As an alternative to existing techniques and algorithms, we investigate the merit of the H-infinity approach to the equalization of communication channels. We first look at causal H-infinity equalization problem and then look at the improvement due to finite delay. By introducing the risk sensitive property, we compare the average performance of the central H-infinity equalizer with the MMSE equalizer in equalizing minimum phase channels

Erdogan, Alper T.

Hassibi, Babak

Kailath, Thomas

Caltech Authors - Main

H” EQUALIZATION OF COMMUNICATION CHANNELS 
Alper T. Erdogan, Babak Hassibi and Thomas Kailath 
Information Systems Laboratory, Electrical Engineering Department, 
Stanford University, 
CA 94305 
ABSTRACT 
As an alternative to existing techniques and algorithms, we inves- 
tigate the merit of the H-infinity approach to the equalization of 
communication channels. We first look at causal H-infinity equal- 
ization problem and then look at the improvement due to finite 
delay. By introducing the risk sensitive property, we compare the 
average performance of the central H-infinity equalizer with the 
MMSE equalizer in equalizing minimum phase channels. 
1. INTRODUCTION 
Equalization is a well studied problem in the area of communica- 
tions. It can be considered as a special case of an estimation prob- 
lem with the data model generally described by a linear model of 
the type shown in Figure 1. The discrete data sequence { b ; }  passes 
through the linear time-invariant channel H ( z ) ,  which causes inter- 
symbol interference (ISI). The observation sequence {y;} is then 
formed by the addition of an unknown measurement disturbance 
{vi} with the output of the communication channel H ( z ) .  The 
purpose is to design an equalizer K ( z )  which estimates bi-d from 
the observations {yi}, where d 2 0 represents a possible delay. 
IVi n 
b, Y 
H K 
CHANNEL NOISE EQUALIZER DECISION 
Figure I : Linear Data Model 
Various structures and methods have been proposed to recover 
transmitted data from their filtered and noise corrupted versions, 
and each method has its own advantages and disadvantages in 
terms of performance and complexity [5 ] .  All these current tech- 
niques make some assumption about the underlying statistics and 
structure of the model. In many applications, however, the true in- 
formation about the model is not available, and the algorithms use 
some estimate of the model parameters. For example, in mobile 
communications, the channel (and other statistical) parameters are 
often estimated from the observations through the use of certain 
This work W~LF supported in part by DARPA through the Department 
of Air Force under contract F49620-95-1-052S-P0000l and by the Joint 
Service Electronics Program at Stanford under contract DAAH04-94-G- 
00s 8- Poooo3. 
training sequences and therefore always contain errors. Time vari- 
ation of the parameters and the errors due to tracking is another 
important issue. Therefore the question is whether small varia- 
tions from the true model, and small disturbances, can cause large 
degradation in the performance of the algorithms. This brings us 
to the issue of “robustness”. 
In this paper, we address the robustness question by approach- 
ing the equalization problem from the H” estimationJ4, 8, 71 
point of view. The richness of robust H”” theory, and especially its 
stochastic interpretation of risk sensitive estimation, has been the 
basic motivation for our approach. Moreover, the availability of 
fast algorithms [4] is another major driving force for looking into 
H” estimation as an equalization alternative. Finally, and per- 
haps most importantly, the results obtained in this attempt provide 
us with a new and different perspective for the understanding and 
analysis of the equalization problem, as well as for H” estimation 
itself. 
2. H” AND RISK SENSITIVE ESTIMATION 
V 
Figure 2: Setup for linear estimation 
2.1. H” Estimation 
The basic setup for a general linear estimation problem is illus- 
trated in Figure 2. In this setup we assume that H ( z )  and L(z )  
are causal linear time-invariant filters that map the input sequence 
{ b i }  to their respective outputs. The driving input {b;} and the 
additive disturbance sequence {vi} are assumed to be unknown. 
The estimation problem is to design a causal linear time-invariant 
estimator K ( z )  that estimates s,, the unobservable output of L(z) ,  
using the observations {yj, j 5 i}. We will denote such estimates 
by B;l, and the resulting estimation errors by ail; = si - &I,. 
Moreover, let TK (z) denote the transfer matrix that maps the un- 
known disturbances {bi} and { v i }  to the estimation errors {&I;}. 
0-7803-4428-6/98 $10.00 0 1998 IEEE 3489 
The choice of K ( z ) ,  and thereby the estimates l i I i ,  depends 
on our choice of performance criterion. In Hw estimation K ( z )  is 
chosen to minimize the maximum error gain of TK(z), also known 
as the Hm norm of TK defined as 
Problem 1 (Optimal H" Filtering Problem) Find a causal es- 
timator K ( z )  that satisfies infK(,) IITK(z)~~&. MoreoveGfnd the 
min-mar energy gain TZpt. 
There are very few cases where a closed-form solution to the 
optimal H" filtering problem can be found, and in general one 
relaxes the minimization and settles for a suboptimal solution. 
Problem 2 (Suboptimal H w  Filtering Problem) Given -y > 0, 
find, $possible, a causal estimator K ( z )  that guaranlees 
l lTK(Z) l lz ,  I TZ (3) 
This clearly requires checking whether y > yopt.  
It will now be useful to give some flavor of the solution to 
Problem 2. (See [4] for more details). We introduce the following 
so-called Popov function, 
-qL(z)H'(z- ' )  - 7 2  +qL(z)L*(z-*) ' 1 T + g H ( Z ) H * ( Z - * )  -qH(Z)L'(t-') C ( z )  = 
which can be regarded as a certain indefinite generalization of the 
spectral density function, r + qH(z)H'(z- ' ) .  Then a causal es- 
timator, K ( z ) ,  that achieves ~~TK(z)II < y exists if, and only if, 
the Popov function admits a canonicalyactorization of the form 
[ L1l(z) L12(z) ] and L l l ( z )  causal and causally in- 
vertible, and L I Z ( Z )  strictly causal. If this is the case, then all 
with LZl(t) Lzz ( z )  
possible Hm estimators of level 7 are given by 
K ( z )  = (Lzz(z)&(z) - L z i ( z ) )  (Lii(z) - Liz(z)Q(z))-' , 
( 5 )  
where Q(z) is any causal and strictly contractive operator, i.e., 
Q ( z )  is causal and is such that IQ(e3W)12 < 1, for all w E [O, 274 
An important choice results from taking Q = 0, so that 
Kcxn(z) = - ~ l  (~)L;, '(z)  (6) 
which is the so-called "central" filter. 
2.2. Risk Sensitive Estimation 
Although the aforementioned H" estimation formulation is a de- 
terministic one, it has a nice stochastic interpretation which we 
now describe. 
Given the basic model in Figure 2, in risk sensitive filtering, [6, 
91, we assume that the disturbances { b ; }  and {U;} are stationary 
independent Gaussian random processes with variances q and r ,  
respectively. The risk sensitive filtering problem is to find a causal 
K ( z )  that minimizes 
where 0 > 0 is known as the risk-sensitivity parameter. 
The cost function 7 shows that as we increase the value of B 
we put more penalty on large values of error as compared to the 
MMSE estimator, which minimizes E(x&, 1s; - .?;Iz). How- 
ever, the 0 parameter can not be made arbitrarily large. In [2] i t  was 
shown that for any 0 5 -&, the causal K ( z )  that minimizes the 
risk-sensitive cost function is given by the central H" filter corre- 
sponding to 7 = &and with energy weights q and T equal to the 
variances of { b i }  and {U,}, respectively. In the equalization appli- 
cation, only the large values of error, which are greater than the 
threshold for detection, are important. Errors below the threshold 
do not play a role. Therefore one may expect that the risk sensitive 
criterion is a good choice for equalization since it penalizes high 
errors more severely than the MMSE criterion. Nonetheless, as 
illustrated in future sections, it has certain drawbacks. 
7,pr 
3. H" EQUALIZERS 
3.1. The Causal Case 
The equalization problem of Section 1 is a special case of the linear 
estimation setup with L ( z )  = z - ~ .  When d = 0, the equalizer is 
constrained to be causal. In [3], the factorization (4) was explicitly 
obtained and thereby a characterization for all H" equalizers was 
derived. The main results can be summarized as follows. 
If the channel H ( z )  is non-minimum phase: We have 
7:pt = 9,  (8) 
which is the same energy gain obtained from K ( z )  = 0, 
i.e., not equalizing at all! Therefore there is no hope for 
causally equalizing a non-minimum phase channel. 
If the channel H ( z )  is minimum phase: we have 
where Y~pt ,amoothrng is the minimax energy gain for the 
"optimal" smoothing filter which turns out to be the cele- 
brated Wiener smoother. This implies that one can perform 
as well as the non-causal (smoothing) solution! 
In the minimum phase case, in [3], it is shown that the central 
equalizer is given by 
3490 
where thc monic and minimum phase transfer function A(%) 
and the scalar R A  are found from the standard spectral factoriza- 
tion 
We should also remark that another H" optimal equalizer is 
K ( t )  = ( l -&,t)H-l(z) ,  whichissimply ascaled versionofthe 
zero-forcing equalizer. Thus, an appropriately scaled zero-forcing 
equalizer is Hm-optimal. 
Comparison of the error spectra in Figure 3 illustrates that the 
smoother outperfroms all other equalizers at every frequency. The 
causal MMSE ( H ' )  filter has the best average performance among 
the causal equalizers, however the peak value of its spectrum is 
greater than the others. The risk sensitive and the scaled zero forc- 
ing equalizers have peak spectra equal to the smoother, but the risk 
sensitive one has better average performance. 
::. 
0 3 1 - 5  , 
I 
o o s  I 1 5  z z s  a 3 5  0 2  
Figure 3: Comparison of the Error Spectra for the channel H ( t )  = 
1 + .7t-' 
3.2. Finite Delay Case 
I t  is clear from the previous section that we need non-causal equal- 
izer structures to equalize non-minimum phase channels, which is 
equivalent to have finite delay, i.e. d > 0. 
In order to illustrate the effect of delay, it will be instructive 
to look at the special case of equalizing the single zero channel 
H ( z )  = 1 + az-I, a E R. 
We know from Section 3.1 that when d = 0, 
Delay=l: After some algebraic manipulation, the Popov func- 
tion for this case simplifies to 
To attempt to factorize E(%) as in Equation 4 consider the follow- 
ing factorization of the Popov function 
( 1  3) 
observe that the center matrix takes the form of the Popov function 
for the d = 0 case. Indeed by making the following substitutions, 
I?(%) = - (a + g z - ' ) .  i ( z )  = 1 = to ,e = $ and Q = -$ we 
can rewrite the Popov function as 
f +  BA(z)A'(z-') - q A ( z ) L * ( z - * )  
(14) 
It thus follows that we need to distinguish between the two cases 
where I?(%) is minimum phase and where I?(%) is non-minimum 
phase. When I?(%) is minimum phase, which is the case for la1 < 
2. we have 
which after some simplification becomes 
When I?(z) is non-minimum phase, which is the case for la1 2 2, 
Figure 4 shows the optimal value of 7 as a function of a. 
I \ 
- - _ _  - - _  
0 2  O** 10 t 2 3 4 s - -  a 7 0 I O  
Figure 4: Optimal y plot for single delay case. Dashed line refers 
to smoothing value 
Delay > 2: I t  can be similarly shown [ l ]  that, for d 2 2, 
y:,t = Y~pt,smooth,ng. i.e., a delay of two units is sufficient to 
obtain the same H" performance as the smoother in equalizing a 
single-zero channel. 
Unfortunately, there is no known explicit factorization for ar- 
bitrary d > 0 and for general non-minimum phase channels. How- 
ever, in [3] it has been shown that in order to get an improvement 
over yopt = 1, the delay d should be chosen greater than the num- 
ber of non-minimum phase zeros of the channel, i.e., the number 
of zeros outside of the unit circle. 
349 1 
4. COMPARISON OF CENTRAL H" AND H 2  
EQUALIZERS IN EQUALIZING MINIMUM PHASE 
CHANNELS 
In  this section of the paper, we will compare the central Hm and 
the H 2  equalizers in terms of average BERs. Figure 5 shows that 
the H 2  and central H" equalizers have the same average BER 
performances for H ( a )  = 1 + 0.5t-'. 
$ 1 6  
IO4 
1 4  5 6  7 I 9 1 0 1 1 1 2 1 3  
w 
d 
Figure 5: BER vs SNR curve for the channel H ( z )  = 1 + 0.5r-': 
Solid line-Risk Sensitive Equalizer, Dashed Line-MMSE Equal- 
izer 
However as illustrated in Figure 6, the H2 equalizer has better 
average BER performance than the central HO" equalizer for the 
channel H ( t )  = 1 + 0.95~- ' .  
I I 
Figure 6: BER vs SNR curve for the channel H ( t )  = 1+0.95t-': 
Solid line-Risk Sensitive Equalizer, Dashed Line-MMSE Equal- 
izer 
In general, for the various channels that we have studied, when 
H ( z )  and the signal statistics are known exactly, the risk sensitive 
and MMSE equalizers have either similar BERs, or the MMSE 
equalizer outperforms the risk sensitive one. It thus may appear 
that, there is no gain in  using central H" equalizers, compared 
to MMSE ones, in the ideal setup. However, in the face of model 
uncertianity and lack of statistical knowledge, i t  is expected that 
the H" equalizer will have acceptable performance H 2  equalizer 
[]I. 
5. CONCLUSION 
In this paper, we introduced the HO" criterion as an alternative 
method for the equalization of communication channels, which 
concentrates on the worst case performance whereas the previous 
algorithms concentrate on the average performance. We showed 
that causal HO" filter has same worst-case performance as the non- 
causal smoothing filter in equalizing minimum phase channels. 
For non-minimum phase channels we need a number of delays 
at least equal to the number of non-minimum phase zeros of the 
channel. Therefore, study of HO" estimation provides us with 
a rigorous basis for the importance of the concepts of minimum 
phase channels and delay in the equalization problem. 
We looked at the central H", or risk sensitive, filter as a 
choice which has good average performance besides its optimal 
worst case performance . We showed that it has good average 
properties due to its stochastic interpretation although it does not 
appear to be better than the MMSE equalizer in terms of BER in 
the ideal case. 
Formulation of equalizers for more general case and the per- 
formance under modeling errors is the area that we are currently 
pursuing. 
6. REFERENCES 
A. T. ERDOGAN, B. HASSIBI, AND T. KAILATH, On H" 
Equalization of Communication Channels, submitted to IEEE 
Tr. SP., (1997). 
K. GLOVER AND J. DOYLE, State-space formulae for all sta- 
bilizing controllers that satisfy an H" norm bound and rela- 
tions to risk sensitivity, System and Control Letters, I 1 (1 988), 
B. HASSIBJ, A. T. ERDOGAN, AND T. KAILATH, Equaliza- 
tion with an H m  Criterion, submitted to IEEE Tr.Inf.Theory, 
(1997). 
B. HASSIBI, A. SAYED, AND T. KAILATH, Indejnite 
Quadratic Estimation and Control: A Unified Approach to H 2  
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New York, 1998. 
J . G. P ROA KIS, DIGITAL COMMUNICATIONS. McGR AW- 
HILL, 1995. 
J.L.SPEYER, J. DEYST, AND D. JACOBSON, Optimization 
of stochastic linear systems with d i t i v e  measurement and 
process noise using exponential performance criteria, IEEE 
Transactions on Automatic Control, I9 ( I  974). pp. 358-366. 
P. KHARGONEKAR ND K. NAGPAL, Filtering and smooth- 
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trol, 36 (1991), pp. 151-166. 
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3492 


H^∞ equalization of communication channels

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H^∞ equalization of communication channels

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