INTERPOLATED TREE ORTHOGONAL MULTIPLEXING SCHEME

FOR COGNITIVE RADIO DESIGNS

_______________

A Thesis

Presented to the

Faculty of

San Diego State University

_______________

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

in

Electrical Engineering

_______________

by

Asmita P. Mahalle

Fall 2011

iii

Copyright © 2011

by

Asmita P. Mahalle

All Rights Reserved

iv

DEDICATION

To my husband Abhay for his unfailing support and my parents-in-law for their

inspiration.

v

If everything seems under control, you’re just not going fast enough. - Mario Andretti

vi

ABSTRACT OF THE THESIS

Interpolated Tree Orthogonal Multiplexing Scheme for Cognitive Radio Designs

by Asmita P. Mahalle

Master of Science in Electrical Engineering San Diego State University, 2011

The demand for bandwidth has increased with the advent of high definition television

and broadband internet accessibility all over the world. But even though the available bandwidth is sparse, studies have shown that the significant portion of the spectrum allocated to licensed services show little usage over time. Hence, to meet the ever increasing bandwidth requirements there is a need for designing communication systems that efficiently uses the available spectrum.

This thesis focuses on the methodology, design and implementation of a communication system called as interpolated tree orthogonal multiplexing (ITOM). This frequency division multiplexed (FDM) channelization scheme produces transmission signals with attractive characteristics regarding time and frequency localization which can efficiently use the available bandwidth. The ITOM tree based structure consists of shaping filters that delivers shaped signals to an interpolation tree consisting of half band filters. All filters are based on low-pass prototypes centered at multiples of the quarter sample rate. The Thesis also discusses other multichannel formation schemes like discrete wavelet transform (DWT) and orthogonal frequency division multiplexing (OFDM) and compares these two techniques with ITOM, based on the channels formed by each of these schemes. All the filters involved has been modeled and simulated in MATLAB to demonstrate the results.

vii

TABLE OF CONTENTS

PAGE

ABSTRACT ............................................................................................................................. vi

LIST OF TABLES ................................................................................................................... ix

LIST OF FIGURES ...................................................................................................................x

ACKNOWLEDGEMENTS .................................................................................................... xii

CHAPTER

1 INTRODUCTION .........................................................................................................1 1.1 Objective of the Thesis ......................................................................................1 1.2 Interpolated Tree Orthogonal Multiplexing (ITOM) .........................................1 1.3 Unique Features of ITOM ..................................................................................2 1.4 Chapter Organization .........................................................................................2

2 FREQUENCY SPECTRUM ANALYSIS .....................................................................3 2.1 Scenario Where ITOM can be Used ..................................................................3 2.2 Spectrum Sensing...............................................................................................3 2.3 Importance of Sub-Channels ..............................................................................5

3 MULTI-CHANNEL FORMATION SCHEMES ..........................................................6 3.1 Discrete Wavelet Packet Transform (DWT)......................................................6 3.2 Shaped OFDM Channelizer Scheme .................................................................7 3.3 Limitations of DWT and OFDM Schemes ........................................................8

4 FILTERING TECHNIQUES USED IN ITOM ...........................................................10 4.1 FIR Interpolation ..............................................................................................10 4.2 FIR Decimation ................................................................................................11

5 POLYPHASE FILTER STRUCTURES USED IN ITOM .........................................13 5.1 Polyphase Interpolators ....................................................................................13 5.2 Polyphase Decimators ......................................................................................15

6 ITOM TREE STRUCTURE ........................................................................................19 6.1 ITOM’s Capability to Form Compact Sub-Channels ......................................19 6.2 ITOM’s Capability to Form Variable Bandwidth Sub-Channels ....................28

viii

6.3 Channel Allocation ..........................................................................................32 7 SUMMARY .................................................................................................................34

7.1 Conclusion .......................................................................................................34 7.2 Future Scope ....................................................................................................34

BIBLIOGRAPHY ....................................................................................................................36 APPENDIX

MATLAB CODE ...............................................................................................................37

ix

LIST OF TABLES

PAGE

Table 5.1. FIR Filter Coefficient Set .......................................................................................13 Table 5.2. Arrangement of 4-Path Polyphase Filter Coefficient .............................................13

x

LIST OF FIGURES

PAGE

Figure 2.1. Unevenly occupied spectrum with active primary users and empty frequency bands or white spaces which can be opportunistically used by ITOM scheme. ...............................................................................................................4

Figure 2.2. Secondary users making opportunistic use of the unused primary user bandwidth in a timely manner........................................................................................5

Figure 3.1. Cascade of half band filters in a DWT modulator and demodulator. ......................6 Figure 3.2. Spectra of spectral bands of 16-point DWT. ...........................................................7 Figure 3.3. Shaped OFDM modulator and demodulator. ..........................................................8 Figure 3.4. Spectra of shaped OFDM using alternate frequency bins. ......................................8 Figure 4.1. Interpolator block where ‘H’ assumed to have a finite impulse response

(FIR). ............................................................................................................................10 Figure 4.2. Shaped pulse 1:2 up sampled by zero packing and an overlapping half

band filter to prevent the up-sampling induced aliased spectrum above the lowpass cutoff frequency. ............................................................................................11

Figure 4.3. Decimator block where ‘H’ assumed to have a finite impulse response (FIR). ............................................................................................................................12

Figure 5.1. Block diagram of polyphase interpolator showing that the up-sampling process is performed after filtering. .............................................................................14

Figure 5.2. Block diagram of polyphase decimator showing that the down-sampling process is performed before filtering. ..........................................................................14

Figure 5.3. Block diagram of shaping filter in terms of prototype filter. ................................17 Figure 5.4. Block diagram of matched filter in terms of prototype filter. ...............................18 Figure 6.1. Modulation of shaped spectra using half band filters H0 and H1. ........................19 Figure 6.2. Spectra of 4 times oversampled shaping filters each centered on multiples

of quarter sample rate...................................................................................................20 Figure 6.3. The 1:4 up sampled shaped spectra are added together. .......................................20 Figure 6.4. The 1:4 up sampled shaped spectra are further 1:2 up sampled and

selectively filtered using half band filter H0 or H1. ....................................................21 Figure 6.5. Efficient sub-band formation using frequent upsampling and filtering

operation. .....................................................................................................................21

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Figure 6.6. Half band filter spectrum with -60dB sidelobes suppression and the transition bandwidth centered on quarter sample rate. ................................................22

Figure 6.7. 1:4 up sampled shaped spectrum spanning four quarter bands on the unit circle. ............................................................................................................................23

Figure 6.8. S0 spectrum is 1:2 up sampled and filtered using half band filter. A low pass or a high pass half band filter is used to select the desired spectrum. .................23

Figure 6.9. Spectrum of half band filter scaled at dB with a normalized frequency axis. Each of the four overlapped half band filters is centered at the quarter sample rate. ..................................................................................................................24

Figure 6.10. Input to output as the ITOM 16 tree is traversed, each half band translation level in modulator operates at twice the bandwidth and sample rate as the previous branch. .................................................................................................25

Figure 6.11. Up sampling and filtering operation of the half band filters at translation level 1. ..........................................................................................................................26

Figure 6.12. Up sampling and filtering operation of the half band filters translation level 2. ..........................................................................................................................27

Figure 6.13. Spectrum of adjacent equal bandwidth signals occupying the spectrum with minimum interference to adjacent band, providing compact sub-channels. ........27

Figure 6.14. ITOM 16 modulator output with channel 1 OFF. ...............................................28 Figure 6.15. ITOM 16 modulator output with channel 6, channel 11 and channel 13

OFF. .............................................................................................................................28 Figure 6.16. ITOM 16 demodulator block diagram. ................................................................29 Figure 6.17. Block diagram of ITOM 16 modulator with X12, X13, X14 and X15

unused. .........................................................................................................................30 Figure 6.18. Unoccupied ch12, ch13, ch14 and ch15. .............................................................30 Figure 6.19. Signals of different bandwidth can be merged together to form unequal

bandwidth non-adjacent sub-bands. .............................................................................31 Figure 6.20. Nonadjacent and unequal width frequency bands at transmitter output as

well as empty frequency bins. ......................................................................................31 Figure 6.21. ITOM 16 modulator paths traversed from input to output of the

modulator. ....................................................................................................................32 Figure 6.22. ITOM modulation and demodulation filters sequence while traversing

different depths of the tree structure. ...........................................................................33

xii

ACKNOWLEDGEMENTS

I offer my deepest gratitude to my research advisor Prof. fredric harris, whose critical

observations and suggestions has helped me at every point wherever I was stuck. I also thank

him for the numerous discussions we had together and for his unwavering support and

encouragement without which this thesis would not have been completed successfully.

I also express my sincere thanks to Dr. Santosh Nagaraj and Dr. Christopher Paolini

for being a member of my graduate committee and also for spending time to review the

thesis.

1

CHAPTER 1

INTRODUCTION

1.1 OBJECTIVE OF THE THESIS This thesis describes the design and implementation details of Interpolated Tree

Orthogonal Multiplexing (ITOM) which is a channelization scheme. The thesis also

discusses other Multi-channel formation schemes and compares them to ITOM scheme,

based on the channels formed by them.

1.2 INTERPOLATED TREE ORTHOGONAL MULTIPLEXING (ITOM)

We know that the transmission of higher data rate needs more bandwidth resulting in

the congestion of spectrum. In addition to this when the primary users (high priority users) in

a licensed spectrum are not using the bands allocated to them at all times, leaves the

spectrum unoccupied [1]. Hence there is a need for techniques that can utilize the limited

bandwidth resource in a spectrally efficient manner. The ITOM scheme of sub-channel

formation can make efficient utilization of frequency spectrum in the above described

scenario by occupying the unused bands, thus increasing the bandwidth efficiency.

ITOM is a Frequency Division Multiplexed (FDM) modulation scheme in which

sub-channels are built through an interpolation tree which allows the utilization of empty

frequency bands without disrupting other signals. This makes it suitable for Cognitive Radio

Designs. Many techniques have been suggested in the past based on advanced modulation

and demodulation schemes that can potentially reduce spectrum congestion [2]. But ITOM is

unique in the way that it allows access to different bandwidth channels at different depths of

the tree with compact channelization capabilities. This thesis is limited to the research and

study of cognitive decision-making algorithms and focuses mainly on ITOM in which

sub-channels are built through an interpolation tree. These sub-channels can be utilized to

occupy the empty frequency bands for efficient utilization of the spectrum.

2

1.3 UNIQUE FEATURES OF ITOM The transform based techniques use complex phase rotators or DFT to select the

center frequency, but ITOM uses half band filters to select center frequency. Also ITOM has

the same computational workload as the transform based techniques. In the Modulator, with

each successive stages from input to output the number of filters decreases but the sample

rate increases and in Demodulator the number of filters increases from input to output but the

sample rate decreases. Hence the work load remains same in each stage of Modulator and

Demodulator.

1.4 CHAPTER ORGANIZATION Chapter 2 discusses the motivation for bandwidth efficiency. It introduces to terms

such as dynamic spectrum access, spectrum sensing and spectrum mobility and need for

efficient sub-channel formation. Chapter 3 explains alternative multi channel formation

schemes like DWT and Shaped OFDM channelizer and their limitations. The simulation

outputs of each of the two techniques are provided along with the block diagram. Chapter 4

discusses the basic filtering techniques such as interpolation and decimation. These basic

filtering techniques are the basis on which the entire structure of ITOM is built. In Chapter 5

Polyphase interpolation and Polyphase decimation is described. In Chapter 6 the block

diagram of the ITOM 16 modulator and demodulator is introduced along with the frequency

spectrum of the outputs at each level of modulation. ITOM’s capability to form compact sub

channels of equal bandwidth and its capability to form variable bandwidth channels can be

clearly understood by the Simulation outputs. Finally the relation between ITOM input nodes

in a generalized form is given. Chapter 7 summarizes the final conclusions about the

Interpolated Tree Orthogonal Multiplexing scheme accounting its efficient implementation

and unique features and future scope.

3

CHAPTER 2

FREQUENCY SPECTRUM ANALYSIS

When available bandwidth is limited, dynamic spectrum management techniques

allow the use of allocated spectrum in an opportunistic manner. Hence it is crucial to

understand the occupancy of the spectrum and how ITOM scheme of sub-channel formation

can make efficient use of available spectrum in a scenario where huge part of the licensed

spectrum is unoccupied by the primary users, creating spectral holes.

2.1 SCENARIO WHERE ITOM CAN BE USED In a licensed spectrum, the primary users (PU) are high priority users in a given

frequency band (e.g. cell phone provider, TV station, emergency services, etc.) and their

access can only be controlled by their base-station. The secondary users (SU) are low priority

users, who take advantage of cognitive radio techniques, to ensure non-interfering

co-existence with the primary users. A SU is assumed to have the capabilities to

communicate with not only the base-station but also other SUs; this is to resolve the

contention when two SU try to use the same unoccupied band of frequency at the same time

[3].

PU’s own different parts of the spectrum but may not be active at a particular time. In

Figure 2.1 the frequency bands indicate that the PU is currently using its spectrum and

consequently this frequency band cannot be used by any SU. Figure 2.1 also shows empty

frequency bands (white spaces), left vacant by PU’s. These bands can be accessed by SU’s

opportunistically. This new framework of spectrum access can be implemented using

Cognitive Radios (CR). Most fundamental role of a CR is to discover spectrum opportunities

by detecting existence or return of PU’s in the channel.

2.2 SPECTRUM SENSING The ability of a cognitive radio to access the white spaces that dynamically appear is

predicated upon its ability to detect the white spaces. Spectrum sensing is a technique which

reliably senses the spectral environment over a wide bandwidth. It detects the presence or

4

Figure 2.1. Unevenly occupied spectrum with active primary users and empty frequency bands or white spaces which can be opportunistically used by ITOM scheme.

absence of legacy users and uses the spectrum only if communication does not interfere with

any primary user. A fundamental requirement for the SU is to continually monitor the

presence of PU. The SU using the unoccupied PU spectrum would have to immediately clear

the sub-channel and find a new free sub-channel once the PU appears again. Because of the

fluctuating nature of the available spectrum SU’s traverse across multiple cells having

heterogeneous spectrum availability also called as spectrum mobility.

The existence check of a PU is done in a timely manner. In Figure 2.2, the PU’s

occupancy of the channel is shown in gray between the frequency bin 1 and frequency bin

10. Note that the PU is not active at all times. This leaves the channel vacant. SU’s

occupancy of the same channel is shown at the bottom. When the primary users vacates the

channel the secondary user occupies it, and the secondary user in turn vacates the channel on

return of the primary user. The time ΔT1 is the time taken for the secondary user to observe

that the channel is actually free and to take action to use it. The time ΔT2 is the time taken

for secondary user to observe that the primary user is back and to subsequently vacate. If the

observation process is very long resulting in large ΔT1, then this can lead to very inefficient

use of white space. In some cases the window of opportunity to occupy the PU channel can

be completely lost. If ΔT2 is very large the amount of interference caused to the PU may be

unacceptable. The timings ΔT1 and ΔT2 are constrained by the regulators of the channel.

Spectrum sensing and spectrum mobility techniques referred in this chapter

Am

plitu

de(d

Bm

) Empty Frequency Bands Active Primary Users

Frequency Spectrum

5

10

PU PUPU

SU

20

30

40

T1T2

Time bin

Freq

uenc

y bi

n

PU PU

PU PU

SU Figure 2.2. Secondary users making opportunistic use of the unused primary user bandwidth in a timely manner.

works through a wireless software suite comprising of policy based rules and database

engines that drive algorithms for frequency agility and cognitive decision-making.

Now that we know that there exists scattered unoccupied bands of frequency over the

entire spectrum and there exists techniques like spectrum sensing and spectrum mobility

which can sense these unoccupied bands, we will look at the multi-channel formation

schemes that can utilize these unoccupied bands. In the subsequent chapters we will also see

how ITOM scheme can make the best use of unoccupied bands as compared to other

schemes discussed.

2.3 IMPORTANCE OF SUB-CHANNELS Since the unoccupied spectral holes are scattered over the entire bandwidth,

sub-channel formation using ITOM offers two advantages. If a PU appears during the

lifetime of a SU link it would impact very few of the sub-channels used by the SU Link [3].

Also, ITOM creates the effect of spectral notching by leaving the sub-channels vacant to

bracket the already occupied PU channels. Hence it is important that the sub-channels are

dynamic in the sense that they have the ability to change their spectral shape in time

corresponding to available spectra, while at the same time minimally interfering with

occupied bandwidth.

6

CHAPTER 3

MULTI-CHANNEL FORMATION SCHEMES

3.1 DISCRETE WAVELET PACKET TRANSFORM (DWT) DWT scheme uses a cascade of half band filters that form a branching tree like

structure. The composite impulse responses of each branch constitute an orthogonal set of

sampled data waveforms [4]. The half band filters as shown in Figure 3.1 perform the dual

task of spectral shaping and interpolation. It uses a set of half band low pass and half band

high pass filters in each subsequent stages. Repeated application of up sampling and half

band filtering passes one while suppresses the remaining spectral replicate.

HP

LP

HP

HPLP

LP

LP

HP

X0(n)2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

X1(n)

X2(n)

X3(n)

X4(n)

X5(n)

X6(n)

X7(n)

+LP

HPHP

+LP

HPLP

+LP

HPHP

+HP

LP

+

+

LP

LP

HP

X0(n) 1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

X1(n)

X2(n)

X3(n)

X4(n)

X5(n)

X6(n)

X7(n)

+HP

LP

HP

LP

HP

LP

Figure 3.1. Cascade of half band filters in a DWT modulator and demodulator.

The up sampling and filtering operation at each level causes the transition bands of

the filters to alias with each other. The energy of the adjacent bands spill into each other and

hence the branch responses of DWT do not form a set of spectral bands with compact

support. Similarly, in a DWT demodulator the transition band of the down sampled filters

alias their transition bandwidths back into the filter pass-band [5].

The top spectrum in Figure 3.2 shows enabled spectral bands of a 16-point DWT. The

middle spectrum shows that when these bands are disabled, empty spaces of similar width

are not created. The overlapping enabled and disabled bands in the bottom spectrum clearly

7

Figure 3.2. Spectra of spectral bands of 16-point DWT.

show that the energy of adjacent bands spill into each other. Hence DWT cannot form

compact adjacent channels.

3.2 SHAPED OFDM CHANNELIZER SCHEME Shaped OFDM channelizer is formed by a cascade of an inverse DFT and a

polyphase partition of a prototype low-pass filter. See Figure 3.3 for block diagram of

Shaped OFDM channelizer. The DFT defines the center frequencies of the channels by dual

task of up-sampling and complex sinusoid signal generation. The polyphase partitioned

prototype filter performs the shaping of the individual time series. To assure zero inter

symbol Interference (ISI), the prototype shaping filter must be a square-root Nyquist pulse

prior to its polyphase partition [4].

The DFT converts a single input sample presented to input frequency bin ‘k’ of an N

point DFT into an output sequence containing N samples of a complex sinusoid exhibiting

‘k’ cycles per interval of length N [5]. Similarly in the Demodulator a polyphase filter is used

to time align the partitioned and resampled time series in each path, DFT is used to phase

align and separate the multiple base-band aliases.

8

N-PNTIFFT

h0(n)

h1(n)

h2(n)

h3(n)

hM-2(n)

hM-1(n)

From Input Mapping

Polyphase Partition

N-PNTFFT

h0(n)

h1(n)

h2(n)

h3(n)

hM-2(n)

hM-1(n)

Polyphase Partition

To Output Mapping

Figure 3.3. Shaped OFDM modulator and demodulator.

In order to avoid the adjacent channel interference (ACI) to the immediate two

spectral neighbors the adjacent channels are skipped and only alternate frequency bins are

used. It can be clearly seen in Figure 3.4 that the shaped OFDM spectrum exhibits compact

support. This is due to the contained bandwidth of the shaping filters. However, Shaped

OFDM channelizer cannot form channels of variable bandwidth.

Figure 3.4. Spectra of shaped OFDM using alternate frequency bins.

3.3 LIMITATIONS OF DWT AND OFDM SCHEMES The DWT scheme discussed in Section 3.1 cannot partition the spectra of the signal

into a set of compact adjacent spectral intervals. The OFDM scheme discussed in Section 3.2

cannot form a set of variable bandwidth sub-channels. However, for opportunistic use of a

frequency band in unoccupied intervals, these two properties are very important. As

9

compared to this, ITOM scheme is unique in a way that it can form compact sub-channels as

well as it can form channels of variable bandwidth.

10

CHAPTER 4

FILTERING TECHNIQUES USED IN ITOM

ITOM receives shaped signals at the input and with the help of half band filter

induced aliasing, translates each signal to a corresponding sub-channel, while traversing

through the path of tree. This chapter discusses the various filtering techniques used in

building the ITOM translation tree.

4.1 FIR INTERPOLATION The FIR Interpolation block resamples the discrete-time input at a rate M times faster

than the input sample rate, where the integer M is the interpolation factor parameter. This

process as shown in Figure 4.1 consists of two steps:

• The block up samples the input to a higher rate by inserting M-1 zeros between samples.

• The block filters the up sampled data with a FIR filter to prevent imaging in the frequency band above the lowpass cutoff frequency.

X(n) H Y(n)1:M

Figure 4.1. Interpolator block where ‘H’ assumed to have a finite impulse response (FIR).

The up sampler converts the Nyquist interval, the observable frequency span, from

the input sample rate to a span ‘M’ time as wide, the output sample rate. Zero packing the

input series, effectively decreases the distance between input samples without modifying the

spectral content of the series. The wider Nyquist interval, spanning ‘M’ input Nyquist

intervals, presents ‘M’ spectral copies of the input spectrum to the FIR filter [6]. The

amplitude of each of the M copies is 1/M of the amplitude of the input signal spectrum. The

zero packing creates a higher-rate signal whose spectrum is the same as the original over the

original bandwidth, but has images of the original spectrum centered on multiples of the

original sampling rate.

11

As shown in Figure 4.2 up-sampling adds to the original signal M-1 undesired

spectral images which are centered on multiples of the original sampling rate. The primary

reason for filtering after up-sampling is to remove these undesired spectral copies. By

eliminating M-1 spectral copies, the bandwidth reduces by a factor of 1/M, the filter gain

precisely compensates for the attenuation of the input spectra due to the zero packing of the

input series [6]. The result of interpolation is as if the signal is originally sampled at the

higher rate.

Figure 4.2. Shaped pulse 1:2 up sampled by zero packing and an overlapping half band filter to prevent the up-sampling induced aliased spectrum above the lowpass cutoff frequency.

4.2 FIR DECIMATION The FIR Decimation block filters and down samples an input signal. The FIR

Decimation block resamples the discrete-time input at a rate M times slower than the input

sample rate, where the integer M is specified by the Decimation factor parameter. This

process as shown in Figure 4.3 consists of two steps:

• The block filters the input data using a FIR filter to avoid aliasing of frequencies above the pass band into the pass band.

• The block down samples the filtered data to a lower rate by discarding M-1 consecutive samples following every sample retained.

Even though conceptually up sampling occurs before filtering in FIR interpolators

and down sampling occurs after filtering in FIR decimators, the two operators filtering and

12

X(n) H Y(n)

M:1

Figure 4.3. Decimator block where ‘H’ assumed to have a finite impulse response (FIR).

resampling can be interchanged in both interpolators and decimators according to the noble

identity [6]. The process proceeds by first filtering and then up sampling in interpolators and

down sampling and then filtering in decimators as discussed in Chapter 5.

13

CHAPTER 5

POLYPHASE FILTER STRUCTURES USED

IN ITOM

All polyphase filters designed in this thesis are based on multirate filters that alter the

sample rate of the input signal by FIR interpolation and FIR decimation during the filtering

process. In the ITOM modulator, tree based structure consists of shaping filters that delivers

shaped signals to an interpolation tree consisting of half band filters. In the ITOM

demodulator the tree structure consists of half band decimators acting as a channelizer which

passes required sub-channels to a bank of match filters.

5.1 POLYPHASE INTERPOLATORS In the case of a 1:M FIR interpolation filter, M-1 samples packed between the

successive input samples are zeros. Each packed zero gets multiplied by a coefficient and

summed with the others. However, this adding-and-summing processing has no effect when

the data sample is zero. Hence to interpolate by a factor of M, we arrange the prototype filter

coefficients as given in Table 5.1 into M-path polyphase partition as given in Table 5.2 and

calculate M outputs for each input using M different sub-filters derived from the original

filter as shown in Figure 5.1 and Figure 5.2.

Table 5.1. FIR Filter Coefficient Set Filter Coefficients h0 h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15

Table 5.2. Arrangement of 4-Path Polyphase Filter Coefficient

H0(Z) h0 h4 h8 h12

H1(Z) h1 h5 h9 h13

H2(Z) h2 h6 h10 h14

H3(Z) h3 h7 h11 h15

14

H1

H0

H2

HM -1

X(n) Y(n)

Figure 5.1. Block diagram of polyphase interpolator showing that the up-sampling process is performed after filtering.

H1

H0

H2

HM -1

X(n)

Figure 5.2. Block diagram of polyphase decimator showing that the down-sampling process is performed before filtering.

15

Figure 5.1 shows a filter with output commutator which accomplishes the task of

simultaneous interpolation and up conversion to the Mth Nyquist zone. A direct

implementation of filter h(Z) with N taps will need N multiplications and N −1 additions per

output, while polyphase implementation will only need (N/M) multiplications and (N/M) -1

additions per output.

The Z-transform of a FIR filter with delayed filter coefficients can be represented as

one dimensional coefficient set by Equation 5.1 [6]. This one-dimensional array can be

converted into a multi-dimensional array with M rows representing the sample rate change.

The structure of the polyphase filter coefficients can be defined by Equation 5.2 [6]. After

factoring of a Z-r term from the rth row and rewriting of Equation 5.2, the polyphase

decomposition of a FIR filter H(Z) can be expressed in Equation 5.3 [6]. Equation 4.3

describes the output of a polyphase FIR filter where the input data is delayed by Z0, Z-1,

Z-(M-1) and processed and summed at the outputs of the filter H0(Z), H1(Z), ….,HM-1(Z).

(5.1)

(5.2)

(5.3)

We have seen in Section 4.1 that when we zero pack the filter coefficients, we get

spectral copies of baseband input signal. Rather than extracting the spectral copy at baseband

from the replicated set of spectrum we can directly extract one of the spectral translates by

using band pass filter. The band pass filter is simply an up-converted version of low pass

filter.

(5.4)

The kth multiple of 1/Mth of output sample rate represents the center frequency of the

up converted filter. The z-transform of such a filter is shown in Equation 5.4.

5.2 POLYPHASE DECIMATORS Decimation is sample rate reduction operation by retaining one sample in every M

samples and down sample only the retained samples. By down sampling by a factor of M:1,

16

every multiple of output sample rate aliases to baseband. This down-sampling causes the M:1

spectral folding, effectively translating the M multiples of output sample rate to the

baseband.

The constraint that the center frequencies be limited to integer multiples of output

sample rate, assures aliasing to baseband by the sample rate change. A complex heterodyne

shifts the signal to the center frequency depending on the offset Δθ rad/sample times the

multiple of output sample rate [6]. The down converted center frequencies located at integer

multiples of output sample frequency are frequencies that alias to zero frequency. The phase

coherent summation of output of M-path filters separate aliases residing in each path, while

simultaneously destructively cancelling the remaining aliased spectral components. A direct

implementation of filter H(z) with N taps, will need M*N multiplications and M*(N − 1)

additions per output, while polyphase implementation will only need N multiplications and

N − 1 additions per output.

The commutator performs an input sample rate reduction by commutating successive

input samples to selected paths of the M-path filter. Sample rate reduction occurring prior to

the filtering causes spectral regions residing at multiples of the output sample rate to alias to

baseband. By delivering consecutive samples to the M input ports of the M-path filter

performs a down sampling operation.

All filters in this thesis are designed by assembling the prototype filter coefficients

into a polyphase structure that supports the desired sample rate change. Filter coefficients are

created using remez algorithm and assembled as 4-path polyphase filters forming 4 quarter

band filters.

Each category of filter contains four spectrally shifted versions of their corresponding

prototype filters. See Appendix for the MATLAB implementation of each category of filters.

(5.5)

where k = {0,1,2,3}.

17

The z-transform of the heterodyned filters in Equation 5.5 can be written in terms of

4-path polyphase partitions from prototype filter. In case of shaping filter,

(5.6)

.

The Equation 5.6 shows that the rth polyphase partition of the kth filter is equal to the

rth polyphase partition of the prototype filter multiplied by the complex constant jkr.

Figure 5.3 and Figure 5.4 show the block diagram of polyphase shaping and match filters in

terms of their prototype filter.

X(n) Yk(n)

S0

S1

S2

S3

jk

-1k

-jk

Figure 5.3. Block diagram of shaping filter in terms of prototype filter.

18

Yk(n)

M1

M2

M3

M0

X(n)

jk

-1k

-jk

Figure 5.4. Block diagram of matched filter in terms of prototype filter.

The polyphase filters structures in Figure 5.3 and Figure 5.4 have same weight as the

prototype filters except there is a change in sign and storage location of the weights.

19

CHAPTER 6

ITOM TREE STRUCTURE

ITOM system accepts externally generated wave-shapes and processes them through

a tree based structure of polyphase half band filters to create a multi-channel modulator and

demodulator. The shaping filters play an important role in ITOM by band-limiting the input.

The translation tree which is the unique and defining feature of ITOM consists of 1:2 half

band interpolators in the modulator. The half band filters receive shaped signals and through

selective up-sampling induced aliasing translates each signal to a corresponding sub-channel.

In the demodulator the translation tree consists of 2:1 half band decimators which act as a

channelizer passing required sub-channels to a bank of match filters. Each sub-band can be

separated by their corresponding matched filters. ITOM uses low pass and high pass filters as

well as two more half band filters centered at the positive and negative quarter sample rate,

also called as Hilbert transform filters. Similarly, the shaping and match filters are also

centered at multiples of the quarter sample rate.

6.1 ITOM’S CAPABILITY TO FORM COMPACT SUB-CHANNELS

In the modulator each successive branch of the tree performs 1-to-2 up-sampling of

the externally shaped input and selectively filter the aliased spectra to create the desired

output spectrum as shown in Figures 6.1 to 6.5.

+S1

S2H1

+S0

H0

+

S3X0(n) 1:2

1:2

1:4

1:4

1:4

1:4

X1(n)

X2(n)

X3(n)

Z(n)

Figure 6.1. Modulation of shaped spectra using half band filters H0 and H1.

20

Figure 6.2. Spectra of 4 times oversampled shaping filters each centered on multiples of quarter sample rate.

Figure 6.3. The 1:4 up sampled shaped spectra are added together.

21

Figure 6.4. The 1:4 up sampled shaped spectra are further 1:2 up sampled and selectively filtered using half band filter H0 or H1.

Figure 6.5. Efficient sub-band formation using frequent upsampling and filtering operation.

Once the externally shaped baseband signal is inserted into the branch ports of the

tree, the half band filters interpolate and selectively choose which spectral replica to insert

into the unused bandwidth.

The spectral and time domain characteristics of the ITOM structure are

completely described by the half band filters. The half band filters do the spectral

translation to the desired channel by changing the sample rate at different depths of the tree

in ITOM.

A low pass half band filter has a half amplitude gain response located at the plus and

minus quarter sampling rate [6]. The transition bandwidth is centered on the quarter sample

22

rate, with the end of the pass band and the start of the stop band being equi-distant from the

quarter sample rate. Figure 6.6 shows low pass half band filter with a sample rate of fs. This

filter is designed with the sidelobe suppression of -60dB and we can see that the amplitude

gain response is -30dB around plus and minus quarter sampling rate. In this thesis, half band

filter based on polyphase interpolation and decimation are used which occupies spectrum at

four quadrants of unit circle as shown in Figure 6.7.

Figure 6.6. Half band filter spectrum with -60dB sidelobes suppression and the transition bandwidth centered on quarter sample rate.

As each branch of the modulator tree is traversed, each branch’s output is interpolated

and the translated spectrum forms the up-sampled spectral output as shown in Figure 6.8.

In this thesis all the half band filters are implemented as polyphase interpolators

and decimators. These half band filters as shown in Figure 6.9 are used to selectively

filter the aliased spectra created by the up-sampling and down-sampling process at each

branch.

As shown in Figure 6.10 once all paths of the first branch are processed, each of the

data series from the first branch is forwarded into the second branch of the modulator tree for

processing. By carefully implementing successive branches of polyphase interpolating half

band filters and applying appropriate half band filter centered at quarter sample rate the

unoccupied spectrum can be populated as shown in Figures 6.11 to 6.13.

The ITOM filters can be classified into four different categories: shaping filters,

match filters, interpolating half band filters and decimating half band filters. They are based

on their respective prototype low-pass filters. Each category of filter contains four spectrally

shifted versions of the corresponding prototype filter.

Every branch of the tree has a shaped spectrum at the input, formed by external

shaping filters. There are a total of four sets of shaping filters feeding the branches of the tree

23

S1

S0S2

S1

S2S3

S3

S0

1:4 upsampled spectrum of prototype filter S showing 4 copies S0, S1, S2 and S3 at each quadrant of unit circle

Filtered S0 spectrum

Filtered S1 spectrum

Filtered S2 spectrum

Filtered S3 spectrum

Figure 6.7. 1:4 up sampled shaped spectrum spanning four quarter bands on the unit circle.

Spectrum occupying the quarter band on a unit circle

1:2 upsampled spectrum

S0

Spectrum filtered with H0

Spectrum filtered with H1

Figure 6.8. S0 spectrum is 1:2 up sampled and filtered using half band filter. A low pass or a high pass half band filter is used to select the desired spectrum.

24

Figure 6.9. Spectrum of half band filter scaled at dB with a normalized frequency axis. Each of the four overlapped half band filters is centered at the quarter sample rate.

based modulator. The bandwidth of each of these filters is limited to less than one quarter

span of the output sampling rate and these spans are centered at the four cardinal directions

on the unit circle. Each of these filters can be implemented as a polyphase interpolator as

described in Section 5.1.

The spectrums shown in Figure 6.12 and Figure 6.13 in blue are the shaped spectrum

at the output of four shaping filters and the spectrums in red are the half band filters H0, H1,

H2 and H3. The addition and interpolation of the shaped spectrum gives subsequent multiple

channels at each level.

Inside the tree every branch is characterized by its own bandwidth and sample rate.

Hence Figure 6.12 shows the equal bandwidth sub-channels over the interval of -8 to +8

25

+S1

S2H1

+S3

S0H2

+S1

S2H3

+S3

S0H0

+S1

S2H1

+S3

S0H2

+S1

S2H3

+S0

H0

+

+

+

+

S3

H0

H1

H2

H3

+

X0(n) 1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

X1(n)

X2(n)

X3(n)

X4(n)

X5(n)

X6(n)

X7(n)

X8(n)

X9(n)

X10(n)

X11(n)

X12(n)

X13(n)

X14(n)

X15(n)

Y(n)

Figure 6.10. Input to output as the ITOM 16 tree is traversed, each half band translation level in modulator operates at twice the bandwidth and sample rate as the previous branch.

which has 2 times more sampling frequency as its previous stage. This continuous

interpolation using half band filters allows simultaneous bracketing of utilized bands of

spectrum while occupying the unused spectrum without interfering with adjacent channels.

The addition of all the branches of the tree after sufficient interpolation gives spectrum of

individual channels adjacent to each other. Figure 6.13 shows the spectrum of the individual

26

Figure 6.11. Up sampling and filtering operation of the half band filters at translation level 1.

channel 0 through channel15 obtained by pruning process of half band filters. Figures 6.14

and 6.15 shows ITOM 16 output with few channels OFF.

The tree based channelizer demodulator is a tree based structure of spectrally

overlapped sub-bands created by polyphase half band and quarter-band filters. The

demodulator is able to accept a data set, reduce the sample rate and bandwidth, and

implement the process of selectively removing overlapping sub-bands. These sub-channels

provide the means to form compact channels.

The main processing task of the tree in the demodulation process as shown in

Figure 6.16 is performed by a 4-path polyphase half band filter. As the input time series is

processed through each branch of the tree, a 2-to-1 down-sampling and a 2-times bandwidth

27

Figure 6.12. Up sampling and filtering operation of the half band filters translation level 2.

Figure 6.13. Spectrum of adjacent equal bandwidth signals occupying the spectrum with minimum interference to adjacent band, providing compact sub-channels.

28

Figure 6.14. ITOM 16 modulator output with channel 1 OFF.

Figure 6.15. ITOM 16 modulator output with channel 6, channel 11 and channel 13 OFF.

reduction occurs. Each branch of the tree is characterized by its own bandwidth and sample

rate. Performing pruning of select spectrally overlapping polyphase filter outputs will provide

the means to deliver compact channelization. The final branch of the tree is processed with

polyphase matching filters with 4-to-1 down sampling and bandwidth reduction and it will be

the final processing performed by the tree.

6.2 ITOM’S CAPABILITY TO FORM VARIABLE BANDWIDTH SUB-CHANNELS

ITOM has a unique capability of forming signals of different bandwidth which are

non-adjacent to each other. Figure 6.17 depicts an ITOM 16 modulator tree where input

branches X12 to X15 are left unused. This gives a spectrum with input signals X0 to X11

only as shown in Figure 6.18.

In Figure 6.19 X12(n), X13(n) and X14(n) are left unused while X15(n) is 1:4 up

sampled and further up sampled by a factor of 2 directly at level 2 of the modulation tree.

29

M1

M2G1

M3

M0G2

M1

M2G3

M3

M0G0

M1

M2G1

M3

M0G2

M1

M2G3

M0G0

M3

G0

G1

G2

G3

X0(n)2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

2:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

4:1

X1(n)

X2(n)

X3(n)

X4(n)

X5(n)

X6(n)

X7(n)

X8(n)

X9(n)

X10(n)

X11(n)

X12(n)

X13(n)

X14(n)

X15(n)

Y(n)

Figure 6.16. ITOM 16 demodulator block diagram.

Clearly the last branch shows that the signals of different bandwidth can be added

together. The unique feature of ITOM shown in Figure 6.20 merges signals of different

bandwidth to form a variable bandwidth non adjacent band which is not offered by any other

multi-channel modulation scheme.

30

+S1

S2H1

+S3

S0H2

+S1

S2H3

+S3

S0H0

+S1

S2H1

+S3

S0H2

+S1

S2H3

+S0

H0

+

+

+

+

S3

H0

H1

H2

H3

+

X0(n) 1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

X1(n)

X2(n)

X3(n)

X4(n)

X5(n)

X6(n)

X7(n)

X8(n)

X9(n)

X10(n)

X11(n)

X12(n)

X13(n)

X14(n)

X15(n)

Y(n)

Figure 6.17. Block diagram of ITOM 16 modulator with X12, X13, X14 and X15 unused.

Figure 6.18. Unoccupied ch12, ch13, ch14 and ch15.

31

+S1

S2H1

+S3

S0H2

+S1

S2H3

+S3

S0H0

+S1

S2H1

+

S3

S0 H2

S1

+S0

H0

+

+

+

S3

H0

H1

H2

+

X0(n) 1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:2

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

1:4

X1(n)

X2(n)

X3(n)

X4(n)

X5(n)

X6(n)

X7(n)

X8(n)

X9(n)

X10(n)

X11(n)

X12(n)

X13(n)

X14(n)

Y(n)

S2X15(n) H31:21:4

Figure 6.19. Signals of different bandwidth can be merged together to form unequal bandwidth non-adjacent sub-bands.

Figure 6.20. Nonadjacent and unequal width frequency bands at transmitter output as well as empty frequency bins.

32

x0

x1

S3

S0

x2

x3

S1

S2x4

x5

S3

S0

x6

x7

S1

S2

x8

x9

S3

S0

x10

x11

S1

S2x12

x13

S3

S0

x14

x15

S1

S3

H0

H1

H2

H3

H0

H1

H2

H3

H0

H1

H2

H3 Y

Figure 6.21. ITOM 16 modulator paths traversed from input to output of the modulator.

6.3 CHANNEL ALLOCATION In ITOM16 there are 16 input nodes, one level of shaping filters and 2 translation

levels. In general a 2P input ITOM system will have P-2 translation levels. Each input signal

has a unique path through the tree passing through a unique sequence of interpolaters. The

first filter to act on the ith input signal is the kth shaping filter where k = (i − 1) mod 4. The

remaining filters in the path are translation filters; their appearance also follows a basic

pattern throughout. In general, as shown in Figure 6.21 , at the pth level of the modulation

tree the ith path passes through Hk where k = [i/(2p) mod 4]. Similarly in the demodulator the

kth half band decimator passes ith input signal and the kth matching passes the ith input signal

where, i = input signal, k = spectral periodicity of each filter and P = Half band translation

level.

33

Channel allocation within ITOM, tells which input port winds up at which frequency

in the spectrum on the Modulator side and also tells signal at which frequency comes out of

the corresponding output port on the Demodulator side, as shown by the sequence in

Figure 6.22.

Figure 6.22. ITOM modulation and demodulation filters sequence while traversing different depths of the tree structure.

34

CHAPTER 7

SUMMARY

7.1 CONCLUSION The efficiency of the polyphase interpolators and decimators derives from the fact

that the subsequent multiplication and addition operations in each level of the tree are greatly

reduced and hence reducing the implementation cost of the entire tree structure.

DWT does not provide compact support due to aliasing of transition band from one

stage of filtration to another and OFDM channelizer does not support spectral sub-channels

of varied width as described in chapter 3. As opposed to this ITOM modulator and

demodulation tree structure described in chapter 6 provides compact spectral support

necessary to effectively divide the spectrum into multi-channel bands. The final output

spectrum created by the ITOM 16 modulator demonstrated the modulator’s ability to notch

spectral gaps which allows the modulation process to utilize unused spectrum without

interfering with adjacent channels. The multiple-bandwidth feature of the modulator is shown

by applying an externally shaped pulse into the second level of the tree. Thus the simulation

results show that the spectrum of compact sub-channels of equal bandwidth and spectrum of

compact channels of unequal bandwidth non-adjacent bands can be formed using ITOM

scheme.

The channel allocation gives the sequence of filter index numbers while traversing the

ITOM tree from input of modulator to output of demodulator. This is especially important

when the number of channels required is high.

7.2 FUTURE SCOPE The ITOM 16 tree can be modeled using Simulink software to allow the simulation of

alternate filter structures within ITOM and to measure the efficiency of the system. The

hardware implementation of the ITOM tree structure can be considered and ITOM core can

be designed in VHDL or Verilog written manually or auto generated using HDL coder. The

HDL coder can auto generate a testbench to verify the design output. Recursive Shaping and

35

Half band filters can be used to reduce workload [7]. Also a mix of tree based and other

channelization schemes can also be considered to see the performance [8].

36

BIBLIOGRAPHY

[1] McHenry, A. NSF Spectrum Occupancy Measurements Project Summary. Vienna, VA: Shared Spectrum Company, 2005.

[2] Simon, M. K. Bandwidth Efficient Digital Modulation with Application to Deep Space Communications. Danvers, MA: Wiley-Interscience, 2003.

[3] Brodersen, Robert W., Adam Wolisz, Danijela Cabric, and Shridhar Mubaraq Mishra. CORVUS: A Cognitive Radio Approach for Usage of Virtual Unlicensed Spectrum. Berkeley, CA: University of Berkeley, 2004.

[4] harris, f. j. and Erik Kjeldsen. “A Novel Interpolated Tree Orthogonal Multiplexing (ITOM) Scheme with Compact Time-Frequency Localization: An Introduction and Comparison to Wavelet Filter Banks and Polyphase Filter Banks.” IEEE Transactions on Microwave Theory and Techniques, 51, no.4 (2006): 1395-1412.

[5] harris, f. j., Chris Dick, and Michael Rice. “Digital Receivers and Transmitters Using Polyphase Filter Banks for Wireless Communications.” IEEE Transactions on Microwave Theory And Techniques, 51, no.4 (2003): 1395-1412.

[6] harris, f. j. Multirate Signal Processing for Communication Systems. Saddle River, NJ: Prentice Hall, 2004.

[7] harris, f. j., Elettra Venosa, Xiaofei Chen, and Markku Renfors. “Cascade Linear Phase Recursive Half-Band Filters Implements the Most Efficient Digital Down-Converter.” Paper presented at the Software Defined Radio Conference (SDR-2011), Washington DC, December 6-9, 2011.

[8] harris, f. j., Elettra Venosa, Xiaofei Chen, and Chris Dick. “An Efficient Channelizer Tree for Portable Software Defined Radios.” Paper presented at the Wireless Personal Mobile Communications (WPMC-2011), Brest, France, October 4-7, 2011.

37

APPENDIX

MATLAB CODE

38

Matlab code for implementing ITOM filters described in Chapter 5. %%Each of the function can be used according to the ITOM 16 block diagram. %%given in Chapter 6.

A.1 Polyphase Shaping Filter %%%%%%%%%%Polyphase Shaping Filter function [y] = shaping(x, h, k) lh = length(h)-1; N = lh/4; Nx = length(x); hh =reshape(h(1:lh),4,N); reg = zeros(1,N); s = j.^(k*[0:3]); m =1; for i = 1:Nx reg = [x(i) reg(1:N-1)]; s0(m) = s(1)*reg * hh(1,:)'; s0(m+1) = s(2)*reg * hh(2,:)'; s0(m+2) = s(3)*reg * hh(3,:)'; s0(m+3) = s(4)*reg * hh(4,:)'; m = m+4; end y = s0;

A.2 Polyphase Half band Interpolator

%%%%%%%%%Polyphase Half band interpolator function [y] = halfband_i(x, hh, k) Nx = length(x); N = length(hh); reg = zeros(1,N); m =1; for i = 1:Nx reg = [x(i) reg(1:N-1)]; y(m) = reg * hh(1,:)'; y(m+1) = (j^k)* reg * hh(2,:)'; m = m+2; end y = y;

39

A.3 Polyphase Half band Decimator %%%%%%%%%Polyphase Half band Decimator function [y] = halfband_d(y,hh,k) N = length(hh); Nx = length(y); reg = zeros(2,N); m =1; for i = 1:2:Nx-1 reg = [[y(i+1); y(i)] reg(:,1:N-1)]; s(m) = reg(1,:)*(hh(1,:).')+(j^k)*reg(2,:)*(hh(2,:).'); m = m+1; end y = s;

A.4 Polyphase Match filter %%%%%%%%Polyphase Match filter function [y] = matchf(x,h,k) l = length(x); lh = length(h); N = lh/4; hh = reshape(h,4,N); reg = zeros(4,16); z = zeros(1,1000); s = j.^(k*[0:3]); m =1; for i = 1:4:(l-3) reg = [x((i+3):-1:i)' reg(:,1:15)]; z(m) = sum(s.*sum((reg.*hh).')); m = m+1; end y = z;