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