Data Directed Estimation Techniques for Single-Tone Hf Modems

DATA DIRECTED ESTIMATION TECHNIQUES FOR SINGLE-TONE HF MODEMS

by Frank M. Hsu

GTE Government Systems CorporationCommunication Systems Division

77 "X' StreetNeedham Heights, MA 02194-2892

ABSTRACT

This paper is concerned with data directed estimation (DDE)techniques for use in single-tone HF modems. The proposedscheme uses least-squares adaptive techniques for dealing withfading and multipath radio HF channels. The DDE is based on

direct estimation of channel parameters and data blocks.Training data is imbedded into the transmission in the form ofblocks of training bits alternated with blocks of source data at

a fixed (though selectable) fraction of the transmission.

Two DDE techniques, i.e., linear and nonlinear, are discussedin this paper. In the linear DDE, the source data block isestimated by a Levinson recursive algorithm. In the nonlinearDDE, only the end symbols of the data block are kept, quan-

tized, and subtracted from the simultaneous equations. Thesymbols left in the block are estimated by a Levinson algorithmagain, and so forth. The procedure described here is used recur-

sively until all the known symbols are obtained. In both linearand nonlinear DDE techniques, the channel is estimated by a

steepest-descent algorithm.

As with a linear equalizer, the linear DDE lacks the abilityto cope with the fading dispersive HF radio channels. However,the nonlinear DDE works well under the same channel condi-tions. The analysis shows that the nonlinear DDE can achieveperformance similar to the decision-feedback equalization (DFE)technique with known channel conditions. The channel estimatorused in the DDE techniques will converge faster than the DFE(see Section VII) if a similar adaptive algorithm is utilized. Asingle-tone HF modem based on the DDE technique has beenimplemented on a digital signal processor at GTE.

I. SUMMARY

Commercial and military demands for greater communicationreliability, lower error rates and higher data rates in the transmissionof digital information over dispersive media, such as HF ionosphericchannels, are continually increasing. There are two classicalapproaches for dealing with fading and multipath. The traditionalconfiguration for an HF modem uses multitone modulation (Kineplex)with guard time inserted between adjacent symbols for avoiding theintersymbol interference. The transmitter based on multitone tech-niques must be capable of supporting a high peak-to-average power

ratio in the range of 8-9 dB. This means that the average transmittedpower is about 8-9 dB below the maximum deliverable power of the

transmitter. Consequently, the power loss in a traditional multitonemodem is enormous. The amount of signal energy available for bitdecisions is further lowered by using the guard time since only a

portion of each received signaling element is used in the integrationinterval. Significant spectral efficiency is lost by the requirement ofseparating the tones by an amount larger than l/T, where T is thesymbol interval. A spectral efficiency of 0.8 bps/Hz is typical in4-phase multitone DPSK modems. Finally, the performance of(uncoded) multitone transmission schemes can be seriously degradedwhen spectral nulls in the channel fall on or near one of the tone

frequencies used. [1]

A second configuration for an HF modem uses single-tone M-aryPSK modulation and a data directed estimation (DDE) demodula-tion techniques. In contrast to the multitone case, a single tone modemin principle suffers from none of the above consequences. Both higherdata rates and low peak-to-average ratio can be achieved by a singletone modem.

For clarity, some possible modulation alphabets, block parameters,data rates, percentage of training, throughput and maximum allowablemultipath spread for an HF modem using the DDE techniques are

listed in Table 1. The block parameters, M and N, denote the sizesof the block for source data and training bits respectively. The modemproposed here uses a 2,400 keying rate and a 3 kHz bandwidth. FromTable 1, it is clear that the single tone modem using the DDE tech-niques may achieve a data rate up to 1 bps/Hz or higher.

Table 1. Modulation Alphabets, Block Parameters, Data Rates,Percentage of Training, Throughput and MaximumAllowable Multipath Spread for HF Modem Using the DDETechniques with 2,400 keying Rate in 3 kHz Bandwidth.

12.4.1CH85CH2202-0/85/0000-0271 $1.00 IEEE

271

MaximumPercentage Allowable

Modulation Block Data of Through- MultipathAlphabets Parameters Rates (bps) Training put Spread

44 M=20, N=20 2,400 50% 50% 8.3 mspsk M=32, N=16 3,200 33'h% 6623 % 6.6 ms

M=36, N=12 3,600 25% 75% 5.0 ms

84 M=20, N=20 3,600 50% 50% 8.3 mspsk M=32, N=16 4,800 3351% 662A% 6.6 ms

M=36, N=12 5,400 25% 75% 5.0 ms

164 M=20, N=20 4,800 50% 50% 8.3 mspsk M=32, N=16 6,400 3351 % 6625 % 6.6 ms

M=36, N=12 7,200 25% 75% 5.0 ms

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Two DDE techniques, i.e., linear and nonlinear, will be discussedin this paper. In the linear DDE, the source data block (bo, b1

, bM-l) is estimated by a Levinson recursive algorithm. In thenonlinear DDE, only b0 and bm_ obtained from a Levinsonalgorithm are kept and b1 to bM-2 discarded in the first step of estima-tion. The estimates, bo and bM 1 are then quantized to the nearestsymbols, bo and bM 1, respectively. The decisions, bo and bM1, aresubstracted from the simultaneous equations. The M-2 symbols (bl,b2, . . . , bM 2) left are estimated by a Levinson algorithm again,and so forth. The procedure described here is used recursively untilall the unknown symbols are obtained. In both linear and nonlinearDDE techniques, the channel is estimated by a steepest-descentalgorithm.

II. MODULATIONS

It is mathematically convenient to represent the digital communica-tion system in a baseband formulation. All signals and filters in thispaper are written in complex-valued, lowpass equivalent form.Suppose an information sequence {In} modulates a basic transmit-ting filter pulse s(t) at a rate I/T. Then the total transmitted signal is

00

s(t) = E Iks(t-kT), (2-1)k=0

where T is the duration of the signaling interval and Ik is the time-independent information symbol that is transmitted in the intervalkT <t < (k+1)T. The information symbols in (2-1) may be complex-valued in general. Is is convenient to assume that the symbols {Ik}are statistically independent, with zero-mean and unit variance.

Three types of modulation fit the mathematical formulationpresented above. First, Ik can take one of M real values, equallyspaced about zero, in a pulse-amplitude-modulated (PAM) signal.Second, an M-ary phase-shift keying (MPSK) signal is obtained ifone allows Ik to take any one of the M values exp[2?ri(m-1) /M],where m =1, 2, . . ., M. Finally, a hybrid PAM-PSK modula-tion also fits the formulation in (2-1).

III. A DISCRETE-TIME CHANNEL

A problem encountered in high-speed serial data transmission overmultipath radio channels is intersymbol interference. The intersymbolinterference is directly caused by multiple propagation modes or pathswith different path delays. The multipath spread of a long distanceradio link (e.g., 4,000 miles) is about 5 milliseconds. Specifically,the transmitted symbol rates on such a channel are of the order of2,400 symbols per second. Hence, intersymbol interference willextend over 12 symbols or so.

The HF channel is modeled at baseband as a tapped delay line withtime-varying coefficients followed by additive zero mean whiteGaussian noise (WGN), n(t), with variance a2. The tapped delay linehas taps spaced at the symbol interval (T) and nonzero tap coeffi-cients corresponding to discrete multipath propagation. The tapcoefficients are assumed to be independent complex Gaussianrandom variables {fi, i = 0, . . . , L} with zero mean and fixed

variances, t[ If 12]. (Note: The sign t indicates an ensemble average.)The tap coefficients are generated by filtering white Gaussian noisethrough a filter whose bandwidth (B) is on the order of the fade rate.

IV. DATA-DIRECTED ESTIMATORS

The proposed DDE techniques use single-carrier transmission andleast-square adaptive techniques for dealing with fading and multipath.The DDE techniques are based upon direct estimation of channelparameters and data blocks. Training data is imbedded into thetransmission in the form of training blocks alternated with sourcedata blocks at a fixed (though selectable) fraction of the transmission.

a) Signal Structure

Three types of modulations (i.e., PAM, MPSK and hybrid PAM-PSK) discussed in Section II are suited for the DDE techniques.Generally, the MPSK modulation is favored for transmission on theHF radio channels. The decision block of the DDE depicted in Figure1 consists of three subblocks, one block with N known symbols (ao,a,, . . ., aN- 1), one adjacent source data block with M unknownsymbols (bo, bl, . . ., bm-1) and another block with N knownsymbols (aN, aN+I, . . , a2N 1) which is adjacent to the previoussource data block. Both a's and b's are the data symbols (i.e., IK)which are described in Section II. A data frame consists of one trainingblock (i.e., block with N known symbols) and one adjacent sourcedata block. Signals are transmitted in successive frames. The infor-mation throughput, (I, and fraction of training, (T, are given by

M-M+N

NM+N

(4.1)

(4.2)

Known Unknown KnownN Symbols | M Symbols N Symbols

a2N-1 . . . aN+IaN bM_1 . . . bibo aN-I . . . alaO l

1 Frame I{ - 1 Frame

Decision Block

Figure 1. Signal Structure of Data-Directed Estimator

For example, if we use 80-PSK modulation, 2,400 keying rate andset M and N equal to 32 and 16, respectively, then the percentageof training is 33 /3 % and the source data rate is 4,800 bps. Table1 lists some possible modulations and data rates.

b) Channel Equations

The HF radio channel can be modeled as a discrete-time channelmodel as given in Section III. Generally, the channel weights canbe written in vector form as

f' =fL fL- I . * fo ] (4.3)

where the tap spacing is one symbol interval, T, LT is the multipathspread of the channel (N > L) and t denotes the transpose. Since

12.4.2272


the channel is very slowly time-varying, we may assume that the chan-nel is fixed in a decision block. Then the received signals withoutadditive noise are given by

ftft

ft

Al

B

A2

where

Alt = [ao a, . . . aN- 1]

Bt = [bob, . . . bM-1]

A2t = [aNa N+1 . . . a2N-11

(4.4)

fofl fo 4

fL-1 fL-2 fO

A3t = [aN-LaN -L+1 . . . aN-1J

M+L

I

(4.11)

(4.12)

and

A4t =[aN aN+L . . . aN+L-1] (4.13)

In (4.8), M1I M2 and M3 are the channel matrices which depend(4.5) on the channel parameters (weights). These matrices are known once

the channel is estimated from the data symbols. In (4.8), M2A3 and(4.6) M3A4 are postcursors and precursors, respectively, due to the train-

ing symbols. These terms can be substracted from the received signals.(4.7) If the white noise is added to (4.8), then the channel equations become

r' is a column vector with M +2N -L elements and 0 representselements with zero values.

In (4.4), the equations with a's and b's are given by_11= MIB + M2 A3 +M3 A4 (4.8)

where

1- M ,|

(4.14)r=-M1B + M2A3 + M3A4 + n

where

n1t = [nN-L nN-L+t . . . nM+N-11 (4.15)

The channel equations are used for solving the data symbols.

c) Channel Parameters Estimation

For a time-varying channel, the received signal can be represented as

fofIf2

fofI fo

fL . . 0fI fo

fI fL- I

fL

f2l J

M+L

fL l

J. f

rk = fki Ik + nk (4.16)

where

(4.17)

M+L (4.9) Ik= [Ik k+l . . .k+L] (4.18)

k represents the symbol time and I stands for a or b. The channelparameters (weights) can be updated by a least-squares algorithm.The least-squares criterion for choosing fk is to minimize the meansquare error which is given by

El = t [ Irk - rk ] (4.19)

where

rk = fk1 Ik (4.20)

represents an ensemble average and Ik is an estimate of the datasignal. The symbol estimate, Ik, is equivalent to ak and bk, where

(4.10) bk is the symbol decision of the source data, for training and source

data symbols, respectively. The symbol decision, bk, which is ob-tained by the data estimator, may be assumed to be equal to bk. Clear-ly, errors in the detection of the data sequence (i.e., bk * bk) will

12.4.3273

fofI

fL

fo

ffL-I . . .

fL . *

fL .

4'

M42 = fL fL 1 .

fL

L

ft

fil = [fL (k) fL-l (k) . . . fo (k) Ik


result in degraded performance of the channel estimator. The chan-nel weights can be estimated by a steepest-descent adaptive methodgiven by

(4.21)-fk = fk-1 + AekIk

where

ek = rk - rk (4.22)

In (4.21) and (4.22), ek is the error signal and A is a smallconstant 0 < A < < 1. During the estimation of the data sequence,{ bk, k = 0,1, . . ., M-1}, the channel weights are frozentemporarily. The channel weights can be updated again once theunknown symbols are determined.

with respect to B yields to

B = (Ml* Ml)-l Mi*tC (4.27)

Equation (4.27) may also be obtained by the orthogonality prin-ciple given as

Mi* (M1B-) = 0 (4.28)

Both Ml and C are known, given the channel parameters attainedfrom the channel estimators. Therefore, the data, B, can be obtainedby using (4.27). However, the matrix pseudo-inverse shown in (4.27)is burdensome in terms of computation.

To avoid this problem, let us define the channel correlation matrix as

Based on simplified models, the Differential Doppler can be trackedindependently. The term "Differential Doppler" refers to the differencein mean Doppler shift between the paths. The term "Doppler Spread"refers to the RMS bandwidth of the Rayleigh fading process associatedwith each path. The Differential Doppler phase shifts {{e,e=0,1, . . . , L} can be estimated adaptively by

I'f,k = Of, k-i + 61 'm (fe, k I ek)

RM = Ml*tMl

and substituting (4.9) into (4.29), we have

RM =

(4.23)

f=o, 1, . . . , L

where Im (x) denotes the imaginary part of x and Al is a smallconstant 0<A1 < <1. The channels weights, {ftekf=0,1, . . . ,L}, can be estimated by least squares method withDifferential Doppler correction as given by

f' =-f ei frk f=0,1, L-k k f,k .=0.l.

To T1 T2 . .

T To 1 *. . . T

(4.24) where

where {fek' f=0,1, . . . ,L} are the channel weights estimatedaccording to (4.21).

d) Linear Data-Directed Estimator

Once the channel has been synthesized we may use the channelinformation for solving the unknown symbols. The data can beestimated by using the least mean square error (LMSE) criterion. In(4.14), if the precursors and postcursors introduced by the trainingsymbols are subtracted from the received signals, then the channelequations become

r1 - M2A3 -M3A4 = M1B + n1= C

Minimization of the mean square error (MSE) defined as

E2 = (MlAB, 5)*t (Ml B-O

LTo0= F, If, 12

i=O

L-1lTi = - fi+ifi

i=O

TL =fLf O

Te = 0 for f > L

(4.25)

(4.26)

12.4.4

274

(4.29)

(4.30)

(4.31)


The matrix RM is a Toeplitz matrix. Equation (4.27) can be rewrit-ten as

RM BM = GM (4.32)

bM -1,M-Iwhere

GM = M1 *tC

G'M = [g1 92 . . . gM]

T0o- [T1T2 T. . TM-I]

(4.33)

(4.34)

Ct = [CN-L CN-L+1 . . . CM+N-1]

and

L

| = - fi* CN-L+ii=O

L

92 E fi* CN-L+I+ii=O

L

lgM = fi* CM+N-L-+i=O

(4.35) e) Nonlinear Data-Directed Estimator

It is well known that multipath radio channels frequently possessnulls in their time-variant spectral characteristics. A linear filter (orestimator) would be inadequate to mitigate the intersymbol interferenceon these channels. Frequently, the estimates

BM = [130b1 . . . bM-1] (4.44)

obtained by linear data-directed estimation (LDDE) indeed have a(4.36) nonuniform variance distribution. The variance of the estimates of

the end symbols (i.e., bo and bM_l) is sometimes a hundred timessmaller than that of the middle symbol estimate (i.e., bM/2). Usual-ly, the variance peaks at the middle symbol and decreases at bothends symmetrically. Because the intersymbol interference due to train-ing symbols is subtracted from the simultaneous equations as givenin (4.25), the end symbols, bo and bM-1, do not have postcursors andprecursors, respectively. For clarity, (4.25) is rewritten as

If the order of simultaneous equations given in (4.32) is reversed,then (4.32) becomes

RMBIM = GM,

C-n, =

(4.37)

where

(Gr)t = [g_gM1 *g1o

(GMr) = [gM gM_1 .. . g1]

(4.38)

(4.39)

and r denotes the order reverse. Eqs. (4.37) can be solved recursivelyby the Levinson algorithm as given by

OM-2,k = OM-3,k - FM-2,M-2 X M-3,M-3-kk=0, 1, . . . ,M-3

(4.40)

fobofobI + f1bo

fObL + flbL-I + . . . + fLbofobL+l + flbL + . . . + fLbl

fobM-i + flbM-2 + .

flbM 1 + .

MTM-I - [T1 T2 *. TM-2]

fM2,M-2 =

9 tr- [rr2 * * 'M-2]

E'M-3,M-3

L OM-3,0

M-3,0

L M-3,M-3

(4.41)

It is clear that bo and bM-l should have the best estimates in theLMSE solutions. Therefore, it is proposed that only bo and bM-lalone are kept and b1 to bM-2 discarded in the first step of theestimation procedure. The estimates, bo and bM-1 are then quan-

tized to the nearest symbols, § and bm-1, respectively. The decisions,

bo and bM-1, are subtracted from (4.45) The new channel equationsare given by

(4.46)M1(M 2) BM-2 + n1(M 2) = CM-2

bM- 1k =bM-2,k -bM-,M-1 M-2,M-k-2, k=O, * *, M-2 (4.42)

12.4.5275

bM-2,M-2

bM-2,0

M-2,0

0 M-2,M-2

(4.43)

(4.45)

* . + fLbM-L-1. . + fLbM-L

fLbM-I

=--4

.0

gm [TI T2 ... TM-j


where M1(M-2) is a (M+L-2)-by-(M-2) matrix given as (4.9),_1(M-2) is a noise vector with (M-2) elements and

CM-2 = C N-L+IC'N +C'N-L+2

C'NcNC N+l

C'M+N-L-IC M+N-L

C'M+N-2

CN-L+l -f1OCN-L+2 -f2b

CN -CNCN+I

CM+N_CM+N-

L-1 - fObM-1-L - flbM-1

CM+N-2 fL-1bM-1

(4.47)

The solution of BM-2 according to LMSE criterion is given by

R* -RM-2 BM-2 = GM-2 (4.48)

where R is given in (4.30),

Consequently, the NDDE ensures that the performance will not bedegraded in each step of estimation if the feedback symbol decisionsare correct.

V. COMPUTATIONAL COMPLEXITY OF THE DATA-DIRECTED ESTIMATORS

In the DDE techniques both channel and data are computed by usingdata-directed symbols - training symbols or information symbols inthe form of symbol decisions.

If the channel is estimated by the steepest descent method (4.21)in the DDE techniques, then the computational requirements per block(M + N symbols) of the DDE techniques are listed in Table 2.

TABLE 2. Computational Requirements of the DDE Techniques

Complex Multiplication Reciprocal Storage

LDDE M2+ 6LM + 3M + 2LN + 2N M 3M + 3L

1 LM2 - i L2M + 9 M2 + 1 M3 +2 4 8 12

NDDE I1 + IL2 + 23 LM + 2LN + M 1 M2+5M+3L12 8 4 4

2 M + 2N--1L12 12

BM'2 = [b'1 b'2 . . . bM-2] (4.49)

LE f*C'i N-L+l+ii=O

Lv, { c,i N-L+2+i

i=O

LE fi*CM+N-L-2+ii=O

VI. THE MINIMUM MSE PERFORMANCE OF THE DATA-DIRECTED ESTIMATORS

The theoretical minimum mean square error (MSE) performancefor the data-directed estimators (DDE'S), linear equalizer (LE) anddecision-feedback equalizer (DFE) can be evaluated. It is shown in(4.27) that the solution of the data symbols is given by

(6.1)B = (MI *tM)- M1*I(4.50)

where

C = M1B + n1

The square errors of a decision block may be expressed as

PM = (B -B)*1 (B - B)

= nl*'M1 (M,*tM )-l (Ml*tM_)-l Ml*tnl

(6.2)

(6.3)

Equations (4.48) can be solved by the Levinson algorithm given in(4.40) to (4.43) again. It should be mentioned that only G given in(4.50) and B given in (4.42) and (4.43) should be recomputed in thisstep. The other terms are unchanged and need not be computed again.The procedure described here is used recursively until all the unknownsymbols are obtained.

The method proposed above is actually a nonlinear data-directedestimation (NDDE) technique. Since the amount of intersymbol in-terference is reduced in each step of estimation, the variance of b1(or bM-2) is either equivalent to or less than that of bo (or bM-1).

The square errors given in (6.3) may be rewritten as

PM = e e

where

et= [eoeI . . . em-1]

and

e = (M1 *t4MI) M1 *n

12.4.6

276

and

GM-2

g M-2

(6.4)

(6.5)

(6.6)


Suppose that the inverse of the channel correlation matrix is denoted as

M)1-= (MI*1M1) -

p -111

P-121

,-1

M,1

p -112

p -1"22

P-1M,2

p -11,M

p -1* . . 2,M

p -1M,M

(6.7)

Then the minimum MSE of the first symbol can be expressed as

E2 (eo) = [Ieo 21]

= t [ P1-M *tnl CL -lMl*tn,) *t ]

where

p -1 1 p -1 p -1-1 =[ 11 , 12 X . . . 9 1,M]

Equation (6.8) can be rewritten as

E2 (eo) = P11 Ml*' [ n,nl*t ] M1 P1 *t

= a2 P-1 R* P -I*tM -1

= Or2 [ 1, O, O . . . o ] PI-I*t

= a2 P-111

where a2 is the noise variance. Similarly, the minimum MSE of theith symbol can be obtained as

E2 (ei-1) = a2 Pii 1

The minimum MSE of the DDE can be computed easily if the chan-nel is given. For example, let us use a maximum dispersive channel[2] as given by

[c 'c c

32 c (6.12)

Then the minimum MSE of each symbol in the LDDE (20) at SNR= 20 dB is listed in Table 3. The shorthand notation LDDE (M) is

(6.8)

used to indicate M unknown symbols in the LDDE. Table 3 clearlyindicates that the estimation variance of the symbols in each block

has a bell-shaped distribution. Usually, the variance peaks at the middlesymbol (or symbols) and decreases to both ends symmetrically. Thevariation of

TABLE 3. The Minimum MSE of Each Symbol in the LDDE (20) at

SNR = 20 dB

variance in each block could be very large. In Table 3, the varianceof symbol #1 is 66.7 times smaller than that of symbol #10. Con-sequently, the linear DDE will not perform well in typical HF radio

(6.9) channel conditions. If the NDDE (20) is used in the same channelconditions as given in (6.12), then the minimum MSE of each symbolin the block at SNR = 20 dB is listed in Table 4. The variance inthe NDDE (20) is computed by assuming that the feedback symboldecision is correct. Errors in the detection of the end symbols willresult in degraded performance of the NDDE. Table 4 clearly showsthat the variance of the NDDE has a bowl-shaped distribution. Thevariance peaks at both ends and decreases to the middle symbol sym-

(6.10) metrically. Clearly, the nonlinear data-directed estimator outperforms

the linear counterpart greatly.

TABLE 4. The Minimum MSE of Each Symbol in the NDDE (20) at

SNR = 20 dB

Symbol MSE Symbol MSENumber Number

1 .0498 11 .01802 .0489 12 .02863 .0477 13 .03504 .0463 14 .03935 .0446 15 .04236 .0423 16 .04467 .0393 17 .04638 .0350 18 .04779 .0286 19 .048910 .0180 20 .0498

(6.11)

12.4.7277

Symbol SymbolNumber MSE Number MSE

1 .0498 11 3.322 .212 12 3.163 .506 13 2.854 .916 14 2.425 1.41 15 1.936 1.93 16 1.417 2.42 17 .9168 2.85 18 .5069 3.16 19 .21210 3.32 20 .0498


The minimum MSE of the linear and decision feedback equalizerscan also be evaluated. [3] It is shown that the corresponding minimumMSE obtained with the LE is given by

KE2 (2K+1) = 1 - E cjf

j=-K(6.13)

where

K (6.14)E cjAmj = fm*, m = -K, . ,-1, 0, 1, . ,K

j=-K

LAmj = f,*Mfj_m+. +U2 bjm (6.15)

i=O

and 2K+1 is the total number of taps. Equation (6.13) can berewritten as

E2 (2K+1) = 1 - f*r (R2K+1 ± a2I) IL (6.16)

where

(2K+1) elements-L =>"0, * * "* fo f1. . . fL °' ] (6.17)

K elements

R2K+I is a (2K+1) -by- (2K+l) matrix given in (4.30) and I isan identity matrix.

For a DFE, the corresponding minimum MSE is given by

NE2 (N+1) = 1 - E ajfj (6.18)

j=0

and the DFE has (N+1) feedforward and L feedback taps. Equation(6.18) can be expressed as

E2 (N+1) = 1 - f* (RN+1 + a2 - H*tH)-1 fD

(N+1),elementsfDt = [fo f1 * * *, fL, 0, . . ., 0]

fl f2 . . .

f2 f3 - * fI

fL 0 . . .

0 0...

0 0...

fL 0 . . .

FL 0 0 . . .

00

. . . . .0

. . . . .0

.0. . . .O

(6.21)

(6.22)

(6.23)

and RN+1 is a (N+1) -by- (N+1) matrix given in (4.30). Theminimum MSE performance of a LDDE(M), a NDDE(M), aDFE(M,N) and a LE(M) is shown in Figures 2 to 7. For simplicitythe shorthand notation LDDE(M) is used to indicate M unknownsymbols (size of source data block) in the LDDE, the NDDE(M)indicates M unknown symbols in the NDDE, the DFE(M,N) indicatesM forward taps and N feedback taps in the DFE, and the LE(M)indicates M total taps in the LE. In Figures 2 to 7, the same max-imum dispersive channel given in (6.12) is used except two differenttap spacings of the channel are shown in these figures. In general,the performance of four techniques discussed in this paper dependson the tap spacing, the number of taps in the equalizers or block sizesin the estimators. The linear equalizer and linear data-directedestimator perform poorly in a maximum dispersive channel (withspectral null in channel characteristics). On the other hand, thenonlinear equalizer (DFE) and nonlinear data-directed estimatorperform superbly in the same maximum dispersive channel. TheNDDE can achieve performance equivalent to the DFE.

where

N, ajBmj = fm*, m=0, 1, . . ., N

j=O

mBmj = E fti fi+j-m + a2± jm

i=O

12.4.8278

(6.19)

(6.20)


us

0

4

0

r

cr

'az

,,

2

SNR (dB) 7193-82

Figure 2. LE (5), DFE (5, 2) and LDDE (5)

0*u

4cr

a4)z

Z.

SNR (dB) 7194 82

Figure 3. LE (11), DFE (11, 10). and LDDE (11)

0a:a:

wJ

a4nz

w

SNR (dB)

Figure 4. LE (21), DFE (21, 2), and LDDE (21)

cr

cta

z:

5-

SNR (dB)

Figure 5. LE (21), DFE (21, 2), and LDDE (21)

12.4.9279

7195-82

7196-82


go lo-1

LuLcJDa:a

E 0-

io-

m

50

w

zS

z

E

SNR (dB)

SNR (dB)

Figure 6. LE (21), DFE (21, 2), and NDDE (20)

VII. CONCLUSIONS

Both linear and nonlinear data-directed estimators (DDE) for digitalcommunication over dispersive channels are investigated in this paper.The proposed schemes use single carrier (or tone) transmission andleast-squares adaptive techniques for dealing with fading and multipath.However, in contrast with earlier methods, which use decision-feedback equalization (DFE) [4], [5] and symbol-by-symbol equalizeradaptation, the DDE techniques are based upon direct estimation ofchannel parameters and data blocks using data-directed signals. Train-ing data is imbedded into the transmission in the form of training blocksalternated with source data blocks at a fixed (though selectable)fraction of the transmission.

As with a linear equalizer, the linear DDE lacks the ability to copewith the fading dispersive HF radio channels. However, the nonlinearDDE (NDDE) works well under the same channel conditions.

The minimum MSE calculation indicates that the NDDE willperform as well as the DFE with known channel conditions. It isrecognized [3] that the spread of eigenvalues of the channel correla-tion matrix may slow down the convergence of a DFE. However, thereis no spread in the DDE case if uncorrelated data is used. It is

Figure 7. LE (21), DFE (21, 2), and NDDE (20)

conjectured that the channel estimator used in the DDE techniqueswill converge faster than the DFE if a similar adaptive algorithm isutilized. A single tone HF modem based on the DDE techniques hasbeen implemented on a digital signal processor at GTE.

References

[1.] Anderson, P.H., Hsu, F.M. and Sandler, M.N., "A NewAdaptive Modem for Long Haul HF Digital Communicationsat Data Rates Greater Than 1 bps/Hz," MILCOM, Boston, Mass.17 Oct. 1982.

[2.] Austin, M.E., "Decision-Feedback Equalization for DigitalCommunication Over Dispersive Channels," Lincoln Lab.Technical Report 437, 11 Aug. 1967.

[3.] Proakis, J.G., "Advances in Equalization for Intersymbol Inter-ference," in Advances in Communication Systems, Theory andApplications, edited by Viterbi, A.J., Academic Press, 1975.

[4.] Hsu, F.M., "Square-Root Kalman Filtering for High-Speed DataReceived Over Fading Dispersive HF Channel," IEEE Trans.Inf. Thy., Sept. 1982.

[5.] Giordano, A.A., and Hsu, F.M., Least Squares Estimation withApplications to Digital Signal Processing, John Wiley & Sons,1985.

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Data Directed Estimation Techniques for Single-Tone Hf Modems

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