TRADE-OFF BETWEEN SNR AND ISI IN MULTI-CHANNEL ZERO FORCINGEQUALIZERSTimothy F. Settle, Michael D. Zoltowski, and Venkataramanan Balakrishnan

Purdue UniversityElectrical and Computer Engineering Department

West Lafayette, Indianasettle@ecn.purdue.edu, mikedz@ecn.purdue.edu, ragu@ecn.purdue.edu

ABSTRACT past samples (not symbols) from each channel, encompass-ing a fraction of the mainlobe of the symbol waveform [3].Multi-channel linear equalizers are commonly employed inEqualization with sampling L times per symbol with L > 1digital communication systems to mitigate channel distor-will be referred to as sample-spaced equalization.tions and thereby reduce inter-symbol interference (ISI).

There are several advantages that accrue from the factTraditional multi-channel linear equalizer design for lin-that we have access to multiple channel outputs. For exam-ear time-invariant channels requires synthesis of equalizingple, it is well-known that in the case of a single channel, iffilters such that the frequency response of the overall sys-the channel frequency response contains a null in the base-tem has unit magnitude and linear phase. In this work, weband, a linear equalizer will amplify noise power in theinvestigate the efficacy of deliberately allowing equaliza-vicinity of the null. This problem is commonly referred totion errors, i.e., allowing deviations from unit magnitudeas noise enhancement. However, this is not true with mul-and linear phase of the overall frequency response, withtiple channels; here, noise enhancement arises only whenthe goal of increasing the signal-to-noise ratio (SNR) in thethe channels have a spectral null at or near the same fre-symbol decision statistics. Our approach relies on numeri-quency. Another advantage with multiple channels is thatcal convex optimization based on linear matrix inequalities.equalization can be achieved with FIR filters whose lengthOur investigations reveal that as the allowable equalizationroughly equals that of the channel impulse responses; inerror is increased, the mean-square error (MSE) of the post-contrast, exact equalization of a single channel almost al-equalizer signal constellation decreases despite the attendantways requires an IIR equalizer.increase in ISI. This implies that the trade-off between SNR

Referring to Figure 1, designing a linear equalizer forand ISI in the symbol decision statistic weighs more heavilylinear time-invariant channels requires the equalizing filterin favor of SNR improvement for a surprisingly large rangecoefficients to effect a frequency response for the overallof allowable equalization error.system from s(n) to y(n) which has unit magnitude and1. INTRODUCTIONlinear phase. We however, propose that by deliberately per-To our knowledge, in all previous equalizers proposed formitting equalization errors, i.e., allowing deviations from

narrowband digital communication systems that use Nyquistunit magnitude and linear phase of the overall frequency re-pulses, the decision statistic for a given symbol value is asponse, yields performance gains under certain conditions.linear combination of past and future symbol-spaced sam-In particular we seek to improve the signal-to-noise ratioples. The total number of samples required depends on (SNR) in the symbol decision statistics beyond that achievedthe effective duration of the tails of the symbol waveform.with exact equalization design methods.These samples may be taken across multiple channels which

We will make use of the bit-error-rate (BER) of the sym-may be either different antenna outputs or virtual channels bol decisions as a performance parameter to substantiate ouras synthesized in fractionally-spaced equalization. We will

claims. We demonstrate that as we increase the allowablerefer to this type of equalization as symbol-spaced equal-

equalization error, an attendant improvement in the BERization.of the symbol decisions is achieved. In other words, byIn many scenarios the multi-path delay spread, , is a permitting an increase in ISI we are able to achieve a sig-fraction of the symbol time interval. In some cases, over-nificant decrease in the power of the additive noise in the

sampling permits one to compute the equalized decisionsymbol decision statistic, while having negligible impact on

statistic for a given symbol value from a small number ofthe signal power in the symbol decision statistic.This research work is supported by the National Science Foundation

under grant no. MIPS-9708309.1

2. PROBLEM FORMULATION 2.2. ZERO FORCING EQUALIZATIONFor sample-spaced zero-forcing equalizers, exact equaliza-2.1. SIGNAL MODELtion requires the g (n)s to satisfy the equationi

N

c

X

h (n) g (n) = (nD); (2)RFi i K

i=1

where () is the Kronecker delta function. This relation-K

ship implies that only a fixed time delay is imposed on thetransmitted symbol waveform sequence and therefore ISIis eliminated given that we are using square-root Nyquistpulses p () for the symbol waveform.sr

Due to the fact that we are sampling with L d2(1 +)e, we must down-sample the outputs of the matched fil-ters by L in order to permit symbol-spaced processing forsymbol-spaced equalizer design. Assuming sample-spacedestimates of the channel impulse responses are available,symbol-spaced (aliased) channels may be constructed as fol-lows,

h (n) = p (n) h (n) p (n);

i sr RFi

sr

a

h (n) = h (nL):

i

iFigure 1: Multi-Channel communication model.In the above development, the frequency response of eachConsider the baseband multi-channel model for a digi-h (n) is band-limited to the frequency response of p(n) =ital communication system shown in Figure 1, where each

p (n) p (n), which has a raised cosine spectrum withsr

srof the N channels is realized through a separate antennac

excess bandwidth parameter . The symbol-spaced chan-element at the receiver. For the model shown in Figure 1,anels, h (n), are the down-sampled versions of the corre-ithe sampled version of the transmitted signal is

sponding h (n)s. Exact equalization for a symbol-spacedi

1

X equalizer would then require the g (n)s to satisfy,i

s(n) = b(k)p (n kL): (1)sr

N

c

k=1

X

a

h (n) g (n) = (nD): (3)i K

i

The symbol time interval is designated as T and b(k) isi=1

a sequence of discrete information-bearing symbols. As-sociated with each information symbol is the transmitted 2.3. SNR OF THE SYMBOL DECISION STATISTICwaveform p (n) having a square-root raised cosine spec-sr The SNR of the symbol decision statistic can be written in

trum with excess bandwidth parameter . The sampling the following forms for sample-spaced and symbol-spacedinterval is T = T =L. In equation (1), L is a positive in-s equalization respectively [2]:

teger which dictates the size of the sampling interval. Note2that Nyquist sampling would require L d2(1 + )e.Efjb(k) + (m)j g

; (4)PThe discrete-time channels are modeled as FIR filters Nc

H

g R g

vv i

i

i=1having impulse responses h (n), i = 1; : : : ; N , each ofRFi c

2length N [3]. The received signal at each antenna elementh

Efjb(k) + (m)j g

: (5)Pis corrupted by an additive zero-mean white noise processN

c

H

g g

i

i

i=1

u (n), with u (n) and u (n) being uncorrelated when i6= j.i i j

In (4) and (5), scales the information symbol and the termThe output of each antenna is passed through a matched

(m) is the residual ISI. Since b(k) is constant, if we canfilter having an impulse response p (n). In turn, thesr

constrain 1 and (m) 0 in (4) or (5), then it followsmatched-filter outputs are passed through a bank of FIRthat minimizing the denominator of (4) or (5) improves theequalizing filters having respective impulse responses g (n),i

SNR of the symbol decision statistic.i = 1; : : : ; N , of length N . These outputs are summedc g

to generate a signal that is sampled at the symbol rate to 2.4. OPTIMIZATION FRAMEWORKgenerate the symbol decision statistics. In this section and the sections to follow, we will refer to

two well known metrics from system theory. They are theH norm and the H norm. The book [1], outlines a1 2

g n1( )

p nsr( )Modulators n( )

u nNc( )

u n2

( )

u n1( ) y n( )

p n*

sr(- )

p n*

sr(- )

p n*

sr(- ) g nNc( )

g n2( )

h nRFNc( )

h nRF2( )

h nRF1( )

x n( )

r n1( )

r n2( )

r nNc( )

Transmitter

Channels

Receiver

rigorous definition and treatment of these norms, the reader with C and D being affine functions of the filter coef-E E

is referred to it for more details. ficients of g (n); : : : ; g (n).1 N

c

In line with the comments made in Section 1, we will From standard results in system theory [1, x2.7.3] thedeliberately allow errors in equalization. To this end, we constraint in Problem (6), i.e., kE(z)k ", can be shown1

P

N

c

D Hdefine E(z) = z H (z)G (z) as the equal- to be equivalent to the LMI, in P = P ,RFi i E

E

i=1

ization error transfer function for sample-spaced equalizer2 3

P

H H H

N

c

D a

A P A P A P B C

E E E E Edesign and E(z) = z H (z)G (z) for symbol-E E E

i

i

i=1

H H 2 H

4 5

B P A B P B " I D

0:

E E E Espaced design. We will measure the size of E(z) using itsE E E

C D I

E E

H norm, i.e., the size of the equalization error will sim-1 (8)ply be the maximum magnitude of the equalization error

We denote the matrix in (8) as Q(P ;C ;D ).E E Efrequency response. Constraining kEk imposes a bound1

The objective function, f(g ; : : : ; g ), in (6) may be1 N

con jE(!)j uniformly over frequencies. With these prelimi-rewritten in a form suitable for LMI optimization as fol-naries, we consider the following optimization problem.Tlows. Let g = [D j C ] . By introducing a slacki G G

i i

P

N

c

H variable , we can minimize f(g ; : : : ; g ) by requiring1 NMinimize: f(g ; : : : ; g ) = g R gc

1 N vv i

c

i

i=1 (6) f(g ; : : : ; g ) and minimizing . From standard1 NSubject to: kEk ";c

1

results in system theory [1, x10.1.1], the constraint with the optimization variables being the coefficients of thef(g ; : : : ; g ) can be shown to be equivalent to the LMI,1 N

c

equalizing filter impulse responses.2 3

[D j C ] [D j C ]

G G G GThe above optimization problem will yield a set of equal-1 1 N N

c c

H

6 7

Dizing filters with the lowest weighted H norm (sample-1

G

2 1

6 7

R 0

H

vv

6 7

Cspaced design) or H norm (symbol-spaced design) for aG

2

1

6 7

0:. . .6 7.given delay D, equalization error bound ", and equalizer . . . .6 7.. . .

6 7

filter length N . We will study the interaction betweenH

g

4 5

D

G

1

N

c

0 Rthese parameters with the MSE of the post-equalized sig-H

vv

C

G

N

cnal constellation and the BER of the symbol decisions as a (9)measure of performance. We will denote the matrix in (9) as F(;C ;D ). NoteG G

i i

2.5. LMI FORMULATION that this same matrix can be used for symbol-spaced designaThe numerical solution of Problem (6) is through convex by replacingR withR and using the equalization errorvv

vv

optimization based on linear matrix inequalities (LMIs). function defined for symbol-spaced equalizer design.LMIs are convex constraints requiring an affine combina- The matrices F() and Q() are affine functions of PE

tion of symmetric matrices to be positive-definite. LMI- and the equalizer filter coefficients. We may now write ourbased optimization has come to be recognized as a valu- original multi-channel equalizer optimization problem as aable tool in the numerical solution of problems from system LMI optimization problem.and control theory. There are effective and powerful algo-

Minimize: rithms for the solution of LMI optimization problems, i.e.,HSubject to: P = PE

Ealgorithms that rapidly compute the global optimum, with (10)F(;C ;D ) 0

G G

i inon-heuristic stopping criteria. We refer the reader to theQ(P ;C ;D ) 0:

E E Ebook [1] and the references therein for details.The transformation of Problem (6) as an LMI optimiza- This problem can be very efficiently solved using stan-

tion problem requires a suitable state-space description of dard numerical techniques [1]. In this section we devel-the various blocks in our communication system. It turns

oped the design equation for a sample-spaced zero-forcingout that for the FIR filters in our system, the appropriate

equalizer based on LMI optimization. The simulation re-state-space realization (A;B;C;D), is such thatA is nilpo-

sults in the next section will compare five versions of thetent, B is a column vector with a one in the first position,

zero-forcing equalizer. These versions are: Sample-Spacedand the C and D matrices are constructed as follows, Minimum-Norm (Sas-ZF-MN), Sample-Spaced LMI (SaS-

ZF-LMI, developed in this section), Sample-Spaced Win-C = [h ; : : : ; h ];

1 N 1

g (7) dowed LMI (SaS-ZF-WLMI), Symbol-Spaced Minimum-D = h ;

0

Norm (SyS-ZF-MN), Symbol-Spaced LMI (SyS-ZF-LMI).Each of the design methods based on the minimum-normwhere fh ; h ; : : : ; h g are the impulse response coeffi-0 1 N1

solution yield zero-forcing equalizers which achieve exactcients. Given state-space realizations for H (z), G (z), andi i

D equalization, i.e., the equalization error is zero. Whereasz of the form shown in (7), it is straightforward to write a

the LMI based designs deliberately allow a non-zero equal-state-space realization for E(z) with (A ;B ;C ;D ),E E E E

ization error. The reader should consult [2] for details onthe other design methods not covered in this paper.

3. SIMULATION RESULTSWe now compare the five different design methodologieslisted in Section 2.5, using 16-QAM modulation. For eachsimulation the symbol rate is 40 KHz, L = 5, i.e., T =s

5s, the symbol waveform has the square-root raised cosinespectrum with = 0:35, the sample-spaced channels aremodeled as a three tap FIR filters with unity energy gain andthe symbol-spaced channels are modeled as seven tap FIRfilters. Two channels are used in all simulations (N = 2).c

For the sample-spaced designs the length of each equalizeris three and for the symbol-spaced designs each equalizerhas a length of seven. The design parameter D is chosensuch thatD = ((N +N 1)=2), where () is a functionh g

which rounds its argument to the nearest integer. Our resultsare summarized in Figures 3, 4, and 5 shown below.

Figure 4: Equalized Spectrums.

a aFigure 2: H , H , H , andH Frequency Responses.RF1 RF2

1 2

Figure 5: BER Curves for 16-QAM Modulation.

[2] T. F. Settle, M. D. Zoltowski, and V. Balakrishnan.Design of Multi-Channel Equalizers via Linear Ma-trix Inequalites: Trade-Off Between SNR and ISI. InProc. University of California San Diego Conferenceon Wireless Communications with IEEE Communica-tions Society, San Diego, CA., March 1998. Acceptedfor publication.

[3] T. A. Thomas and M. D. Zoltowski. Space-TimeProcessing for Interference Cancellation and Equaliza-tion in Narrowband Digital Communications. In Proc.IEEE Vehicular Technology Conference, pages 160164, Pheonix, AZ, May 1997.Figure 3: MSE versus ".

4. REFERENCES[1] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrish-

nan. Linear Matrix Inequalities in System and ControlTheory, volume 15 of Studies in Applied Mathematics.SIAM, Philadelphia, PA, 1994.

8 6 4 2 0 2 4 6 8 10x 104

0.5

1

1.5Equalized Spectrum

Frequency in Hz

Mag

nitu

de

SaSZFMN SaSZFLMI SaSZFWLMI

1.5 1 0.5 0 0.5 1 1.5 2x 104

0.8

0.9

1

1.1

1.2

Equalized Spectrum

Frequency in Hz

Mag

nitu

de

SySZFMN SySZFLMI

3 2 1 0 1 2 3x 104

8

6

4

2

0

2HRF1 and HRF2 Frequency Responses

Frequency in Hz

Magn

itude

in dB

HRF1HRF2

2 1.5 1 0.5 0 0.5 1 1.5 2x 104

8

6

4

2

0

2Ha1 and H

a2 Frequency Responses

Frequency in Hz

Magn

itude

in dB

Ha1Ha2

10 15 20 25 30 35108

107

106

105

104

103

102

101

100BER versus SNR, (16QAM)

SNR (dB)

BER

SaSZFMN SaSZFLMI SySZFMN SaSZFWLMISySZFLMI

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3513.5

13

12.5

12

11.5

11

10.5

10

9.5

9MSE versus

e

MSE

(dB)

SaSZFMN SaSZFLMISySZFMN SaSZFWLMISySZFLMI