Let’s Assume the System Is Synchronized

Abstract

It is amazing how many papers on radio systems, networks, error correcting codes, and related topics contain a version of the sentence “Let’s assume the system is synchronized.” Alright, let’s assume the system is synchronized. But I have a few questions: Who did it? How did they do it? Who will do it in the next decades as many of us retire from the field? An important one is; where are they acquiring the skills required to negotiate and navigate the future physical layers?

Full text

Let’s Assume the System is Synchronized
fred harris
San Diego State University & Aalborg University
fred.harris@sdsu.edu
Abstract
We arm our graduate students with the best
tools and our best vision to enable them to in-
vent, to improve, and to create the wireless
world of tomorrow. To better compete and suc-
ceed in the coming decades we give them the
best education in signal processing and com-
munication systems. We do, don’t we? Perhaps
not! This paper calls attention to a gaping void
and suggests that we fill that educational gap.
1. Introduction
It is amazing how many papers on radio
systems, networks, error correcting codes, and
related topics contain a version of the sentence
“Let’s assume the system is synchronized.”
Alright, let’s assume the system is synchro-
nized. But I have a few questions: Who did it?
How did they do it? Who will do it in the next
decades as many of us retire from the field? An
important one is; where are they acquiring the
skills required to negotiate and navigate the fu-
ture physical layers?
This brings us to the question of “What do
we mean by synchronize”? Its etymology starts
with Chronos, (also Khronos and Chronus) the
ancient Greek Immortal Being (Χρόνος) per-
sonified in the modern age as Father Time. We
thus form synchronize from the Greek prefix
syn, meaning “together with” and chronos
which we interpret as “time”.
With the industrial revolution and the ascen-
dancy of the high speed railroad came the need
to synchronize clocks in adjacent towns in or-
der to maintain arrival and departure timeta-
bles. This helps protect trains from cornfield
collisions. Today, the higher speed transport
and commerce of communication signals places
an even greater premium on the measurement
of time and the alignment of remote clocks and
oscillators.
When discussing the importance of syn-
chronization in my modem design class I pre-
sent the needle point shown in figure 1 and re-
mind the students that Momma’s middle name
is synchronizer! If the radio is not synchronized
no other system can operate! Not the matched
filters, not the equalizer, not the detectors, not
the error correcting codes, not the decryption,
not the source decoding!
Figure 1. The Synchronizers’ Needle Point
At the waveform level, synchronization entails
the frequency and phase alignment of remote
oscillators for carrier acquisition and tracking,
for modulation symbol timing, for chip align-
ment and hopping boundaries of spread spec-
trum modulation overlays.
What have we missed by assuming the sys-
tem is synchronized? We skipped a challenging
and most interesting part of the process. We
have skipped the task of estimating, in a short
time interval, the unknown parameters of a
known signal in the presence of noise. We have
replaced the task of processing a noisy wave
shape for the easier task of processing a binary

data stream with unknown errors. We have also
skipped making Momma happy!
2. Source of the Problem
Figure 2 presents a very simple model of a
communication system. Here input bits are sub-
jected to a number of transformations at the
modulator with matching inverse transforma-
tions at the demodulator. In total, the input bit
stream to the modulator is converted to the ra-
dio frequency wave form at the modulator out-
put. A communication system is seen through
the eyes of the beholder and different eyes see
different systems. Some see the system from
the Shannon perspective, as a discrete modula-
tor and demodulator connected to a discrete
channel as shown in figure 3. In Shannon’s
model, the discrete modulator and demodulator
perform a set of transformations on discrete
sequences as shown in figure 4.
Data
Transforms
Data
Transforms
Waveform
Transforms
Wavef orm
Transforms
Sp ectral
Transforms
Sp ectr al
Transforms
Bandwidth
Limited
Channel
AWGN
Bits
Bits
M -Ar y
Alphabet
M -Ar y
Alphabet
Baseban
d
Wav ef o rm
Baseband
Wa ve fo rm
Radio Frequency
Wa veform
Radio Frequency
Waveform
Digital
Digital
Analog
Analog
Modulator
Demodulator
Figure 2. Simple Communication System
Data
Transfo rms
Data
Transfo rms
Waveform
Trans forms
Wavef orm
Trans forms
Sp ectr al
Transfo rms
Sp ectr al
Trans forms
Channel
AWGN
Bits
Bits
M-Ary
Alphabet
Baseband
Wav ef o rm RF
Discrete
Demodulator
Discrete
Modulator
Discret e Channe
l
Figure 3. Discrete Communication System
Bits
Bits
Bandwidth
Reducing
Bandwidth
Preserving
Bandwidth
Expanding Channel
Source
Encoding
Channel
Coding
Channel
Decoding
Source
Decoding
Encryption
Decryption
Data Transformations
Figure 4. Shannon’s Communication System
What Shannon did was brilliant: his model
abstracted the system. His model separated the
communication system from the physical sys-
tem and the discipline blossomed under this
separation. Truly remarkable advances in
communication systems can be traced to devel-
opments in this model space. I fear that since
there are no wave shapes in the Shannon model
we have lost sight of their importance to the
process. The model’s success inadvertently
relegates the wave shapes in the physical sys-
tem to a secondary status in the academic
community.
At the other extreme of the model space is
the hardware model that emphasizes the signal
conditioning and signal processing of commu-
nication waveforms exiting the modulator and
entering the demodulator. Such a model is
shown in figure 5. In this model the discrete
transformations emphasized in the Shannon
model are located in the processing blocks ti-
tled Bit-Map at the input to the transmitter and
at the output of the receiver.
The transmitter of the physical layer model
explicitly shows the shaping filter, the up-
converter, and the output power amplifier
which perform the base band spectral shaping,
the linear RF spectral transformations, and non-
linear spectral transformations respectively. We
note the non-symmetry of the transmitter and
the receiver. The receiver contains many more
subsystems than does the transmitter. These
subsystems are seen to be servo control loops
that participate in the signal conditioning re-
quired to demodulate the input waveform.
These loops estimate the unknown parameters
of the known input signal and invoke corrective

BITS
Input
Cloc k
Carrier
Reference
Bits
Timing
Timing
Control
Equalize
Control
Carr ier
Control
Phase
Control
Gain
Contr ol
Carr ier
DDS
DDS
Bit
Map
DETECT
Noise Int er f er en ce
Chan ne
l
Non I deal
Channel
Bit
Map
Shaping
Filter
Matched
Filter
Band Edge
Filter
Equalizer
Amplitude
and
Phase
Transmitter
Receiver
Amplitude
Quantized
Amplitude
Baseband
Waveform
Baseband
Wavef orm
Carrier
Wavef orm
Carrier
Waveform
Carrier
Waveform
VGA LNA
PA
SNR
Estima te
SNR
SNR
SNR
SNR SNR
Figure 5. Physical layer model of Transmitter and Receiver.
signal processing and signal conditioning op-
erations to ameliorate their degrading effects on
the demodulation process.
A partial list of these processes includes the
following subsystems: i.) an AGC loop to esti-
mate and remove the unknown channel attenua-
tion, ii.) a carrier recovery loop to estimate and
remove the unknown frequency offset between
the input signal’s nominal and actual carrier
frequency, iii.) a timing recovery loop to esti-
mate and remove unknown time offsets be-
tween the receiver sampling clock and the op-
timal sample positions of the matched filter
output series, iv.) an equalizer loop to estimate
and remove unknown channel distortion re-
sponsible for inter-symbol interference, v.) a
phase recovery loop to estimate and remove
unknown carrier phase offset between input
signal and local oscillator, and vi.) an SNR es-
timator to supply important side information to
the just enumerated subsystems.
The closing question in this section in this
section is “Where do your students learn the
science and engineering embedded in the sub-
systems of figure 5? Where do they learn the
new DSP based techniques that have sup-
planted legacy analog designs? We now exam-
ine an overview of some of the signal process-
ing tasks performed during the synchronization
process.
3. Phase Lock Loop, Timing Recovery:
The function of the various loops in the re-
ceiver is to estimate the various unknown pa-
rameters of the received noisy signal. The esti-
mate ˆ
θ
is based on observations of a sampled
data sequence y(n) derived from the output of a
matched filter that acts to reduce the effects of
the additive noise.
Shaping
F ilter
Matched
Filter Bank
h(t)
h(t, )τ
1
h(t, )τ
2
h(t, )τ
3
h(t, )τ
M
δτ(t- )
x(t) x(t- )
τ
r(t)
y(T- )
τ
1
y(T- )
τ
2
y(T- )
τ
3
y(T- )
τ
M
n(t)
......
......
Select Largest
T
T
T
T
Unknow n
Channel
Delay
d(n) d(n)
^
Figure 6. Bank of Matched Filters Parameter-
ized over Time Delay Variable τ.
The estimate can be obtained from a bank of
matched filters parameterized over the un-

known variable such a time delay τ. This option
is shown in figure 6. The outputs of the filter
bank at specific symbol time increments nT are
subjected to a detector and are averaged to ob-
tain stable statistics The smoothed outputs are
compared and the filter with the largest output
magnitude is the one matched to the signal time
delay τk.
In modern receivers, the filter bank is avail-
able to the receiver as the paths of an M-path
polyphase filter. Rather than operate all the
paths simultaneously, they are operated sequen-
tially in response to side information which
guides a state machine to the peak of the corre-
lation function. This side information is the
slope at the output of each hypothesized filter
selection. The system selects any filter in the
bank and tests the hypothesis this it is the cor-
rect filter. It does that by forming the derivative
at the test point. In legacy receivers the deriva-
tive was estimated from early and late matched
filters bracketing the test point in question. In
modern receivers the derivative is formed by a
derivative matched filter bank. The derivative
τ
τ
τ
τ
1
τ
2
τ
3
C( )τ
S( )=C( )ττC( )τ
C( )τ
.
.
C>0
C<0
Figure 7. Correlation Function showing Slope
at Various Test Points and Detector S-Curve
formed as product c( .
τ)c(τ)

at selected points of a correlation function is
shown in figure 7 for positive valued and for
negative valued correlations. Note that for a
positive valued correlation a positive slope in-
dicates the peak is ahead of the test point and a
negative slope indicates the peak is behind the
test point. Since the slope has the reverse polar-
ity when the correlation value has a reverse po-
larity the information residing in the slope must
be conditioned on the polarity of the amplitude.
The amplitude conditioning of the slope is em-
bedded in a detector S-curve formed as the
product of the amplitude and the slope.
The state machine is designed to move the
hypothesis test point in the direction that sets
to zero. Note this happens at two loca-
tions! If c( is zero, we are at the peak of the
correlation function and the system is a maxi-
mum likelihood estimator and if c( is zero, we
are at the zero of the correlation function and
the system is denoted the Gardner or minimum
likelihood estimator. The state machine oper-
ates as a servo system called a phase locked
loop whose block diagram is shown explicitly
in figure 8. The figure shows the standard com-
ponents of the PLL timing loop. These include
the matched filter bank and the derivative
matched filter bank as well as the loop filter
which averages through the received noise and
modulation noise and the phase accumulator
that selects the candidate hypothesis filter from
the filter bank. Since the loop contains 2-
integrators it is a type-2 loop, able to track a
ramp input, a frequency offset, with zero steady
state error.
c(τ)c(τ)

τ)

τ)
Two other components shown in the block
diagram, derived from the ML equations, are
particularly interesting. These are the SNR
scale factor (2Eb/N0) that serves to tell the loop
the quality of the signal it is processing. If the
signal sample has a low SNR the input to the
loop filter should be scaled in proportion to its
quality. This of course reflects the philosophy
of all matched filter processes. The second
component is the TANH that conditions the

Matched
Filter
Polyph ase
Polyph ase
Derivative
Matched
Filter
Tanh( )
Phase Accumula tor Loop Filter
2
E
N
B
0
2E
N
B
0
CLK
r(t, )τr(nT, )τy(nT+ T, )δτ
y(nT+ T, )δτ
.
δ()Tk
δ()Tk
Inte ger
Part
K
I
K
P
Z
Z
-1
-1
k
Figure 8. DSP Based Polyphase Filter Bank Maximum Likelihood Timing Recovery Loop
amplitude from the matched filter as it interacts
with the derivative. At high SNR the TANH
faults to the sign of the input signal as a condi-
tional correction to the derivative information.
At low SNR the TANH defaults to a unity
gain applied to the matched filter output to
avoid possible errors in the sign decision of the
amplitude as the conditional correction of the
derivative. The SNR gain term in the loop filter
throttles the loop bandwidth in response to the
SNR. At low SNR the loop filter reduces its
bandwidth so that it has to average over a
longer interval to obtain stable control signals.
Conversely, at high SNR the loop filter in-
creases its bandwidth since it can obtain stable
control signals by averaging over shorter time
intervals.
The problem with these two components is
that most receivers do not have real time back-
ground SNR estimators operating to feed the
quality assessment of data samples to the loop
filter. Thus most receivers replace the TANH
with its small SNR gain and operate as sub-
optimal systems. We can do better than that!
Figure 9 shows the transient response ob-
served at the output of the phase accumulator
as it moves from an initial filter path to the cor-
rect filter path in the polyphase filter bank. The
red curve is the accumulator content while the
blue overlay curve is the integer part of the ac-
cumulator that defines the index pointer to the
selected path of the M-path filter. Figure10
shows the eye diagram of the signal at the input
and output of the timing recovery loop.
Figure 9. Transient Response of Phase Accu-
mulator During Timing Acquisition
Figure 10. Eye Diagrams at Input and Output of
Timing Recovery Loop

4. Phase Lock Loop, Phase Recovery:
In many modulators the signal formed by
the shaping filter is amplitude and phase modu-
lated in accord with the input bit mapping
process. The amplitude and phase terms are
represented in Cartesian coordinates and de-
scribed as a complex base band signal. The
quadrature components of the signal are up-
converted or amplitude modulated on the quad-
rature components, the cosine and sine, of a
radio frequency carrier.
At the demodulator the process is reversed
and the radio frequency carrier is down con-
verted by a pair of cosine and sine quadrature
sinusoids. The frequency and phase of the up-
converter and the down-converter oscillators do
not match by virtue of manufacturing toler-
ances, age and temperature related drift, and
Doppler offsets due to velocity vectors between
platforms.
The signal obtained at the output of the
quadrature down converter is monitored and
applied to a phase detector to obtain a measure
of the phase misalignment. Here we quickly
describe the process by which the phase lock
loop aligns the local oscillator phase with that
of the phase of the received signals’ underlying
carrier. First we examine the phase detectors
for a binary phase shift key BPSK signal. What
do we do with the samples of the I – Q pair,
which are time aligned with the correlation
peaks of their matched filters, to obtain infor-
mation about their phase θ? What we do is ex-
amine a legacy solution formed by their prod-
uct I⋅Q. As seen in figure 11, the product is
proportional to the sine(θ/2) which is the ex-
pression for an S-curve phase detector. The de-
tector has two zero value references at 0° and at
180° which of course is responsible for the
two-fold ambiguity in the synchronized phase
lock.
-0.5 -0.4 -0.3 -0.2 -0. 1 00.1 0.2 0.3 0.4 0.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Δ-φ
S-Curve Product detector x*y
-0.5 -0.4 -0.3 -0.2 -0. 1 00.1 0.2 0.3 0.4 0.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Δ-φ
Inputs to Product Detector
Figure 11. I⋅Q BPSK Phase Detector and S-
Curve
Figure 12. SGN(I)⋅Q BPSK Phase Detector and
S-Curve
-0.5 -0.4 -0.3 -0. 2 -0.1 00.1 0.2 0.3 0.4 0.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Δ-φ
Inputs to Product Detector
cosine
sgn(cosine)
-0.5 -0.4 -0.3 -0. 2 -0.1 00.1 0.2 0.3 0.4 0.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Δ-φ
S-Curve Product Detector sign(x)*y

Matched
Filter
Polyph ase
Polyph ase
Matched
Filter
Tanh( )
Phase Accumu lator Loop Filter
2E
N
B
0
2E
N
B
0
CLK
r(t, )θ
r(nT, )θ
x(nT, )θ−θ
y(nT, )θ−θ
Inte ge r
Pa rt
K
I
K
P
Z
Z
-1
-1
exp(-j )θ
^
^
^
2-to-1
2-to-1
Figure 13. Maximum Likelihood Phase Recovery for BPSK Receiver
Matched
Filter
Polyphas
e
Polyph ase
Matched
Filter
Tanh( )
Tanh( )
Phase Accumulator Loop Filte r
2E
N
B
0
2E
N
B
0
CLK
r(t, )θ
r(nT, )θ
x(nT, )θ−θ
y(nT, )θ−θ
Inte ger
Pa rt
K
I
K
P
Z
Z
-1
-1
exp(-j )θ
^
^
^
2-to-1
2-to-1
+
-
Figure 14. Maximum Likelihood Phase Recovery for QPSK Receiver
An alternate phase detector is the SGN(I)⋅Q
which is shown in figure 12 where we see a
more linear S-curve spanning a wider range of
input phase offsets. These correspond to the
small signal to noise ratio and the large signal
to noise ratio approximations to the TANH de-
scribed in the previous section. Thus it will
come as no surprise to see the structure shown
in figures 13 and 14 of a BPSK and a QPSK
receiver implementing Maximum Likelihood
phase recovery.
Incidentally, we make an interesting obser-
vation with respect to figure 13 which is the
ML phase estimator for the BPSK receiver.
Suppose we leave off the SNR scaling factors
from the two paths, and further replace the
TANH of the upper path with a wire (or a unity
gain path) and similarly replace the 2-to-1

down sampler on each path with a wire. What
would we have? Give up? We would have a
Costas loop phase recovery system. The Costas
loop is embedded in many legacy receivers.
Having discarded the items listed earlier we
quickly conclude that the Costas loop is far
from an optimum phase recovery process. In
fact it is easy to verify that the sub-optimal loop
performs quire poorly at low SNR. As we
bravely move forward to the next decade, let us
not carry legacy suboptimal solutions with us.
Figure 15 presents the S curves for the ML
phase detector of the QPSK receiver for a range
of scaled SNR. Note the change in slopes at the
zero crossings of the S-curve. These slope
changes alter the loop gain of the PLL and are
responsible for SNR dependent loop bandwidth
of the acquisition system.
Figure 15. S-Curve for QPSK Phase Detector
for Range of input SNR.
Figure 16 shows the trajectory of the QPSK
constellation during Phase acquisition. The tra-
jectory has the appearance of a comet with its
trailing tail because the figure plotted the con-
stellation points near the end of the trajectory in
red to emphasize the final state of the trajec-
tory.
As observed in the section on timing
recovery, most receivers do not have real time
background SNR estimators operating to feed
the quality assessment of data samples to the
loop filter. Thus most receivers replace the
TANH with its large SNR gain and operate as
sub-optimal systems. We again note that we
can do better than that!
Figure 16. Constellation Trajectory while PLL
Acquires Phase
5. Phase Lock Loop, Frequency Recovery:
If the frequency offset between the local
oscillator used in the final down converter and
the center frequency of the input signal is suffi-
ciently small the phase detector and the phase
lock loop of the previous section can acquire
and de-spin the input signal. On the other hand,
if the frequency offset is significantly larger
than the bandwidth of the PLL loop filter the
loop will not successfully acquire and de-spin
the signal. In this event an acquisition aid must
be invoked to assist the phase lock loop.
In legacy receivers a lock detector is inter-
rogated after a preset time-out to see if the ac-
quisition aid should be invoked. If invoked, the
local oscillator is slowly swept through the
likely range of frequency offsets till the loop
acquires the signal. The acquisition occurs
when the frequency offset is small enough for
the frequency error signal to pass through the
narrow bandwidth loop filter. The lock detector
terminates the sweep upon detection of the ac-
quisition.
In modern receivers a maximum likelihood
frequency estimators perform the task of fre-

quency acquisition. To reduce phase jitter due
to the frequency lock loop noise injection, this
loop is disabled when the system acquires final
phase lock.
Here we see two scenarios; One in which the
input spectrum is centered between the two
band edge filters and one in which the input
spectrum has shifted towards one and away
from the other band edge filter. When centered,
the two band edge filters collect the same en-
ergy from the input spectrum and their average
energy difference is zero. When offset, the two
band edge filters collect different amounts of
energy from the input spectrum and their en-
ergy difference contains a DC term propor-
tional to the frequency offset. The energy dif-
ference of the band edge filters is formed as the
difference of the conjugate products of the time
series from each band edge filter. This differ-
ence, proportional to the frequency difference,
is the input signal to the frequency lock loop
loop-filter. A block diagram of the frequency
lock loop with its associated lock detector is
shown in figure 18.
The frequency lock loop is based on a ML
frequency estimator. When we take the deriva-
tive of the output of matched filter with respect
to the unknown frequency offset we obtain the
frequency derivative matched filter. The fre-
quency derivative filter, often called the band
edge filter, is a sensitive detector of frequency
offsets of the input signal’s spectral mass. How
this is accomplished is visualized with the aid
of figure 17.
Spectrum:
Freq uenc y Matched Filter
Spectrum:
Freq uenc y Matched Filter
Spectrum:
Centered Input Signal
Spectrum:
Shifted Input Signal
Spectrum: Respon se to
Centered Input Signal
Spectrum: Response to
Non Centered Input Signal
ω
ω
ω
ω
Figure 19 shows the phase profile of the
frequency lock loop operating on an input sig-
nal with a frequency offset equal to 2.0% of the
signal symbol rate. We see here the phase of
the input carrier offset and the phase response
of the loop. In steady state the two phase pro-
files have the same slope, hence same fre-
quency but with a phase offset. We also see the
band edge energy difference which returns to
zero as the signal acquires frequency lock. The
Figure 17. Frequency Centered and Offset In-
put Spectra Interacting with Band Edge Filters
Energy
Difference
Band Edge
Filters
Δθ
^
{R(n)e }
j(n)θjnΔθ
e
Phase Accumulator Loop Filte r
Inte ge r
Pa rt
K
I
K
P
Z
Z
-1
-1
exp(-j )θ
^
Digital Frequency
Fr eq ue nc y
Lock
Detection
Figure 18. Maximum Likelihood Frequency Recovery for QAM Receiver

lower subplot of figure 19 shows the frequency
offset of the loop DDS relative to the known
input frequency offset.
Figure 19. Phase Profiles of Input and Output
of Frequency Lock Loop and Energy Differ-
ence Control Signal and Frequency Error Pro-
file
6. Concluding Comments
We have commented on a gaping void in
the communication sequences we teach to the
next generation of communications and signal
processing engineers. A red alarm flashes every
time I see a paper starting with the comment
“Let’s assume the system is synchronized”. I
think we are doing the next generation of engi-
neers a disservice by not presenting an impor-
tant aspect of communications residing in the
physical layer.
We have presented a very light overview of
the synchronization process performed in many
radio communication systems. The particular
processes we examined were timing recovery,
phase recovery, and frequency recovery. We
avoided derivations and emphasized concepts
easily illustrated with figures and discussions
based on senior level undergraduate electrical
engineering curriculum. What we have not
discussed would fill a few small textbooks. We
did not ask nor answer the question where does
the energy reside in the modulation signal that
the synchronizer accesses to perform its tasks.
Do you know? It resides in the excess band-
width of the modulation spectrum. We might
wonder what systems do not have excess
bandwidth. We might and the answer is stan-
dard OFDM! How do we synchronize in
OFDM? Answer: we acquire on preambles and
track on pilots! What about other modulation
formats such as Offset or Staggered Quadrature
Amplitude Modulation (O-QPSK), or Vestigial
Sideband (VSB), or Gaussian Minimum Phase
Shift (GMSK), O-QPSK OFDM, or Shaped
OFDM (S-OFDM), and on and on and on?
7. Bibliography
[1]. Rice, Michael, “Digital Communications:
A Discrete Time Approach”, Prentice-
Hall, 2009.
[2]. Myer, Heinrich, Marc Monocle, and Stefan
A. Bechtel, “Digital Communication Re-
ceivers: Synchronization, Channel Esti-
mation, and Signal processing”, John
Wiley & Sons, 1998.
[3]. harris, fred, “Multirate Signal processing
for Communication Systems”, Prentice-
Hall, 2004.
[4]. Proakis, John, “Digital Communications”,
Third Ed., McGraw-Hill,1995.
[5]. Ramanmurthy, Arjan and fred harris, “An
All Digital Implementation of Constant
Envelope, Bandwidth Efficient GMSK
Modem using Advanced Digital Signal
Processing Techniques”, WPMC-06,
Sept. 2006, San Diego, CA
[6]. Ramanmurthy, Arjan and fred harris, An
all Digital Implementation of M-VSB
Modem Using Advanced Digital Signal
Processing Techniques”, SPAWC-06,
July 2006, Cannes, France.
[7]. harris, fred, Chris Dick, and Michael Rice,
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Communications”, Microwave Theory
and Techniques, MTT, Vol. 51, No. 4,
April 2003, pp 1395-1412.