Blind Equalization with Differential Detection for Channels with ISI and Fading Eloise Tse A thesis subrnitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Cornputer Engineering University of Toronto @ Copyright By Eloise Tse (1997)
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Blind Equalization with Differential Detection for
Channels with ISI and Fading
Eloise Tse
A thesis subrnitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Electrical and Cornputer Engineering
University of Toronto
@ Copyright By Eloise Tse (1997)
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Blind Equalization wit h Different ial Detection for
Channels with ISI and Fading
Eloise Tse, M.A.Sc.
Graduate Depart ment of Electrical and Computer Engineering
University of Toronto, 1997
Supervisor: Professor Pas S. Pasupathy
Using coherent detection with carrier tracking and adaptive equalization with train-
ing sequence can achieve good performance in equalizing time-varying channels at the
expense of complexity and feasibility. Thus, differential detection and blind equaliza-
tion, which eliminate PLL and training sequence, are proposed. Decision feedback is
dso added to equalize null and fading channels. In this thesis, Godard and Modified
Constant Modulus Algorithms (MCMA) axe used. New systems are set up by com-
bining coherent and noncoherent detection with these two algorithms. It is found that
the use of noncoherent detection degrades the system performance. For MCMA, as it
can track the carrier, neither noncoherent detection nor PLL is required. Contrarily,
Godard needs either noncoherent detection or PLL to correct phase error. Thus,
the proposed system combining differentid detection, blind equalization and decision
feedback c m indeed equalize different channels, though the robustness of the system
is compromised. Further investigations must be done to deal with this problem.
Acknowledgment s
First and foremost, thanks to my supervisor, Professor Pas S. Pasupathy, for bis
unlimited patience and guidance.
Special thanks to Ali Masoomzadeh-Fard, for helping me to start my work,
and for his patience when answering my never-ending questions.
Last but not least, my family and friends, whose understanding and support
5.1 Coherent System with Godard for LMS equalizer (CG) . . . . . . . . 31
5.2 Godard with Differential detection for LMS equalizer (DG-1 ) . . . . . 33
5.3 Differential detection with Godud for LMS equalizer (DG-2) . . . . . 34
5.4 SER vs SNR c u v e for coherent systems for Channel A . . . . . . . . 41
5.5 SER vs SNR curve for Godaxd systems for Channel A . . . . . . . . . 42
5.6 SER vs SNR c u v e for MCMA systems for Channel A . . . . . . . . . 43
. . . 6.1 Coherent Godard with feedback path for LMS equalizer (CGFB) 46
6.2 Differential Godard with feedback path for LMS equalizer (DGFB) . 48
6.3 Coherent MCMA with feedback path for LMS equalizer (CMFB) . . 52
. . . . . . . . . . . . . 6.4 SER vs SNR c w e for systems for Channel B 57
6.5 SER vs SNR cuve for systems for Channel C . . . . . . . . . . . . . 60
. . . . . . . A.1 Multiple signal paths due to reflections in fading channels 67
Chapter 1
Introduction
When detecting the data sequence from the received signal, two types of detection
can be used. One is coherent detection, the other is noncoherent detection. If coher-
ent detection is used, schemes such as phase-locked loop (PLL) are used to recover
the absolute phase of the received signal. However, this complicates the hardware
especidy when the data transmission rate is high and the phase is varying rapidly.
Thus, noncoherent detection, a simple structure where no phase tracking is required,
is desirahle. In mobile and data communications, differential detectiou is a commonly
used signaiLing scheme. Bandwidth efficient scheme like dgerent ia l quadriphase shift
keyzng (DQPSK), which utilizes both in-phase and quadrature axes, is used (11. How-
ever, with differential detection, nonlinear intersymbol interference (ISI) is generated.
Simple equalization methods are no longer feasible. Different schemes have been em-
ployed to deal with this non-linearity (21. However, this nonlinear component together
wi t h the charnel dis tortions, degrade the performances of the noncoherent receivers,
and make them inferior to the coherent decision feedback scheme. Furthermore, the
complexity for some of the schemes is very high. AU of these structures require some
knowledge about the channel characteristics which is not known for most practical
channels. Therefore, an efficient equalization algorithm is required for noncoherent
detection. In this thesis, new structures which combine blind equalization with dif-
ferential detection are presented. Also, to equalize null and fading channels, decision
feedback is added to these new structures.
Among the already known noncoherent receiver structures, sever al adapt ive solu-
tions are proposed for DQPSK. First, a linear equalizer is placed before the nonlinear
detector without decision feedback. This method fails to equalize fading channel
which requires decision feedback. Without decision feedback, nulls in the channel
response give rise to noise enhancement at the input of the slicer. Hence, poor perfor-
mance results. Then, the situation is remedied by introducing two modified adaptive
equalizers. In [2], these proposed algorithms are simulated. Acceptable performances
result . However, a testing sequence is still needed in the above systems where a known
sequence is transmitted and received before the actual data is sent. The equalizer
adapts to the channel by minimizing the enor between the known sequence and re-
ceived sequence. This known sequence is, thus, c d e d the training sequence. Usually,
a pseudo-noise (PN) sequence is used for this application. However, training sequence
imposes a certain amount of delay which must be taken into account if the actual
data transmission is short. Also, by using the training sequence, it is assumed that
the channel characteristics do not deviate a lot fiom its initial state after the sequence
is sent. This is certainly not true for time wying channels such as mobile channels.
In order to track the channel response, the training sequence has to be transmitted
periodically for the adaptation of the equalizer to the time variations. Sometimes
the transmission of such sequence may even be impossible in some communications
chasnels. This is why blind equalization is proposed. With blind equalization, no
such sequence is needed. Instead, the equalzer estimates and adapts itself imme-
diately when the data is received. This rnethod equalizes the channel based on the
information ob tained from the received signal, contrary to previous algorithms. Since
existing blind algorithms are applied to coherent detection, the challenge is to apply
differential detection to blind equalization. These two features together with decision
feedback result in new systems which c m be applied to mobile communications.
Among numerous blind equalization dgorithrns [3, 4, 51, two in particular are
of interest. They are Godard Algorithm (Constant Modulus Algorithm CMA) and
Modified Constant Modulw Algorithm (MCMA) from the Stochastic Gradient class.
The transmitter and channel mode1 are presented in Chapter 2. In Chapter 3, several
already proposed differential structures are discussed. Next, an overview of blind
equalization is given. Then in Chapter 5, the two blind equalizers are combined with
coherent and differential detections. Simulation results for channei without nuil are
presented. In Chapter 6, decision feedback is added to some of the structures in
Chap ter 5. The resulting systems are simulated for a charnel wit hout null, with nulis
and with fading. Lastly, conclusions about ail the systems and results are drawn.
Chapter 2
Coherent and Noncoherent
Systems
In order to compare the performances of the proposed equalizers, complex equivdent
baseband systems wit h coherent and diRerentia.1 detections are discussed.
2.1 Coherent source
For coherent systems, QPSK is used as the source, providing four possibilities ((1
+ j), (1 - j), (-1 + j), ( 1 - j)} for a = x + y . The symbols ai are shaped by
a raised cosine filter g( t ) , go through the channel p ( t ) (assuming it is symmetric)
and comipted by zero mean additive white Gaussian noise (-4WGN) n(t) with power
spectral density N 4 2 . Then they are received by a lowpass filter h(t) . At output of
this filter, the received signal is,
where û is an unknown angle introduced by the bandlimited channel, J is the memory
of the channel, f ( t ) is the convolution of g(t), p(t) and h( t ) , R(t) is the noise at the
output of the lowpass filter, and 1/T is the data transmission rate. Sampling r(t)
every T seconds yields,
where ri = r(iT), fj = f(jT) and hi = fi(iT). To equalize different channels, Least
Mean Square LMS equalizer with decision feedback is used in Figure 2.1. After ri has
passed through the forward equaiizer,
where 8 represents convolution, c,- and N are the tap coefficients a ~ d length of the
fornard equalizer respectively, and f i i is the noise after the equalizer. As for the
feedback equalizer, its output is,
where âi is the estimated symbol from the slicer, di is the tap coefficients of the
backward equalizer, and M is the length of backward equalizer. The s u r n of the
forward and feedback equdzers' output is fed to the slicer,
QPSK a i raised Channel Source cosine g(t) p(t) .
Figure 2.1: Coherent Decision Feedback equalizer ( D F E )
The slicer used is a complex quantizer which maps its input to the closest QPSK
constellation point. For conventional DFE, the error which is fedback to the equalizer
is defmed as the difference between the output and input of the slicer,
where âi and ai are the output and input of the slicer. In training mode, âi = ai; in
other words, known and correct data are fedback to adjust the equdizer's coefficients.
With LMS algorithm, the equalizer adjusts its tap coefficients y with the following
cri terion,
where p is the step size, V, is the gradient with respect to y, and Ji = ei in ( 2.6).
Thus, the coefficients of the fonvard and backward equalizers are adjusted using (see
Appendix B. 1 ),
If a fractionally spaced equalizer (FSE) is used instead of the symbol rate spaced one,
the performance of the equalizer improves significantly for the fading channel [6]. To
rnodie a symbol rate equalizer into a $-rate fractiondy spaced one, the sampling
rate of the forward equalizer is doubled (and/or doubling the number of taps).
2.2 Noncoherent source
For our noncoherent systems, differential phase shift keying (DPSK) is used. After
transmission through the communication channel, an unknown phase is often intro-
duced to the received signal. To compensate for this unknown phase, differential
encoding is combined with phase shift keying (PSK). We assume that the unknown
phase m i e s slowly, so that it is constant within a two-bits intenal. Instead of encod-
ing the information into absolute phase, the information is encoded by the change in
phase. Thus, for transrnitted symbol with the form Ak = e j O k , #& is determined by,
with the information embedded in Ah. Some common implementations are clifferen-
tial binary phase shift keying (DBPSK) and DQPSK where differential encoding is
combined with BPSK and QPSK.
To detect this differential scheme, coherent detection and noncoherent detection
can be used 161. To detect coherently, the actual phase of the received symbol is
Figure 2.2: Differentid detection scheme for DQPSK
determined. Then, the change in phase is caiculated by subtracting the phase of the
previous syrnbol from it, giving A*. Howeves, this scheme can be very difficult to
implement; especially when the phase characteristics of the channel vary so rapidly
that tracking of the carrier phase is an impossible task. Even if tracking can be done,
PLL has to be used, which complicates the hardware.
However with noncoherent detection, the phase is detected differentially. In other
words, only the change in phase between the previous and present symbols is of
interest. From Figure 2.2, ài contains Ak rather than the absolute phase. Thus,
the unknown phase is eliminated through multiplication without employing complex
hardware. However, this detection induces a 3 dB penalty compared to coherent
detection due to an increase of noise at the slicer input [6] (except for DBPSK which
is essentially the same as BPSK), shown by the bit error rate (BER) for DPSK,
1 Eb BER = Zexp(--)
N o
where Es, No are bit energy and noise power spectral density respectively. However,
this penalty can be easily compensated.
The differential scheme is particularly useful in fading channels where PSK is pre-
ferred. In t hese channels, the phase characteristics change rapidly. Ins tead of taking
Diff. & hardlirniter encoder
Channel cosine g(t)
A a i
4 sticer a a i ~ i . f f decoder
T
Figure 2.3: Linear Differential equalizer (DIFF)
on the difficult task of tracking this rapidly changing phase, DPSK with differential
detection is used. Therefore, with DQPSK, Figure 2.3 is setup. After the differential
encoder,
where ai , bi aze the QPSK and DQPSK symbols'.
The symbols bi are then transmitted through the same mode1 as the QPSK scheme.
Therefore, at output of the lowpass filter h(t) , the received signal is,
where f ( t ) is the convolution of the raised cosine filter g ( t ) , channel p ( t ) and lowpass
filter h( t ) , B is an unknown angle, J is the memory of the syrnmetric channel, and
n(t) is the noise at the output of the lowpass filter. Again r ( t ) is sampled every T
ltf ai, bi are cornplex, a hardlimiter is used to keep their magnitude to 1.0.
seconds to give,
where ri = r ( iT) , fj = f(jT) and f i i = fi(iT). After passing through the forward
equalizer,
where ci, N are the tap coefficients and length of the fonvard equalizer. The data is
then recovered by decoding the output of the equalizer hi using a diflerentid decoder.
This operation involves a nonlinear process,
This is where, if there is residual ISI, nonlinear ISI is generated. Since the data is
differentidly encoded, the function which is used to correct the equalizer's coefficients
must be derived by taking the gradient in ( 2.7). Same as before, Ji = ei where ei is
( 2.6) . It is found that ei has to be multiplied by the delayed output of the equalizer A
bi-l (see Appendix B.1). Therefore the equalizer's coefficients are adjusted by the
following equation, A
q + l = ci + bi-lei. (2.16)
This equation is different from ( 2.8) since ai is the output of the differential de-
coder. Then Z i is passed through the slicer which maps the data into one of the four
possibilities within the QPSK constellation.
In order to implement a decision feedback path, the estimated data âi must be
re-encoded into 6i before feeding into the backward equalizer,
Summing up the forward and feedbadc equalizer's output, bit input to the decoder is
rnodified into,
where the first convolution is ( 2.14), M, di are the length and tap coefficients of
backward equalizer. Same gradients are found for the LMS adaptations,
The above details the transmit ter, channel mode1 and two conventional sys tems,
while the blind receiver structures will be discussed in details in later sections. Next.
several previously proposed differential systems are discussed.
Chapter 3
Equalizat ion wit h Different ial
Detection
In most communication applications, the channel effects can be modeled by a discrete
linear time-invariant filter. Thus, the received signal is the convolution of the channel
impulse response and the input signal (after the transmit ter filter) as in ( 2.2) and
( 2.13). In order to compensate for this channel distortion, minirnize the noise, and
estimate the transmitted signal, a deconvolution is needed. There are many different
approaches as to how this deconvolution is done, such as linear equalization with zero
forcing (ZF) and mean squared error methods (MSE), decision feedback equalization,
adaptive equalization with LMS and recursive least square (RLS) algorithms, and
nonlinear equalization.
Previously [2], some work has been done on dxerential detection with adaptive
decision based equalizer on selective fading channel models. In this model, DQPSK,
as discussed in the last chapter, is used. The same transrnitter and channe1 mode1
Figure 3.1: Linear equalizer (LE) placed before differential detec tor
in Section 2.2 are used. The raised cosine filter has a roll-off factor of 1. Also, it is
assumed that the chamel characteristic is time-limited.
When differentid detection is used, nonlinear processing is done as shonm in Fig-
ure 2.2. Thus, any equalizer placed after this nonlinear processor has to deal with
nonlinear ISI. In order to incorporate a linear equalizer with differential detection, it
has to be placed before the nonlinear processor as in Figure 3.1. This way, the equal-
izer deais with linear (assuming that the channel is linear time-invariant ) rather than
nonlinear ISI. For channels with spectral n d s , such as fading channels, this equalizer
fails. This is due to the noise enhancement produced when the equalizer tries to invert
the channel effects. To combat this noise enhancement, decision feedback is required.
Therefore, another scheme with decision feedback is proposed. After the slicer, the
recovered data is re-encoded and fedback to a feedback equalizer 171. h this case
though, the fedback data does not have the unknown phase which the received data
has when entering the forward equalizer. Therefore, this decision feedback can only
be added to those equdzers which cari track phase variations and compensate for the
unknown phase. However, if this is so, there is no need to use differential detection;
since the whole purpose of noncoherent detection is to avoid phase tracking.
Thus, another proposal is to put the equalizer after the nonlinear detector [2].
Unlike the method above, here the equalizer deals with nonlinear ISI rather than
linear ones:
where 3, ta are memory of the channel and system
term and taking the last two terms as noise, the
be shown that the first term contains the desired
Using this result, an equalizer is setup to calculate
time delay. Ignoring the second
first term is expanded. It c m
response and nonlinear ISI [2].
the nonlinear ISI, resulting into
the structure in Figure 3.2. Same as linear equalizer, a coefficient adaptation, such
as LMS or RLS can be added to improve the performance. One advantage of this
nonlinear equalizer is that i t can equalize channels wi th spectral nulls, though this is
traded off with more complexity.
Then, the above linear and nonlinear methods are combined [2]. First, a linear
equalizer is placed before the differential detector, which is then followed by a non-
linear equalizer. Here, the predetection equalizer deals with the precursor ISI, while
the postdetection one deals with the postcursor ISI. This separation reduces the corn-
plexity of the nonlinear equalizer, which deds with both precürsor and postnirsor
ISI at the same time. Only half of the amount of nonlinear ISI is dealt with in this
system.
In a fading channel, its response is usually UiSU10wn. Thus, matched filtering can-
not be done with T-sampling. Also, an excess bandwidth pulse is used in practice for
transmission. To avoid aliasing, T-sampling again is insuscient. IR order to improve
ri- * 1 I
Cornpute ISI s Figure 3.2: Nonlinear equalizer mode1 for DPSK
-
the performance, FSE is proposed. Instead of sampling every T (a symbol duration)
seconds, the samples are taken every seconds where n is an integer. Simulations
are done in (21 using T-spaced equalizer on lineu phase and nonlinear phase channel.
It is found that for nonlinear phase channels, the input folded frequency spectnun
is varying too rapidly for the T-spaced equalizer to tradc. Rather, with FSE, better
performance results. FSE can be implemented into the stnictures discussed earlier
by replacing the original sampler at the receiver by a $ sampler.
Through simulations [2], it is found that the linear equalizer performs better than
the nonlinear equalizer when little ISI is present. However, if spectral n d s are present,
the linear equalizer fails. Indeed, decision feedback is necessary for the equalization
of such channels. Thus, the nonlinear structure is able to equalize the null channel in
the simulation. As for the combination of linear-nonlinear stnictures, its performance
is almost the same as the nonlinear one, but with less complexity. However, these
Equalizer Y i
Slicer
A a i
'
equalizers' performances are still worse than that of the coherent DFE.
Another fact that needs to be considered is that ail of the techniques mentioned
above require either the input signal or the channel impulse response be known.
However, this is generdy riot available. Consequently, blind equalization is proposed.
Chapter 4
Blind Equalizat ion
Other names for this technique are blind deconuoktion and self-recouering equaliza-
tion. It is cailed blind or self-recovering as no training sequence is sent to assist the
adaptation of the equalizer to the channel. Even with little or no knowledge about
the input sequence and the channel, blind equalization can estimate the transmit ted
signal from the received signal. Thus, in situations where the channel characteristics
are time varying, with blind equalization, no extra delay is induced by the training
sequence for periodic update of equalizer's coefficients. One of the simplest blind
equalization algorithm is the decision directed algorithm. As implied by its name,
this algorithm adjusts the equalizer's coefficients by minimizing the error between the
output and input of the slicer. Due to its simplicity, it is unable to equalize channels
which sufTer severe ISI.
Assume that the equivalent baseband channel has sampled impulse response, fi,
identical and independently distributed (iid) input data, ai, sampled received data, ri,
and comipted by white Gaussian noise ni. The basic procedure for blind equalization
is to estimate fi from ri. Then deconvolve fi with ri to obtain âi. The following are
some of the more sophisticated methods which give better performances than the
decision directed method.
4.1 Probabilistic Approach
The f i s t class uses probabilistic methods based on maximum-likelihood criterion (ML)
or maximum a-posteriori estimation principles (MAP) . By M L criterion, the trans-
mitted data and the channel characteristics are estimated based on the maximization
of p(r If, a), the joint probability density function of received vector (r) conditioned on
channel impulse response (f) and input vector (a). Since this function is Gaussian in a
Gaussian noise channel, the rnaximization of the function is equal to minimization of
the exponent. Therefore, the metric for minimization is simplified to the following [3],
where N is the length of the data block, L is the channel response vector length, and
A is the data matrix,
al O 0 ... O
a* ai O O
O
a ~ a ~ - 1 a ~ - 2 --• a N - L 4
Since both f and a are unknown, it is diffidt to find the solution using this metric.
There are two approaches in finding the vectors f and a through the minimization
of the metric. One is by averaging the metric over d possible data sequences, and
thus, finding p(r1f). The f which maximizes this function is the solution. It is found
to be (31,
Once this optimal f is known, the most likely transmitted sequence â can be found
through Viterbi algonthm (VA) using the metric defined above.
The second method is to estimate the data and the channel impulse response
simultaneously. This can be done by calculating the estimation of the channel impulse
response for every data sequence. Then the sequence which gives the minimum metric
is selected. Generalized Viterbi algorithm (GVA) devised by Seshadri (1991) cm be
used in h d i n g the most probable sequence of data (31. If conventional VA is used,
computational complexity grows exponentially with the length of the data sequence.
In GVA, through the fkst L (where L is the length of data sequence) stages of the
trellis, the search is the same as the original VA. All the data sequences and their
corresponding channel impulse response estimates are stored. After that , K, instead
of one, surviving sequences and their channel estimates per state are retained. To
reduce the complexity, at each state, the channel estimates are updated recursively
by LMS algorithm. This method has pretty good performance at moderate SNR with
K = 4 [3], though its complexity is even higher than conventional VA.
Another approach is called the Quantized-channel algorithm [3,4]. In this method,
the channel impulse response assumes a
for this response is found by VA. And
certain value. Then the optimal data sequence
the initial channel estimate is updated using
this detected sequence. The algorithm repeats until the most likely data sequence is
found, or in other words, until the algorithm converges.
This class of blind equalization has the disadvantage of high computational com-
plexity due to the use of VA. However, they can be usefid for constellations that are
approximately Gaussian distributed. Also, this algorithm is optimal as it uses VA.
4.2 Steepest Descent Approach
The second class, based on the steepest descent, is called the Stochastic Gradient al-
gorithm. In this method, the equalizer's coefficients are initially set to certain values.
Thus, the output of the equalizer is the convolution of the received signal, channel
impulse response and the equalizer impulse response. This output includes three
elements-the desired response, the noise component and residual ISI (or sometimes
called convolutional noise). Then least MSE criterion is employed to estimate the de-
sired signal. The desired signal is a nonlinear function of the equalizer's output. This
noniinear function can be with memory or memoryless. The error is then caiculated
and fedback to the adaptive LMS equalizer as shown in Figure 4.1. There are many
different algorithms in this class. The difference between them lies in the nonlinear
function used. Some of the common algorithms are shown in Table 4.1 [3, 41. Note
that contrary to non-blind methods where the error is defined as the difference be-
tween the detected data and actual data, this equalizer calculates the error between
received data and output of the nonlinear function.
The main concern for this class is its convergence. Convergence is reached when
the average gradient of the cost function equals to zero. Typically, slow convergence
Figure 4.1: General adaptive structure for S teepes t Descent Approach
is expected when LMS is used, though it is easier to implement. To be able to
converge, the algorithms must satisfy the Bussgang property [3,4]. By this property,
the autoconelation of the equalizer's output ai equals to the cross-correlation between
the equalizer's output and the output of the nonlinear function g( i i i ) :
Thus, this class is sometimes known as the Bwsgang algorithm. As the nonlinear
functions in Table 4.1 are generdy multimodal, the LMS method may converge
to local equilibrium points rather than the desired point where the MSE is truly
minimized.
4.3 Higher Order Statistic Approach
The third class uses second and higher-order statistics of the received signal to equalize
the channel. It is known as the polyspectra approach. Recall that for Gaussian
distribution, only its &st and second order statistics are meanin@. As a result,
Godard (CMA p = 2)
Sato
GSSA
Benveniste-Goursat
S top-and-go
Nonlinear fvnction g(ai)
Table 4.1: Table of nonlinear functions for Steepest Descent Method
for signals that are comipted by Gaussian noise (provided that signals themseives
ore not Gaussian distributed), only their second order statistics are afFected. The
noise free higher order statistics can then be used to recover the transmitted signals.
Furthemore, this approach can be applied to channels with non-minimum phase
response whose true phase characteristics are not available in second order statistics.
The higher order statistics involved are nth order cumulants. Given a zero mean
sequence y@), its second and third order cumulants are dehed as follows [4],
And the polyspectra of the sequence is then the m-dimensional Fourier transform of
the (rn + l ) th order cumulant. One of the very cornmon polyspectra is the power
spectrum. It is the one-dimensional Fourier transform (1-d FT) of the second order
cumulaat, commonly known as autocorrelation. Some of the others which are used
in this class are bispectrum and trispectrurn, the two-dimensional FT of the third
order cumulant and three-dimensional FT of the fourth order cumulaut respectively.
Anotker function which is of interest is the inverse m-dimensional FT of the logarithm
of its mth-spectmm. Thus, for m = 2, 3, this function is known as biceptrum and
tricepstrum respectively. Proposed so far are three techniques, the pûrametric ap-
proach, the nonlinear least-squares optimization approach and the polycepstra-based
approach. The second approach is an adaptive one which rninimizes the cost function
derived from higher order statistics. And the third one calculates directly the poly-
cepstra of the received sequence, and uses its result to approximate the coefficients
of the equalizer.
An algorithm using the third approach mentioned above called Tn'cepstmm Equal-
ization Algorithm (TEA) has been proposed by Hatzinakos and Nikias (1991). In this
method, the tricepstrum and trispectrum is calculated. In (41, it is found that tricep-
strum is related to the minimum and maximum phase characteristics of the channel.
Therefore, by calculating the tricepstrum at different values of q, 1, r3, a set of equa-
tions is set up. Solving these equations, the characteristics of the channel can be
found. From the derived characteristics, the coefficients of the equalizer are then
calculated. Thus, any type of equalizer can be useci since channel characteristics and
equalizer's coefficients are found separately. This algorithm can also be implemented
adap tively by adj us ting the maximum and minimum phase charac teris t ics found from
the above procedure using LMS.
The problem with this approach is that large amount of data and high complexity
are involved due to the computation of higher order statistics, especially for the TEA
algorithm. However, TEA does have an advantage over the nonlinear least squared
approach, since the TEA adaptive algorithm guarantees convergence to the absolute
minimum of the cost function.
4.4 Sequence Estimation Approach
Most of the approaches discussed in the previous sections, except for the steepest
descent approach, estimate the channel response first. Then, using this estimate,
the channel effects are inverted to find the transmitted sequence. These methods
are thus, more applicable to situations such as imaging, where channel identification
is necessary [8]. However, for equalization in communication channels, recovering
the transmit ted sequence is more important t han channel identification. Also, some
channels are not identifiable due to the presence of spectral nulls. Therefore, a class
of algorithm which directly estimates the transmitted sequence arises [9, 101.
in the first algorithm [9], the transmitted sequence is estimated through the ex-
amination of the received signal. From this, the second order statistics of the source
are estirnated, and VA is employed to find the data sequence. Before the algorithm is
discussed, several assumptions are made. First, the channel response is assumed to
Se finite with length d symbol i n t e d s . And for N receivers, the channel response
forrns a N x d matrix with f d column r a d ; in other words, it has a finite impulse
response. The input is zero mean and iid. The correlations between symbols from
different receivers and different time instances are zero. The noise is assumed to have
zero mean, and the noise between different receivers are independent. Lastly, the
input and the noise are also independent.
In order to estimate the correlation of input symbols, the diamel ha . to be or-
thogonalized. This is done by Mahalanobis Orthogonalization. For received signal
where T is the transform matrix, which when multiplied by the channel response
matrix gives an orthogonal matrix. Consequently, when no noise is present, the
correlation of the orthogonalized preserves the correlation of the source. Thus, the
correlation of the source can be recovered using ri. Taking noise into account, the To
which gives the optimal input correlation estirnate is,
where d2 is the estimated variance of the noise; and ô2, A:, II,' are found from the
singular value decomposition (SVD) of the correlation of the received signals riri-1.
The whole algorithm is the following [9]. The SVD of &(O)' is computed to fmd
the impulse response dimension estimates d and noise variance estimates B2. With
dl â2, and ri, the optimal transform matrix T', is found. Then the correlation of the
transformed data yi is computed. Using the metric,
where &(i) = ~ L ~ a ~ - k - ~ ; VA is applied to find the transmitted sequence (Fig-
ure 4.2). By simulating this algorithm [9], good performance is obtained. Since the
estimation of the source correlation is simpler than channel identification, this al-
gorithm is less complex. Also, this algorithm applies to both single and multiple
receivers structures. For fading channels, spatial diversity can improve performances
of the receivers. Thus, this multiple receivers structure is very convenient .
Figure 4.2: Block diagram for blind sequence estimation
The second method uses MAP to estimate the transmitted sequence [IO]. Assum-
ing the input alphabet is finite, a channel response is calculated for every possible
sequence. Using MAP of the input sequence as the cost function, the most likely
sequences are selected. To reduce complexity, at each stage, only the most likely
sequences are retained; though, this only gives an approximation.
The mode1 used in this method is a time va,rying channel with additive noise. In
matrix form,
Ri = AiVdF + Ni (4.10)
where R, N, d are the received vector (d x 1), noise vector (d x l), and length of the
channel response; while,
%y MAP, p(a:lri) is computed for a.ll possible data sequences. Then only K sequences
with the largest ~ ( a : l r ~ ) (in other words, most probable) are retained. The computa-
tion repeats. Note that k counts the number of sequences, and takes on a value from
1 to M i where M is the alphabet size.
For this algorithm, convergence and complexity depend on the identifiabili ty of
the channel and the properties of the input. If input is from a finite alphabet, and is
persistently exciting of order 2d - 1, the channel and the data sequence is identifiable.
And the complexity is bounded by the first time instant (that is, Mi0 where to is the
system time delay) if the input is persistently exciting of order d where d is the length
of the channel impulse response [IO]. The performûnce of this algorithm shows fast
convergence, low BER and good tracking proper ties.
4.5 Application
BLind equalization can be applied to transmission monitoring, deblurring of astro-
nomical images, multipoint network communications, echo canceling in wireless tele-
phony, digital radio links over fading channels, and identification of the channel re-
sponse [4,11,12,13]. Among al1 types of blind equalization mentioned above, the ML
method is optimal though its computational complexity is very high due to the use
of VA. Therefore, this algorithm is suitable for channel where the span of ISI is short;
while other approaches, though suboptimal, can deal with Channel with a long span
of ISI. If tracking of carrier phase is required, then the steepest descent method can
provide this tracking along with equalization. Moreover, blind equalization is most
applicable to channels where the transmission of a training sequence is impossible.
By modification, new algorithms can be derived for a p a r t i d a r application. But
so far, blind equalization has only been applied to coherent detection. Thus, in the
following chapters, two blind equalizers, Godard and MCMA, are used with the LMS
algori thm and differential detection to equalize different channels.
Chapter 5
Linear Godard and MCMA
Systems
5.1 Linear Coherent Godard Algorithm
The Godard cost function selected for LMS is independent of carrier phase, and is
the dispersion of order p [14],
where R, is a positive real constant,
Then this cost function is used in the LMS algorithm to adjust the equalizer's coeffi-
cients ci by the following equation,
where p is the stepsize, and VCi is the gradient with respect to ci. One of the most
common application is with p = 2 and is used here.
When Godard is used with the LMS equalizer, the cost function D(2) replaces the
conventional error to be evaluated for the adjustment of tap coefficients [14]. In order
to expand ( 5.3), the gradient V, of the cost function magnitude squared with respect
to tap coefficients has to be evaluated. This gives the necessary error expression ei.
Thus, assuming coherent detection is used, the Godard cost function is,
where âi is the output of the equalizer, and
where ai is the source symbol [14]. Assume that QPSK source is used and it is
followed by a hardlimiter. Therefore, lail is always 1. As a result, the constant R2 in
the cost function is found to be 1. After calcdating the gradient, the error fed into
the equalizer is (see Appendix B.2),
With this expression, the coherent system CG using Godard with LMS is setup in
Figure 5.1. The coefficients c,- are adapted with the following equation,
where p is the step size, ri is the complex conjugate of the received signal in ( 2.2)
and the remaining expression is e; in ( 5.6). This Godard algorithm belongs to
the Stochastic Gradient class (see Section 4.2). Godard is blind in the sense that
QPSK Source a i raiseci Channet & hardimiter cosine g(t) ~ ( t )
1 Slicer
Figure 5.1: Coherent System with Godard for LMS equalizer (CG)
it does not require prior knowledge of the Channel or transmission of any training
sequence. The cost function used is a nonlinear, multimodal function. This implies
that it contains local and global minima. When the equalizer converges, there is a
possibility that it converges to a local rather than global minimum. There are many
papers written on the convergence issue of Godard [15, 5, 161. Simulation by Godard
(1980) shows that this algorithm converges with only an order of magnitude more
iterations than the equalization scheme with a training sequence. And the smailer the
step size, the longer the convergence period. In [14], it is found that the convergence
of the cost function depends on the initial tap values of the equalizer. It must be
setup such that the energy at the output of the equalizer must be sufticient for it to
converge to a global minimum. Therefore, the center tap c, m u t be initialized to a
value greater than the threshold below [14],
where a; is the source symbol and po is a sample of the channel response with the
largest magnitude. By exaaining the cost function, D(*) is dso found to be phase
blind as it takes the absolute squared of the information Ci. In other words, the phase
information is not used when the cost function is evaluated for ( 5.3). And carrier
recovery is independent of the convergence of the system. This blindness leaves the
equalized data with a phase error if phase rotation or frequency offset is introduced
during transmission through the channel. To eliminate these distortions, the carrier
must be tracked with a PLL. Thus, Godard has proposed a joint equalization and
carrier recovery structure in [14]. The carrier phase is tra&d by,
where p+ is the step size, ej = r ; - âi, zi = Ziexp-j" is the equalized output with
phase error correction, and âi is output of the slicer. Next section, an altemate
structure is proposed to compensate this phase blind problem.
5.2 Linear Differential Godard Algorithm
Ins tead of using PLL to correct the residue phase error, two new structures are pro-
posed here using differential detection. To combine Godard with different ial encoding,
two stmctures DG-1 and DG-2 are possible. In DG-1, the output of the equalizer ii is used for the evaluation of Godard cost function before it is fed into the differential
decoder as shown in Figure 5.2. The only clifference between this and Figure 5.1 is
QPSK Source ia i Diff. b i a . Chanel & hdlirnittr encoder cosint g(t) ~ ( t )
A a i
4- Diff g i LMS
S licer r- I
decoder n-1
[cilo
e
Figure 5.2: Godard with Differential detection for LMS equalizer ( D G 1 )
the addition of differential encoder and decoder. At the decoder,
where hi, ai are the input and output of the decoder. Since the differential decoder (a
nonlinear processor) is placed after the calculation of the Godard cost function, the
residue ISI is linear and the error ei is found to be the same as CG'S (see Appendix
B.2), with replacing âi in ( 5.6). That is, for LMS equalizer, the tap coefficients ci
are adjusted as,
where p is the step size and rt is the complex conjugate of the received signal in
( 2.13).
In DG-2, the signals are differentidy decoded before they are used to evaluate
the enor as shown in Figure 5.3. Hence, the decoded symbol ai is used to calculate
QPSK Source & hrudlimitcr
Figure 5.3: Differentid detection with Godard for LMS equdizer (DG-2)
A a i -
the error, and it contains nonlinear residue ISI. The gradient in ( 5.3) has to be re-
evaluated. It is found that the error in ( 5.6 ) , multiplied by the delayed version of A
the equalizer's output, bi - l , forms the new error (see Appendix B.2)'
This is consistent with the conventional linear differential equalizer which multiplies
the conventional error by the delayed equalizer output [l?]. Therefore, equation for
the adjustment of tap coefficients is modified into,
" i
SLiccr
The difference between ( 5.14) and ( 5.7) is the multiplication of bi-L . With these two structures, the phase blind problem of Godard is solved as dif-
ferential detection, which doesn't detect any information fiom the absolute phase, is
used. Therefore, even if 6 # O in ( 2.13), the systems c m still converge. However,
several dBs are traded off for the use of differential detection which introduces error
propagation and noise enhancement. In the simulation, the trade off is determined.
Diff. c n d a
b i
9
;yT-p
r
raised Chruincl cosine g(t) PO)
a i *
Godard
4 .
Diff LMS
decoda n-1
[cib
lowpass "
And the performances of DG-1 and DG-2 are compared to investigate whether the
positioning of the differential decoder is of importance. Since Godard is blind, no
training sequence is required to bring the equalizer into convergence. However at low
SNR, when most of the symbols are in enor, the error propagation and the blindness
of Godard slow down the rate of convergence, and decrease the performance of DG-1
and DG-2. As mentioned in the last section, convergence of the equalizer is related
to the initial tap values. This sensitivity of the cost function to the change in initial
tap values continues to be an issue when differential detection is used. There exists
a threshold for the initial value of the center tap above which the equalizer is able
to converge, same as the CG, as shown in the simulation. Next, the second blind
equalizer, MCMA, is discussed.
5.3 Linear Coherent MCMA
In this algorithm, the Godard cost function is split into real and imaginary parts [Ml.
With the equalizer's output splits into real R;,R and imaginary âcr, the cost function
becomes,
where
To implement into the LMS dgorithm, assuming coherent detection, the gradients
( 3 ) 2 Vci of IDiTRl and 1 DI: l2 are evaluated. Thus, the following error expressions are
found [18] ,
where ai is the output of the equalizer. With this error ei, the LMS equalizer adjusts
its coefficients with ( 5.11). MCMA exhibits similar properties as Godard with less
complexity. Therefore, to setup the coherent MCMA system (CM), the Godard block
in Figure 5.1 is replaced by MCMA block. Both Godard and MCMA are sensitive
to the variation of the initial tap values. Thus, convergence is an important issue,
and depends upon the initial set ting of the tap values. However, contrary to Godard,
MCMA cost function is not phase blind. This is achieved by splitting the Godard cost
function into real and imaginary parts to form the MCMA cost function in ( 5.15).
As a result, it is sensitive to both modulus and phase of the equalizer's output.
Even without the carrier tracking loop, phase recovery is done sirnultaneously with
equalization; though there will be a limitation as to how fast it can track [18]. This
certainly improves over Godard as no PLL is required for carrier recovery. In fact,
MCMA performs just as good as Godard with PLL as shown in [18]. To eliminate the
limitation in carrier tracking, differential MCMA systems are proposed and discussed
next.
5.4 Linear Differential MCMA
When setting MCMA up with differential detection, simila structures as Godard's
are derived. To observe the behavior of differential MCMA, the detector is placed
after and before the calculation of the error. Similar to previous sections, the gradient
of magnitude squared of MCMA cost function is evaluated to form the error ei for
LMS equalizer. The ei are similar to that of the differential Godard systems. Thus,
to setup D M 4 and DM-2, simply replace the Godard block in Figure 5.2 and 5.3
with MCMA. For DM-1, where differential decoder is placed after the calculation of
ei , the data used in calculating ei carries linear residue ISI. The tap coefficients are
adjusted using ( 5.11) with ei in ( 5.17) replacing ai with Bi (output of the equalizer).
For DM-& the symbols bi are decoded into âi before the calculation of ei. As a result,
the residue ISI is nonlinear, and the error ei in ( 5.17) muse be multiplied by a delayed
version of the equalizer's output &i-i.
Just as CM, the two differential systems are sensitive to the wiat ion of the
initial tap values. The center tap must be initialized above a threshold in order for
the equalizer to converge which is confirmed by simulation. Also, the convergence
rate of the equaiizers is slow due to error propagation at the differential decoder.
However, this propagation is only of one symbol duration, and thus, is not severe.
Since MCMA is able to track the carrier by itself, difFerentia1 detection is a redundant
operation. And by switching from coherent to differential detection, several dBs of
SNR are traded off as shown next. In the next section, the simulation results for all
of the above systems are discussed.
1 Sampling frequency 1 19200 bps 1 1 Nurnber of samples per symbol 1 8 ( Baud rate 1 2400 1 1 Raised cosine roll off factor 1 0.5 1 1 Number of taps of raised cosine filter 1 33 1