Adaptive Equalization Techniques for HF Channels

238 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. SAC-5, NO. 2, FEBRUARY 1987

Adaptive Equalization Techniques for HF Channels EVANGELOS ELEFTHERIOU, MEMBER, IEEE, AND DAVID D. FALCONER, FELLOW, IEEE

Abstract-Data transmission at rates of 1.2 kbits/s or higher through voiceband ionospheric channels is subject to impairment from severe linear distortion, fast channel time variations, and severe fading. In this paper, we have focused on the performance of DFE (decision feed- back equalization) receivers for communication over 3 kHz bandwidth HF channels. We describe the results of simulations for a wide range of fading rates on simulated and real recorded HF channels, using fractionally spaced DFE receivers. Both LMS (least mean square) and FRLS (fast recursive least squares) adaptation algorithms with peri- odic restart were evaluated, and both ideal-reference and decision-di- rected operation was observed. The results indicate that FRLS adap- tation yields s\uperior performance to LMS in rapid fading conditions, but that this performance advantage diminishes at low signal-to-noise ratios. Also, fade rates greater than about 1 Hz produced relatively high error rates, irrespective of which adaptation method was em- ployed. Finally, a novel modification of the simple LMS algorithm which improves its tracking ability was evaluated. This involved pre- ceding the LMS DFE receiver with an adaptive lattice whitening filter.

I. INTRODUCTION

T HE high-frequency, nominally 2-30 MHz portion of the spectrum has been of great interest for many years

for long-distance radio communications in many military and civilian applications. Despite the inherent difficulties associated with the nature of the transmission environ- ,merit, which had rendered HF receiver structures compli- cated during the past, recent advances in VLSI technology and contemplated signal processing devices are very promising for high-frequency , high-data-rate communi- cation links with moderate to high reliability.

Physically, the HF channel is characterized as- a mul- tipath time-varying environment producing time and fre- quency spreads. The typical time dispersion or multipath spread is measured to be between 2-8 ms and strongly depends on the range (physical path length) [l], [2]. The frequency spreads are usually less than 1 Hz, but higher values are possible on certain transauroral paths.

The major limitations on high-data-rate HF transmis- sion’are a result of the nonideal characteristics of the me- dium, such as linear distortion, rapid channel variations, and severe fading as well as bandwidth constraints. The existence of several distinct more or less independent propagation paths on an HF channel reduces the likeli-

Manuscript received February 5 , 1986; revised August 4, 1986. This work was supported in part by the Department of Communications of Can- ada under Contract 03SU82-00134 and by the Natural Sciences and Engi- neering Research Council under Grant A5828.

E. Eleftheriou is with the IBM Research Laboratory, CH8803 Riischli- kon, Switzerland.

D. D. Falconer is with the Department of Systems and Computer En- gineering, Carleton University, Ottawa, Ont., Canada K l S 5B6.

IEEE Log Number 8612172.

hood that a deep fade occurs in which all paths fade simul- taneously. However, this inherent diversity advantage of multipath is accompanied by severe frequency selectivity. Thus, serial data transmission through voiceband HF channels at a rate on the same order as or higher than the channel bandwidth is considered ‘‘high speed’ ’ and gen- erally requires powerful equalization techniques.

Because of severe frequency selectivity, a nonlinear equalization technique such as decision feedback equal- ization (DFE) or maximum likelihood sequence estima- tion (MLSE, also known as the Viterbi algorithm) should be employed at the receiver [3]. In spite of its effective- ness, the computational and storage requirements of the MLSE receiver limit its application. The computational complexity of the MLSE receiver increases exponentially with the duration of the channel’s impulse response. Fur- thermore, a.comparison study between DFE and MLSE receiver performance on real recorded HF channels [4] reveals, that there is no spectacular advantage by using a Viterbi decoder rather than a DFE. These results argue for the adoption of a decision feedback equalizer; Its moderate complexity and its ability to perform well on channels with spectral nulls make the DFE appropriate for equalization of a time-variant multipath channel such as the HF.

The efficiency of a DFE receiver in demodulating sin- gle carrier bandwidth efficient modulation schemes for high-speed data transmission over narrow-band iono- spheric channels has been demonstrated in [ 5 ] . There are a number of papers [6]-[9] dealing with the performance of DFE receivers for communication over, fading disper- sive HF channels which,concentrate on the tracking prob- lem in a time-varying environment presenting and testing different fast adapting RLS schemes. Apart from the high- complexity issue concerning the adaptation algorithms themselves (Kalman, square-root Kalman, and lattice), the simulated channel models used are somewhat simplified. The transmission medium is modeled as a T-spaced tapped delay line with usually 2-3 T-spaced time-varying coef- ficients, followed by additive zero-mean white Gaussian noise. Such a model encompasses a matched filter and a symbol rate sampler and is referred to as the equivalent discrete-time white-noise filter model [3]. As a result, the effect of larger impulse responses (not necessarily with T-spaced components), the existence of a fixed low-pass filter instead of a whitening matched filter, and the sub- sequent sensitivity to the symbol clock sampling phase of T-spaced equalizer structures have not been considered. A more realistic and flexible HF channel simulator has

0733-8716/87/0200-0238$01 .OO 0 1987 IEEE

ELEFTHERIOU AND FALCONER: EQUALIZATION TECHNIQUES FOR HF CHANNELS 239

been developed in [ 101. In this reference, the performance X (n

of a T / 2 fractional spaced DFE operating with the most computational intensive square-root Kalman adaptation algorithm has been studied. Ji, tl B

In this paper, we have focused on the performance of DFE receivers for communication over 3 kHz HF chan-%y

c

nels. The behavior of the DFE receiver has been tested not only on simulated data, but also on real recorded HF data. The decision feedback equalization structure utilizes a T / 2 , rather than a T-spaced forward transversal filter. At this point, we should emphasize that sampling a@sub- sequent equalization at a rate 2 / T (two samples per sym- bol clock are shifted in the forward section) is appropriate not only as a means to eliminate the sensitivity to sample timing phase, but also to compensate effectively for an extensively wide range of delay distortion and consider- ably improve severe amplitude distortion [ 131-[15].

The transversal DFE used in our study adapts according to a fast recursive least square (FRLS) or fast Kalman algorithm [l 13. This adaptation algorithm' imposes a somewhat lower computational burden than Kalman, square-root Kalman, or lattice alternatives, while offeiing equivalent tracking capabilities. However, it has unfavor- able stability properties. Stability problems are eliminated from our transversal FRLS algorithm by modifying it through the use of a periodic restart procedure, introduced in [ 161 and [ 181, and described further in Section IV.

Finally, a further novel scheme has been tested which involves the cascade of a T / 2 adaptive predictor followed by a T/2-forward spaced LMS decision feedback equal- izer. The predictor operates as a whitening filter and aims to produce an approximate equal eigenvalue autocorrela- tion matrix. The whitening of the input signal seen by the LMS DFE causes it to converge and track faster.

11. SERIAL DATA TRANSMISSION OVER HF CHANNELS In this study, we have been concerned with 2400 bit / s

transmission of QPSK modulated signals over 3 kHz wide HF channels. For a linearly modulated QPSK data signal passed through a linear channel, the channel's bandpass output waveform can be represented as the real part of a complex waveform:

r ( t ) = h Re a,h( t - nT; t ) *ej2?rfof + vc( t ) ] L ( 1 )

where h ( 7 ; t ) represents a complex-valued time-varying equivalent low-pass impulse response accounting for transmitter and channel filtering. It is apparent that h ( 7 ; t ) in its dependence on the age variable T includes and describes the time variability of the transmission medium. The asterisk denotes complex conjugate. The data sym- bols { a, } are statistically independent discrete-value complex random variables from a finite set; for the case of QPSK, each a,, takes one of the four possible values l /&e jen , 8, = ~ / 4 , 3 a / 4 , 5 ~ / 4 , 7 a / 4 . The symbol interval is T seconds and the parameter fo represents the

tp3q c b *

N2

Fig. 1 . Decision feedback equalization receiver.

carrier frequency. Finally, the complex v J t ) represents channel additive noise. After a demodulation stage in the receiver and low-pass filtering, the baseband channel out- put can be represented by the complex-valued baseband waveform

x ( t ) = C a,h(t - n ~ ; t>* + v ( t ) ( 2 )

where v ( t ) is a low-pass filtered version of the additive noise v, ( t ) . The real and imaginary parts of x ( t ) are out- puts of the cosine and sine demodulators, respectively. The complex demodulator output x ( t ) is assumed to be sampled at a multiple of the symbol rate 1 / T which is above the Nyquist rate. Most useful data signals, includ- ing those in our simulations, have excess bandwidths of less than 100 percent, and therefore, a sampling rate of 2 / T is appropriate [13]-[ 151. The resulting samples are the input to the equalizer.

A typical DFE consists of a forward and a feedback transversal filter. The inputs to the forward section are channel output samples at T / 2 or T second intervals for a symbol rate of 1 / T and the inputs to the feedback filter are previous data symbol decisions 6,. A transversal de- cision feedback equalizer is shown in Fig. l . Referring to Fig. 1, the input z ( n ) = z ( n T ) at time nT into the quan- tizer is

n

A

~ N I - 1

z ( n ) = c c f * ( n ) x nT - i - i=O ( 3

Nz

+ C c!* (n) d,-i = C * ( n ) X ( n ) (3) i = l

where the N-dimensional complex partitioned vectors C( n ) and X ( n ) at time n have been defined as

C ( n ) ' (c fo(n) , c f ( n ) , * ' 9 c { N z - l ( n ) , c ! ( n ) , T

c m , * - 9 c&<.)) (4)

and

240 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. SAC-5, NO, 2 . FEBRUARY 1987

(superscript T denotes transpose). Also, N = 2N1 + N 2 . In a time-varying environment, previous studies [2], [5], [6] have indicated that the ability of the adaptive equal- ization algorithm to track the time-varying channel re- sponse is at least as important as the choice of the equal- ization method itself.

111. RLS ALGORITHMS IN DECISION FEEDBACK EQUALIZATION

Recursive least squares (RLS) filter adaptation algo- rithms with faster tracking capability than that of the LMS algorithm are available for channel equalization of rapidly time-varying channels such as HF radio channels. They are applicable to' both transversal and lattice filter struc- tures. Here we briefly restate the RLS criterion applied to a decision feedback equalizer, which results in superior convergence and tracking properties in comparison to the typical LMS algorithm. In particular, a recursive least squares adaptive DFE yields at any time nT that coeffi- cient vector C ( n ) which minimizes the weighted cumu- lative square error

n

J ( n ) = c X " - k j d k , - C ( n ) * X ( k ) I 2 k = O

+. 6XnC(n)* ANC(n). ( 6 ) '

The exponential weighting factor A is a positive constant close to but less than 1 , while the parameter 6 ( 6 > 0) ensures stability at the startup of the algorithm. Finally, A N is an N X N diagonal matrix defined as A N Diag ( 1, X-', x-2 , . . . h-N+ 1 ).

The, in general, superior tracking properties of the RLS algorithms compared to those of the LMS algorithm have been studied in depth in [17] and [ 181. We have shown that regardless of adaptive system structure and indepen- dently of the variations of the environment, the conver- gence of the lag error vector is governed by a unique time constant. This time constant does not depend on the ei- genvalue spread of the autocorrelation matrix and is given by

symbol intervals. On the other hand, the convergence and tracking ca-

pability of the- LMS algorithm can be severely degraded by channels with a large eigenvalue spread. In decision feedback equalization of a time-varying channel where the eigenvalue ratio can take on any realistic value, such be- havior explains the performance advantage we obtain by using RLS algorithms instead of the simple LMS algo- rithm.

Efficient methods which minimize recursively in time and order J ( n ) in single-channel applications are avail- able. Such algorithms are discussed in [7], [ 111, [ 121, and [19]. In the case of a decision feedback equalizer, and especially when a T/2-spaced forward section is em- ployed as opposed to the T-spaced feedback section, one

must resort to the so-called "multichannel" versions of the RLS algorithms with different sampling rates. The FRLSmultichannel transversal schemes possess the low- est complexity of all multichannel least squares algo- rithms [ l l ] , [12]. In particular, if we assume M parame- ters in each channel such that the dimensionality of the input signal vector is N = Mp, then the number of com- putation and storage requirements rises in proportion to Mp2 or Mp where p denotes the number of new entries at each iteration [ll], [12], [19] (for a fractional spaced DFE, p = 3). The main problem with these algorithms is numerical instability due to the propagation of roundoff errors [ 121.

The square-root Kalman [7], [lo] is the most stable (in a floating-point arithmetic environment) RLS transversal filter. It has the disadvantage of requiring matrix opera- tions. Therefore, the number of calculations is propor- tional to N 2 and grows very fast with increasing N. Es- pecially in a fractional spaced implementation, one would expect an additional increase in computational burden and memory requirements.

Multichannel lattice filters are known to exhibit very good numerical stability properties [9], [20]. Their com- plexity, although proportional to Mp or Mp 3, is still quite intensive in comparison-to that of the FRLS transversal multichannel algorithms because of large proportionality constants and a larger number of required division oper- ations.

While the square-root Kalman transversal adaptive fil- ter or the various lattice adaptive filters have been advo- cated for equalization of fading HF channels [6], [7], [9], [lo] primarily for their numerical stability properties, we have primarily focused our attention on the simpler frac- tional spaced FRLS transversal DFE. Numerical stable propagation of the algorithm is ensured by any one of the efficient restart techniques developed in [ 161 and [ 181 de- scribed in the next section.

IV. FRLS TRANSVERSAL DFE WITH AN EFFICIENT PERIODIC RESTART

The fast transversal algorithm, with its derivation, for computing the Kalman gain vector sequences and subse- quently for updating the decision feedback equalizer tap coefficients can be found in [ll]. The general principles and basic ideas behind the restart procedures for FRLS transversal adaptive filters have been presented in detail in [ 161 and [18]. Here we shall briefly describe the appli- cation of the most efficient one, restart procedure 1 , in decision feedback equalization.

Following a restart at time noT, all inputs x (nT ), x (nT - T / 2 ) , and d, - are taken to be zero for n I no. In other words, for n > no, the N-dimensional input vector for purposes of adaptation is taken to be z(n). whose components consist of

ELEFTHERIOU AND FALCONER: EQUALIZATION TECHNIQUES FOR HF CHANNELS 24 1

where 5 ( n ) is a three-dimensional vector ( p = 3 ) with components

$ ( n ) = x ( n T ) , x nT - - an-* . '3 (9)

According to restart procedure 1 [16], during the reini- tialization period, the desired outputs used for FRLS DFE adaptation should be

( ( 2'>. ,. ) T

Li:, = 2, - C,,,(n) * X(n) ' - ' (10)

where C,,, ( n ) is the current set of tap coefficients from an auxiliary LMS algorithm and X( n ) ( - ) is an N-dimen- sional signal vector defined as

x ( n ) ' - ) X ( n ) - a@). ( 1 1 )

A slightly different approach, but within the framework of restart procedure 1 , is to avoid the auxiliary LMS al- gorithm. In this case, in place of C,,, ( n ), we use the not zeroed set of tap coefficients C( n ) . Thus, the useful quan- tizer input during restart will be

z ( n ) = C ( n ) * X ( n ) (12)

while (10) again provides the desired response during the same period, i.e.,

an = 2, - q n > * X ( n ) ? (13 )

In both cases, following a restart at time noT, based on method 1 , the FRLS DFE minimizes

J ( n ) = hn-ki;k - c(n)* $ ( k ) 1 2 + 6Xn-"O(C(n) k = n o

I * + ~ x " - " o ( c ( ~ ) - c(no))* ( ~ ( n ) - ~ ( n , ) ) .

(14)

If the variations of the environment are slow with respect to the baud interval T, and the reinitialization period is sufficiently small, then approximately

' (c(n) - c(nO))* AN (c(n) - c ( n O ) ) .

(16) Therefore, modification of the desired response mini- mizes the disruption associated with the step function ef-

TABLE I COMPARISON OF Two RESTART PROCEDURES

Number of Symbol Errors Observed

Restart Procedure 1 Restart Procedure 1 Period of Initiation with Auxiliary no Auxiliary LMS

of Restart" LMS Algorithm Algorithm

300 88 1 143 1 500 99 1 1193 750 866 1141

1000 852 108 1

aRestart interval = 30 symbols.

interval is sufficiently short so that the momentary reli- ance on the slower adapting LMS algorithm or on the non- zeroed set of tap coefficients of the restarted FRLS DFE should cause little reduction in the adaptive filter's track- ing capability.

Here we shall compare the error rate performance of the two options in restart procedure 1 discussed previously, applied to a DFE with N = 20 coefficients. The unknown time-varying channel is a simulated 0.15 Hz Rayleigh fading channel with a 4T symbols interval multipath spread, while the signal-to-noise ratio is set to 20 dB. In all cases, the restart interval is 30 symbols and the simu- lation runs are 200 000 symbols long. Table I lists the number of observed symbol errors as a function of the period (in symbol intervals) of initiation of a restart. We can see that in the case where an auxiliary LMS algorithm is used in connection with restart procedure 1, then the number of symbol errors is almost insensitive to how fre- quently we invoke the restart. However, in the case where we do not use an auxiliary LMS DFE, the error rate de- creases as the period of initiation of a restart increases.

On the basis of simulation results like these, we con- cluded that a valid and efficient procedure for periodically restarting the FRLS adaptation algorithm applied to a DFE receiver for HF channels is restart procedure 1 aided by an LMS algorithm. This is the restart procedure employed in all FRLS simulations described henceforth.

V. DFE RECEIVER PERFORMANCE OVER SIMULATED HF CHANNELS

A. HF Channel Simulator The baseband HF channel simulator has been modeled

as a tapped delay line with time-varying coefficients, fol- lowed by additive zero-mean white Gaussian noise. The resulting signal plus noise passes through a low-pass filter with cutoff frequency at 1200 Hz to form an output signal. The time-varying tap coefficients are assumed to be in- dependent complex zero-mean Gaussian random vari- ables. They have been generated by passing white Gauss- ian noise generated at 9600 samples / s through low-pass filters with 3 dB bandwidths on the order of the fade rate. The filters used to generate the Rayleigh fading path com- ponents are two-pole Butterworth-type filters with a se-

fects on the filter's input signal vector. The-FRLS restart lectable bandwidth as described in [6].

242 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. SAC-5, NO. 2, FEBRUARY 1987

I I 8 *

I Square Root Kalman Algorithm

2 x 10.3 I .92 .93 .b4 .95 .96 .97 9 8 .99

Exponential Weighting Factor X

Fig. 2 . Error performance for simulated Rayleigh fading channel as a function of X. SNR = 20 dB; fade bandwidth = 0.15 Hz; four paths; multipath spread = 7T; DFE with 18 forward and seven feedback tap coefficients; ideal reference mode.

B. Ideal Reference Mode

Through simulations, we have observed the bit error rate performance of the FRLS receivers as a function of the signal-to-noise ratio in a variety of Rayleigh simulated HF channels for different fade rates and multipath spreads. The DFE operates in ideal reference mode, i.e., the data symbols fed back and used for adaptation are the correct ones. The results of these simulations shed light on the inherent capabilities of the DFE equalizer and its adap- tation methods. The bit error probabilities have been es- timated over 200 000 data symbol long (400 000 bits) simulation runs. In all cases, the SNR is defined as the average transmitted power at the input of the receiving low-pass filter over the average additive-noise power at the .same point.

To display the effect of the exponential weighting pa- rameter X on the tracking capability of the adaptation al- gorithm, and to compare the performance of the square root Kalman algorithm [7] to our' periodically restarted FRLS algorithm, we have simulated both types of DFE adaptations on a 0.15 Hz fading channel with four paths, a multipath spread (MPS) of 7T, and a 20 dB SNR. The DFE has 18 forward T/2-spaced coefficients and seven feedback coefficients. Fig. 2 shows the measured bit error rate as a function of X. For the FRLS algorithm, the re- start interval is 30.

These results show that the relatively simple periodi- cally restarted FRLS algorithm is stable and yields very little in error performance to the much more complex square-root Kalman algonthm. It is also evident from these results and from analytical results in [ 181 that the choice of X represents a compromise between the steady-state er- ror and the tracking capability of the adaptation algo- rithm.

Fig. 3 shows the bit error rate as a function of the SNR

1 o - ~ i 5 10 SNR 15

(dB) 20

Fig. 3. Error performance versus SNR for a three-path, Rayleigh fading channel; MPS = 4T; ideal reference mode.

. . for various fade rates. The channel used in this investi- gation has three equal strength (in the mean-square sense) 2T-spaced paths, i.e., an MPS of 4T. The length of the decision feedback equalizer has been fixed to the total N = 20 coefficients; 16 T/2-spaced forward coefficients, and four feedback coefficients. For the slowest fading channels, i.e,, 0.08 and 0.15 Hz, the exponential weight- ing factor X was set to 0.97, while in the other two cases, we chose X = 0.94. In the same figure, we present results for a 0.15 Hz fading channel for an LMS DFE. In all cases, the LMS step size was approximately 1 / N for .equalizer inputs with unit variance. This value gave the most favorable tracking performance. The superiority of the FRLS DFE is evident. In general, the FRLS exhibits much better performance at moderate fade rates, while in the case, of 1.1 Hz fading bandwidth or higher, it bottoms out. These results are generally in close agreement with those presented in [ 101. There, under similar channel con- ditions and using an uninterrupted square-root Kalman al- gorithm, error rate versus SNR curves have been obtained which are approximately 5-10 times better in every case (even in the case of an LMS DFE). This leads us to con- clude that this small difference in performance should be attributed to the different filter structure used in limiting the bandwidth of the tap coefficient's. driving noise pro- cesses of the HF channel model, rather than to the slight suboptimality in tracking performance introduced by re- starting the FRLS DFE.

The tracking performance of the FRLS algorithms un- der restarting conditions is an important issue and de- serves further analytical and experimental investigation, especially at high SNR's. It may not be as crucial, though, in an HF modem which will work in a decision-directed mode. In this case, as we shall discuss later, deep fades will induce crash situations, and thus, retraining the equal-

ELEFTHERIOU AND FALCONER: EQUALIZATION TECHNIQUES FOR HF CHANNELS 243

IO-^ L----J 5 10 SNR 15 20 IO-^ 5 1 10 I SNR 15 I 20 I

(d 6) (dB) Fig. 4. Error performance versus SNR for a four-path Rayleigh fading ~ i ~ , 5 , performance versus SNR for a three-path Rayleigh fading

channel; MPS = 7T; ideal reference mode. channel; MPS = 4T; fade bandwidth = 0.15 HZ.

izer or even restarting it would be a necessary measure regardless of numerical instability problems.

Another factor which contributes to slow tracking be- havior and subsequently to high bit error rates is the num- ber of coefficients of the DFE structure. In our experi- ments, in both simulated and real HF channels, the total number is in the range of 20-30 in order to cover all pos- sible multipath spreads. This is in contrast to what has been used in [lo] where the maximum number is 14 or 15 tap coefficients.

Fig. 4 depicts the bit error rate versus SNR for a much more dispersed in time channel and for various fade rates. In this case, the number of paths is four with MPS = 7T. The length of the DFE has been increased to N = 25 tap coefficients (18 T/2-spaced taps and seven feedback taps). Again, the FRLS DFE receiver shows good perfor- mance, especially at the low fade rates.

C. Decision-Directed Mode For a sufficiently deep fade, the receiver’s error rate can

become high enough that the sequence of decisions used for filter adaptation becomes unreliable; then, the adapted equalizer tap coefficients may wander far from their op- timal values, and catastrophic error propagation occurs. Such “crashes” for decision-directed adaptation were ob- served during deep fades in our simulation: the decision- directed FRLS algorithm usually failed to reconverge within a reasonable time following a crash.

Therefore, subsequent simulations which used the re- ceiver’s decision { d,, } for equalizer adaptation and for decision feedback equalization were modified in the fol- lowing way, called the decision-directed mode. Periodi- cally, the receiver’s decisions were replaced by the por- tion of the known pseudorandom data sequence during that period.

In the case of the FRLS DFE receiver, the training se- quence coincided with each restart period. The decision- directed mode has been tested in the case of a 0.15 Hz fading bandwidth for the 2T-spaced three-path Rayleigh fading channel. The results are shown in Fig. 5. The pair (530, 30) indicates that an ideal reference training se- quence of 30 data symbols was transmitted once every 530 symbol intervals. This is referred to as a 5.6 percent ideal reference training sequence. Obviously, by increas- ing the percentage of retraining, the equalizer will ap- proach the ideal reference case. More simulation results on the decision-directed mode as well as discussion on the limitations posed by crashes will be presented in the next sections in connection with the real HF data.

VI. A WHITENING PREDICTOR AS A FRONT END FOR AN LMS DFE

It is well known that fast convergence and tracking with an LMS algorithm are not possible when the eigenvalues of R are very different in magnitude. Especially, in a dy- namically changing environment, such as the HF multi- path fading channel, the eigenvalue ratio of the autocor- relation matrix might take on a variety of values. The eigenvalues can become equal if we apply an orthonormal transformation on the autocorrelation matrix R . This is essentially the fundamental principle behind the devel- opment of adaptive lattice equalizers. Basically, the adap- tive lattice algorithms generate a set of orthogonal signals which can be used as inputs to equalizer tap coefficients

In a DFE case, the inputs into the feedback section are usually uncorrelated, being past symbol decisions. Thus, reduction of the overall eigenvalue spread calls for de- correlation or whitening of the forward section input sig- nal. This can be achieved either by using all order back-

W I .

Fig. 6 . Lattice whitening filter followed by an LMS DFE.

ward residuals of a lattice predictor for orthogonalization of the feedfoMard input signal vector or by using only the highest order residual for whitening the channel input signal stream. The second approach is the one'we shall discuss in the sequel.

Fig. 6 shows the combination of a lattice filter in cas- cade with an' LMS adapted decision feedback equalizer. The lattice filter has as input the sampled channel output, while its highest order normalized backward residual feeds

. the forward section of the DFE. Since the lattice structure operates as a prediction-error filter, it essentially "whit- ens" the data stream seen by the LMS DFE. Such a whit- ening reduces 'the eigenvalue ratio and causes the LMS algorithm to track faster. Nevertheless,. the use of the highest order residual rather' than all' order residuals in order to decorrelate the input signal vector elongates the overall impulse response'to be equalized. So one would expect that, in general; a longer DFE structure might be necessary'in order to achieve a specific mean-square er- ror. ,Simulation experience, however, indicates 'that a small number of lattice stages is more than adequate for the necessary whitening without the need of increasing the length of the DFE significantly.

Implementation of this simple concept requires a gra- dient or' FRLS lattice predictor followed by a conven- tional LMS DFE, both adapted separately, The compu- tational advantage of this approach is more apparent in a fractional spaced implementation of the receiver. In this case, the whitening predictor exploits the shifting prop- erty of the sampled channel output since the desired train- ing sequence is available at the same sampling rate as that of the input signal. Thus, a single-channel predictor adapted at the Same rate as the input sample stream can be used, notwithstanding the fact that the adaptive pre- dictor's input is a cyclostationary (period T ) random pro- cess.

Fig. 7 shows,the bit error rate performance, in the ideal reference mode, of the receiver with a seven-stage lattice predictor followed by a transversal LMS DFE compared to a conventional LMS DFE and to the FRLS transversal DFE with periodic restart. The channel used is a three- path 2T-spaced Rayleigh fading channel model. As we can see, the simple whitening lattice approach offers a probability of error which is supeiior to the transversal LMS DFE, but inferior to that of the FRLS DFE.

In conclusion, employing a whitening predictor as a

Fade rate 0.1 5Hz

5 I O SNR 15 i0

(dB)

Fig. 7. Error performance versus SNR for a three-path Rayleigh fading channel; MPS = 4T; ideal reference mode.

front end to LMS equalizers, linear or decision feedback, significantly improves convergence and tracking behavior at the expense of a slight increase in complexity. Such an approach appears to be suitable for a variety of applica- tions, and especially to those where computationai burden is a very critical factor.

VII. DFE RECEIVER PERFORMANCE ON REAL HF CHANNELS

This part of our work deals with computer simulation of receivers which process digitally recorded received waveforms from real HF channels. These recordings were made and provided to us by the Communication Research Centre (CRC) of Canada.

A. Transmitted signal and Characteristics of the Recorded Data

The transmitted signals were generated by QPSK mod- ulating a length 1023 pseudorandom pattern. With two bits per symbol, the QPSK carrier frequency was 1505.107 Hz and the baud rate was 1200. The approxi- mately 2.4 kHz bandwidth signal thus generated was sub- sequently up converted to the desired RF frequency in the HF band. Two separate recordings were made under dif- ferent propagation conditions, the tapes being denoted MDA010 and'MDAOl1.. Transmission was over a 90 km path, and the conditions and other information on channel characteristics are found in [4]. Channel MDA010's fade rate was typically about 0.15 Hz, while that of MDAO11 was typically about 0.5 Hz'.

The front end of the simulated receivers included sine and cosine demodulation followed by .low-pass filtering. The low-pass filters are identical raised cosine filters with cutoff bandwidths of 1200 Hz. The sampling frequency of the output complex baseband signal used for equal- ization is 2 / T .

ELEFTHERIOU AND FALCONER: EQUALIZATION TECHNIQUES FOR HF CHANNELS 245

n I LMS Adaptation

a

-7

Ll, L, L_-__

/ FRLS Adaptation

=' m a 10.3!

m 0 4 8 12 16 20 8 a NUMBER OF SYMBOL ERRORS IN

BLOCK OF 100 DATA SYMBOLS

Fig. 8. Error statistics for FRLS and LMS DFE on channel MDAOl 1 (de- cision-directed mode).

(I) U

W FRLS Adaptation

> t =' -3 . . m 10 2 0 4 6 12 16 20

8 a NUMBER OF SYMBOL ERRORS IN BLOCK OF 100 DATA SYMBOLS

Fig. 9. Error statistics for FRLS and LMS DFE on channel MDAOlO (de- cision-directed mode)?

B. Pegormance Comparison of FRLS and LMS Adaptation Algorithm

A fractional spaced DFE receiver was simulated to test its performance by using the complex baseband 2.4 kbit QPSK-modulated, recorded signals. Preliminary simula- tions established the choice of the number of equalizer tap coefficients: 20 T/2-spaced forward tap coefficients and seven feedback tap coefficients, resulting in a total num- ber of N = 20 f 7 = 27.

The simulated DFE equalizers, adapted according to the LMS algorithm and also according to the FRLS algorithm with periodic restart procedure 1, were compared with re- spect to the error rate. The error rates were measured as the probability distribution of the number of symbol er- rors occuiring in blocks of 100 transmitted symbols. The totai number of transmitted symbols in each simulation was 30 000 (300 blocks).

Figs. 8 and 9 show the block error distribution for chan- nels MDAO11 and MDA010, respectively, in the deci- sion-directed mode described in Section V-C. A known reference sequence of 30 symbols was transmitted at in- tervals of 330 symbol intervals; thus, 9.1 percent of the data was reference. For the FRLS algorithm, the training sequence coincided with each restart period. Fig. 8, for

all/ I II LMS ADAPTATION IN DECISION-DIRECTED DFE

II FRLS ADAPTATION

0 I I 1 oow ZObW 30000

ELAPSED SYMBOL INTERVALS

Fig. 10. SNR versus time for channel MDAOlO (SNR averaged with 500- symbol time constant).

14

SNR id61

12

10

8

r

I BOW BE

I1 1111 I E I n EFRLS ERROR BURSTSIN DECI~ION-DI\RECTED DFE ADAPTAT!O'

rd,B ea0 0 E3 LMS

ADAPTATlOl I

) 10000 20000

ELAPSED SYMBOL INTERVALS 30000

u

Fig. 11. SNR versus time for channel MDAOll (SNR averaged with 500- symbol time constant).

the faster fading channel MDAO11, shows a significant superiority of the FRLS algorithm over the LMS algo- rithm. However, for the slower fading channel MDAO10 shown in Fig. 9, the two adaptation algorithms yield vir- tually identical error statistics.

Similar results, not shown here, were observed for the ideal reference mode. It is worth noting that in all cases, the decision-directed mode yielded virtually the same fraction of error-free blocks as the ideal-reference mode of operation.

Figs. 10 and 11 show. the occurrences of blocks con- taining one or more errors for the two channels. Isolated erroneous blocks are shown as vertical lines; groups of two or more consecutive erroneous blocks are shown as rectangles. Superimposed on these blocks is the instan- taneous signal-to-noise ratio measured using an adaptive channel identification algorithm [4]. The very bursty na- ture of the decision eirors is apparent. The times of oc- currence of error bursts coincide with dips in the chan- nel's signal-to-noise ratio.

In conclusion, we should again mention that',as far as the real data simulations are concerned, the restart.FRLS algorithm displayed markedly superior performance over the LMS algorithm only in channel MDAOll. Other mea- surements [4] showed that MDA010 exhibited less time variation and slower fading than MDAOll apart from one deep sustained fade shown in Fig. 10.

C. Summary In this paper, we have focused on reception techniques

for data transmission over 3 kHz band-limited fading mul-

246 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. SAC-5, NO. 2, FEBRUARY 1987

tipath HF channels. The relatively long impulse response which can occur due to multipath propagation argues for the adoption of a DFE receiver instead of the more pow- erful, but complex, MLSE receiver. We have employed fractio’nal spaced equalizers to reduce the sensitivity to choice of sampling phase and to function more effectively on heavily distorted channels.

The error rate performance advantage of the FRLS DFE over the LMS DFE was evident in all Rayleigh fading simulated channels. However, in the case of the recorded real data from two channels, the’ difference in perfor- mance was less pronounced, probably because it. was dominated by prolonged periods of severe fading and low signal-to-noise ratio.

The problems of instability due to accumulating round- off errors in FRLS transversal algorithms was met and dealt with by means of an efficient periodic restart pro- cedure. The resulting FRLS adaptation of the DFE yielded very similar performance to a more complex square-root Kalman DFE. As well as the periodically applied restart procedure for the FRLS adaptation algaiithm, it was found necessary to modify the normal decision-directed mode of operation by periodically inserting a training sequence. Although a small portion of the transmitted data capacity was thereby lost, periodic retraining allowed the adaptive receiver to recover effectively from the effects of severe fading.

In addition, another new concept has been tested in which a fractional operating whitening predictor is used as a front end for a fractional spaced LMS adapted DFE. Although the performance of such a scheme in an HF con- text is better than the simple LMS DFE, it is still inade- quate in comparison to the FRLS DFE. The reduced over- all complexity offered by the above-mentioned structure encourages further study in different applications where computational complexity is crucial.

Concluding, the experimental results on both simulated and real data indicate that fade rates approximately as high as 0.5 Hz form an upper limit regarding an acceptable error rate performance of a bit-by-bit adapting serial DFE receiver. For fading bandwidths higher than that, even the

, ‘powerful fast adapting RLS algorithms reach their limi- tations, resulting in poor performance. Deep fading and low signal-to-noise ratio on HF channels further limit their performance. Further improvements would only be avail- able from coding and diversity techniques.

ACKNOWLEDGMENT We are’grateful to D. Clark of the Communications Re-

search Centre for providing the recorded HF data and for supporting and encouraging this project in its early stages. We are also grateful to R. Tront for providing [lo], to P. Driessen for advice on computer processing of the re- corded data, to M. Tobis for early simulation assistance, and to J . Cioffi, H. M. Hafez, J. Proakis, and A. U. H. Sheikh for helpful discussions.

REFERENCES [11 B. Goldberg, “300 kHz-300 MHz MF /HF,” IEEE Trans. Commun.

[2] N. M. Maslin, “High data rate transmission over HF links,” Radio Electron. Eng., vol. 52, pp. 75-87, Feb. 1982.

[3] J. G . Proakis, Digital Communications. New York: McGraw-Hill, 1983.

[4] D. D. Falconer, A. U. H. Sheikh, E. Eleftheriou, and M. Tobis, “Comparison of DFE and MLSE receiver performance on HF chan- nels,” IEEE Trans. Commun., vol. COM-33, pp. 484-486, May 1985.

[5] H. E. dePedro, F. M. Hsu, A. A. Giordano, and J. G. Proakis, “Sig- nal design for high-speed serial transmission on fading dispersive channels,” in Proc. NTC’78, Birmingham, AL, Dec. 1978.

[6] F. M. Hsu, A. A. Giordano, H. E. dePedro, and J. G . Proakis, “Adaptive equalization techniques for high-speed transmission on fading dispersive HF channels,” in Proc. NTC’80, Houston, TX, Dec. 1980.

[7] F. M. Hsu, “Square-root Kalman filtering for high-speed data re- ceived over fading dispersive HF channels,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 753-763, Sept. 1982.

[SI F. Ling and J . G. Proakis, “A generalized least squares lattice al- gorithm and its application to decision-feedback equalization,” in Proc. ICASSP’82, Paris, France, May 1982.

[9] -, “Adaptive lattive decision-feedback equalizers-Their perfor- mance and application to time-variant multipath channels,” IEEE Trans. Commun., vol. COM-33, pp. 348-356, Apr. 1985.

[lo] R. Tront, “Performance of Kalman decision-feedback equalization in HF data modems,” M.Sc. thesis, Univ. British Columbia, Canada, Dec. 1983.

[ l l ] D. D. Falconer and L. Ljung, “Application of fast Kalman estimation to adaptive equalization,” IEEE Trans. Commun., vol. COM-26, pp.

[12] J. M. Cioffi and T. Kailath, “Fast recursive-least-squares transversal filters for adaptive filtering,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp. 304-337, Apr. 1984.

[13] G. Ungerboeck, “Fractional tap-spacing equalizer and consequences for clock recovery in data modems,” IEEE Trans. Commun., vol.

[14] S. U. H. Qureshi aqd G. D. Forney, “Performance and properties of a T/2 equalizer,” in Proc. NTC’77, Dec. 1977.

[15] R. D. Gitlin and S. B. Weinstein, “Fractionally spaced equalization: An improved digital transversal equalizer,” Bell Sysr. Tech. J . , vol. 60, pp. 275-296, Feb. 1981.

[16] E. Eleftheriou and D. D. Falconer, “Restart methods for stabilizing FRLS adaptive equalizers in digital HF transmission,” in Proc. GLOBECOM’84, Atlanta, GA, Dec. 1984.

[17] -, “Steady-state behavior of RLS algorithms,” in Proc. IEEE ICASSP’85, Tampa, FL, Mar. 1985.

[18] -, “Tracking properties and steady state performance of RLS adaptive filter algorithms,” IEEE Trans. Acoust., Speech, Signal Processing, to be published.

[19] J . M. Cioffi, “Fast transversal filters for communications applica- tions,” Ph.D. dissertation, Stanford Univ., Stanford, CA; 1984.

[20] F. Ling and J . G . Proakis, “A generalized multichannel least squares lattice algorithm based on sequential processing stages,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp. 381-389, Apr. 1984.

I211 E. H. Satorious and S. T. Alexander, “Channel equalization using adaptive lattice algorithms,” IEEE Trans. Commun., vol. COM-27, pp. 899-905, June 1979.

1439-1446, Oct. 1978.

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Technol., vol. COM-14, pp. 767-784,’ Dec. 1960. Greece.

ELEFTHERIOU AND FALCONER: EQUALIZATION TECHNIQUES FOR n F CHANNELS 247

David D. Falconer (M’68-SM’83-F’86) was born in Moose Jaw, Sask., Canada, on August 15, 1940. He received the B.A.Sc. degree in engi- neering physics from the University of Toronto, Toronto, Ont., Canada, in 1962 and the-S.M. and Ph.D. degrees in electrical engineering: from M.I.T., Cambridge, in 1963 and 1967, resp@- tively.

After a year as a Postdoctoral Fellow at the Royal Institute of Technology, Stockholm, Swe- den, he was with Bell Laboratories, Holmdel, NJ,

from 1967 to 1980, as a member of the Technical Staff and later as Group Supervisor. During 1976-1977 he was a Visiting Professor at Linkoping University, Linkoping, Sweden. Since 1980 he has been at Carleton Uni- versity, Ottawa, Ont., Canada, where he is a Professor in the Department of Systems and Computer Engineering. His interests are in digital com- munications, signal processing, and communication theory.

Dr. Falconer is Editor for Digital Communications for the IEEE TRANS- ACTIONS ON COMMUNICATIONS and is a member of the Communication The- ory Technical Committee. He is also a member of the Association of Pro- fessional Engineers of Ontario, Sigma Xi, and Tau Beta Pi. He was awarded the IEEE Communications Society Prize Paper Award in Communications Circuits and Techniques in 1983 and in 1986.