APPLICATION OF CEPSTRUM ANALYSIS
AND HOMOMORPHIC FILTERS
TO INTERFERENCE REJECTION
by
Daniel Lee Spellman
B.S., University of Nebraska, 1983
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
1993
: i : S
This thesis for the Master of Science
degree by
Daniel Lee Spellman
has been approved for the
Department of
Electrical Engineering
by
Tamal Bose
Date /ni%3
Spellman, Daniel Lee (M.S., Electrical Engineering)
Application of Cepstrum Analysis and Homomorphic
Filters to Interference Rejection
Thesis directed by Professor Joe E. Thomas
ABSTRACT
A recurring problem encountered in signal
processing is the rejection of a signal which is
interfering with the recovery of a desired signal.
Traditionally, this is accomplished with band pass
filters centered around the bandwidth of the desired
signal. Unfortunately, if the interfering signal
has frequency components within the bandwidth of the
desired signal, they are not rejected by the
bandpass filter and can degrade the desired signal.
The approach taken by this paper is to utilize
homomorphic filters to perform interference
rejection. Homomorphic filters operate in the
quefrency domain. This domain is generated by the
cepstrum transform. In the quefrency domain, the
harmonic content of signals can be isolated and used
in filtering operations.
A brief history of the cepstrum transform and
its applications is presented. The mathematical
background is developed using a very simple signal.
iii
Then, models of typical interfering signals are
developed. The models used simulate commercial
television broadcast signals and satellite telemetry
signals. The individual models and the result of
combining them into an interference model are then
analyzed in the quefrency domain. The complications
which arise, the results of the homomorphic
filtering process, and the feasibility of using this
approach are discussed.
This abstract accurately represents the content of
the candidate's thesis. I recommend its
publication.
Signed
Joe E. Thomas
CONTENTS
CHAPTER
I. INTRODUCTION ............................... 1
II. OVERVIEW OF CEPSTRUM ANALYSIS
AND HOMOMORPHIC FILTERS .................... 5
III. DEVELOPMENT OF A TELEVISION MODEL ... 33
IV. DEVELOPMENT OF
A SATELLITE TELEMETRY MODEL ............... 68
V. CEPSTRUM OF INTERFERING SIGNALS .... 74
VI. CONCLUSIONS.......................... 96
APPENDIX
A. CEPSTRUM TRANSFORM PROGRAM................ 100
B. INVERSE CEPSTRUM TRANSFORM PROGRAM ... 106
C. SATELLITE TELEMETRY MODEL .......... Ill
D. TELEVISION MODEL ......................... 118
REFERENCES..................................... 125
V
CHAPTER I
INTRODUCTION
More information is being transmitted today via the
airwaves than ever before. Everything from the
proliferation of cellular phones to the increasing number
of communication satellites contributes to this trend.
There is no reason to anticipate any change. When High
Definition Television is introduced, the allocated
bandwidth of a television broadcast may increase from six
MegaHertz to 20 MegaHertz. This will undoubtedly make
the problem worse. Transmission bandwidth is becoming a
precious commodity. Signals are being placed as close
together in the frequency spectrum as possible.
Most of the transmission standards in use today were
developed when analog filters were the only means of
limiting bandwidth. Digital signal processing techniques
in use today provide significantly greater dynamic range
than those available in the past. As a result, the stop
band attenuation specifications are less stringent than
what is achievable with today's techniques. By today's
standards, most commercial broadcasts leak significant
power into their stopbands.
Interference problems can occur when two signals are
1
near one another in the frequency domain or when power
leaking into the stopband of a broadcast signal
encroaches into the bandwidth of a low power signal.
Often demodulators cannot deal with interfering frequency
components which are near one another. This results in
the corruption of the information which is being
transmitted.
Traditionally, interference rejection problems have
been solved using bandpass filters. As the frequency
components of the interfering signal become closer to the
frequency components of the desired signal, the
transition bandwidth of the bandpass filter must be made
narrower. This makes the filter more difficult to design
and build. When the frequency components of the
interfering signal overlap the frequency components of
the desired signal, multiple bandpass filters with very
narrow transition bandwidths are required. This further
complicates the interference rejection process. In the
worst case, some frequency components of the interfering
signal coincide with frequency components of the desired
signal. In this case, bandpass filtering cannot remove
the coincident components of the interfering signal.
This paper uses a technique called homomorphic
filtering to attempt interference rejection. Homomorphic
filtering occurs in the quefrency domain. The quefrency
2
domain is generated using the cepstrum transform. The
cepstrum transform allows the harmonic content of a
signal to be isolated and analyzed. If two signals have
different harmonic contents, it should be possible to
differentiate between them in the quefrency domain even
if they occupy the same frequency bandwidth. The
interfering signal can then be removed from the quefrency
domain and the desired signal will be recovered by the
inverse cepstrum transform. The mathematical concepts
involved in the cepstrum transform are discussed in the
first part of this paper. A simple application of
homomorphic filtering is also completed in order to
explain the process and to verify the techniques are
working.
Models of two typical interfering signals are then
developed. Commercial television broadcast signals are
involved in many interference problems because of their
large bandwidth. Consequently, one of models used in
this paper simulates commercial television broadcasts.
The other model simulates a satellite telemetry signal.
This type of signal is usually lower power and often the
victim of interference. Commercial television broadcasts
are characterized by many harmonic spectral lines
extending across the full bandwidth of the signal. The
satellite telemetry signal has fairly well defined
3
sidebands with little harmonic content. The decisions
and difficulties encountered during the development of
these models will be discussed in detail.
After the individual models are developed, they will
be added to form an interference model. The quefrency
domain representations of the individual models as well
as the model of the interfering signals will be generated
using the cepstrum transform. These representations will
be discussed in detail. The portion of the quefrency
domain corresponding to the television model will be
removed and the result of the homomorphic filter process
will be generated using the inverse cepstrum transform.
This result will be discussed and conclusions as to the
feasibility of this technique are presented.
4
CHAPTER II
OVERVIEW OF CEPSTRUM ANALYSIS
AND HOMOMORPHIC FILTERS
In 1963, a paper was published by Bogert, Healy, and
Tukey entitled "The Quefrency Alanysis of Time Series for
Echoes: Cepstrum, Pseudoautocovariance, CrossCepstrum,
and Saphe Cracking" [1]. This paper proposed a heuristic
approach for finding the echo arrival times in a
composite signal. The authors had observed that the
power spectrum of a signal containing an echo had a
periodic component related to the echo arrival times.
When the logarithm of the power spectrum was observed,
the periodic component was even more pronounced. If an
inverse Fourier transform was performed on the logarithm
of the power spectrum of a signal containing an echo, the
magnitude of the result would contain a peak at the echo
delay. The units of measure of this new function were in
time. They called this function the cepstrum,
interchanging letters in the word spectrum. The authors
also coined many other terms to describe the
characteristics of this new function. These terms are
associated with the traditional terms used in signal
processing as follows:
5
quefrency.................frequency
saphe....................phase
gamnitude.............. .magnitude
liftering.................filtering
rahmonic.................harmonic
repiod...................period
Today the two most widely used terms are cepstrum and
quefrency. The cepstrum proposed by Bogert, et al.,
discarded all phase information and was, therefore, not
reversible. This type of cepstrum analysis has since
been renamed the power cepstrum. The complex cepstrum
was an outgrowth of homomorphic system theory. The
concept of homomorphic systems was proposed by Oppenheim
in 1967 [3]. Homomorphic systems are nonlinear in the
classical sense, but do satisfy a generalization of the
superposition principle. This will be discussed in
greater detail later. The complex cepstrum retains the
phase information present in the original signal and is,
thus, reversible. Since the process is reversible,
signals can be transformed into the quefrency domain,
filtered in the quefrency domain to remove one or more
components of the composite signal, and transformed back
to the time domain. Therefore, the complex cepstrum can
be used for wavelet recovery from a composite signal as
well as determining the echo delay. This paper is
6
concerned with cepstrum filtering techniques. These
techniques require that the cepstrum transform be
reversible. Since the power cepstrum is not reversible,
this paper will focus on the complex cepstrum.
Consider a stable sequence x[n] whose ztransform
expressed in polar form is:
X{z) = \X{z) eJZJf(z) . (2.1)
The magnitude of the ztransform is indicated by
X(z)and the phase angle of the ztransform is indicated
by /x(z) Since the input sequence is stable, the region
of convergence of its ztransform includes the unit
circle, and the Fourier transform of x[n] exists and is
equal to X(z)b=ejco. According to [4], the complex
cepstrum corresponding to x[n] is defined to be the
stable sequence x[n] whose ztransform is:
3Â£{z) =log [X(z) ] , (2.2)
where log[X(z)] indicates the complex logarithm of the
sequence. The complex logarithm of a complex sequence,
X(z), is defined in [4] as:
Jt(z) =log[Z(z) ] =log[X(z) eJ'ZJf(2)] =logX'(z) \+j/.X(z)^2'3^
The imaginary part of equation (2.3) is unique only to
7
within integer multiples of 2n since the phase angle of
the polar representation of a complex number is unique
only to these limits. Since we require the complex
cepstrum to be stable, the region of convergence of X(z)
must contain the unit circle, and the complex cepstrum
can be represented by the inverse Fourier transform of
X(z)fe=cjco. Since the region of convergence for the z
transforms of both the original sequence and the cepstrum
sequence include the unit circles, Fast Fourier
transforms (FFTs) are used to implement the ztransforms
required to generate a cepstrum sequence.
Figure 2.1. System to generate cepstrums.
Figure 2.1 illustrates the process required to
generate the complex cepstrum. Typically, the input
8
sequence is real. The output of the first ztransform
is, therefore, complex and conjugate symmetric. The
output of the complex logarithm will also be complex and
conjugate symmetric. Special steps are required to
insure the result of the complex logarithm transform is
conjugate symmetric. These steps will be discussed
later. The output of the final inverse Fourier transform
will be real.
In order to illustrate the processes involved in
generating the complex cepstrum of a sequence and some of
the traditional applications, an example will be used.
This example is based upon the example shown in Oppenheim
and Schafer's DiscreteTime Signal Processing text [4].
The example1 sequence is made up of the convolution of two
9
sequences:
x[n] =v[n] *p[n] ;
(2.4)
10
where:
p[n] =6 [n] +P6 [nN0] +p26 [n2N0] ,
(2.5)
and
v[n] =jb0w[n] +b1w[nl] ,
(2.6)
with:
w[n] ={cos (Bn) cos [0 (n+2) ]} u[n]
4sin20
(2.7)
These sequences are shown in Figure 2.2 through
Figure 2.4 with the following parameters:
b0 = 0.98,
b, = 1.00,
0 = r = 0.9,
6 = 77 / 6,
N0 = 15.
Excluding the possible effect of windowing
functions, the Fast Fourier transform and inverse Fast
Fourier transform required to implement the cepstrum
process are well defined. Windows are typically applied
to data records when performing analysis of a sampled
signal in the frequency domain. A window in this sense
11
consists of a sequence of numbers which are multiplied by
the sequence of numbers in the data record being
analyzed. In general, the sequence of numbers included
in the data record are generated from samples of a
continuous signal. The continuous signal may or may not
be finite in time. The data records are necessarily
finite. Therefore, the data records are, in general, a
truncated representation of a continuous signal. A
simple truncation of an infinite length signal is
equivalent to multiplication by a rectangular window.
The multiplication of a window times the data record in
the time domain corresponds to a convolution of the
frequency response of the data record and the frequency
response of the window. The frequency response of the
rectangular window sometimes produces undesirable
effects. Windows such as the Hamming, Hanning, and
Blackman windows are used to compensate for that
truncation. This compensation usually takes the form of
lowering the amplitude of the sidebands of the frequency
response of the window at the expense of the width of the
main lobe. The widened main lobe creates a smearing
effect on very narrow band signals.
Windows are typically used to make the results of
the Fourier transform more easily interpretable. Windows
have been found to degrade the capabilities of
12
homomorphic filters when they are used to extract
wavelets from a composite signal containing echoes. If a
windowing function is applied to x[n] as shown:
y[n] =x[n] xw[n]
(2.8)
The ztransform of the equation would yield:
Y(z) =X(z) *W(z)
(2.9)
It is clear from equation (2.9) that the complex
logarithm would not be useful in separating X(z) from
W(z). This would result in the window distorting the
sequence obtained for X(z).
It is preferable to avoid windows when possible. In
the current example, the data sequence decays rapidly and
can be considered finite. Therefore, no windows are
required. If the data sequence did not decay, some type
of windowing would be useful to help the analyze the
results of the Fourier transform and the cepstrum
transform. It has been found that an exponential window
produces the least undesirable side effects. This type
window is defined by the following equation:
Since the focus of this paper is the effect of
homomorphic filters, windows are not useful. Therefore,
they will not be discussed further.
(2.10)
13
Figure 2.6. Magnitude of spectrum of v[n] and p[n].
The magnitude of the frequency spectrum of x[n] is
shown in Figure 2.5. The spectral magnitudes of v[n] and
p[n] are shown in Figure 2.6. These results were
14
obtained by using a 2048 point Fast Fourier transform.
The amplitude of the spectrums are shown using a linear
scale rather than the traditional logarithmic scale.
Therefore, the spectrum of the x[n] sequence is equal to
multiplying the spectrums of v[n] and p[n]. The linear
scale was used in this case to emphasize the effect of
the complex logarithm described later. It is clear from
these figures that a traditional filter to remove the
wavelet echoes would be very difficult to design and
implement. It is important to note that the spectrums of
v[n] and p[n] have much different harmonic properties.
The spectrum of p[n] shows many harmonics, while the
spectrum of v[n] has no harmonics. As we shall see, this
translates into a difference in the quefrency domain.
The next step in obtaining the cepstrum of a
sequence is to perform a complex logarithmic transform on
the sequence generated by the Fast Fourier transform.
Figure 2.7 illustrates the operations required to
calculate the complex logarithm of a sequence. As
described in equation (2.3), the real part of the complex
logarithm of a complex value is equal to the natural
logarithm of the magnitude of the complex value.
Figure 2.8 shows the real part of the complex logarithm
of the x[n] sequence. When Figure 2.8 is compared with
Figure 2.5, it becomes clear that one of the effects of
15
Figure 2.7. System to generate complex logarithms.
Figure 2.8. Real part of complex logarithm of spectrum of
x[n].
the complex logarithm transform is to flatten the
spectrum. This is sometimes called spectral whitening.
16
The complex logarithm transform is considered a severe
spectral whitener. Other types of whitening are
sometimes used when the whitening effects of the complex
logarithm are considered too severe. These include an
unwhitened transform and a moderately whitened transform.
The unwhitened transform consists of some form of mean
removal in the frequency domain. The moderately whitened
transform is made up of the square root of the power
spectrum. These transforms are not necessarily
reversible and cannot be used in cepstrum filtering
techniques. Therefore, they will not be examined in
detail in this paper.
The imaginary part of the complex logarithm is equal
to the phase angle of the complex value. As mentioned
17
earlier, the complex logarithm must have a region of
convergence that includes the unit circle. This requires
that both the real and imaginary parts of the complex
logarithm be continuous. Most inverse tangent routines
return values which are modulo 2ir. The values returned
range between n and +n. This means that the phase angle
of the complex logarithm must be "unwrapped." Figure 2.9
shows the phase angle of the spectrum of the x[n]
sequence before it is "unwrapped." Notice that twelve
jumps of 2n occur between zero and n radians of
normalized frequency. The process of "unwrapping" the
phase will remove these jumps.
Figure 2.10. Phase angle of spectrum of x[n], unwrapped.
One method used to unwrap the phase breaks the
continuous phase angle into two parts as shown in
18
equation (2.11):
<> [n] =$ [n] +2nk (2.11)
where 0[n] is the continuous phase angle, $[n] is the
modulo 2tt phase angle returned by the arctangent routine,
and k is an integer. This method requires that
differences between successive samples of the continuous
phase are less than tt. If the difference between $[n]
and $[nl] is greater that n, it is assumed that the
continuous phase, 0[n], actually decreased to $[n] minus
27r. Similarly, if the difference between Â§[n] and $[nl]
is less that tt, it is assumed that the continuous phase,
0[n], actually increased to Â§[n] plus 2ir. This method is
the simplest to understand and is used throughout this
paper. Figure 2.10 shows the phase angle of the spectrum
of the x[n] sequence after it has been "unwrapped."
Notice that the continuous phase sequence shown in
Figure 2.10 contains a linear phase term. The sequence
begins at zero when [oj=0] and ends at 2n when [co=27r].
This linear phase term results in a sequence which is not
conjugate symmetric. Conjugate symmetry must be achieved
in order to satisfy the stability criteria described
earlier. Therefore, the linear phase term must be
removed before the cepstrum is calculated. Removing the
linear phase term corresponds to simply shifting the
19
Figure 2.12. Continuous phase angle of x[n] with no linear
phase term.
input sequence in time by some number of samples. This
can be seen by examining the ztransform of a shifted
20
sequence. If we let
y[n]=x[nn0] (2.12)
and the ztransform of x[n] be defined as
x[n] Â£ X(z) , (2.13)
then
Y(z) =z'nX{z) .
(2.14)
If we let z = e^,
Y(ejv>) = e~jan,>X(ej<*) (2.15)
where X(eja>) is the Fourier transform of x(n) The linear
phase term is represented by ej"n0. The continuous phase
of y[n] with Dj = 1 is compared with the continuous phase
of x[n] in Figure 2.11. Notice that the slope of the
phase sequence has changed. The continuous phase angle
with the linear phase term removed is shown in
Figure 2.12.
Once the phase is unwrapped and the linear phase
term is removed, the cepstrum can be generated by
performing the inverse Fourier transform on the complex
logarithm. The cepstrum of x[n] is shown in Figure 2.13.
21
Figure 2.14. Cepstrum of v[n] and p[n].
The cepstrums of v[n] and p[n] are shown superimposed in
Figure 2.14.
Some important observations can be made about the
22
cepstrums shown in Figure 2.13 and Figure 2.14. Both the
real and imaginary parts of the cepstrum are shown in
Figure 2.13. Notice that the magnitude of the imaginary
part of the cepstrum is always very near or equal to
zero. The spectrum of v[n] contains no harmonic
components. In the quefrency domain, this results in
most of the energy being very close to the origin. In
contrast, the spectrum of p[n] contains many harmonic
components. This results in little energy near the
origin and impulses at intervals of fifteen time units
from the origin. Fifteen time units is the delay between
the impulses which make up the p[n] sequence. This
corresponds to the echo delay in the composite signal.
Figure 2.15. Inverse cepstrum generation system.
As mentioned earlier, the generation of a cepstrum
23
from an input sequence is reversible. Figure 2.15 shows
the system required to generate a time domain sequence
from a quefrency domain sequence. As with the cepstrum
generation system, Fast Fourier transforms are used to
implement the ztransforms required for the inverse
cepstrum generation. If no windows were used to generate
the original cepstrum, the Fast Fourier transform pairs
used are straight forward and will not be described in
any greater detail at this time. If windows were used in
the original cepstrum generation, then the effect of the
windowing function must be removed in the inverse
cepstrum process. Windows are not used in this paper and
will not be described further. The complex exponential
is less common than the Fast Fourier transforms and will
be described slightly more. The complex exponential is
defined as:
X(z) =exp[j?(z)] = @Ke [Â£()]
As shown in equation (2.16), the exponential of the real
part of the input to the complex exponential becomes the
magnitude of the output. The imaginary part of the input
becomes the phase angle of the output. Note that it is
not necessary to rewrap the phase angle of the output
24
since the following equality holds:
A e^=Aej{a*2ltk) (2.17)
where A is the magnitude of the complex number, w is the
phase angle of the complex number, and k is an integer.
Since the cepstrum transformation is reversible, the
sequence generated by the cepstrum transformation can be
modified and then transformed back to the time domain.
To understand how a modification of the cepstrum of a
sequence affects the sequence itself, it is useful to
understand the concept of homomorphic systems. Linear
systems are well understood and can be defined as a
system that satisfies the superposition principles:
T[ax(n) ] =aT[x(n) ] , (2.IB)
T[x{n) +y(n)] =r[x(n)] +T[y(n)] , (2.19)
T[ax{n) +by(n) ] =aT[x(n) ] +bT[y(n) ] , (2.20)
where a and b are scalar terms, x[n] and y[n] are
sequences, and T[] is the linear transform. The
sequences, x[n] and y[n], can be generalized to
continuous signals. However, for this application,
sequences are appropriate. In a linear transform, the
25
transform of a scalar times a sequence is equal to that
scalar times the transform of the sequence. Similarly,
the transform of a sum of sequences is equal to the sum
of the transforms of the sequences. Equation (2.20)
shows how these two properties can be combined.
Homomorphic transformations satisfy a generalization of
the superposition principle. In homomorphic
transformations, operations applied to input sequences
are mapped to operations which can be applied to output
sequences. These operations are not necessarily the
same. A common example of a homomorphic transform is the
logarithm:
loq[x(n) x y(n) ] =log [x(n) ] +log [y(n) ] (2.21)
The logarithm transform maps multiplication of the input
sequences to addition of the output sequences. The
logarithm transform is nonlinear, but does satisfy the
criteria for a homomorphic transform. It is the complex
logarithm transform used in the cepstrum generation
system which makes it a nonlinear homomorphic system.
The Fast Fourier transform and the inverse Fast Fourier
transform are both linear.
It is interesting to explore the mapping of the
convolution operation in the time domain into the
26
quefrency domain by the cepstrum transform. Referring to
Figure 2.1, the cepstrum transform is made up of a
Fourier transform, a complex logarithmic transform and
an inverse Fourier transform. Convolution in the time
domain is mapped into multiplication in the frequency
domain by the Fourier transform:
v[n] *p[n] ^V(e^u) xP(eJU) (2.22)
Multiplication is mapped into addition by the complex
logarithm transform:
log [V(eJ<1>) xP(eJ) ] =log[^(eJ") ] +log [P(eJ') ] =?(eJ') +P(e^)
(2.23)
Since the inverse Fourier transform is linear, addition
of the input sequences map into addition of the output
sequences:
?(e>) +P(eja)F^v[n] +p[n] (2.24)
Summarizing, the cepstrum transform maps convolution in
the time domain into addition in the quefrency domain:
v[n] *p [n] c^>V[ri] +P tn] (2.25)
The cepstrum of the x[n] sequence shown in Figure 2.13 is
equivalent to the addition of the cepstrums on v[n] and
p[n] shown in Figure 2.14.
27
If the exact nature of one of the signals convolved
together to form the composite signal is known, an
obvious technique to remove that signal would be to
subtract the cepstrum of the known signal from the
cepstrum of the composite signal and perform the inverse
cepstrum transform on the result. Given equation (2.25),
we can define y[n] as:
Performing a Fourier transform on y[n] yields the complex
logarithm of the spectrum of y[n]:
Taking the complex exponential of both sides of equation
(2.27) returns the spectrum of y[n]:
Finally, taking the inverse Fourier transform of both
sides of equation (2.28) yields a filter which can be
implemented as a convolution:
where p'[n] is equal to the inverse cepstrum transform of
p[n]. This procedure works quite well when the
composite signal is made up of the convolution of two
signals. However, this procedure will not work when the
y[n] =x[n] p [n] ='0[n] +j3[.n] p [n] =0[n] (2.26)
*(e>) y(e*)xP(e*) v{ci}
P(ejf) P(eJ")
(2.28)
y[n] =x[n] *p/[n] =v[n]
(2.29)
28
exact nature of the signal to be removed is unknown. In
addition, the procedure is not useful when the input
sequence in the time domain is not made up of two
convolved sequences. Furthermore, this procedure does
not require cepstrum analysis to be used. The filter
sequence, p'[n] could be found from the inverse Fourier
transform of 1/P(e>) if precautions to avoid division by
zero were taken. The focus of this paper is to provide a
means of rejecting interfering signals. Interfering
signals are typically described by the addition of two
signals which are close to each other in frequency.
Therefore, this method of developing filters will not be
explored further.
29
Referring to Figure 2.13 and Figure 2.14, another
method to separate the component signals in the composite
30
signal presents itself. Since the two components of the
composite signal are separated in quefrency, a filter
like function can be applied to the cepstrum sequence to
attenuate either the information contained in the
impulses corresponding to the p[n] sequence or the data
near the origin corresponding to the v[n] sequence.
Modifying data in the quefrency domain by applying a
filterlike function is sometimes called "liftering." A
filter in the quefrency domain is equivalent to
multiplying a filter sequence times the x[n] sequence.
Since the units of the quefrency domain are equivalent to
sample times, the characteristics of the "lifters" are
described slightly different than characteristics of
filters in the frequency domain. Instead of being
described as lowpass or highpass, "lifters" are
described as shortpass or longpass. Shortpass
"lifters" pass data that occurs earlier in time and
attenuate the signals that extend past the cutoff time
of the "lifter. Longpass "lifters" pass data that
occurs later in the data sequence and attenuate the
signals which happen early. The "lifter" applied to x[n]
in this example is shown in Figure 2.16. This "lifter"
is characterized as a shortpass "lifter" and will be
used to remove the impulses associated with the p[n]
sequence. The result of the inverse cepstrum transform
31
of the "liftered" cepstrum is compared with the original
v[n] sequence in Figure 2.17. It is interesting to note
that the "liftered" version of x[n] corresponds almost
identically to the v[n] sequence until approximately the
fifteenth sample. At that time, the liftered sequence
begins to be attenuated. This corresponds to the length
of the short pass filter defined in Figure 2.16.
32
CHAPTER III
DEVELOPMENT OF A TELEVISION MODEL
One of the more common sources of interference is
television broadcast signals. In order to study the
application of homomorphic filtering to rejection of
interfering television broadcast signals, it is necessary
to develop a model of a television signal. Since the
purpose of this paper is not to study television signals,
the model will be developed as simply as possible.
Although monochrome television signals are no longer
used, the standard [7] still exists and is the most
simple to develop. The standards used for the model are
those which were used in television broadcasts from
commercial television stations in the United States.
Similar to a motion picture, a moving television
picture consists of individual still pictures called
frames. Since the two dimensional information contained
in each frame is being transmitted via a one dimensional
medium, signal amplitude, some form of scanning technique
must be employed. The standard [7] for monochrome
television broadcasts divides each frame into 525 lines.
The lines are then scanned one by one. The aspect ratio
of width to height which was used for monochrome
33
television broadcasts is four to three. Therefore, if we
assume that the horizontal resolution of a television
picture is the same as the vertical resolution, each line
can be viewed as 700 picture elements. These picture
elements are sometimes referred to as pixels. The entire
frame is made up of 367,500 pixels. Thirty frames per
second are transmitted in a standard monochrome
television broadcast. This equates to 11,025,000 pixels
transmitted every second. An approximation of the
bandwidth required to broadcast television signals can be
made by assuming that the rise time of the system must
equal the time required to transmit one pixel. If an Re
type response is assumed, the bandwidth required for the
video portion of television broadcast signals, B^, is
3.86 MegaHertz as shown in equation (3.1).
Btv~ (0.35)/ tr= (0.35) (11,025,000) =3.86MHz (3.1)
In practice, a bandwidth of 4 MegaHertz is considered
adequate.
Large carrier amplitude modulation is used to
transmit the video portion of a television signal. Use
of both sidebands would require 8 MegaHertz per channel
for just the video portion of the television signal.
However, a 6 MegaHertz bandwidth allocation for each
34
PMiiwcantor
Figure 3.1. Simplified TV Spectrum
channel including the audio signal has been established
[7]. In order to limit the bandwidth required for the
video portion of the television signal, a form of
vestigial sideband transmission is used for the video
portion of the signal. Attenuation of the lower sideband
begins at 0.75 MegaHertz below the video carrier and is
completely band limited to 1.25 MegaHertz below the video
carrier. The audio portion of the television signal is
transmitted using frequency modulation with a peak
frequency deviation of 25 kiloHertz. The audio portion
of the signal is centered 4.5 MegaHertz above the video
carrier. This combination of the vestigial sideband
modulated video signal and the frequency modulated audio
signal results in a total band width of 6 MegaHertz. A
35
simplified spectral diagram of a television signal is
shown in Figure 3.1. In order to keep the model as
simple as possible, the audio portion of the television
signal will not be included in the development of the
model.
In order to prevent aliasing of the sampled
television signal, a sample rate of at least 12 MegaHertz
must be used. The lowest frequency component associated
with the sampled television signal is due to the vertical
synchronization pulse train. In order to avoid the
appearance of flickering, television signals use a
technique called interlaced scanning. Interlaced
scanning means that half of the 525 horizontal lines are
scanned during each vertical scan. The vertical scan
then begins again and the missing lines are filled in.
This vertical synchronization pulse train, therefore,
occurs sixty times every second or at a frequency of 60
Hertz. If Fast Fourier transforms are to be used in the
analysis of the signal, and the signal is sampled at 12
MegaHertz, a data record of 200,000 samples must be
captured in order to provide a resolution of 60 Hertz.
This very large number of samples is unachievable with
the hardware and software available to develop the model.
The hardware used to develop the simulation models
36
was a IBMPC compatible computer based on an Intel 80386
microprocessor with four Megabytes of random access
memory. The software used to develop the simulation
models consisted of the Turbo C compiler, version 2.0,
developed by Borland International, Incorporated and the
C/Math Toolchest, version 1.0.1 developed by MIX
software. The addressing technigue used by the Intel
80386 employs a data segment address and a data offset.
The final address of a byte of data is determined by
adding the data offset to the data segment address which
has been left shifted four bits. The data offset value
is 16 bits long. Therefore, a single data segment can
contain 65,535 bytes of data if the entire range of the
data offset is used. In order to simplify the array
indexing scheme, arrays defined in Turbo C are limited to
one data segment. The maximum amount of memory which can
be allocated for an array should be equal to 65,536
bytes. The Fast Fourier transform routine included in
the C/Math Toolchest requires an array of complex numbers
to contain the initial data record as well as all of the
intermediate results. The array length must be equal to
an integer power of two. This array is dynamically
allocated from the memory heap. In the default
configuration provided by the C/Math toolchest, each
complex number consists of two double precision floating
37
point numbers. Each double precision floating point
number uses eight bytes of memory. A memory allocation
of 65,536 bytes should be capable of holding 4,096
complex numbers. Given a sampling frequency of 12
MegaHertz, a 4,096 point Fast Fourier transform will
provide a frequency resolution of approximately 2.9
kiloHertz. Since the vertical synchronization pulse
train of a television picture occurs at a 60 Hertz rate,
this resolution is not adequate to study the low
frequency characteristics of the television signal.
In order to study the low frequency components of
the television signal model, a low frequency model was
developed in parallel with the high frequency model. The
characteristics of both models were compared throughout
the development process in order to understand the
contributions of both the high and low frequency
components. For the low frequency model, a resolution of
at least half of the lowest frequency component must be
used. In order to provide a resolution of at least 30
Hertz, the sampling rate must be less than 123 KiloHertz
given a 4096 point Fast Fourier transform. The frequency
of the horizontal synchronization signal is 16 kiloHertz.
In order to study the contributions of the horizontal
synchronization signal, a usable bandwidth of at least 16
kiloHertz must be provided. This will require a sampling
38
frequency of at least 32 kiloHertz. In order to satisfy
both of these criteria while focusing on the low
frequency components, a sampling frequency of 25
kiloHertz was chosen for the low frequency model. This
will allow a resolution of approximately 12 Hertz and a
usable bandwidth of 50 kiloHertz. It will not be
possible to study the effects of the four MegaHertz data
included in each line scan or the vestigial sideband
modulation with the low frequency model since the usable
bandwidth of the model is much less than the required
bandwidth of the signal.
In order to study the vestigial sideband modulation
with the high frequency model, a higher sampling
frequency than the minimum required sampling frequency of
12 MegaHertz was used. A sampling frequency of 20
MegaHertz was chosen. This will allow the six MegaHertz
bandwidth to be centered at five MegaHertz or 7r/4 radians
of normalized frequency. This is the center of the
usable bandwidth of the model. Given a 4096 point Fast
Fourier transform, a sampling frequency of 20 MegaHertz
provides a resolution of approximately 4.9 kiloHertz.
This is adequate to study the effects of the horizontal
synchronization signal as well as the higher frequency
components.
During the development of the models, an
39
idiosyncracy was discovered in the memory allocation
function provided by the Turbo C compiler. This
idiosyncracy caused the memory allocation function to
return a pointer with an address offset of 8 bytes when
65,536 bytes of memory are requested. This offset
prevented the allocated memory from being used as a 4096
point array of complex numbers. The next largest
possible array which conforms to the constraints imposed
by the Fast Fourier transform algorithm would be capable
of holding 2,048 complex numbers given each complex
number requires 16 bytes. A 2,048 point Fast Fourier
transform would allow a resolution of approximately 24
Hertz in the low frequency model. This will reduce the
resolution of the Fast Fourier transform by half. This
resolution is not adequate to study the effects of the 60
Hertz components. If the sample rate is reduced, the
usable bandwidth of the Fast Fourier transform is also
reduced. In order to provide a resolution of 15 Hertz
given a 2048 point Fast Fourier transform, the maximum
sampling frequency is approximately 30 kiloHertz. This
provides a usable bandwidth of only 15 kilohertz. This
is not enough bandwidth to study the effect of the
horizontal synchronization signal. Thus, it did not
appear that a 2,048 point Fast Fourier transform was
adequate to study the low frequency components of the
40
television signal model.
Two solutions to this problem were explored. The
first solution considered was to find a way to allocate
memory so that the initial address of the block of memory
did not include an offset. This would make the block of
memory suitable for an array. Several memory allocation
functions were used to allocate memory for the array.
All of these functions returned a pointer with an offset
of eight bytes. This forced a conclusion that indicates
that the eight bytes are used by the Turbo C compiler to
help manage its memory allocation scheme. This indicates
that the capability to allocate an entire data segment
was not achievable with the Turbo C compiler. The second
solution was to reduce the number of bytes required to
contain the array. The default data type used for each
complex number in the array is a structure consisting of
two double precision floating point numbers. The use of
double precision floating point numbers reduces the
quantization noise in the models. The amount of
quantization noise present in the models is not as
important as the resolution which can be achieved with
the Fast Fourier transforms. Therefore, a model was
developed which used single precision floating point
numbers. This reduced the amount of memory required for
each complex number from 16 bytes to 8 bytes. This
41
reduction enabled the array of 4096 complex numbers to be
stored in 32,768 bytes. Changing the default data type
required that the C/Math Toolchest software package be
recompiled.
The characteristic of television signals which makes
them difficult to remove with conventional filtering
techniques is the wide bandwidth and the existence of
several harmonic frequency components. Since the model
is to be as simple as possible, the contributions of all
of the components of the television signal must be
identified. Therefore, the models were developed in
several steps. First, the horizontal synchronization
signals were incorporated into the models in order to
identify its effect upon the spectrum of the signal.
Next, the vertical synchronization signal was added and
the spectrum generated. In this way, the contributions
of the horizontal and vertical synchronization signals
can be isolated. The limited bandwidth of the low
frequency model does not allow any further enhancements.
The next addition to the high frequency model was the
higher frequency components of the signal which represent
the brightness of each pixel in the line scans. The high
frequency model was further developed to simulate the
effects of vestigial sideband modulation.
The first generation of the high frequency model and
42
Figure 3.2. First generation TV models, time domain.
the low frequency model of a television signal are
compared in Figure 3.2.
The high frequency model shows the details of the
horizontal synchronization signal used for broadcasting
television. The amplitude of the signal corresponds to
the modulation levels used. The pulses have a period of
63.5 microseconds. The modulation level used for the
main pulse is 100 percent. On either side of the main
pulse are signals called the front porch blanking signal
and the back porch blanking signal. The modulation level
used for the front porch blanking signal and back porch
blanking signal is 75 percent. These levels are used for
blanking the cathode ray tube electron beam during the
time needed to perform the horizontal retrace associated
43
with each horizontal synchronization pulse. Each
horizontal synchronization pulse is used to trigger the
horizontal sweep circuitry. The brightness of each pixel
during the horizontal sweep is determined by the
modulation level between the back porch blanking signal
and the front porch blanking signal. An inverse
modulation scheme is used such that 70 percent modulation
corresponds to reference black and 12.5 percent
modulation corresponds to reference white. The
television picture which would be produced by the signal
shown in Figure 3.2 would be completely white.
The details of the horizontal synchronization signal
are not available in the low frequency model. In fact,
it is apparent that the low frequency model cannot even
be trusted to capture every horizontal synchronization
pulse. The main purpose of the low frequency model is to
study the impact of the vertical synchronization signal
upon the composite television signal. If both the low
frequency model and the high frequency model are studied
together, the main contributors to the composite
television signal can be identified. No vertical
synchronization signals or information about the picture
has been included in the first generation of the model.
The magnitudes of the frequency spectrums of the
high frequency model and the low frequency model are
44
TREQ DOMAIN MODEL. 1st GEHER
B 5BBB 1BBBB 15BBB 2OB00 25BBB
FREQUENCV
Figure 3.3. First generation TV models, frequency domain.
Figure 3.4. First generation high frequency TV model,
frequency domain.
compared in Figure 3.3. The frequency resolution
available from the low frequency model is 400 times
45
higher than the resolution of the high frequency model.
Likewise the usable bandwidth available from the low
frequency model is 400 times less than the useable
bandwidth of the high frequency model. The entire
bandwidth of the high frequency model is shown in
Figure 3.4. The values of the samples in each model are
always greater than zero. This translates into a DC
offset and is represented by a spike in the frequency
domain at zero Hertz. This is evident in the spectrums
of both the high frequency model and the low frequency
model. Based upon what is known about the frequency
domain representation of a series of periodic pulses, we
would expect to see a spike in the frequency domain at
the pulse repetition rate and at integer multiples of the
pulse repetition rate with little energy detected in
between these pulses. The high frequency model seems to
satisfy this criterion. However, the smearing effect of
the Fast Fourier transform and the frequency resolution
of only 4.9 kiloHertz does not allow the expected spikes
in the high frequency model to be highlighted.
The frequency spectrum of the low frequency model
does not seem to satisfy the known qualities of a
periodic pulse. In the spectrum of the low frequency
model, spikes seem to be present at integer multiples of
approximately 390 Hertz. An explanation for this
46
phenomenon is needed. The models developed to simulate
the television signal consist of a sequence of pulses.
Each of these pulses have a rise time which is less than
one sample period. This is interpreted as a rise time of
zero by the Fast Fourier transform. The harmonically
related frequency components of the frequency domain
representation of a periodic pulse train with rise times
of zero will extend to infinitely high frequencies.
These frequencies can certainly cause the Nyquist
sampling criteria to be violated. It is possible that
the spikes at integer multiples of 390 Hertz are a result
of aliasing of the high frequency components of the
periodic pulse train.
In order to improve the models to the extent that
aliasing is no longer a major factor, the signals must be
band limited to one half of fs in some way. The approach
taken for band limiting the signals was to provide
samples of the signal produced for the television model
at much higher rates than desired, use a digital filter
to band limit the signal appropriately, then decimate the
signal to the desired data rate.
This technique required the development of digital
filter design tools. The algorithm described by Parks
and McClellan [5] for optimal linear phase finite impulse
47
response filters was chosen. The filter design program
requires that the edges of the passband and the stopband
be defined. The region in between the passband and the
stopband is assumed to be the transition band. In order
to preserve as much of the usable bandwidth as possible,
the transition bandwidth when translated to the desired
frequency was limited to 0.2 times the sampling
frequency, fs At first glance, this does not seem to be
a very stringent requirement. However, before the
transition band is translated to the desired sampling
frequency, the width must be reduced by a factor of one
over the decimation rate. If the decimation rate is
twenty, then the width of the transition band must be
0.01 times fs. This is a reasonably steep filter. Given
an equal number of coefficients, filters with very narrow
transition bands typically have more passband ripple and
less stop band attenuation than filters with wider
transition bands.
The width of the transition band affects the filter
performance and the decimation rate drives the
requirement for the width of the transition band.
Therefore, higher decimation rates will result in worse
filter performance in terms of passband ripple and stop
band attenuation. Three filters were designed based upon
48
DECIMATION FILTER RESPONSE
50
0.000000
O.Â£29318 1.25Â£Â£3fi 1.094954 2.513ai2
NORMALIZED FREQUENCV (Ml)
3.14159C
Figure 3.5. Frequency response of decimation filters.
Table 3.1. Decimation filter specifications.
DECIM. RATE PASS BAND TRANS. BAND STOP BAND MAX. PASS BAND RIPPLE (dB) MIN. STOP BAND ATTEN. (dB)
5 ofs 0.08fs 0.08fs 0.12fs 0 12fs lf. 0.024 70
10 Ofs 0.04fs 0.04fs 0.06fs 0.06fs if9 0.32 44
20 of 0.02fs 0.02fs 0.03f, 0.03fs Ifs 1.7 34
decimation rates of five, ten, and twenty. Each of these
filters consist of 127 coefficients. The frequency
responses of the three filters are shown in Figure 3.5.
The filter specifications are tabulated in Table 3.1.
49
Each filter was designed so that the passband after
decimation ends at 0.4 times f8 and the stopband after
decimation starts at 0.6 times f8. This results in a
transition band after decimation from 0.4 times f8 to 0.6
times f8. Aliasing may occur in this region, but should
not affect the passband. The passband was chosen to
extend to almost one half of fs in order to preserve as
much bandwidth as possible. This allows more
characteristics of the models to be observed.
The filter response of the three filters is not
equiripple. This indicates that the filters are not
optimal. This is typically the result of truncation
errors in the filter design program. For long filters,
such as the ones specified here, 32 bit floating point
math is not sufficient to discriminate between different
coefficients which are returned. Shorter filters
generally benefit more from the optimization process.
However, the filter performance obtained from a 127
coefficient filter is better than that achieved from a
filter with less coefficients even though it is not
optimal.
It is obvious from Figure 3.5 that the filter
required for a decimation rate of five will provide
superior performance. However, the decimation rate of
50
Figure 3.6. Low frequency TV model obtained by decimation
by five.
five requires the use of a sampling frequency that is
onefourth the sampling frequency used for a decimation
51
rate of twenty. Higher decimation rates allow the use of
higher sampling frequencies. As can be observed in
Figure 3.4, the frequency spectrum of the television
models rolls off naturally at higher frequencies. The
higher sampling frequencies will cause less aliasing
since the aliased signals are images from frequency
components which are further out in frequency. This
complex relationship was studied with the low frequency
model of the television signal. The results obtained by
decimating by five and decimating by twenty are compared
in Figure 3.6 and Figure 3.7, respectively.
FREQ DOMAIN MODEL, Es=3B.01kHz
10 
B SBBB LB00B 13000 20000 23000
FREQUENCE (Hz)
Figure 3.8. Low frequency TV model sampled at 50.01
kiloHertz.
Even though better filter performance was achieved
by the filter used for the decimation by five, the
52
aliasing present in the result was worse. For this
reason, the model obtained by decimation by twenty is
used for the remaining study of the low frequency model.
Even though most of the aliasing encountered in the
initial model has been removed by the new technique, some
unexplained frequency components are still noticeable.
These components occur at approximately 7.9 kiloHertz and
23.6 kiloHertz. It is desirable to find out whether
these components are attributable to the model of the
signal or if they are continuing examples of aliasing.
One way to determine if these frequency components result
from aliasing is to determine if their location in
frequency is dependent upon the sampling frequency. If
the frequency components are a result of aliasing, a
slight change in the sampling frequency will result in a
change in the location of the components. If the
frequency components do not change their location when
the sampling frequency is changed, they can be attributed
to the low frequency model of the signal. Figure 3.8
shows the frequency domain representation of the low
frequency model sampled at 50.01 kiloHertz instead of 50
kiloHertz as was shown earlier. As can be seen, the
frequency components at 7.9 kiloHertz and 23.6 kiloHertz
of the model that was sampled at 50 kiloHertz have moved.
We must, therefore, conclude that these components of the
53
model are the result of aliasing. The frequency
components at 0 Hertz, and 15.7 kiloHertz are basically
unchanged. Therefore, these must be components of the
low frequency model.
Figure 3.9. Second generation high frequency TV model,
frequency domain.
The frequency domain representation of the filtered
and decimated high frequency model is shown in
Figure 3.9. At low frequencies, there is no significant
difference between the frequency characteristics
resulting from the nondecimated model and the decimated
model. This is to be expected since the magnitude of the
frequency components of the high frequency model roll off
at higher frequencies. Since the higher frequency
components have less magnitude, the aliased images
54
produced by these components have much less magnitude.
The only apparent difference between the decimated
version of the model and the nondecimated version of the
model is at frequencies near 10 MegaHertz. The non
decimated version of the high frequency model does not
roll off past about 30 dB. The decimated version of the
high frequency model continues to roll off to 35 dB.
This can be explained by the presence of aliased noise in
the nondecimated version. In the decimated version, the
filter attenuates much of the high frequency noise in the
model. This also attenuates the aliases produced by high
frequency noise. Since there are no external noise
sources present in the model, quantization noise accounts
for most of the noise present.
Figure 3.10 shows the time domain representation of
both the high frequency model and the low frequency
model. The effect of the filtering and decimation upon
the models is much more pronounced in the low frequency
model. The resulting signal resembles a sine wave more
than the pulse train shown in the high frequency model.
This is expected since all of the high frequency
components which comprise a square wave have been removed
from the low frequency model. Only the first harmonic
remains in the low frequency model. This would be a
severe limitation if the purpose of the low frequency
55
Figure 3.10. Second generation TV models, time domain.
model was to model the shape of the signal. Fortunately,
this is not the case. The purpose of the low frequency
model is to model the contribution of the low frequency
components to the television model. In particular, the
contribution of the vertical synchronization waveform is
to be studied with the low frequency model. This
waveform has not yet been added to the television model,
but will be added in the next generation of the model.
The effect of the filtering and decimation upon the
high frequency model is much less evident. The only
significant effects evident upon the high frequency model
are the slight ringing and small undershoot or overshoot
present at each transition of the signal. This effect is
sometimes called filter ringing. The original signal
56
allowed the pulses generated in the television model to
reach full scale. When ringing occurs, the overshoot of
the fullscale value causes the amplitude of the signal to
exceed the fullscale value.
TIME DOMAIN MODEL, 3rd. GDfER.
D9190
Figure 3.11. Third generation low frequency TV model, time
domain.
The addition of the vertical synchronization pulse
interval to the low frequency model is shown in
Figure 3.11. The low frequency model does not have
enough resolution in the time domain to adequately
illustrate the shape of the signal. In the low frequency
model, the television signal appears as a combination of
a 16 kiloHertz sine wave and a pulse train at a
repetition rate of 60 Hertz. The details of the signal
are lost.
57
Figure 3.12. Third generation TV models, time domain.
It is not possible to illustrate the addition of the
vertical synchronization pulse interval with the high
frequency model. The length of time included in the high
frequency model is not long enough to show the entire
vertical synchronization pulse interval. Therefore, a
model with a sampling rate in between the sample rates of
the low frequency model and the high frequency model has
been developed to show the shape of the signal. This
model uses a five MegaHertz sampling frequency.
Figure 3.12 shows the low frequency model overlaid with
this new model. The pulse interval associated with
vertical synchronization consists of two equalizing pulse
intervals and a vertical sync pulse interval. Each
equalizing pulse interval is made up of six short pulses
58
at twice the horizontal synchronization rate. The
vertical synchronization pulse interval is made up of six
long pulses at twice the horizontal synchronization
rate. After the vertical synchronization pulse interval
and equalization pulse intervals, three horizontal
synchronization pulses are shown.
An interesting difference between the two models is
the delay between them. This can be explained by
considering the filters used to band limit the signals.
Each filter requires 127 data samples. The time between
samples in the low frequency model is 100 times longer
than the time between samples in the model sampled at
five MegaHertz. The filter in the low frequency model
operates on data which extends 100 times further than the
model sampled at five MegaHertz. Therefore, the filter in
the low frequency model begins to operate on samples much
earlier than the filter in the other model since it sees
those samples much earlier.
As stated earlier, the low frequency model cannot
adequately represent the pulses in the equalization pulse
intervals or the vertical synchronization pulse interval.
This is true because of the low frequency model is band
limited to 20 kiloHertz. The pulses are occurring at a
rate of approximately 32 kiloHertz. Once again, this is
a severe limitation of the low frequency model. However,
59
Figure 3.13. Third generation low frequency TV model,
frequency domain, full bandwidth.
FREQ DOMAIN MODEL., 3rd CEMER.
10
20
< 30
40
50
1920 2BO0
FREQUENCE
4800
Figure 3.14. Third generation low TV frequency model,
frequency domain, limited bandwidth.
the low frequency model is to be used to study the low
frequency components of the television signal. These seem
60
of be adequately represented in the low frequency model.
The spectrum of the full useable bandwidth of the low
frequency model is shown in Figure 3.13.
The spectrum of the very low frequency components
included in the model is shown in Figure 3.14. Upon
close inspection, it can be seen that there is a
frequency component at 60 Hertz and at multiples of 60
Hertz. This is expected since there are two vertical
synchronization pulse intervals for each frame of the
television picture. A new frame is broadcast 30 times a
second. Hence the 60 Hertz frequency component. It can
also be seen from Figure 3.13 that the harmonics of the
60 Hertz component are attenuated significantly by the
time the 15.7 kiloHertz component associated with the
horizontal synchronization pulse occurs.
The quality of a television signal which makes it
difficult to remove with traditional filters is the large
bandwidth of the signal. The signal is made up of
several harmonic components. The harmonic components due
to the horizontal synchronization pulses have much
greater amplitude the components due to the vertical
synchronization pulse interval. Only one of the harmonic
frequencies is visible in the spectrum of the low
frequency model due to the low useable bandwidth
available.
61
The next enhancements planned for the television
model are the addition of data which represents the
picture and translation of the signal to a higher
frequency prior to applying the filter required for
vestigial sideband modulation. Both of these
enhancements require more bandwidth than the low
frequency model can support. Therefore, further
enhancements to the low frequency model will not be
pursued. The vertical synchronization pulse interval
does not contribute significantly to the bandwidth of the
television signal and cannot be supported by the high
frequency model. Therefore, the vertical synchronization
pulse interval will not be used in the high frequency
model.
The next addition to the high frequency television
model is a simulation of the data present in each line
scan of the television signal. This data will be
represented by a sequence of random numbers that have
been band limited to 4 MegaHertz. The time domain
representation of this signal is shown in Figure 3.15.
Note that the edges of the synchronization pulses have
more overshoot, undershoot and ringing than the previous
models had. This is caused by further limiting the
bandwidth imposed upon the model. The magnitude of the
frequency spectrum of this signal is shown in
62
1.230
1.123
i.eaa
0.073
Â£ 0.750
0.623
0.500
0.375
0.250
0.125
o.eee
.00000008 .00004093 .00000190 .00012203 .00016300 .0002047!
TIME
Figure 3.15. Fourth generation high frequency TV model,
time domain.
frequency domain.
Figure 3.16. Note that the 4 MegaHertz band limit is
very obvious. Also note that the spectrum is almost flat
63
except for the frequency components at very low
frequencies. This is due to the random nature of the
line scan data simulation. A purely random signal will
have equal frequency components across the spectrum. The
low pass filter used limits the spectrum to 4 MegaHertz,
but within the 4 MegaHertz window, the spectrum is flat.
Figure 3.17. Fifth generation high frequency TV model,
time domain.
The final enhancement which will be added to the
high frequency television model is amplitude modulation
and application of a vestigial sideband filter. The
amplitude modulation was accomplished by multiplying the
previously generated baseband signal by a sinusoid which
was oscillating at 3.5 MegaHertz. The result was then
64
filtered by the vestigial sideband filter. The time
domain representation of the model is shown in
Figure 3.17. The frequency domain representation of the
model is shown in Figure 3.18. The large frequency
component at 3.5 MegaHertz is characteristic of large
carrier amplitude modulation. The large carrier makes it
easier to demodulate the signal since the carrier can
easily be found and used to synchronize the receiver.
Vestigial sideband modulation is characterized by the
asymmetric appearance of the frequency spectrum.
Vestigial sideband modulation differs from single
sideband modulation by the inclusion of a portion of the
lower sideband. This is done to insure that the low
65
frequency components of the television signal are not
distorted by the transmission process. This is important
for television transmission since the horizontal and
vertical synchronization information is contained mostly
in the low frequency components.
The signal illustrated in Figure 3.17 and
Figure 3.18 will be used in the remainder of this paper
to provide interference to received signals. The signal
model differs from a genuine television signal in three
major ways. First, there are no vertical synchronization
pulses included in the model. These were found to be
unnecessary. The vertical synchronization pulse interval
does not contribute significantly to the large bandwidth
of the television signal or the harmonic content. The
large bandwidth is the characteristic of television
signals most difficult to deal with in interference
rejection. The second major difference between the model
and a genuine television signal is the absence of the
audio signal. In a genuine television broadcast, the
audio carrier is located 4.5 MegaHertz above the video
carrier. The audio signal is frequency modulated onto
the audio carrier with a maximum bandwidth of 50
kiloHertz. Because of the limited bandwidth of the
modulated audio signal, it also does not contribute
significantly to the overall bandwidth of the television
66
signal and was not included in the model. The final
major difference between the model and a genuine
television signal is the nature of the line scan data.
The method of simulating the line scan information is
very conservative. In a genuine television signal, the
frequency components at the higher end of the passband
would be lower than those near the carrier. The line
scan data generated for the model is essentially flat
across the passband. One characteristic of images is the
existence of more low frequency information than high
frequency information. The higher frequency components
could be attenuated with a filter but this would make the
model more complex and does not provide any particular
advantages. In addition, the line scan information does
not contribute to the harmonic content of the television
signal. On this basis, it is not critical to the
operation of the model and will not be discussed further.
67
CHAPTER IV
DEVELOPMENT OF
A SATELLITE TELEMETRY MODEL
In order to study the application of homomorphic
filtering to interference rejection, it is necessary to
develop a model of a signal which is susceptible to
interference. One common victim of interference is
satellite telemetry signals. Satellite telemetry is
transmitted from satellites orbiting the earth. The
power needed to broadcast these signals is typically
generated by solar cells mounted on the satellite. This
limits the power available for the transmitters. In
addition, the distance from the transmitter to the
receiver is very great. This further weakens the signal
received at the satellite ground station.
Typically, large dish antennas are required to
collect enough power to reliably receive the satellite
telemetry signal. After the antenna, a low noise
amplifier is used to boost the signal strength. The
signal is then downconverted to approximately 70
MegaHertz. This frequency is, coincidently, in the
television broadcast band. Television signals can,
therefore, interfere with satellite telemetry signals. A
satellite telemetry signal will be modeled in order to
68
study the effect of interference and the application of
homomorphic filters to remove that interference.
1.2 1.8 B.8 \ 0,6 I 8.4 1 e.a TIME DOMAIN MODEL, BIT STBZAH
, ' I 1 ' 1 1 ' ' 1 ' ' I'''1 1 ' ' ' ' 1 1 1 ' '
. . i . . , . i . . , . i . . . . . i ....
.080880 .080098 .808180 .088198 .808281 TIME (e>
Figure 4.1. Telemetry bit stream, time domain.
Satellite telemetry consists of a string of digital
data which is modulated on a carrier and transmitted to
the satellite ground station. For most commercial
communication satellites, the data rate is approximately
one kilobit per second. The television model which
provides interference is much more complex than the
telemetry model. Consequently, the television model will
drive the parameters used for the telemetry model. A 20
MegaHertz sample rate was used in order to accommodate
the six MegaHertz bandwidth specified for commercial
television broadcasts. The model was limited to 4096
samples by the hardware used to develop the models. In
69
order to sample two bits of satellite telemetry data at
20 MegaHertz, 40,000 samples are required. This is not
achievable with the hardware and software available to
develop the models. The model was therefore modified so
that the hardware and software could support it. The bit
rate used for the model was increased to 10 kilobits per
second. This allows two bits to be sampled in 4,000
samples given a 20 MegaHertz sampling rate. The digital
bitstream used for the satellite telemetry model is shown
in Figure 4.1.
Figure 4.2. Telemetry BPSK Subcarrier, time domain.
The one kiloHertz bitstream used for commercial
satellite telemetry is typically biphase shift key
modulated onto a 32 kiloHertz subcarrier. The typical
subcarrier frequency, 32 kiloHertz, is not high enough to
70
support a 10 kilobit per second data rate. Consequently,
the subcarrier frequency used in the model was increased
to 320 kiloHertz. The time domain representation of the
subcarrier is shown in Figure 4.2. Biphase shift key
modulation uses two sine waves which are 180 degrees out
of phase. The signal phase depends upon the value of the
digital bit stream. A 180 degree phase change is
noticeable at 0.0001 seconds in Figure 4.2. This is due
to the transition in the bit stream from zero to one at
Figure 4.3. Telemetry PM carrier, time domain.
Once the subcarrier has been generated it is used to
phase modulate a carrier. The carrier frequency was
chosen to place the signal carrier in the middle of the
frequency band occupied by the television model. An
71
expanded view of the resulting time domain representation
is shown in Figure 4.3. This view is centered around the
instant of time where the digital bit stream transitions
from zero to one. The point where the bit transitions is
no longer readily apparent. The frequency domain
representation of the signal is shown in Figure 4.4. The
carrier is characterized by as a peak at five MegaHertz.
The subcarrier is characterized by the peaks located at
320 kiloHertz intervals and centered at five MegaHertz.
A commercial satellite typically contains at least
two telemetry downlinks. These downlinks are typically
separated by one MegaHertz. Therefore, the bandwidth of
the transmitted satellite telemetry must be limited to
one MegaHertz. Since the model used for this paper uses
72
a data rate of 10 kilobits per second rather than the
typical one kilobit per second, the bandwidth of the
signal must be increased by a factor of ten. Ten
MegaHertz is the maximum bandwidth that can be supported
by a 20 MegaHertz sampling frequency. Therefore,
developed model will not be further bandlimited.
Additional, bandlimiting would have been required if
significant aliasing occurred in the model. However,
this was not necessary.
73
CHAPTER V
CEPSTRUM OF INTERFERING SIGNALS
Now that models exist for both a television signal
and a satellite telemetry signal, the cepstrum of each of
these signals can be generated. Once this is done, the
cepstrums can be compared to identify any components in
the quefrency domain which discriminate one signal from
the other. The effect of removing these components from
the signal will then be explored. When the effects of
removing these components are understood, the signals
will be added together as if they were interfering with
one another. The cepstrum will again be generated and
components will be removed in order to study the
application of homomorphic filtering to interference
rejection.
The cepstrum of the telemetry signal and the
television signal were generated using 4096 samples. The
sample rate was 20 MegaHertz. The full scale cepstrum of
the telemetry signal is shown in Figure 5.1. The full
scale cepstrum of the television signal is shown in
Figure 5.2. There are no obvious differences between the
representations of the two signals. The highest
amplitude components of both cepstrums are clustered
74
Figure 5.1. Full scale cepstrum of telemetry model.
i ....
T
U
1
2
3
.00810
ClFftHIH Of IV MOdal
T' rmTi
i L
: 1 : 1 ! f**
... i.... i i . .. 1 
.00084 .00082
TINE (sea)
.80088
Figure 5.2. Full scale cepstrum of TV model.
around the center of the cepstrum. Since there are
significant differences in the time domain
representations and the frequency spectrums of the
75
telemetry and television signals, differences in the
cepstrum are expected. More detailed analysis of the
cepstrums is provided in the following paragraphs.
Figure 5.3. Expanded cepstrum of telemetry model.
The cepstrum of the telemetry model between 0
microseconds and 20 microseconds is shown in Figure 5.3.
Since the telemetry model uses a carrier frequency of 5
MegaHertz, the cepstrum should show an oscillation with a
period of 0.2 microseconds. This oscillation is visible
in the cepstrum shown in Figure 5.3.
The carrier frequency of the telemetry signal is
phase modulated with a 320 kiloHertz biphase signal
modulated at one kiloHertz. Some change in the cepstrum
should be detectable with a period of 3.125 microseconds.
The cepstrum contains beats every 1.563 microseconds.
76
This is inconsistent with the expected results. However,
the 320 kiloHertz signal is biphase modulated and can be
thought of as two 320 kiloHertz signals which are 180
degrees out of phase. These two signals are modulated on
and off alternately. When considered in this way, two
phenomena should exist and be offset from each other by
one half of the 3.125 microsecond period. The cepstrum
in Figure 5.3 satisfies this criterion.
Figure 5.4. Expanded cepstrum of television model.
The cepstrum of the television is expanded in
Figure 5.4 and Figure 5.5. Based upon what is known about
the television signal, the cepstrum should show an
oscillation at the carrier frequency and some detectable
difference in the cepstrum with a period of 63.5
microseconds since the horizontal synchronization pulse
77
is a significant part of the television signal.
Unfortunately, neither of these expected characteristics
can be detected in the cepstrum.
Since the effect of the harmonics associated with
the horizontal synchronization pulse are the focus of
this paper, the television model may be more complex than
necessary. The television model was simplified in order
to explain the absence of the expected characteristics.
The band limited pseudorandom numbers which were used to
model the linescan information present in a television
signal were removed in order to simplify the model. The
time domain representation of the simplified model is
shown in Figure 5.6. The cepstrum of this model is shown
in Figure 5.7. Expanded versions of the cepstrum of this
78
TINE <**C>
Figure 5.6. Simplified television model (time domain).
Figure 5.7. Cepstrum of simplified television model.
model are shown in Figure 5.8 and Figure 5.9. Activity
around 63.5 microseconds can now be detected as shown in
Figure 5.9. The pseudorandom number used to model
79
model showing high frequency oscillations.
Figure 5.9. Expanded cepstrum of simplified television
model showing activity at 63.5 microseconds.
linescan information can be thought of as a model for
noise. The fact that activity is now detectable at 63.5
80
microseconds indicates that the cepstrum transform is
susceptible to noise. In the cepstrum transform, signals
are whitened by the complex logarithm function. The
effect of this whitening is to reduce the signal to noise
ratio of the input signal presented to the final inverse
Fourier transform. In the case of the television model,
the noise floor was raised above the anticipated
components of the cepstrum.
Although not explicitly shown in any figure, even
greater problems are introduced by noise generated within
the cepstrum and inverse cepstrum transforms. One source
of this type of noise is the quantization effects
inherent in calculations performed with a computer.
Noise introduced before the complex logarithm transform
produces similar effects to those experienced with other
processes. Noise introduced after the complex logarithm
transform and before the complex exponential transform
produces much greater effects. This is a result of the
complex exponential transform. Any noise introduced in
this part of the process will be amplified exponentially
due to the complex exponential transform.
The high frequency oscillation present in the
cepstrum is much higher than the carrier frequency. In
addition, the cepstrum of the simplified television model
shown in Figure 5.7 contains a very large spike at zero
81
Figure 5.10. Complex logarithm output for simplified
television model.
microseconds. These two anomalies can be explained by
examining the output of the complex logarithm function
used to generate the cepstrum. Both the real and
imaginary parts of the complex logarithm are shown in
Figure 5.10. As mentioned earlier, the real part of the
complex logarithm is equal to the natural logarithm of
the magnitude to the Fourier transform. The imaginary
part of the complex logarithm is equal to the unwrapped
phase angle of the Fourier transform with the linear
phase term removed. As shown in Figure 5.10, the
imaginary part of the output complex logarithm transform
has much more amplitude than the real part of the output
of the complex logarithm transform. This is
82
unanticipated and probably an error. The probable error
is a result of the phase unwrapping process.
The phase is calculated using the arctangent of the
quotient of the imaginary part over the real part of the
output of the Fourier transform. The arctangent function
returns a value between tt radians and n radians. In
order to keep the phase function smooth when successive
samples cross the n radian or ir radian boundary, the
phase unwrapping process is used. The phase unwrapping
algorithm assumes that the difference in phase between
successive samples is always less than it radians. If two
successive samples have a difference greater than it, 2it
is either added or subtracted in order to make the
difference less than n radians. If the nature of the
input signal makes it possible for the phase angle
difference between successive samples to be greater than
ir, the algorithm can make mistakes. This appears to be
occurring in Figure 5.10.
Between 5 MegaHertz and 8 MegaHertz, the imaginary
part of the output of the complex logarithm transform
drops in steep steps. This is usually an indication of
errors in the phase unwrapping process. The phase
unwrapping errors cause the imaginary part of the output
of the complex logarithm transform to be negative 647r
radians at 2ir radians of normalized frequency (20
83
MegaHertz). The inverse Fourier transform requires that
the output of the complex logarithm be conjugate
symmetric. To make this happen, the phase unwrapping
algorithm removes the linear phase term. This forces the
imaginary output of the complex logarithm transform to be
artificially amplified and results in a peak at about 5.5
MegaHertz. The peak explains the oscillation in the
cepstrum at approximately 5.5 MegaHertz as well as the
very high amplitude of the cepstrum.
Other methods of unwrapping the phase exist. These
include integrating the phase derivative. This has the
effect of a low pass filter on the phase. This method
was attempted on the complex logarithm transform used for
the television signals, but other phase unwrapping errors
occurred. The only way to effectively stop phase
unwrapping errors is to increase the number of samples
used. This has the effect of increasing the resolution
of the Fourier transform. If the resolution of the
Fourier transform is increased, the chance that the phase
angle will jump more than n radians is decreased.
Unfortunately, given the hardware limitations discussed
earlier, this is not an option. Since there is
detectable activity at 63.5 microseconds from the origin
of the cepstrum of the simplified television model, the
effect of blanking the cepstrum of the simplified
84
television model as well as the cepstrum of the telemetry
model will be studied.
Notice in Figure 5.1 that all the characteristics of
the cepstrum of the telemetry model described earlier
decay quickly and are not observable more than 40
microseconds from the origin. The cepstrum of the
simplified television model has activity centered at
approximately 63.5 microseconds as shown in Figure 5.9.
In order to explore the effects of homomorphic filtering,
the cepstrums of the telemetry model and the television
model will be set to zero in this area and the impact to
the signal recovered by the inverse cepstrum process will
be examined.
Figure 5.11. Filtered cepstrum of telemetry model.
The cepstrum of the telemetry signal with the area
85
beyond 40 microseconds from the origin in both the
positive and negative directions set to zero is shown in
Figure 5.11. When the original cepstrum shown in
Figure 5.1 is compared with the filtered cepstrum in
Figure 5.11, it is apparent that a minimal amount of the
cepstrum was removed. A linear phase term of 0.00460194
radians per sample was removed from the complex logarithm
of the Fast Fourier transform when the cepstrum was
generated from the telemetry model. When the inverse
cepstrum process is applied the linear phase term will be
added. This should keep the result of the inverse
cepstrum transform in phase with the original telemetry
model.
Llftered. Telenetrv Model
2.0
1.6
1.2
B.8
B.4
B.4
B.S
1.2
1.6
2 O
.'eaaBa .BBaa6 .00012 .00010
TIME < CBS)
Figure 5.12. Liftered telemetry model time domain.
The time domain representation of the result of the
86
Figure 5.13. Liftered telemetry model spectrum.
inverse cepstrum transform is shown in Figure 5.12. The
spectrum of the signal is shown in Figure 5.13. The
amplitude of the signal resulting from the homomorphic
filter process is very distorted. However, the frequency
components of the signal seem to be intact although
slightly attenuated. Since the amplitude is severely
distorted, it would be inappropriate to use homomorphic
filtering in a system intended to recover signals which
carry information in the amplitude of the signal.
The filtered cepstrum of the television model is
shown in Figure 5.14. Notice that the activity centered
around 63.5 microseconds has been removed. The time
domain representation of the result of the inverse
cepstrum transform is shown in Figure 5.15. The spectrum
87
Figure 5.15. Liftered television model time domain.
of the result is shown in Figure 5.16. As with the
telemetry model, the amplitude of the result of the
homomorphic filtering process is severely distorted. If
88
Figure 5.16. Liftered television model spectrum.
the intent was to recover the television signal, the
homomorphic filtering technique would be useless.
89
However, the intent is to reduce or remove the harmonic
spectral lines associated with a television signal in
order to reduce the bandwidth that the signal occupies.
The spectrum of the result shows that homomorphic
filtering does indeed reduce the power in the spectral
lines slightly. When an expanded view of the spectrum of
the liftered signal is compared to the spectrum of the
original as in Figure 5.17, it is apparent that the
effect of the homomorphic filter is to smooth the
spectral lines rather than attenuate them.
The next thing examined will be the effect of
homomorphic filters on interfering signals. A set of
interfering signals will be generated by adding the
signal produced by the telemetry model to the signal
90
Figure 5.19. Interfering signals spectrum.
generated by the simplified television model. The time
domain representation of the result is shown in
Figure 5.18. The frequency domain representation is
shown in Figure 5.19. The maximum amplitude of the
telemetry signal is set to 0.01 times the maximum
amplitude of the television signal. Since the television
signal has so much more amplitude, the telemetry signal
is not visible in the time domain representation of the
interfering signal. On the logarithmic scale shown in
Figure 5.19, the carrier frequency of the telemetry
signal is barely visible. Neither of the sidebands of
the telemetry signal are discernible. A homomorphic
filter will be applied to the interfering signal in order
to make the telemetry signal more detectable.
91
Figure 5.20. Cepstrum of interfering signals.
Figure 5.21. Liftered signal with interference.
The cepstrum of the interfering signal is shown in
Figure 5.20. The vertical scale is expanded to show the
activity around 63.5 microseconds. Not shown in
92
Figure 5.22. Spectrum of liftered signal with
interference.
Figure 5.20 is the large beat at zero. Notice that this
cepstrum is very similar to the cepstrum of the
television signal alone shown in Figure 5.14. The
cepstrum of the interfering signals will be blanked in
the same way as the cepstrums of the individual telemetry
and television signals were earlier. The blanked
cepstrum is not shown since it is similar to previous
figures. The results of the inverse cepstrum transform
applied to the filtered cepstrum are shown in
Figure 5.21. The frequency domain representation of the
result is shown in Figure 5.22. The first two sidebands
of the telemetry signal are now visible.
Figure 5.23 compares the spectrum of the output of
93
FREQ
Figure 5.23. Comparison of liftered signal to original
telemetry signal.
the homomorphic filtering process with the spectrum of
the telemetry signal. The telemetry signal used in the
signal with interference has much less amplitude than the
original telemetry signal examined. Based upon this
comparison, it .is clear where the sidebands of the
telemetry signal occur. The effects of the interference
and the homomorphic filter have removed most of the
detail present in the telemetry sidebands. It is
probably not possible to recover the information present
in the sidebands of the telemetry signal extracted from
the interference. However, it is interesting to note
that some recovery of the signal was possible. If the
problems encountered in the cepstrum transform can be
94
solved, the results of the homomorphic filter process
would certainly be improved. Given significant
improvement, it may be possible to extract phase or
frequency modulated signals which are buried within
interfering signals using homomorphic filtering
techniques.
95
