised Cosine

Seconds

Space Frequency Coordination Group

Signal-to-Noise Ratio

Signal Processing Worksystem

Solid State Power Amplifier

Square Root Raised Cosine

Bit Period

Traveling Wave Tube

xiv

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Chapter 1

INTRODUCTION

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_mu

E--

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With the continuing development of communications and increasing users,

frequency bands are becoming more and more congested. In order to cope with this

frequency congestion, methods to increase bandwidth utilization have be studied in

[1]-[6] and [13]-[16]. These methods consider efficient modulation which is

implemented using pulse shaping methods.

Two kinds of pulse shaping methods have been investigated to realize

bandwidth efficiency. One kind of pulse shaping methods is pre-modulation pulse

shaping which performs a low pass filtering before modulation to reduce bandwidth.

Pre-modulation pulse shaping with different modulation schemes, such Pulse Code

Modulation (PCM), Binary Phase Shift Keying (BPSK), Quaternary Phase Shift

Keying (QPSK) and Gaussian filtered Minimum Shift Keying (GMSK) have been

studied in [13]-[16]. The other pulse shaping method is post-modulation pulse

shaping which conducts low pass filtering after modulation. Caballero [6] has used

this method to simulate 8 level Phase Shift Keying (8-PSK) signaling over non-linear

satellite channels.

In [6] post-modulation pulse shaping method is investigated with various

filters. Three kind of filters, consisting of a fifth order Butterworth, third order Bessel

and Square Root Raised Cosine (SRRC) filters, with different Bandwidth-Time (BT)

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product or roll off factor, have been used to reduce bandwidth in the simulations of 8-

PSK over non-linear satellite channels. Because of post-modulation pulse shaping, it

produced a non-constant envelope pulse shaped 8-PSK which can spread the power

spectrum when the signals go through a non-linear Solid State Power Amplifier

(SSPA). The simulations also included measures of symbol error rate (SER) for post-

modulation pulse shaped 8-PSK signaling. Simulation results indicated that there is

trade-off between bandwidth efficiency and SER.

Prabhu and Rowe [1] presented a matrix method for computing the power

spectrum of digital phase modulation in 1974, and studied pre-modulation pulse

shaped pulse train with or without overlapping. Pre-pulse shaping may produce not

only a continuous component but also a discrete line component at some particular

frequencies in power spectral density (PSD). Based on Prabhu and Rowe, Simon

recently obtained explicit expressions for power spectrum when signaling is m-ary

PSK in the presence of intersymbol interference (ISI) produced by pulse shaping.

Martin and Nguyen in [13],[15] and [16], Otter in [14] have studied different

traditional filters (Butterworth, Bessel and SRRC filters) used as pre-pulse shaping

for different modulation types (PCM, BPSK, QPSK and GMSK). By comparing pre-

modulation pulse shaping and post-modulation pulse shaping, Martin and Nguyen

indicated that pre-modulation pulse shaping can overcome disadvantages produced by

post modulation pulse shaping, such as larger, heavy filters which are not compatible

with some mission operations requirements.

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In contrast to using traditional filters by Martin and Nguyen, Greenstein [4]

has examined PSK with different overlapping baseband pulses. Greenstein

investigated triangular, cosine, raised cosine and Nyquest pulse shapes, yielding the

results for continuous and discrete line components of PSD.

Although many studies have investigated pre-modulation pulse shaping and

post-modulation pulse shaping with various filters, post-modulation pulse shaping can

induce a non-constant envelope signaling which can spread the reduced bandwidth.

Pre-modulation pulse shaping can induce power spectral spikes which are not good

for power efficiency and may produce adjacent channel interference. A study is

needed for pre-modulation pulse shaping which produces a constant modules signal

with no spikes

This project conducted simulations on pre-pulse shaped 8-PSK over non-

linear channel. First, three kinds of traditional filters, 5th order Butterworth, 3rd order

Bessel and SRRC, were used as pulse shaping filter applied to 8-PSK phase signals.

Filtered phase signals were then modulated to become pulse shaped 8-PSK signals.

Pulse shaped 8-PSK signals were amplified by a non-linear SSPA. In order to

simulate channel noise, an additive white Gaussian noise (AWGN) was added. At the

receiver, an integrate and dump circuit was used. Power spectra produced by pre-

modulation pulse shaping were compared with those using post-modulation pulse

shaping. In addition, received bit error rate 03ER) was also compared between two

pulse shaping methods. Second, theoretical power spectrum for pulse shaped 8-PSK

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was studied based on [1] and [2]. Third, many pulse waveforms were found for 8-

PSK to reduce bandwidth and reduce or eliminate power spectral spikes. Simulations

and numerical results indicated that trapezoid and weighted raised cosine transit

pulses with more than two symbol periods can get rid of spectral spikes. These results

are in agreement with Simon's recent report [3]. Finally, According to simulation

results, recommendations were given for searching for optimal pulses to reduce

bandwidth without spikes.

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Chapter 2

THEORY

This section will introduce some of the concepts which were used for this project.

First, 8-PSK signals and pulse shaping methods will be described. Butterworth, Bessel and

Raised Cosine Filters are discussed as traditional pulse shaping filters. Second, a detailed

discussion of pulse shaping waveforms and power spectral components of pre-modulation

pulse shaped PSK signals will be presented. Pre-modulation pulse shaped 8-PSK power

spectra consist of continuous and discrete line (spikes) components. Various pulse

shaping waveforms, such as trapezoid and raised cosine are used. Third, since a Solid

State Power Amplifier (SSPA) was used in the simulations, the characteristics of the SSPA

will be studied. Finally, the topic will be an Additive White Gaussian Noise (AWGN)

channel and a matched filter used as receiver.

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2.1 8-PSK Pulse Shaping Methods

In 8 level Phase-Shift Keying (8-PSK), information is contained in the phase of the

signal, and each symbol has 3 bits. The phase of the carrier can

spaced values, such as in Table 2-1 and shown in Figure 2-1.

constellation is shown in Figure 2-1. An 8-PSK signal, S(t), can be expressed as following.

take one of 8 equally-

A plot of 8PSK signal

S(t) = Acexp[j(coc t +_o (t))] = A¢cos[ co_t + (,o(t)l+jAcsin[o_ t + cp(t)] (2-1)

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IMAGINARY

(QUADRATURE)

Ai

011 ! 001

]

Yi I............ :6 oooI

Xi

• I00

111 I01

REAL

('1_ PHASE)

Figure 2-1 8PSK Signal Constellation

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Where the envelope A c is constant in this project, _ =2 zc f_, and fc is the carrier frequency

in hertz, _o is the phase which takes on one of the 8 discrete values (in degrees)

corresponding to the 3 bit Gray code. Table 2-I shows the relationship between Gray

codes and _o.

Table 2-1 Gray Codes and Phase _p

Binary Gray Codes (p (Degree)

000 22.5 °

001 67.5 °,I

011 112-5 °

010 157.5 °

110 147.50

111 247.5 °

101 292.5 °

100 337.5 °

In SPW, the Gray codes and phase q_ can be produced by the design shown in Figure 2-2.

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The phase signals q_(t) can be modulated to 8-PSK signals v(t), where

v(t)=Ace j _p (t). To realize the pre-modulation pulse shaping, the phase signal, (# (t), goes

through a pulse shaping filter before it is modulated. In this project, various filters are

used. In addition, the change of the position of the pulse shaping filters can produce a

change in results. If the pulse shaping filter is placed after the modulator [5], this method is

called as post-modulation pulse shaping. Figure 2-3 shows the modulation for unfiltered

and filtered 8-PSK signals.

Random [ JBinary toData ] "lNumeric

I Random _._.__Binary to tData[ INumeric

Random

Data

JBinary to

"]Numeric

control

xix2 control

T

Number

to

Symbol

_p(t)OUTPUT

Figure 2-2 Producing Phase Signals Block Diagram

Where CI=I, C2=2, and C3=4. The Number to Symbol block functions as Table 2-1, and

completes the conversion from Gray codes to phase in degrees.

2

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= ,,

Phase ]modulator

_1 Ace j(p

8PSK v(t)

(a) Unfiltered 8PSK With Constant Envelope

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Figure 2-3.

Phase (a Modulator

A cejfO I Pulse- Shaping

8-PSK vl (t)

Co) Post-Modulation Pulse Shaping Method

Phase Pulse [Modulator

Shaping _[ Acej(p

8-PSK v2(t)

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(c) Pre-Modulation Pulse Shaping Method

Modulation Schemes for Unfiltered and Filtered Phase Signals.

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2.2 Filters Used for 8-PSK Pulse Shaping

Pulse Shaping Filters are used to narrow bandwidth and improve bandwidth

utilization. In this report several traditional types of filters such as the 5th order

Butterworth, 3rd-order Bessel and Square Root Raised Cosine (SRRC)( a =1) are utilized

in the simulations. Other kinds of filters with overlapping or non-overlapping pulses, will

be introduced in section 2.3.2.

Pulse shaping can induce distortions of 8-PSK signals and increase the risk of Inter-

Symbol Interference (ISI)[4]. The distortions due to the pulse shaping filters make the

implementation of an optimal receiver difficult.

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Butterworth filters: Butterworth filters provide the maximally fiat response in the

passband. Response varies monotonically within the stopband, decreasing smoothly from

f2 =0 and £2 = oo. In the simulations, the 5th order Butterworth filter was used. The Figure

2-4 shows the amplitude response for different orders of Butterworth filters. The cut off

frequency fc corresponds to -3dB amplitude point.

-2O

-40

-6o

-80

-100

Buttorvvorth Filter fc(3dS) = 1 Hz & ordor=2.3.4.5,6.7,8

o. 3: : _ ,_. :.<_'--.. _---..._ .._.... _ i'm :

...... -t! i i i--_:----_ ........ :........... .... -'_ :,-.z _,_ _,_.__z.Z._._.. _._Q_._[_._i_:,.-._ ........ _ ..q-._Q ....... _ ....

: • . : f : \ \ _ _'_. _ "__ --.._ : _-.__

........... ....!-t!.....-.---,.... -i _ ill "q.."-,,_'-., '--.. i _ _ "

Cut-Olaf _ i I i n = S "'--='="-""_:.'_ "_. _ _ "'-_._

..... E_,_,,._.,t_y..4.... -/--"........................ "--":-_ .... _."- -:"_ .... :_-_-

...... _ ..... ["" :.'.... _'"i ........................ r ......... :r--'K .... " ....

Frequency (H:z)

iii

Figure 2-4 Amplitude Response of Various Orders of Butterworth Filters

-0

m pulse R esponse of 5th-O rder B utterwo rth Filte

...... I ......... _.......... i.......... _.......... r .......... i.......

0 1 2 3 4 5 6

Tim e (S ym bol tim e T)

Figure 2-5 An Impulse Response of 5th-Order of Butterworth Filter

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9

All orders of this filter pass through the 3 dB cut-off point. An impulse response of 5-

order Butterwouth filter is shown in Figure 2-5. Since the duration of the impulse response

is more than one symbol period T, it may produce Intersymbol Interference (ISI).

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Bessel Filters: The Bessel type of filter has the slowest transition between

passband and stopband and the passband phase response is better than the Butterworth. It

has approximately linear phase in the passband so that it has a property of preventing

dispersion of the signal. The amplitude for Bessel filters is not as flat in the passband

region as for the Butterworth filters. This project uses a 3rd order Bessel filter for the

simulation. Figure 2-6 shows the amplitude response of various orders of Bessel Filters.

Note that the amplitude is not as fiat in the passband region as for the Butterworth filter.

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-20

-40

-60

-8O

-100

Bessel Filter fc(3dB) = 1 HZ & n=2.3.4.5.8.7.8

-_ ! ; ,-----L-_ : n=2i i

........ _.3_¢__-____-___.:........................ .:._.__.__,_____:! .=_ '-i----/i

10 °

Frequency (I-12:)

Figure 2-6 Amplitude Response of Various Orders of Bessel Filters

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10

Also the transition from the passband to the stopband region is not as rapid as for the

Butterworth Filters for the various orders of filters. An impulse response of the 3-order

Buessel filter is shown in Figure 2-7.

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m,

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1

0.8

0.6

0.4

0.2

0

Impulse Response of 3rd-Order Bessel Filter| ! i

.............. • ................. ,D ................. ,_ .................

!!!iiiiii!iiiii!!!!i!::::::i:::::i:::ii::::::ii:i::::::i1

1 Time (Symbol _me "F) 3

Figure 2-7 An Impulse Response of 3rd-Order of Bessel Filter

Square Root Raised Cosine Filter (a =1) The Raised Cosine filters are used to eliminate

ISI in a linear channel. The transfer function of the Raised Cosine Filter is

RC(f)=

(1 - a)"T, 0 _<lfl <

2T

2[,. L 2a ' _-_)-<1-'1-<2T(1 + a)o, I/F>

2T

(2-2)

where the zeros will occur at t = nT (T is the sampling interval) and ct is the rolloff factor

or the excess bandwidth over the Nyquist Band Filter which can be varied from 0 to 1. The

rolloff factor, o_, determines the bandwidth of a Raised Cosine Filter. Figure 2-8 shows the

I1

m

frequency response of the Raised Cosine Function with different rolloff factors

0.25, 0.5 and 1) and T =1 seconds.

1

0.9

0.8 • ..... _V;--_-"

............. _.................0.7

0.6

0.5

O4

0.3

0.2

0.1

00

..... -1./(2a'-)-w_h._r,_-1:_-_1-s-e-e.:-.....fo!r ct = 0

i i0.2 0.4

Transfer Function of Raised Cosine Filter with r = 0. 0.25.0.5 and 1

= 0250t

_!_ _.-0......i........................; ...... :" - IVT- W'h-_ t'e- _ aq- g_ ..................

for_ -_ I

.... F........ .-rrh_TT ........ _.............

.....!........................

0.6 0.8 1

Normalized Frequency (I-_)

Figure 2-8 Frequency Response: Raised Cosine Characteristics with Different

(t_ = 0,

r It

z_

m,

08

06

-O2

-3

Time I_esponse of F_sised Cosine _'lltef _th r = O. O :_. O _; and 1

............. _............ i........ 7------_ ........ i.........................

.........--_i-;--;---_-_-,-i:----//-----i.....\_............Y:.-_,;-_:-,_---

............_........,,___i....,......._....._ ....i..........:;;.........Ti me (t)

Figure 2-9 Impulse Response of Raised Cosine Filters for Different ct

Figure 2-9 gives impulse response for different rolloff factors. From Figure 2-9, it can

be seen that if the symbols are sampled at the nulls of the sinc function then the symbol

interference from the previous and next symbols will not exist. As mentioned in [17], if

the Raised cosine filters are used the ISI will be eliminated in a linear channel. To

eliminate ISI in a linear channel, a Square Root Raised Cosine filter is used in both the

12

l

transmitter and the receiver. If RC(f') is the frequency response of the Raised Cosine

Filter, the frequency response of Square Root Raised Cosine filter is

SRRC(f)= _ (2-3)

According to [10], the coefficients of SRRC filter can be obtained from the formula (2-4):

[cos((l+a)rc 7)+ sin((1-a)rc )]h(t)=4 a (2-4)t 2

rcVrT[(4a -_) -1]

In this equation, cr is the roll off parameter and T is one symbol period (1/symbol rate).

The Real Raised cosine block in SPW is selected for the project simulation. Ture SRRC

filtering is noncausal and has infinite coefficients, but actually a finite number of

coefficients are needed to determine the FIR tap length. This truncation can cause some

frequency distortion. Since pre-modulation pulse shaping makes the channel no longer

linear, simulation will show that SRRC can not eliminate ISI.

u n lille re p h a s e

i ............... i................ i- .............. J4 ................. . ............... _:................. i ...............

0 500 1000 1500 2000

filte re d p h a s e s a m p le s

4

2

0

0 5001000 1500 2000

sam pies

Figure 2-10 The Waveforms of Phase _pand Filtered Phase _p'

(Butterworth 5th filter is used, BT=I)

The Observation of Pulse Shaped 8-PSK Waveforms. With the use of pulse shaping

filters, the 8-PSK signals change their shapes. The main character of the waveform change

13

is a transient (rise and drop) in the filtered _p' (as shown in Figure 2-10). In Figure 2-10, it

can be seen that the filtered phase (a' has a smooth transition from 0 to 2 n., in sample

range 0 to 500. The transient (rise or drop) makes the big difference between unfiltered and

the filtered 8-PSK shown in Figure 2-11 and Figure 2-12.

"x,,,

Real 1of v(t)

0.5

0

-0 .5

-1

Im ag

of vCt) 1

0 .01 0 .02 0 .03 0 .04 0 .05 0 .06

Tim e s

0.5

-0 .5

-10 0.01 0.02 0.03 0.04 0 05 0.06

Times

-±

IiR

w

Figure 2-11 The Waveforms of Real Part and Imaginary Part of 8-PSK

The filtering process elongates data symbols such that they overlap one another. This

shows the existence of Intersymbol Interference (ISI) introduced by the filtering process.

When the pulse waveforms pass through the bandlimited system, they will be spread or

dispersed which increases error rates and affects power spectrum.

m

m

14

X-.

real part of v2 {t)

1

oo-0 .5

-1 .....

0 500 1000 1500 2000

im a gary part of v2 (t) sam pies

1

-0 .5

-1

500 1 000 1 500 2000

sam pies

Figure 2-12 The Waveforms of Real Part and Imaginary Part of Pre-

Modulation Filtered 8-PSK (512 Samples/Symbol)

r

i

w

2. 3 Power Spectral Components and Pulse Shaping Waveforms

Pre-modulation pulse shaping for phase modulation could produce power spectra

with not only continuous components but also spikes [1]-[4] and [13]-[16]. Prabhu and

Rowe [1] have given a rigorous discussion for components of pulse shaped PSK power

spectra. Simon [2] & [3] have also derived explicit formulas for continuous and discrete

components of pulse shaped PSK based on [1]. Greenstein [4] studied numerous pulse

shapes and compared their efficiency of narrowing bandwidth. In this section, 8-PSK

power spectral components based on [1] and [2] with an assumption of cyclostationary will

be discussed.

2.3.1 Baseband Pulse Shaped 8-PSK Vector-Matrix Notation

The following developments on power spectra of pre-modulation pulse shaped 8-

PSK are based on [1]. A digital angle-modulated carrier of the form can be described as

m

w

_I

_i

m

15

m

s(t)= cos [2 _r fct+ (p (t)]

=Re {exp[j2 n"let+ (,o(t)] }, (2-5)

m

= ..

where (p (t) is a digital angle modulation, and f¢ is the carrier frequency.

define the equivalent complex baseband modulation

v(t)=exp(j _p(t)),

According to Prabhu and Rowe, the spectrum of s(t) is

Ps(f)= _ Pv(f-f¢)+ ¼ Pv(-f-f_),

where Pv is the spectrum of v(t).

For pre-modulation pulse shaped 8-PSK, go(t) takes on one of eight level pulses.

Where p(t) is the

passed through an ISI-producing filter. Denote a column vector g(t) such that

g(t) =[gl(t) g2(t) g3(t) g4(t) gs(t) g6(t) g7(t) gs(t)] x .

If g_(t) is strictly time-limited to a bit interval T, for example, let i=2, then

(o (t) =g2(t) = _ 5p(t). ; 0 < t < T

Using vector notation, ¢ (t) becomes

_0(t) =[0 1 0 0 0 0 0 0]g(t)

For convenience, a column ak vector is defined as:

a k =[ ak (1) ak (2) ak (3) ak (4) ak. (5) ak (6) ak (7) akt8)] T,

and define the basis column vector

For convenience,

(2-6)

(2-7)

g_(t)= 8 [8-(2i-1)]p(t), (i=1,2 .... 8) (2-8)

pulse shaped waveform which may be the result of a rectangular pulse

(2 -9)

(2-1o)

(2-11)

16

m

m

r

and ak takes on only one of

_o(t) may be expressed as

et=[10000000] T

e2=[0 1 0 0 0 0 00 ]T

e8=[0 0 0 0 0 0 0 1 IT

the values el, e2, ... es

oO

(p (t)= Z ak T g(t)],(_ --oo

(2-12)

with the same probability 1 theng,

;-m < t _< m (2-13)

Using the same idea, define two vectors b k and r(t), such that

b k =[ bk (I)bk (2)bk (3)bk (4) bk (5)bk (6)bk (7)bk(8)]T, (2-14)

1

bk also takes on only one of the values el, e2, ... e8 with the same probability _,

r (t) =exp[jg(t) 1= [ exp(jgl(t)) expfjg2(t)) ... exp(jgs(t)) ]g. (2-15)

According to (2-10), the complex baseband 8-PSK modulation may be described as

v(t)=exp[j _o(t)]=exp[jgz(t)]=exp[ j -_ 5p(t)]

--[0 1 0 0 0 0 0 0]r(t) =b2 T r(t)

;0< t < T (2-16)

;0_< t < T (2-17)

Then v(t) can be written as

17

i

w

..+.

v(t)= _ bkTr(t-kT) ;-oo < t < oo (2-18)k=---_o

2.3.2 Power Spectral Components of Pulse Shaped 8-PSK

In order to determine the spectral density of (2-18), the signal r(t) is deterministic,

and bk is a random vector sequence from 2-14. Assume that bk is wide-sense stationary.

Then, define a mean vector M b of bk as follows

Mb=E[bk]. (2-19)

If all signal pulses are equally likely,

1 ITMb=_ [1 1 1 1 1 1 1 1 . (2-20)

The autocorrelation function matrix of b k is

(k-I)=E[bk bin], (2-21)

and the power spectral density matrix of bk is

E(f) = _ e'J2_fnR_ (n). (2-22)/'1=--_

The goal is to find the power spectral density of the baseband 8-PSK signal v(t). The

autocorrelation function of v(t) is given by

Rv(t+ r ,t) = E[v(t+ r )v*(t)] (2-23)

=E{ _ bkTr(t+r-kT)[ _ bkTr(t-IT)] * }k=-n_ I=--_

=E[ _ bkTr(t+r-kT) _ blnr*(t-lT)]k=-co /=-_o

18

--E[£k=--co ]=-_

b£ r r(t+ r -kT) btn r*(t-lT)]

--E[£ £ rT(t+r-kT)bkblnr*(t-lT)]k=-co 1=--.o

=£ £ rm(t+r-kT)E[bkbin]r*(t-lT)]k=--_ 1=--_

=£ £ rV(t+r-kT)_(k-1)r*(t-IT)k=-oo I=-_e

(2-24)

From (2-24), it is easy to show

Rv(t + nT + r ,t + nT)= Rv(t + r ,t), (2-25)

then, v(t) is cyclostationary. According to [9], the time-average autocorrelation function

over a signal period is defined as

T

1iR_(r) =T- r

2

Substituting (2-24) in (2-26),

R_(t+ r ,t) dt (2-26)

T

I=--_ T

2

rV(t+ r -kT) R0 (k-l)r (t-lT) dt.

Let k-l--n and t-IT --x,

1

R_(r)=y

T---IT

£ £ 21 rT(t+l--nT) _ (n)r*(x)dx

n=-=o I=--_ _T_I T2

19

Let t--x, then

rT(t+ r -nT) R, (n)r'(x)dx

=

w

= :

1 _, _ rV(t+r-nT)_(n)r*(t)dtRv( )=7n =-_0 --00

Power spectral density of v(t) is defined as

Pv(f) = S Rv( r )e -j2rc r r d r

= -_ e"J2 rT(t+ r -nT) _ (n)r*(t)dt d 2---00 n= --00 --o0

e-J2 _ f 2"rT(t+ r -nT) R--'_(n)r*(t) dt d r.

Let x = t+ r, then

1 _-'_ _ " 2Z" fx -- S " 7Ff'/"Pv(f) = _ 0=-_ e -J2 rT(x-nT) _ (n) dx e J2 r*(t) dt d r.-oo .-_

(2-27)

(2-28)

L..a

R(f) = 5 e -j2rc ft r(t) dt, is the-00

Thus Pv(f) can be expressed as

Fourier transform of r(t), and it is also a column vector.

! -

RT(f) e-J2z _T Ro (n) R*(f)

1

Te "j2rc fnT _(n) }R*(f) (2-29)

20

w

w

L...d_

| J

w

By using (2-22), Pv(f') becomes

1Pv(f') =_ RT(f) _(fT)R*(f)

since

_(f)= £ e-J2_f_R_ (n).

Introducing covariance matrix Kb (n) = Rb (n)- M b Mb n, then

(f)= £ e-J2 lr _n(_ (n)+ Mb Mb n)/'1 = --¢0

= £ e-J2 _ fnZ (n) + M b Mb tl _ e "j2n"fnn=---_ n=--_

Using Fourier series, we have

e-J2 _ fn = £ _ (f-n)n= --oC} n=-_O

£ e'J2_'frn 1 £.=-_ =-T ,=-_ g(f- ).

(2-30)

(2-31)

(2-32)

(2-33)

Therefore,

e-J2 7/"fn/_ (n)+ M b Mb H

n=--oo n=-oo

= Pbc(f) + Pb, (f) (2-34)

B

where Pbc (f) is continuous spectrum component, and Pbt (f) is the discrete line spectrum

component.

21

=

From (2-30) and (2-34),

m

p_(f) =_ RT(f)i_oc(fT)+ _ (flr)]R*(f)

=1 RT(0_(IT)R*(f)+ 1 RT(f) Po_ (£T)R*(f)T T

=Pv¢(t')+ P,,(t') (2-35)

Pvc(f) is the continuous component of the pulse shaped 8-PSK power spectral density. A

further study on Pvl(f) is given as follows.

I RT(f) Po_t (fT)R*(0e ,(0=

1 RT(f){ 1 _ ;=-_ -_-M b Mb H 8(f- )}R*(f)/3="-o0

1 n

T 2 RT(f)MbMbHR*(f') _ _'(f--_)/1=--o0

m

_1 nT 2 MbTR(f)MbHR*(f) _ _(f-_)

_ 1 Iu 6(f_T)T2 IMb T R(f)r/=---co

_ 1 s 12_ nT282 I,T__.,R_(0 6(f--_). (2-36)

i=1 n=-co

The Pvt(f) is the discrete line component. (2-36) is the result of non-overlapping pulses,

that is they are strictly time-limited to one symbol time. When signal pulses have

overlapping, [1] defined a overlapping index K which stands for maximum signal pulse

22

duration. According to [1] and [2], Pvl(f) for arbitrary index K may be expressed as

w

u

=

u

_1 .. £Pvl(f) T282K I_ Ri(f)[ 2 6(f- ). (2-37)i=1 n=--_

For 8-PSK with overlapping K=2, power spectral line component is

Pv,(f) T28 , I_ R_(t312 8(f- ). (2-38)i=1 n=--_

and

R(f)= j" e-J2 z rt r(t) dt-oo

TR(f) = e-i2.1,

"ei{ ot (t)+g_( t- r) }

e,i[ gl (t)+OM (t--T)}

ei{ oz (t)+ gt (t--T)}

ej[ g_ (t)+oM (t- T)I

eYl uz(t)+ot (t--T) }

ei[ ¢__, (t)+o_ (t--T)}

e;{ o,, .)+o, (t--T) }

e,Jl _M (t)+oH (t-T)]

dr.(2-39)

where M=8 and gi(t) can be obtained from (2-8).

2.3.3 Pulse Shaping Waveforms

Some particular signal pulses (filters) which overlap and do not overlap will be

studied. According to [1], waveforms for different overlapping index K are shown as

follows.

23

K- !

-i o 2

.. ,,"",,//_\, I'"- "'t', X\• X K-7

..' : ..: / \ \

= =

-T 0 f 2

- 0 1 2

, _ K_4

-T 0 T 71' ]'T 4T

Figure 2-13 Pulses with different overlapping index K from [1]

Numerous pulse shapes with different pulse duration are simulated to investigate the effect

on narrowing bandwidth and reducing spectral spikes. In the following expressions, T is

used for one symbol time in seconds. In order to obtain a better effect on reducing spikes,

some pulses will have T duration fiat with different transits.

Cosine with Single-Interval

expressed as

T duration cosine with a single interval can be

P(t) = {

Tcos(_ t/T) ltl - -

2

70 Itl >-

2(2-40)

=

;--d

7

24

T duration cosine pulse shaped and non-pulse shaped 8-PSK phase signals are shown in

Figure 2-14. In Figures2-14 to 2-20, assume one symbol duration T=1.2

w

i

= =

-7

i i

0.8

0.6

0.4

0.2

-0.25

Cosine Signaling with Pulse Duration T

I 1 I 1 I I I I t

....... _- ...... I- ........... t" ...... t" ...... f- ............ I" ...... t" ......

I I I I ] I I I I

....... ;- ...... _ ...... F ...... F ...... r ...... F ...... F ........... T ......I I I I 1 I I I i

I I ! i i I I

I I I I I I I I I__ i

1 I I t I I I I I

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

4

2

00

Phase Signals

i i l _i f\

......... ,'-........ ',................. : ..... 7:7--i -I- --1-ql_ i ._ i i i, ,, /\ , I,/

......... ,......... ,-/- - -- -V - -,- f- - %-- - _....... t- ,-I......I I I I _ I' :/, k/ \_ti

0.5 1 1.5 2 2.5

tI I

I I

l- ........ I.......

i

-/-X -,!_ _,

3 3.5

RealPartofPulseShaped 8-PSK Signals

....... _l .... i_; ....

/i / 1

2 _

W

0.5

0

-0.5

......... :........ -,,r--

......... - ....... l_,,I__' i-"-_

......... :........ t-:-k-.......i ,it,

0 0.5 1 1.5 2 2.5 3

l .......

/,I

3.5

Figure 2-14 T Duration Cosine Pulse Used as Phase Pulse and 8-PSK Signaling

= _

. =

25

Cosine with Double-lnterval Cosine with double interval can be expressed as

cos(_r _T ) Jtl< T

h(t)={ 0 I_>T (2-41)

2T duration cosine pulse used as

2-15.

phase pulse and 8-PSK signaling are shown in Figure

w

L_

0.8

0.6

0.4

0.2

Cosine Signaling with Pulse Duration 2T

I I I ...... t- ............ P ...... P ....................................

, _ , .... L ......................... 7 .......................... 7 ...... r-- , ' '

I I I I I t _-

I II I,- I,,.. ...... I- ...... ;- ...... g- ...... i.- ...........

- , , , ,, : ', ' , __

/ I I a s , I

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

6

4

2

1

0.5

0

-0.5

Phase Signals

,.........,........,____-..__.,____!........!........_........... I

!

........ t I i

' .... _,;.... _'x ' ' -/- ....I __L ....... L__

/ ...... _...... , , ,I

II I I I 1

O0 0.5 1 1.5 2 2.5 3 3.5

RealPartofPulseShaped S-PSK Signals

-: i ....... j .............

.........'......." ......LL_,___u__--_- l_ ::-\

......... ," ....... ; X / ; , ' \ / ', 1', \ J, ; \ / , ' .___'____I__L ........ ;--+, .... --r

................... _---_ .................... ,--\ / , _, ± _' ':\/ ', ', i _ i ;-\7i I "_¢ t I

0 0.5 1 1.5 2 2.5 3 3.5

Figure 2-15 2T Duration Cosine Pulse Used as Phase Pulse and 8-PSK Signaling

= =

26

Raised Cosine with Single-Interval

expressed as

Raised cosine with single interval can be

W

w

w

w

w

w

_=_

w

T

Tcos2(Kt_ Itl -< -

2h(t)= { T

0 It[ >-- (2-42)2

duration raisedcosine pulse used as phase pulse and 8-PSK signalingare shown in

Figure 2-16Raised-Cosine Signaling with Pulse Duration T

I I I I $ I I I I

0.8 ....... ,- ...... '- ...... _ .... " ...... " ...... " ..... _ ...... _ ...... _ ......I I I I I I I i l

= I = ' ___L ...... L ......

0.6 ....... r ...... r .... r ...... _"...... r ...... r ...... r -- ,I I I I I I i I I

0.4 ................................................ _ ...................i t I I I I I

I ; i I i 1 1 I i....... .....I I I I I I I

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

Phase Signals

t i I I _, /\ i......... ........ ........ ........,....... ..........4 ..... , , , , , r, /\ l,'

I I l ! 1 _ _ ,,d i

, , ' ' '_ _ ' /_ I'' ' ' ' I' / \ ' /_ '

2 - -- _,--'_,---.-,- ....... "_........ _ ....... T r-] - - _- -- r"/----_--L'-- --I, / \ ",-- _ ..... / \ '/\ I', t / \ , / \ _, z I ' / _ II_ I

J,/ \ , / \ -,-_-_--,---_-_- ,/ \ !1 \i i l l i

o _"/ \ / \ _" _ ;/ \ z \.r _.., ,0 0.5 1 1.5 2 2.5 3 3.5

1

0.5

0

-0.5

0

Real Part of Pulse Shaped 8-PSK Signals

1 i ih f ,_ {,.,b,____L__ ,_,_ I , \I, , , ____2 ....... L'_........ L_,____,___,__,___,-__,-,,, _____ ....

I' _, I _ \ I ', _ / , \/ d' ' '

...... l-r- -_-- -/- - - --r- _ _ -1........ _ ....... _, _ /t ....

I' _ I , , _ I,......... A- _ -- -I.........................

L:_ __ _ ,, ,, ', I '_,I ! I I |1

0.5 1 1.5 2 2.5

Figure 2-16 T

___L_' I __ILl ....... J

-I--':qA/--,,- 1--_:11-1/-- ',_jl I U V l

3 3.5

Duration Raised Cosine Pulse Used as Phase Pulse and 8-PSK Signaling

_m

w

27

W

w

Raised Cosine with Double-Interval

expressed as

Raised Cosine with double interval can be

2T

c°s2(# "_7:) Itl < Th(t)={ 0 Itl>T (2-43)

duration raised cosine pulse used as phase pulse and 8-PSK signaling in Figure 2-17.

w

L

w

=._

w

0.8

0.6

0.4

0.2

Raised-Cosine Signaling with Pulse Duration 2T

, , _ _ ,, ------_, , , ...... • ......I I ...... ____ ---t- ...... t" ...... _" ..... l- ..... ! I

I I ..................... ! I I

....... F ...... F-- ' i I i ) I

i I- .... I- ...... l- ...... L- .... ;

....... _ ...... I- .... i I I I i ' ....

i i r --, /___L ...... ;-...... r ...... r ...... r .... , '

....... r_ _f---r .... i i L ' ! ',; ; i , ' ' ' ' 0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

4

1

0.5

0

-0.5

Phase Signals

, i, I',- - -___t, I , _ _ _ _ ___"_-.._.._._ ....... -' _-- -/ ..... " - - -'_- - -;-'- - -2" - ,

......... ,- ....... _---7 .... q.... _'.._ _',- - -"---J-;-_ ', ', 7- _' ',IL/

i ] i_ ........ =........

-- ,................ _ .... i I2 -- _', _ ....... 'I i •

' ' _ I IIJ _ I II

00 0.5 1 1.5 2 2.5 3 3.5


I I I I I 1II'_ , , ._ " -,'_ , __'\ _ -- -- --Zr_ ' - .... Jr ..... -/-L',-N ........ ,..... -3

........ L,___\.... r,...... /1-','_...... i-- ,, / i! \ :I I I i -- -- ..... t-" .......,\ ,, ___l_____\..... _....... _.---/--r_-\ ,, 7

_---_ ,-: I _! \ ', ,', I _! \ .... L:..... /--......... ,, ,, /

i,._ - - _.. / .--_ ,l / ,'._ 3.5

0 0.5 1 1.5 2 2.5 3

Figure 2-17 2T Duration Raised Cosine Pulse Used as Phase Pulse and 8-PSK Signaling

28

Weighted Raised Cosine with Double-Interval

double interval can be expressed as

Weighted raised cosine [5] with

where 0.5<k<1.5 2T

are shown in Figure 2-18.

l[cos(x t)]_ 12_.__kk

h(t)= { 2 0

2rot

-cos-_ ] Itl _<Tm

Itl> T (2-44)

duration raised cosine pulse used as phase and 8-PSK signals

I

m

L =

=--

w

=--

w

w

w

=--

0.5

0-0.5

Weighted Raised-Cosine Signaling with k=1.5

I I " I Ii .... r ..... r" ...... r" -

I I I l I I I

I I I I I

I I I I ;

I I I 1 I I 1 I

....... :-..... ,'-...... L...... '- ............. L...... '- ...... ,_..... ,_......I I I

I I I

I I J

-0.4 -0.3 -0.2 -0. 0 O. 0.2 0.3 0.4 0.5

6

4

2

00

Phase Signals

I I I 1 II I t I I I

......... I.........-_ol ...... -I ........ .t ..... 1. ...... I- I

I ._.I 1 I t I I.... _ _ __2 .... 2 .... L__ , , --'rr ' "_- , "_'_- , ...... F-- 7_"-_ ....

I l l I eft" -- I.... t, I _:__ _ _x._;/ _ _X._Z_ __ ., -/- -'_:rF7 ....... __ ..... I.... r: .... \-- _,-t ....... ,....I' ' i

I I I I I I

0.5 1 1.5 2 2.5 3 3.5

1

0.5

0

-0.5


'h ',fh........ L:__....... :_/_q

I'1 '/_........ _:__....... :_/____........ :/ __

i-V-0 0.5 1

I I II I I __ __

---2__ ' rl f' ...... _....... "ir ..... -PI I i I /

f I I I /-----'q ........ "r ....... '4"1" .....

I l I /, A , A _, /

1'.5 2 2.5

, !AA............. TiI

"1...... ::-/......'i VN:,)3 3.5

Figure 2-18 Weighted Raised Cosine Pulse Used as Phase Pulse and 8-PSK Signaling

29

Trapezoid Signaling A trapezoid signal can be expressed as

1

h(t)=

Tltl < -

2

2 T<2- -_ Itl -_ -

0 Itl >T

where 0.5 _<k< 1.5

Itl < T

(2-45)

Trapezoid pulse used as phase and 8-PSK signals are shown in Figure 2-19.

i

===

I

i

==

I

--=

I

0.5

0-0.5

Trapezoid Signaling with Pulse Duration 2T

y I I I II I t I

I I I I I I

I t I l I I

I I I I I I

....... r. ..... r" ...... r- ............ I- ...... r'----

I I I I I

i I I [ I I

I I I i I i

I I I l I I

I I I I I I

I I I I

-0.4 -0.3 -0.2 -0.1 0 0.1

, ",4.. :I

I I I

I [ I

I I I

.......--_ r ..... P ......

i I I

I l i

I [ I

i I i

i

I I I

0.2 0.3 0.4 0.5

Phase Signals

i i i i i/\ i i

I I I I _ _[I t

6 ..... ' '-_- ' ....-'--_-' .... '- - --' .....-- -- -1 ...... I .... i 1--- - - -- I ..... I-- - -

I I / _ t _ I I I i

I I I I I I I

4 ---, ......... / ..... "X,- -_-//- -\-- l_ / ...... __........ r ....... ,....... -_r, T----'? .... _ \_/ _:-- : Jr,_, ---q -- - ----_ut,..... ,_....... __y------x-:--_- - - - - 1

0 1 i i I t _l0 0.5 1 1.5 2 2.5 3 3.5

0

-10


i__ I I t I '1 '

i\ , , A, , /I I I I I I,it , I,/ ',: \// , /, /_ , , / ', '_1 I , I' /

....... , t- -_ ,, 7_ /_,r ..... _r.... l ,, -,F;_ ....

' ',, ., :,',i\ L,0.5 1 1.5 2 2.5 3 3.5

Figure 2-19 2T Duration Trapezoid Pulse Used as Phase Pulse and 8-PSK Signaling

3O

Weighted Raised Cosine Transit Signaling A pulse with weighted raised cosine

transit and flat within one symbol period can be expressed as

i

w

w

w

.I

!

z

TItl< -

2

h(t)=

2

0 Itl >T

Itl_ T

where 0.5_<k< 1.5

The pulse with weighted raised cosine transit used as phase and 8-PSK signals are shown

in Figure 2-20

Weighted Raised-Cosine Transit Signaling with Pulse Duration 2T

, , I , , , , , ,1 I I I I

I I I I I I I I I

I I I ! I I I I I

I I I I I I I I I

U._ ....... r- ..... i- ...... i- ...... r ...... it- ...... r = ...... i'- ----_- ..... 7" ......

I I I I I I I

, , , , , , , , ,I I I I I

I I I I I I I I I

' i ' ' I ' 'I_ I I I I

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

8

6

4

2

00

Phase Signals

I I I /\ I I I I

' : :"-'/:x' ' '...... I....... I ..... 7 ...... 3" ........ T ........ I- ........ I........

I I I I I I I

......... a.......... £ - -..-"x- .... L ........ L ........ i ........

: _7----c---_-/ \ : : :/_........ 7:- 7__, ........ _....... _:.... T--r ....... _,- 7---Tr----X-;........ ,_,,_/ ...... :........ ._....... __:=.... _____.-_/ ..... '__:_._:_

1.1/ I i i _l _V i0.5 1 1.5 2 2.5 3 3.5


0 0.5 1 1.5 2 2.5 3 3.5

Figure 2-20 Weighted Raised Cosine Transit Pulse Used as Phase Pulse

and 8-PSK Signaling

3]

w

2.4 Solid State Power Amplifier (SSPA)

A Solid State Power Amplifier (SSPA) is used in the project. The magnitude and

phase plots of the SSPA were obtained from JPL. From the plot (Figure 8), the SSPA has

two main ranges which include a linear and non-linear region. If the input power is less

than -5 dB, the SSPA is worked in the linear region, and the phase offset is approximately

zero. If the input power is larger than -5 dB, the SSPA is working in the nonlinear region.

This project sets the SSPA to work at saturation level of 0 dB.

.=

w

m

m

0nn

-10O

EL

>

J -20

r_

"5©

SSPA, ESA 10WattsAmplifier:MagnitudeResponseCurve

, ,,i t w ii I i ,, ! ,i I

............ 013.............

iii

-3[? i-30 -25 -20 -15 -10 -5 0

Input LevelPower (riB)

SSPA, ESA 10Watts Amplifier:PhaseResponseCurveI 1 Ii

._--.- .............................. , ,

u._t _.............

n ,-6 ; ; l-30 -25 -20 -I5 -10 -5 0

InputLevelPower (dB)

Figure 2-21 The Output Characteristic of 10 Watts SSPA

32

i--

The main part of the SSPA simulation is an analytical Traveling Wave Tube (TWT)

block which can simulate the nonlinear channel. By looking up the SSPA table file

provided by JPL, the TWT realizes a conversion from input amplitude to output amplitude

and input amplitude to output phase. Figure 2-22 shows the structure of the SSPA.

=

w

w

DTWT

in out

(TABLE)

AVERAGE

COMPLEX

x POWER

reset hold

I' 't

iave_pwr hold

J,SSPA

Figure 2-22 The Structure of SSPA.

w

m

i

w

w

w

2.5 Additive White Gaussian Noise Channel

Additive White Gaussian Noise (AWGN) is added to the signals out of the SSPA to

PSD,(f)

No/2

-fs/2 0 fJ2

Figure 2-23 Power Spectral Density Of White Gaussian Noise.

w

33

r_

simulate channel noise. Figure 2-23 Shows the power spectral density for the AWGN used

in the simulation. Actual white noise has a flat spectrum from -oo to oo. The AWGN is

taken to have zero mean and with variance o- n2 =No/2. N0/2 is the power spectral density of

the white noise, and Fs is the sampling frequency.

W

w

w

m

m

2.6 Receiver and Error Rate Estimation System

There are three parts in the receiver and estimation system, receiver, synchronizer

and error rate estimator. Figure 2-24 shows the block diagram for this system.

.. ...................................... _ .................................

r(t):s(t)+n(t)MATCHED

FILTER' i:

Receiver

DELAY

&

PHASE

METER

Synchronizer

• ...... ? ........................ •

ERROR

"_ RATE

ESTIMATOR

Figure 2-24 Block Diagram for the Receiver Error Rate Estimator

Receiver In an ideal receiver, a matched filter is used. The matched filter is a

linear filter which maximizes the signal to noise ratio (SNR). In this project, an integrate

and dump block (which match only a rectangular pulse) is used as the receiver.

w

34

w

Synchronizer The synchronizer performs a correlation between the reference

signals and received signals (delayed and distorted), and gives an estimate of the number

of samples of delay and the phase between the two samples. The reference signal is then

delayed the same number of samples so that the two signals can be compared to get the

error rate. Figure 2-25 shows the block diagram for the synchronization. The DELAY &

PHASE METER can detect the number of sample delays of the received signals. The

COMPLEX VARIABLE DELAY lets the reference signals be delayed by the same

number of samples as the received signal so they can be compared in the error rate

estimator.

u

w

w

u

reference

8 PSK

I Complex I

Variable

x Delay y

delay

received

8PSK

DELAY & PHASE

Figure 2-25 Block Diagram of Synchronizer

Error Rate Estimator

same delay), are put into the

The reference signal and received signal (now having the

error rate estimator to count the number of bit errors. The

35

main block used to count bit errors is SIMPLE ERROR RATE ESTIMATOR(Figure 2-

26) which compares the reference and the received signals to find bit errors.

=

w

L--

in SIMPLE

ERROR

i'ef RATE

ESTIMATOR

rate

errs

N

reset

TFigure 2-26 Simple Error Rate Estimator Block Diagram

m

w

w

w

36

Chapter 3

PULSE SHAPED 8-PSK SIMULATIONS

w---

This chapter will provide simulation procedures and results of pre-pulse shaped 8-PSK.

Simulations included power spectral density (PSD) and bit error rates (BER). In addition

to simulating traditional filters, some particular 8-PSK waveforms will first be simulated to

eliminate power spectral spikes.

3.1 Simulations on Power Spectral Density

Figure 3-1 shows the procedure for

shaped 8-PSK signal.

obtaining the power spectra for pre-pulse

Random

Data I BasebandFilter

Phase

Modulator

Solid State.I Power

"]Amplifier

] (SSPA)

Spectrum

Analyzer

(FF

w

.... T

Spectrum

Analyzer

(FFT)

Figure 3-1 Block Diagram of Spectrum Analysis

The random data source provides ideal data with a symbol rate (Rs), and the

sample frequency, Fs, is 512Rs Hz. For the pulse shaping, traditional filters and particular

w

37

w

i --

w

filters are used. For Butterworth and Bessel filters, bandwidth (BT) is equal to the cutoff

frequency which corresponds to the -3dB amplitude point. The BT represents the product

of the bandwidth with the symbol time. Traditional filters used in the simulations are in

Table 3-1.

Table 3-1 Traditional Filters Used in the Simulation

Filters

5th-Order Butterworth

3rd-Order Bessel

SRRC

BT=I

BT=i

BTs or roll-off factor ( a )

BT=2

a=l

BT=1.2

BT=2.8 BT=3

BT=2 BT=3

w

,,,,,,t,-

Particular 8-PSK waveforms, such as trapezoid, weighted raised cosine transit signaling,

cosine transit and raised cosine transit signaling, are also investigated to test their ability to

get rid of or reduce spectral spikes. The constant envelope pulse shaped 8-PSK signal

which comes from modulator goes to the input of the Solid State Power Amplifier (SSPA).

The spectrum Analyzer can be put before and after the SSPA. Since the SSPA is

worked at the saturation level and a constant envelope 8-PSK signal is used, the spectrum

of the signals at the input of the SSPA is almost the same as that of the signal output from

the SSPA. The power spectrum is obtained by taking the Fourier Transform of the input

signals of length 131072, using a Bartlett window. The power spectrum is then estimated

by averaging 100 such FFTs.

w

38

w

3.1.1 PSD Simulation Results of Using Traditional Filters

The plots of the power spectra are given in Figures 3-2

following parameters were used to obtain the power spectra.

Data Source •

to Figure 3-11. The

a°

c°

Ideal data with p(0) ( probability of zero )=0.5;

Sample Rate (fs) =512P_ Hz

m

i

m_m

m

E_

m

=__

!

Traditional Filters for Pulse Shaping:

a. None

b. Butterworth 5th Order (BT=1,2,2.8,3)

c. Bessel 3rd Order (BT=1,1.2,2,3)

d. SRRC (a =1, Number of tap length=32*P,_=8192, Bartlett window

Power Amplifier:

European Space Agency (ESA) 10-watt SSPA

SPW Spectrum Analyzer

a. FFT length=512R s

b. Window type: Bartlett

c. Average: 100 FFTs

d. Sampling frequency =512R_ Hz

Spectrum Plots Ranges

+ 10 Rb; + 75 R b

39

=.-.

tall=

=

°I , , , ^/_^ ' ,

!!T .-80

-10 -8 -6 -4 -2 0 2 4 6 8

Frequency (Rb)

(a) Unfiltered 8 PSK + 10 1_

0

m

-75 -60 -40 -20 0 20 40 60 75

Frequency (Rb)

I'

i

ii==w

.i

m

L_

nn"10

(b) Unfiltered g PSK +_.75 1_

Figure 3-2 Unfiltered 8-PSK Power Spectral Density

SSPA Output, +/-10Rb (5th Order Butterworth BT=I ,Ideal Data)0 , , , i_t , , ,

-80 .

-100

* I

-10 -7.5 -5 -2.5 0 2.5 5 7.5 10

Frequency (Rb) With All Spikes

(a) 5th Order Butterworth Filter (BT=I), -+ 10 P,q,

40

.+ ,

0

-100

I 1 1

-75 -60 -40 -20 0 20 40 60


SSPA Output, +/-75Rb (5th Order Butterworth BT=I ,Ideal Data)! I I I I I

75

(b) 5th Order Butterworth Filter (BT=I), +_751%

i i

t

m_

13:1

lll

Figure 3-3 Power Spectra of 8-PSK Pulse Shaped with 5th-order Butterworth filter (BT=I)

0

-20

-40-60

-80

-100

SSPA Output, +/-10Rb (5th Order Butterworth BT=2,1deal Data)I I I ' I I I

- iiiiiiii!i,ii,,, i/ill'fill+iiii_-10

I

-7.5 -5 -2.5 0 2.5 5 7.5 10


(a) 5th Order Butterworth Filter (BT=2), __.1ORb

0

-20

-40

-60

-80

-100

-75 -60 -40 -20 0 20 40 60

SSPA Output, +/-75Rb (5th Order Butterworth BT=2,1deal Data)! ! ! I1[ I , i

L

75


(b) 5th Order Butterworth Filter (BT=2), :t:75R_

Figure 3-4 Power Spectra of $-PSK Pulse Shaped with 5th-order Butterworth filter ('BT=2)

41

0SSPA Output, +/-10Rb (5th Order Butterworth BT=2.8,1deal Data)

w

ID

rn"O

-20

-40

-60

-80

-100

-10

Frequency (tbo) With All Spikes

(a) 5th Order Butterworth Filter (BT=2.$), + 10R_

0

-20

-40

--60

-8O

-100

SSPA Output, +/-75Rb (5th Order Butterworth BT=2.8,1deal Data)I I I =IL I I I

-75I

-60 -40 -20 0 20 40 60 75


w

0

(b)5th Order Butterworth Filter (BT=2.$), + 751_,

Figure 3-5 Power Spectra of 8-PSK Pulse Shaped with 5th-order Butterworth filter

03T=2.8)

SSPA Output, +/-10Rb (5th Order Butterworth BT---3,1deal Data)I I I

rnnO

-20

-40

-60

-80

-100

-10 -7.5 -5 -2.5 0 2.5 5 7.5 10


(a) 5th Order Butterworth Filter (BT=3), + 10P_

42

m

"U

0

-20

-40

-6O

-8O

-100

SSPA Output, +/-75Rb (5th Order Butterworth BT--3,1deai Data)I I I

-75 -60 -40 -20 0 20 40 60 75


(b) 5th Order Butterworth Filter (BT=3), + 75Rb

0

-20

-40

-60

-80

-100

-10

Figure 3-6 Power Spectra of $-PSK Pulse Shaped with 5th-order Butterworth filter (BT=3)

SSPA Output, +/-10Rb (3rd Order Bossel BT=I ,Ideal Data)

-8 -6 -4 -2 0 2 4 6 8 10


m (a) 3rd Order Bessel Filter ('BT= 1), _+10R4,

SSPA Output, +/-75b (3rd Order Bessel BT=I ,Ideal Data)

0 / , , , , , ,-2o ........!............i.............:...........-40 _- ........ ' ............ : ............. i...........

U

n,1.Io

-100

-75 -60 -40 -20 0 20 40 60


Co) 3rd Order Bessel Filter (BT=!), +_.75Rb

Figure 3-7 Power Spectra of g-PSK Pulse Shaped with 3rd-order Bessel filter (BT=I)

75

43

0SSPA Output, +/-10b (3rd Order Bessel BT=1.2,1deal Data)

-8 --6 -4 -2 0 2 4 6 8 10

-20

-40

m --60'10

-80

-100

-10/


(a) 3rd Order Bessel Filter ('BT=1.2), + 10R_

.,=11

t_

blJ

en"10

co

1

-20!

..40 '

-60

-80

-100

-7.:

0

-20

-40

-60

-80

-100

SSPA Output, +/-75b (3rd Order Bessel BT=1.2,1deal Data)

! ! i , t

I 1

-60 -40 -20 0 20


40 60 75

(b) 3rd Order Bessel Filter (BT=I.2), + 75P,_

Figure 3-8 Power Spectra of 8-PSK Pulse Shaped with 3rd-order Bessel filter (BT=1.2)

SSPA Output, +/-10b (3rd Order Bessel BT=2,1deal Data)1 I I I 1 I i I

0 --8 -6 -4 -2 0 2 4 6 8


(a) 3rd Order Bessel Filter (BT=2), + 10R_

0

44

0SSPA Output, +/-10Rb (3rd Order Bessel BT---3,1deai Data)

m

rn

-2O

m -40"O

-6O

--8O

0

-20

-40

-60

-80

-100

-10 -7.5 -=; -2.5 0 2.5 5 7.5 10


(a) 3rd Order Bessel Filter (BT=3), _+]ORb

Figure 3-9 Power Spectra of 8-PSK Pulse Shaped with 3rd-order Bessel filter (BT=2)

SSPA Output, +/-75b (3rd Order Bessel BT=2,1deai Data)I I I I I I

-75

0

J .-.----Lm

-60 -40 -20 0 20


40 60 75

(b) 3rd Order Bess¢l Filter (BT=2), + 75P_

SSPA Output, +/-75Rb (3rd Order Bessel BT--3,1deal Data)I : I t

w

w

, =

........ :. ............ ° ............ _. ........

.... :............. i ............. ! ........

,L .L_

-75 -60 -40 -20 0 20 40 60


75

Co) 3rd Order Bessel Filter (BT=3), + 75Rb

Figure 3-10 Power Spectra of 8-PSK Pulse Shaped with 3rd-order Bessel filter (BT=3)

45

0

-20

-40m"o -60

-8O

-1000

SSPA Output, +I-10Rb (Raised Cosine,a=1 ,Ideal Data)

-7.5 -5 -2.5 0 2.5 5 7.5


(a) Square-Root Raised Cosine (a = l), + I 0R_

10

0SSPA Output, +/-75Rb (Raised Cosine,a=l,ldeal Data)

I I I I

m.,.-,,

0Q"to

-75 --60 --40 -20 0 20 4O 6O


Co)Square-Root Raised Cosine (a =l), + 751_

Figure 3-11 Power Spectra of 8-PSK Pulse Shaped with SRRC filter (a =1)

75

n

46

m

w

: :w

3.1.2 PSD Comparison Between Two Pulse Shaping Methods

The PSD related to pre-modulation pulse shaping will be compared with

simulations using post-modulation pulse shaping. In order to compare the effect of two

methods, a frequency band utilization ratio p [16] can be defined as:

Number of spacecraft with Filtering Accommodated in Frequency BandP = Number of spacecraft without Filtering Accommodated in Frequency Band

An important assumption in deriving this ratio was made: the "spectra from

spacecraft in adjacent channels will be permitted to overlap one another provided that, at

the frequency where the overlap occurs, the signals are at least 50 dB below that of the

main telemetry lobe (1 st data sideband)." [16].

The Power Spectrum plots were used to determine this 50 dB level. For example,

for 8-PSK signals without a pulse shaping, the -50dB point is situated at approximately 35

Rb and using the Butterworth Filter (BT=I), the -50 dB point is approximately at 1.28 Rb,

then the Utilization ratio, p, is equal to

P = 35R_ =27.341.28Rb

From the simulations, the utilization ratios of pre-modulation pulse shaping were

calculated and listed in Table 3-3 along with the utilization ratios for post-modulation

pulse shaping [6]. From the utilization ratio table and Figure 3-12 to 3-13, for the same BT

and baseband filters, pre-modulation pulse shaping has a higher band utilization ratio than

that of post-modulation pulse shaping. Although the pre-modulation pulse shaping method

has a narrower bandwidth, it produces spikes which are discrete line components of power

47

spectral density at particular frequencies. Figure 3-14 gives a PSD comparison of two

methods with 5th-order butterworth filter.

Table 3-2 Utilization Ratio for Post-Modulation Pulse Shaping 8-PSK

Filter Type -50 dB point (Rb) uiiiizationNone 35 I

Butterworth(5th order)BT= 1 2.5 14

Besssel (3rd order) BT=I 2.95 11.86

SRRC rolloff =I 2.5 14

Table 3-3 Utilization Ratio of Pre-Modulation Pulse Shaping 8-PSK

Filter Type -50 dB point (R,o) Utilization

None 35 1

Butterworth(5th order) BT=I 1.28 27

Butterworth(5th order) BT=2 2.43 14.4

Butterworth (5th order) BT=2.8 3.23 10.8

Butterworth (5th order) BT=3 3.45 10.1

Besssel (3rd order) BT=I 1.75 20

Besssel (3rd order) BT=i .2 2.04 17.1

Besssel (3rd order) BT=2 3.22 10.8

Besssel (3rd order) BT=3 4.53 7.7

SRRC rolloff= 1 1.23 28.5

i ,

=___

BT=3

BT=2.8"

BT=2"

BT=I"

Utilization Ratio

0 5 10 15 20 25 30

Figure 3-12 Utilization Ratio of Using 5th-Order Butterworth filters in Pre-Modulation Pulse Shaping

48

BT=3_

BT=2'

BT=1.2"

BT=I'

Utilization Ratio

0 5 10 15 20 25 30

Figure 3-13 Utilization Ratio of Using 3rd-Order Bessel filters in Pre-Modulation Pulse Shaping

-40

m -60-o

-80

-100

SSPA Output, Pulse Shaping filter: 5th Order Butterworth BT=I

, t ! ! [,_l ! ! ! !

.... t_"i %..i 2Post-,PuseShaping ....... : ................ ........... :......... :.......... :.... _r .... i...... • ,. - i ]

......... ! ....... i ...... !........ / ....... :......... _....Pre-,Pulse Shaping ...... ; .........

-8 -6 -4 -2 0 2 4 6 8 I0

Frequency (Rb)

Figure 3-14 PSD Comparison of 8-PSK Between Two Pulse Shaping Methods

(5th Order Butterworth Filter ,BY=l)

3.1.3 Simulation Results of Particular 8-PSK Waveforms

In order to reduce power spectral spikes produced by the pre-modulation pulse

shaping method, some particular filters (8-PSK waveforms) were simulated. Since

Greenstein [4] has already shown that triangular, cosine, raised cosine and Nyquist

waveforms could not eliminate spectral spikes, formula (2-37) indicated that new

49

u

m

waveforms have to be investigated to get rid of spikes. The simulation parameters are the

same as in section 3.1.1 except for different filters (waveforms) as follows.

8-PSK Particular Pulse Shaping Signaling

a. Trapezoid Signaling (2T and 3T duration)

b. Cosine Transit Signaling (2T duration)

c. Raised Cosine Transit Signaling (2T duration)

d. Weighted Raised Cosine Transit Signaling (2T duration)

The above waveforms share a common characteristic, that is to keep fiat within one symbol

time T. Figures 3-15 to Figure 3-19 show these particular waveforms not only can reduce

bandwidth but also can eliminate or reduce spectral spikes.

0.5

0


' ' ' ' ' ' ' '7 ';

I I I I ] I I I

I I I I I I I

I I I I I I I I I

I I i I t I I I I

t I I I I I I I i...... m ...... F---

r --F -F FI I I I I

I I I I 1 I I

I I I I I I t I t

I I I I I I I I I

I I I I

i l I I I I I I I

-0.8 -0.6 -0.4 0.4 0.6 0.8-0.2 0 0.2

Time (T)

7

w

w.

o

-20

nn

"o_40

-60

-5

PSD of 8--PSK used 2T duration trapezoid pulse shaping

I I I I I I I I I

-4 -3 -2 -1 0 1 2 3 4

Frequency (Rb)

Figure 3-15 PSD of 8-PSK Used 2T Duration Trapezoid Pulse Shaping

50

05


I

I I II

----V ....... I II

I I

1 I

I I 1

I I I

I I I

- I

I I

I !

I |

I I

................... "r .........

I i

i i

| I

00 0.5 1 1.5 2Time (T)

2.5

PSD of 8-PSK used 3T duration trapezoid pulse shaping

z :

.......................... i i iI ] I ! I

-10 -8 -6 -4 -2 0 2 4 6 8Frequency (Rb)

Figure 3-16 PSD of 8-PSK used 3T-s duration trapezoid pulse shaping

10

=

0

-20

-4O

0.8

0.6

0.4

0.2

Cosine Transit Signaling with Pulse Duration 2T

I I j I I I I I IIII 11

,, ___J , , , _ --_ ' ,

....... p ...... p ..... p ...... p ...... p ................ _P ...... p ......l I I I l | I I l

1 I l l l I I .... I ...... I ......

....... 7 ............ F ...... F ...... T ...... i"- --7I I I 1 t I I I I................................................

I I I I I

I I I I ...... I ...... I -- I

¢ I

-- - --f ...... r I" r --r

I I I I I/ i i L , _ I , ,-0.8 -0.6 -0.4 ...0.2 0 0.2 0.4 0.6 08 1

Time (T)

PSDI I

i..........i........:.........:.........."..........".........."..........:.........".......... " " i i , J i

I I I I

-8 -6 -4 -2 0 2 4 6 8Frequency (Re)

Figure 3-17 PSD of 8-PSK used 2T-s duration cosine transit signaling pulse shaping

10

5]

w

0.8

0.6

0.4

0.2

0

-20

Raised-Cosine Signaling with Pulse Duration 2T

i t I i I i i l | i i

I [ 1 I I l I I E

I I I I t I I I I ......

....... F ...... F ..... F ...... y ...... F ...... F ...... F ...... y ...... T

I l I I I I I I I

.... _ ...... _ ..... _ ...... _ ...... _ _ ...... _ .... _ _ ......

I I I l I I I

I I I I

I I I I I __l__ F ...... nil ill--...... -- I 1 [ ...... F ...... [

I I I I I

-" -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Time (T)PSD

"o

t I I I I I I I

-8 -6 -4 -2 0 2 4 6 8

Frequency (Rb)

Figure 3-18 PSD of 8-PSK Used 2T Raised Cosine Transit Signaling Pulse Shaping

10

- 5

;l=d

= ;:

m

0.5

0

0

-20

-40

-60

180

-10

Weighted Raised-Cosine

I I_1 i I I I

II I I I I II l I I

I I I I l l

I I l I I I

I I I I I I

l I I I I

] I I I t I

I I I l

II I I I I

I t

43.8 -0.6 -0.4 -0.2 0Time (T)

Transit Signaling with Pulse Duration 2T

I t I

I

I I I

1 I I

____ ...... _ ..... ,_......

I

I I I I

I I

I I I I

0.2 0.4 0.6 0.8 1

PSD

I I I I I I I I

-8 -6 -4 -2 0 2 4 6 8

Frequency (Rb)

Figure 3-19 PSD of 8-PSK Used 2T Duration

Weighted Raised Cosine Transit Signaling Pulse Shaping

10

52

= .

m

3.1.4 Analysis Results of Particular 8-PSK Waveforms

According to formula (2-38) and (2-39), numerical analysis results of PSD for

trapezoid and weighted raised cosine signaling at some particular frequencies have been

calculated and are shown in Table 3-4 and Table 3-5. The discrete line components are

virtually zero, and they can eliminate all spikes. Simon's recent report [3] also indicated

3T duration trapezoid can eliminate power spectral line components. Tables 3-6 and Table

3-7 show that the raised cosine and cosine transit signaling can reduce line components,

but could not get rid of them. Analysis results are consist with simulation results.

r

w

Table 3-4 PSD Line Components for Trapezoid Shaping

iT 0

PSD 0.0819xl 0 "'_

1 2

0.0048xl 0-"': 0.0048x10 -j'_ 0.0098x 10-'_

4

O.O091xlO -j':

Table 3-5 PSD Line Components for Weighted Raised Cosine Shaping

wfT

PSD

0

O. 1736x 10 '_

1 2

0.0217x10 -_ 0.0087xl 0-'_ 0.0164xl 0 -j_ 0.0052x 10 -_z

i

Table 3-6 PSD Line Components for Raised Cosine Transit Shaping

fT

PSD 4.213x10 _ 4.213x10 _

2

4.213x10 _ 4.213x10 _ 4.213x10 _

= =

w

53

Table3-7PSDLine Componentsfor CosineTransitShaping

fff

PSD

0

4.333x10 -_ 4.333x10 -_

2

4.333x10 -_

3

4.333x10 -_

4

4.333x10 -_

From the simulations, the utilization ratios of pre-modulation pulse shaping with these

waveforms were calculated and listed in Table 3-8 and Figure 3-20

Table 3-8 Utilization Ratio of Pre-Modulation

Pulse Shaping 8-PSK with Different Waveforms

i .

Pulse Types -50 dB point Utilization

0 b)None 35 1

2T-s Trapezoid Pulses 3 11.6

3T-s Trapezoid Pulses 1 35

2T-s Pulses with Cosine Transit 7.5 4.6

2T-s Pulses with Raise-Cosine Transit 10 3.5

2T-s Pulses with Weighted Raised-Cosine Transit (k=1.5) 1.8 19.4

w

Weighted Raised Cosine'

Raised -Cosine Transit'

Cosine Transit'

3T-s Trapezoid'

2T-s Trapezoid'

0 1"0 t"5 2"0 2'5 3'0 35

Utilization Ratio

Figure 3-20 Utilization Ratio of Different Pulses in Pre-Modulation Pulse Shaping

54

• J

w

3.2 Bit Errors Rates Simulations

Figure 3-21 shows the system block diagram for pulse shaped 8-PSK end to end

system simulations

w

L.d

r =

Random

Phase

Baseband

Filter

Phase

Modulator

iDATA SOURCE! !PULSE SHAPING! ! MODULATION AMPLIFIER• . ........................ , ........................... • .................................................................

ERROR

RATE

ESTIMATOR

ERROR RATE...................................

AWGN CHANNEL

& INTEGRATEPHASE

& DUMPMETER

SYNCHRONIZER RECEIVER

Figure 3-21 BER Simulation System Block Diagram

m

w

AWGN is added to the signals out of the SSPA to simulate channel noise. The bit errors

are produced due to the ISI and AWGN. This report provides the measure of bit errors

according to bit energy to noise ratio (Eb/No). Figure 3-21 shows the system block

diagram for measuring bit error rates. For different pre-modulation pulse shaping filters

with the same BT, the plots of the BER are in Figure 3-22 to Figure 3-24.

55

m

= <

L.2

The following parameters were used to obtain the bit error rates:

Data source:

a. Ideal data with p(0) (probability of zero)=0.5;

c. Symbol rate Rs=l

b. Sample Rate (fs) = 16Rs

Pulse Shaping Filters:

a. Butterworth 5th Order (BT=1,2,2.8,3)

b. Bessel 3rd Order (BT=I, 1.2,2,3)

c. SRRC( a = 1,Number of tap length=256R s , rectangular window)

Power Amplifier:

European Space Agency(ESA) 10-watt SSPA, Back off--0dB as obtained

from JPL

BER for Pre-Modulation Pulse Shaping Filters with BT=I

1.00E+00

oILl:= 1.00E-04m

1.00E-05 j 1

1.00E-06

-1 1 3 5 7 9 11 13

Eb/No(dB)

•--_fr-Theoretical 8PSK

Butterworth

--B--Bessel

SRRC(alpha=l)

--0.-- Bessel BT=1.2

Figure 3-22. BER Plot for Pre-Modulation Pulse Shaping Filters with BT=I

56

BER for Pre-Modulation Pulse Shaping Filters with BT=2

L ;

w

1.00E+O0

I .OOE-01

i .00E-02

o 1.00E-03

m__ 1.00E-04

1.00E-05

i ' I '

i i

! i

t II : Ii =

I, E i ] \

' ! I I

, II

t I iI _ I , 1 i

L I • ,1.00E-06

-1 1 3 5 7 9 11 13 -

Eb/No(dB)

Figure 3-23. BER Plot for Pre-Modulation Pulse Shaping Filters with BT=2

--:J[--- Theoretical 8PSK

Butterworth

--I!-- Bessel

--_k-- SRRC(alpha=l)

= ,

m

JDm

.on

LLI

in

1.00E+00

1.00E-01

1.00E-02

1.00E-03

1.00E-04

1.00E-05

1.00E-06

BER for Pre-Modulation Pulse Shaping Filters with BT=3

r i

m

k I

-1 1 3

I=

i

f

5 7

Eb/No (dB)

9 11 13

---X---Theoretical 8PSK

--(k-- Butterworth

--I1--- Bessel

--A--- SRRC(alpha=l)

--'0-- Butterworth BT=2.8

Figure 3-24. BER Plot for Pre-Modulation Pulse Shaping Filters with BT=3

57

Although the Utilization ratio (Figure 3-20) for 3T Trapezoid is very high and the power

spectra are no spikes ( Figure 3-6) , this kind of waveform has overlapping within 3

symbol periods. This overlapping introduce a very serious ISI such that the integrate and

dump circuit can not work. Figure 3-25 shows the BER of pre-modulation pulse shaping

by using 3T Trapezoid. Since this BER is not accepted, other kind of receiver should be

used to correct ISI. An equalizer or sequential detector may be used to improve BER

performance.

m

w

m

etl.

tU

m

1.00E+00

1.00E-O1

1.00E-02

1.00E-03

1.00E-04

1.00E-05

1.00E-06 i

-1 1 3

: ,L ,L, r T

:_J, ! i !

i i

T

i i T5 7 9 11 13

Eb/No (dB)

-qY:-- Theoretical 8PSK

_(k--3T-s Trapezoid Pulse

Figure 3-25. BER Plot for Pre-Modulation Pulse Shaping With 3T Trapezoid Signaling

w

ISI Losses Due to Filtering

The ISI losses due to the pulse shaping filters can be determined by the BER

simulations. One needs to note that the receiver here was not an optimal receiver. Since the

2

-.-

. =

m

58

wintegrate and dump was used as receiver, it could not matched the pulse shaped signals.

Furthermore, there was no equalization for correcting ISI. The following tables give the

results of baseband filter ISI losses at 10 -3 BER for theoretical 8-PSK(or BPSK) [7] and the

filtered 8-PSK.

Table 3-9 ISI Loss Measurements at 10 -3 BER (Compared to ideal 8PSK)

Filter Type Loss Loss Loss Loss [,oss

BT=I BT=1.2 BT=2 BT=2.8 BT=3

(dB) (dB) (dB) (dB) (dB)

Butterworth ...........

5th Order 12.5 2.32 1.48 1.48

Bessel .....................

3rd Order 10 2.85 1.20

SRRC(a =1) ...................

4.22

Table 3-10 ISI Loss Measurements at 10 -3 BER (Compared to ideal BPSK)

=z

=

Filter Type Loss

BT=I

(dB)

Loss

BT=I.2

(dB)

Loss

BT=2

(dB)

Loss

BT=2.8

(dB)

Loss

BT=3

(dB)

13.2

Butterworth ...........

5th Order 15.7 5.52 4.68 4.68

Bessel ...........

3rd Order 6.05

SRRC(a =1)

7.42

4.88

-A

As stipulated in [16], the optimum filter should have the smallest BT value and provide

losses<0.4 dB. Thus an ideal matched filter and equalizer should be applied to meet the

= =

59

w

requirement of ISI loss <0.4. The Figure 3-25 shows the

utilization ratio for different filters.

ISI Loss (dB)

relationship of ISI loss and

m

20

15

10

5

iiiiiii iiiiiiiiiii!!!!!!!ii.: : i

5 10 15 2O 25 30

Utilization Ratios

Figure 3-25. Plots for ISI Loss as Utilization Ratios for Pre-Modulation Pulse Shaping

( ...... for 5th Order Butterworth Filter ; ...... for 3rd Order Bessel Filter; * for SRRC filter)

= -

w

w

60

W

w

w

Chapter 4

CONCLUSIONS AND RECOMMENDATIONS

This simulation based investigation on the pulse shaped 8-PSK modulations lead to

the following conclusions.

Pulse shaping is an effective way to reduce bandwidth and cope with the increasing

congestion of frequency band. Pre-modulation pulse shaping can keep constant envelope

and is more efficient on narrowing bandwidth than post-modulation pulse shaping. On the

other hand, pre-modulation pulse shaping using Butterworth, Bessel and SRRC filters can

produce discrete spectral components, and using integrate and dump receiver could not

ideally match pulse shaped signals. Thus the BER simulations by the pre-modulation pulse

shaping method are a best-case scenario. In order to reduce power spectral spikes and

improve BER, specialized 8-PSK pulse waveforms have been studied.

Pre-modulation pulse shaping and post-modulation pulse shaping have different

effects on narrowing bandwidth or frequency band utilization ratio. In general, pre-

modulation pulse shaping had better results for narrowing the bandwidth than post-

modulation pulse shaping did for the same traditional filters and BT values due to the

spectrum spreading effect of a non-constant modules through the non-linear amplifier.

Pre-modulation pulse shaping and post-modulation pulse shaping also have

different effects on producing bit error rates. In general, post-modulation pulse shaping had

lower bit error rates than pre-modulation pulse shaping for the non-ideal integrate & dump

receiver. That means that the narrower the bandwidth, the larger the bit error rate. For the

3rd Order Bessel filter, the BER produced by post-modulation pulse shaping (BT=2) is

61

i

w

w

2_

m

almost the same as that produced by pre-modulation pulse shaping (BT=3). For the 5th

Order Butterworth filters, the BER produced by pre-modulation pulse shaping (BT=2.8 or

3) is between those produced by post-modulation pulse shaping BT=I and BT =2.

Several pulse waveforms have been simulated, the pulse with cosine and raised

cosine can reduce power spectral line components. Furthermore, simulation and analysis

have shown that trapezoid and the pulse with weighted raised cosine transit can get rid of

discrete spectrum. Simon[3] and simulations also indicated that keeping at least one

symbol period flat and changing pulse transit can generate different effect on narrowing

bandwidth with no discrete spectrum. For further study, an improved method may be

found to change the pulse transit such that an optimal bandwidth with no spikes can be

achieved. On the other hand, any such method may also introduce potentially severe ISI.

A matched filter is crucial for receiving the signal to lower the BER. Matched

filters which can match pre- modulation pulse shaped signals are needed so that the BER

can be reduced. Since trapezoid and the pulse with weighted raised cosine transit have been

found to eliminate power spectral spikes, and they have deterministic waveforms which

have at least two symbol period duration, matched filters can be obtained according to

these waveforms.

Pulse shaping methods can introduce ISI and increasing BER. In order to obtain

better performance on BER, in addition to using an ideal matched filter, an equalizer may

be needed to reduce ISI.

i iw

62

REFERENCE

[1] V. K. Prabhu and H. E. Rowe, "Spectra of Digital Phase Modulation by Matrix

Methods," Bell Syst. Tech. J., vol. 53, pp. 899-935, May/June 1974.

[2] M.K. Simon, "On the Power Spectrum of Angle-Modulated PSK Signals Corrupted

by ISI," Private Communication, pp. 1-7, March 1996.

[3] M. K. Simon, "On the Power Spectrum of Angle-Modulated PSK Signals Corrupted

by Intersymbol Interference," TDA Progress Report 42-131, pp. I-10, November

1997.

[4] L. J. Greenstein, "Spectra of PSK Signals with Overlapping Baseband Pulses," IEEE

Trans. Commun., vol. COM-25, pp. 523-530, May 1977.

r_.

i[5] D. Subaainghe-Dias and K. Feher, "Baseband Pulse Shaping for rc/4 FQPSK in

Nonlinearly Amplified Mobile Channels," IEEE Trans. Commun., vol. 42, pp. 2843-

2852, October 1994.

[6] R. Caballero, "8-PSK Signaling over Non-Linear Satellite Channels," Masters Thesis,

New Mexico State University, pp. 1-200, May 1996.

[7]

[8]

P. J. Lee, "Computation of the Bit Error Rate of Coherent M-ary PSK with Gray code

Bit Mapping," 1EEE Trans. Commun., vol. COM-34, pp. 488-491, May 1986.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood

Cliffs, NJ: Prentice-Hall, 1989.

u

[9] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995.

i

63

[lO] SPW- The DSP Framework Communications Library Reference. Comdisco Systems

Product Number: SPW8015, March 1994.

[ 11] Signal Processing WorkSystem DSP Library Reference, Alta Group of Cadence

Design Systems, Inc. Product Number: SPW8015, March 1995.

[12] Signal Processing WorkSystem Interactive Simulation Library Reference. Alta Group

of Cadence Design Systems, Inc. Product Number: SPW8015, March 1995.

m

[13] W. L. Martin and T. M. Nguyen, "CCSDC - SFCG Efficient Modulation Methods

Study, A Comparison of Modulation Schemes, Phase 1: Bandwidth Utilization

(Response to SFCG Action Item 12/32), " SFCG- 13, Ottawa Canada, 13-21 October

1993.

[14] M. Otter, "CCSDC - SFCG Efficient Modulation Methods Study, A Comparison of

Modulation schemes, Phase lb: A Comparison of QPSK, OQPSK, BPSK, and

GMSK Modulation Systems (Response to SFCG Action Item 12/32), "European

Space Agency/ESOC, Member CCSDS Subpanel IE, (RF and Modulation), June

1994.

w

[15] W. L. Martin and T. M. Nguyen, "CCSDC - SFCG Efficient Modulation Methods

Study, A Comparison of Modulation Schemes, Phase 2: Spectrum Shaping (Response

to SFCG Action Item 12/32), " SFCG Meeting, Rothenberg, Germany 14-23,

September 1994.

[16] W. L. Martin, T. M. Nguyen, A. Anabtawi, S. M. Hinedi, L. V. Lain and M. M.

Shihabi, "CCSDC - SFCG Efficient Modulation Methods Study Phase 3." End-to-End

System Performance," Jet Propulsion Laboratory, May 1995.

64

[17] S. Haykin, Communication Systems. New York: John Wiley & Sons, 1994.

65

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