Training sequence and memory length selection for space-time Viterbi equalization

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Training Sequence and Memory Length Selection

for Space-Time Viterbi Equalization

Chih-Sheng Chou and David W. Lin

Abstract: We consider signal and receiver design for space-time

Viterbi equalization for wireless transmission. We propose a search method to find good training sequences, termed min-norm train-ing sequences, for least-square channel estimation. Compared to either a maximum-length sequence or a randomly generated train-ing sequence, the traintrain-ing sequence obtained can drastically re-duce the channel estimation error. We also derive a simple lower bound on the achievable channel estimation error of any training sequence. Knowledge of this lower bound helps the search for min-norm training sequences in that it facilitates a measure of the good-ness of the best sequence examined so far. For operation under the situation with unknown channel response lengths, we propose a simple method to select the memory length (tap number) in the Viterbi equalizer based on the SNR of the received signal. The re-sulting equalization performance is found to be comparable with the case where a preset, fixed memory length is used. However, the proposed method often results in use of a smaller tap number, which translates into a reduction in the computational

complex-ity. Simulation results show that at symbol error rate below10

2

(SNR>5dB) the amount of complexity reduction is of the order

of5%to25%on the average, for typical wireless channels.

Index Terms: Wireless communication, space-time signal

process-ing, Viterbi equalization, decision-feedback sequence estimation, channel estimation, training sequence design, channel length selec-tion.

I. INTRODUCTION

Wireless communication systems are evolving towards higher data rates, which experience more adverse channel effects than at the lower rates of earlier systems. On the other hand, space-time signal processing is known to be able to improve the wire-less transmission performance and has attracted much recent at-tention [1], [2]. In this paper, we consider signal and receiver design for space-time Viterbi equalization in high-speed mo-bile communication. For simplicity, we consider nonspread-spectrum transmission, although the fundamental architecture could be augmented to address spread-spectrum transmission.

Fig. 1 illustrates the structure of the transmission system, where the space-time Viterbi equalizer is composed of a vector channel estimator and a vector-channel Viterbi sequence estima-tor. The system employs a training sequence for the equalizer’s use, but the receiver does not have to know the received signal’s

Manuscript received July 28, 2000.

The authors are with the Department of Electronics Engineering, National Chiao-Tung University, Hsinchu, Taiwan, e-mail: cschou@mail2000.com.tw.

This work was supported by the National Science Council of R.O.C. under grant NSC 89-2219-E-009-013.

1229-2370/00/$10.00 c 2000 KICS

DOA (direction of arrival) or the array manifold vector [1], [3] (also known as the array propagation vector [4]). The adaptabil-ity of the antenna array consists in the Viterbi equalizer’s abiladaptabil-ity to adapt its channel model through the channel estimation unit.

First, we consider the design of the training sequence for channel estimation. A search method which combines exhaus-tive search and a Newton-like algorithm for finding the training sequence is proposed. By this search method, we can get a rea-sonably good training sequence. A simple lower bound on the achievable channel estimation error of any training sequence is also derived. Secondly, we consider using dynamic tap num-ber selection in the Viterbi equalizer according to the channel SNR condition. Simulation results show that the approach can offer significant reduction in the receiver complexity while not sacrificing transmission performance.

In Fig. 1, leta i be the

i-th transmitted (baseband) symbols, p(t)the pulse shape,u(t)the impulse response of the receiver’s front-end filters, M the number of elements in the receiver’s antenna array, andb(t)the vector channel impulse response. We have x (t)= 1 X i= 1 a i p(t iT)b(t)u(t)]+n(t) 4 = 1 X i= 1 a i r(t iT)+n(t) (1)

whereT is the symbol period andr(t)is the combined impulse response of the pulse shaping filter, the vector channel, and the receiver’s front-end filters. Upon sampling ofx(t), we obtain

x(k)=R a(k)+n(k) (2)

where

R= r(kT)r( k+1]T)r( k+2]T)] (3) i.e., the overall channel impulse response matrix, and

a(k)= a k a k 1 a k 2 ] 0 (4) where 0

denotes the matrix transpose operation. For conve-nience, we let each row ofRhave unit energy; that is, the sum of squared values of each row ofRis equal to one.

Consider tentatively the transmission of the training sequence only. Let the training sequence have lengthL. Let q be the length of the sampled vector impulse responser(kT), that is, q is the number of columns inR. Then the center section of the received data, where there is full convolution between the

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Channel Estimation i Up_Convert . . . . . . b(t) x(t)

a

i Up_Convert . . . . . . b(t) x(t) Down Convert Down Convert Down Convert ... x(k) _. _. _. x(k) . . . . . . p(t) #M #1 p(t) #M #1 u(t) u(t) Vector Viterbi Sequence Estimator

Space-time Viterbi Equalizer

Detected symbols

a

Fig. 1. A wireless transmission system with space-time Viterbi equalization.

overall channel impulse response and the training sequence, has lengthL q

+1

. And we may express it in a matrix form as [1]

X

=

R G

+

N

=

2 6 6 6 4 x 11 x 12 x 1L q+1 x 21 x 22 x 2L q+1 .. . ... .. . x M1 x M2 x ML q+1 3 7 7 7 5 (5)

whereGis the training data matrix given by G

=

g 0 1 g 0 2 g 0 q

]

0 (6) with g i

=

a q i+1 a q i+2 a L i+1

]

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andN is a matrix of noise samples. We assume the noise into the different antennas to be zero-mean white Gaussian, uncorre-lated, and with an identical variance

2 .

In what follows, Section II characterizes the training sequence for least-square channel estimation. It also derives a simple lower bound on the estimation error for any training sequence. It then proposes a search method to find training sequences yield-ing the minimum estimation error (called the min-norm trainyield-ing sequences). Section III presents an algorithm to dynamically select the memory length (tap number) in the Viterbi equalizer according to the SNR condition. Section IV presents some sim-ulation results and Section V gives the conclusion.

II. MIN-NORM TRAINING SEQUENCES

Let R

^

denote the channel response estimate. Given the re-ceived dataX and the training data matrixG, consider least-square estimate of the channel response as

min

^ R kX RGk

^

2 F (8) wherekk

1 k 2 F : (10) We now show that minimization of the mean-square estimation error is equivalent to minimization of the following quantity:

q X i=1

1

j i j 2 (11) where

iare the singular values of G.

For this, note that the least-square estimation error is given by kR

^

R k 2 F

=

kNG H

(

GG H

)

1 k 2 F

=

trf

NG H

(

GG H

)

1

]

)

=

M 2

I (where I denotes an identity ma-trix) by the earlier uncorrelated AWGN assumption. There-fore, the mean-square estimation error is proportional to the sum of the eigenvalues of

(

GG

H

)

1

. But since the eigenvalues of

(

GG H

)

1

are equal to the inverses of the eigenvalues ofGG H

, or equivalently the inverses of the squared magnitudes of the sin-gular values ofG, minimization of the mean-square estimation error requires minimization of the sum-square inverse singular value (11) as claimed above.

For convenience, term the sum (11) as the normalized error

norm and term the training sequences yielding the minimum

normalized error norm as min-norm training sequences. We have the following result.

Theorem 1: Let each symbol in the training sequence have

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havenrows and`columns (where in the earlier discussion we hadn

=

qand`

=

L q

+ 1

). Then the normalized error norm (11) is lower bounded byn=`.

Proof: Since each symbol in the training sequence has

unit magnitude and the dimension ofGisn`, the diagonal elements ofGG

H

are all equal to`andtrfGG H

g

n X i=1 j i j 2 (17) whereis the Lagrange multiplier. Taking the partial deriva-tives ofy with respect to j

i

j, i

= 1

n, and setting the results equal to

0

, we get

@y @j i j

= 2

j i j 3

+ 2

j i j

= 0

8i: (18) Hence j 1 j 4

=

j 2 j 4

=

j n j 4

= 1

: (19)

=

`, then the rhs is equal to unity. 2 SinceGis Toeplitz in actual systems,j

i

jare not independent variables as assumed in the foregoing proof. For common mod-ulation schemes (BPSK, QPSK, etc.), furthermore, they are not even continuous variables. Therefore, the lower bound may not be reachable in actual systems. The reachable minimum value under these conditions is yet unknown and subject to further study. It is also interesting to note that the foregoing results on optimal training sequences in the uncorrelated AWGN environ-ment do not depend on the number of antennas. Therefore, an optimal training sequence is optimal for an array-based receiver regardless of the number of antenna branches in the array, as long as the length of channel response stays the same.

Change Set i=1

Set Sequence 0 = Sequence n to a value which yields the smallest

normalized error norm

elements of Sequence i-1

New sequence is labeled as Sequence i

i = i + 1 i == n ? NO YES NO NO YES YES

Sequence n is better than current best sequence? Sequence n == Sequence 0 ?

Randomly generate Sequence 0

Replace the current best sequence by Sequence n

i-th

Fig. 2. Flowchart explaining the search procedure for min-norm se-quence.

A. Search for Min-Norm Training Sequences

Since no simple constructive methods exist for obtaining the min-norm training sequences, we resort to a search approach to find sequences with small normalized error norms. However, exhaustive search is often impossible. In the simulation study reported later, for example, we make use of a training sequence of 29 QPSK symbols. In this case, there are

4

28

possible se-quences! We therefore employ Newton’s method with multiple initial points. The search method generates an initial data se-quence at random and modifies the sese-quence in a way to move the point toward a direction of descent till it reaches the bottom of a valley. Since the result may be a local minimum, the pro-cedure is repeated a number of times with a number of random initial points, in hope to find the global minimum or a better lo-cal minimum. Because the normalized error norm has a lower bound (equal to 1 when the training data matrix is square), a possible stopping criterion for the search is when we have ob-tained a sequence whose associated normalized error norm is within a certain tolerance of the lower bound. A disadvantage of this method is that we cannot predict the required search time. But this is not a serious drawback, since the search is conducted in the stage of system design and not in the stage of its actual operation.

Since the training sequence is composed of discrete numbers, the Newton steps are somewhat different from that in the case of continuous numbers. The search procedure is illustrated in Fig. 2. We summarize the search procedure as follows:

1. Randomly generate a sequence and call it Sequence

0

. 2. Fori

= 1

tonwherenis the length of the training

se-quence, do the following: Change thei-th element of Se-quencei

1

to a value which yields the smallest

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normal-ized error norm, i.e., trf(GG H

) 1

g. Call the resulting sequence Sequencei.

3. Check if Sequencenis the same as Sequence0. If so, then compare the associated normalized error norm with that of the current best sequence; replace the current best se-quence by Sese-quencenif the latter has a lower error norm; go to Step 1. Otherwise, rename Sequencenas Sequence 0and go to Step 2.

We conducted a search over QPSK sequences of length 29 and found that the following sequence yielded the lowest normalized error norm when the sequence is formed into a1515square training data matrix:

11j111j11j1111

j1j1jjj1j11jjjj:

The value of the normalized error norm is1:058, which is very close to the lower bound of1. In fact, it can be shown that, since the training data matrix is of an odd dimension, it is impossible to have a sequence with unity normalized error norm under ei-ther BPSK or QPSK. We state the previous result in the form of a theorem below. As a result, the above sequence may be one of the best sequences of length 29 under QPSK modulation.

Theorem 2: When the training data matrix is square and of

an odd dimension, it is impossible to have a sequence with unity normalized error norm under either BPSK or QPSK.

Proof: Consider eigenvalue decomposition of the matrix

GG H

asEVE H

whereV is the diagonal matrix of the eigen-values of GG

H

andE is the corresponding matrix of eigen-vectors. The proof of Theorem 1 shows that, ifGachieves the lower bound of normalized error norm, then all the eigenvalues

ofGG

H

are equal (and they are equal to the column dimension ofG). In the case of a squareGof dimensionn, therefore, we would haveGG H = EVE H = nEE H =nI, which implies that the rows ofGare orthogonal. But such orthogonality is im-possible between any BPSK or QPSK sequences of odd lengths, because an odd number of1(in the case of BPSK) or an odd number of1andj(in the case of QPSK) cannot sum to zero. Thus the corresponding sequence cannot have unity normalized

error norm. 2

An interesting question is whether some of the well-known kinds of sequences in communications, such as the maximum-length (ML) sequences [6], would be optimal or nearly optimal. While a theoretical derivation is not conducted in this work, the simulation results presented later show that ML sequences can yield significantly higher channel estimation error than a sequence which minimizes the normalized error norm. ML se-quences are also less flexible in their lengths, which can only be equal to2

n

1wherenis an integer, whereas a min-norm sequence can be of any length. The simulation results presented later also show that a random training sequence can be much worse, as expected.

Incidentally, an independent study on training sequence de-sign was reported by Crozier et al. [12]. Among other differ-ences, they focused on single-antenna reception under (primar-ily) BPSK modulation. In addition, our derivation of the min-norm condition is more direct and succinct.

III. DYNAMIC SELECTION OF VITERBI EQUALIZER’S MEMORY LENGTH

Intuitively, the memory length (number of taps in the channel response model) of the Viterbi equalizer should vary with the length of the channel response. However, previous study shows that when the receiver input SNR is low, use of fewer taps in the Viterbi equalizer may yield a better performance than using more. This is because the truncation of some small-valued taps can reduce the channel estimation error when the SNR is low [11]. Moreover, use of fewer taps in Viterbi equalizer can also reduce the computational complexity. We now propose a simple method to determine the memory length of the Viterbi equalizer, under the condition that the input SNR is known. By selecting memory length this way, we can achieve the same performance as using a larger number of taps.

Consider again the received training signal (5) and the least-square estimate of the channel response (9). From (9) we have

^ R=XG H (GG H ) 1 =R+NG H (GG H ) 1 : (22)

Considering the use of QPSK modulation, then the input SNR can be defined as SNR= kR k 2 F EfkNk 2 F g : (23) Let the SNR in ^ R, denoted ^ SNR, be defined as ^ SNR= kR k 2 F EfkNk 2 F gkZk 2 F = SNR kZk 2 F (24) whereZ=G H (GG H ) 1 .

Assume the original channel containsqtaps, but only thek-th to the(k+L

win

1)-th taps are selected in the Viterbi equalizer; that is, onlyL

wintaps are used in the Viterbi equalizer. From (22) we can find that the noise variances in all the estimated taps are the same; this is due to that each column ofZhas the same sum-squares value for its elements. Therefore, we estimate the noise energy in each estimated tap, denotedE

n, as E n = ^ P ( ^ SNR+1)q (25) where ^ P 4 =k ^ Rk 2

Fis the energy of the estimated channel ^ R. The total estimation error over the length of channel modeled in the Viterbi equalizer can be estimated as

E s =L win E n : (26)

And the channel truncation error can be estimated as E t =( ^ P ^ P win )(qL win )E n (27) where ^ P

winis the energy in the selected taps of the estimated channel ^

R.

The total channel estimation error can thus be estimated as E s +E t =2L win E n +( ^ P ^ P win )qE n : (28)

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In the rhs, sinceP

^

andq E

n do not vary with L

win, we may chooseL

winby performing the following minimization:

min

L w in (

2

L win

^

(

GG H

)

1

g. In the simulation re-ported below, we assume for simplicity thatkR k

2

Fand the noise power are known exactly.

IV. SIMULATION RESULTS

In the simulated space-time Viterbi equalizer, we employ the delayed-decision sequence estimation architecture [8] wherein the channel response estimate is provided by a channel response estimator. To evaluate the performance of various techniques, 100 channel responses were generated according to the model of [7]. Then the average performance of these techniques over this set of channels was obtained. The model of [7] covers differ-ent propagation environmdiffer-ents, including urban, suburban, and hilly. We generated channels with rms delay spread between about 1 and 10s. The QPSK signals (a

i

2f

for the MLSE part and

0

and

1

for the DFE part. We see that the performance of these two schemes is near the same.

1 2 3 4 5 6 7 8 9 10−4 10−3 10−2 10−1 100 SNR [dB]

Symbol Error Rate(SER)

3 antennas, 100 channel average, 104 symbols/channel run

randomly generated sequence maximum−length sequence min−norm sequence

Fig. 3. Symbol error rates from using different training sequences. The tap numbers in the MLSE and the DFE parts are4and1,

respec-tively. 1 2 3 4 5 6 7 8 9 10−4 10−3 10−2 10−1 100 SNR [dB]

Symbol Error Rate (SER)

3 antennas, 100 channel average, 104_{symbols/channel run}

Lv=4,Ldffe=1

Dynamic taps selection, Lv=3~4, Ldfe=0~1

Fig. 4. Symbol error rates from using dynamic tap number selection and a fixed tap number.

Finally, consider the comparative computational complexity between using a fixed tap number and using dynamic tap num-ber selection. We useL

vto denote the tap number in the MLSE part of the equalizer, andL

dfe denotes the tap number in the DFE part of the equalizer. AndL

winis equal to the sum of L v andL dfe. We use

4

L v 1

to give a rough estimate of the com-plexity for each channel, since this value gives the number of metrics computed in MLSE. In the case with a fixed tap num-ber,L

v

= 4

because the number of taps in the MLSE part is

4

. In the case with dynamic tap number selection, L v is ei-ther

3

or

4

depending on the channel condition. The ratio in computational complexity of these two schemes, averaged over the

100

test channels for each simulated SNR value, is shown in Fig. 5. We see that at symbol error rates below

10

2

(SNR >

5

dB), dynamic tap number selection can reduce the

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compu-1 2 3 4 5 6 7 8 9 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 SNR [dB] Complexity Ratio

Computational complexity ratio between dynamic tap number selection and Lv=4,Ldfe=1

Complexity ratio: Lv=3~4,Ldfe=0~1 vs. Lv=4,Ldfe=1

Fig. 5. Ratio in computational complexity between using dynamic tap number selection and a fixed tap number.

tational complexity by about5%to25%compared to using a fixed tap number, while achieving the same performance.

Note that, with dynamic memory-length Viterbi equalization, the hardware complexity is still determined by the largest al-lowed memory length. But the reduced computational complex-ity will result in reduced power consumption in system opera-tion.

V. CONCLUSION

We considered signal and receiver design for space-time Viterbi equalization in non-spread-spectrum mobile communi-cation at high data rates, where the Viterbi equalizer is of the delayed-decision sequence estimation variant. The examined system structure employed a training sequence for channel esti-mation and used the estimate in space-time Viterbi equalization. We proposed a search method to find the best training se-quences (called the min-norm training sese-quences) for least-square channel estimation. Simulation results show that such training sequences can achieve a much better performance in channel estimation and signal transmission than using a maximum-length sequence or a randomly generated sequence in training. We also derived a simple lower bound for the achiev-able channel estimation error of any training sequence. One in-teresting aspect of the theoretical results is that the optimality of a training sequence does not depend on the number of an-tenna branches used in the receiver. Interpreted another way, the receiver may freely choose the number of antennas to use with-out concern for the optimality of the training sequence. This is certainly beneficial to system design.

For the situation with unknown channel response lengths, we proposed a simple method to select the memory length of the Viterbi equalizer based on the receiver input SNR. Simulation results show that the performance of the Viterbi equalizer under this method is comparable to that using a fixed memory length, even though the memory length (tap number) in the former case is often shorter than that in the latter. On the average, at symbol

error rates below10 2

(SNR>5dB), the reduced tap number translates into a reduction of about5%to25%in computational complexity of the Viterbi equalizer.

Least-square channel estimation method requires matrix in-versions, in principle. When the dimension of the matrices or the number of such inversions is high, the computational com-plexity can be a concern in practical implementation. However, since the matrices that need be inverted are solely a function of the training sequence and hence are known in advance, they can be pre-computed and stored. Thus matrix inversions do not con-stitute a part of the per-sample computational complexity, but only some matrix multiplications.

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Chih-Sheng Chou was born in Pingtung, Taiwan, in

1970. He received the B.S. and M.S. degree in elec-trical engineering from Tatung University, Taipei, Tai-wan, in 1992 and 1994, respectively. He is currently working towards the Ph.D. degree at National Chiao-Tung University, Hsinchu, Taiwan. His research inter-ests include signal processing and wireless communi-cations.

David W. Lin received the B.S. degree from National

Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1975, and the M.S. and Ph.D. degrees from the Uni-versity of Southern California, Los Angeles, in 1979 and 1981, respectively, all in electrical engineering.

He was with Bell Laboratories during 1981–1983, and with Bellcore during 1984–1990 and again during 1993–1994. Since 1990, he has been a Professor in the Department of Electronics Engineering and the Cen-ter for Telecommunications Research, National Chiao Tung University. He has conducted research in digi-tal adaptive filtering and telephone echo cancellation, digidigi-tal subscriber line and coaxial network transmission, speech and video coding, and wireless communi-cation. His research interests include various topics in communication engineer-ing and signal processengineer-ing.