IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO. 3, ,MAY 1988 537

Detection of Baseband Signals Using a Delta Modulator

Ahshstruet -The performance of a digital delta modulator when used as a repeater for binary non-return-to-zero signals between a noisy channel and an error-free link is analyzed. A Markov chain is used to model the accumulator output (observation process) of the device, and a Bayes test with probability of error criterion is formulated to evaluate its performance in additive band-limited Guassian channel noise. The key design parame- ters of sampling rate and accumulator range are varied for computation of performance degradation relative to the optimal matched filter. Numerical results are presented that demonstrate the feasibility of this technique.

I. INTRODUCTION

CLASSICAL problem in digital communications is A the analysis and design of receivers that must perform a binary detection on the individual bits of a transmitted NRZ (non-return-to-zero) digital data signal. This is tradi- tionally performed by an analog integrate-and-dump de- vice, which under conditions of additive white Gaussian channel noise can be analytically shown to be optimal in the sense that the probability of error is minimized. Here we consider the detection problem where a digital delta modulator is used as a repeater between an analog signal- plus-noise channel and an error-free digital link. This channel may have a signal present that is from a set of many types, e.g., voice or digital data of various rates. Assuming there is no prior knowledge of the signal type present in the channel, we are considering the case when this signal is NRZ digital data. (Obviously, given that this is known, any detector applied to the delta modulated output will have degraded performance.) The analysis in this paper is motivated by the potential of obtaining a simple device for the detection of binary NRZ signals with a tolerable performance degradation from optimal. The device that implements this technique is called a delta detector.

The investigation of the use of a delta modulator struc- ture for detection appears to fall between two areas of research. On one hand, a wide variety of aspects of delta

Manuscript received August 12. 1985; revised August 18. 1986. This paper was presented at the IEEE International Symposium on Informa- tion Theory, Brighton, England, June 24-28, 1985.

P. G. Flikkema is with Techno-Sciences, Inc., Suite 620. 7833 Walker Drive. Greenbelt, M D 20770.

L. D. Davisson is with the Department of Electrical Engineering. University of Maryland, College Park, MD 20742.

IEEE Log Number 8821207.

modulation have been analyzed. Representative is the analysis of a basic delta modulation system by ONeal [l], where signal-to-quantizing noise ratios for Gaussian inputs are developed. The evaluation of the properties of a delta modulator in a general setting has been performed by Fine [2]. The other area of research is the investigation of optimal detection strategies for sampled and quantized signals. A good point of departure is the paper by Kassam [3]. The more specific case of the digital matched filter, in which the integration is approximated by a discrete-time accumulation of quantized samples, has also received con- siderable attention [4], [ 5 ] .

The analysis presented here shows that the accumulator output of the delta detector can be modeled as a finite-state homogeneous Markov chain for which the transition and stationary probabilities can be found. Then, using the accumulator outputs of the delta detector as observations of the signal plus noise, a Bayes test with a probability of error criterion is formulated to determine the performance relative to that of the matched filter. Two cases are ex- amined. In the first, the entire history of the accumulator output for each bit is used as the observation, providing a performance upper bound for the technique. Then a scheme in which only the final accumulator output for each bit is processed for detection is examined to determine the per- formance of a simpler approach. Finally, the analysis is implemented by computer to give numerical results as a function of the key parameters of sampling rate and accu- mulator range.

11. DELTA DETECTOR MODEL

A block diagram of the system under consideration is shown in Fig. 1. It demonstrates that the delta detector is exactly the same as a delta modulator in that it implements a one-step predictor to estimate the next sample of the incoming signal.

Referring to Fig. 1, the quantization step size (or scaling factor) is A , and the quantized unscaled error process { FnT, } is a sequence composed of + 1’s and - 1’s. In the sequel, the sampling interval 7 will be assumed under- stood; therefore, FnT, = F,. Using this convention, the error process is

e, = x, - x,,

where in is the estimate (or predicted value) of x,,. The (1)

0018-9448/88/0500-0537$01.00 01988 IEEE

538 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO. 3, MAY 1988

r - - - - - 1 I Delta Detector

I 1

&--q--$ mulator I bl decision

-----I

Fig. 1. Delta detector system block diagram.

quantizer is represented by s g n ( . ) so that the quantized error is

+1, e , > 0 zn=sgn(en)=( -1, en<O.

the rate of 1/T, so that

x , = x ( t = nT, ) = s, + w, (8)

(2) where {s,} and { w , } are the filtered and sampled signal and noise processes, respectively. To define the sampling

The accumulator holds the previous unscaled estimate ZnP1 and is followed by a delay of one sample. Therefore,

3, = Znp1 + (3)

2, = A * 3, (4)

The estimate of x, is

from which it follows by substituting (3), (4), and (1) that 2, = 2,- + A * C,, -

( 5 ) - - x , - ~ - enPl + A-Z, ,_ , . The predicted value 2, thus differs from the previous actual sample x n W 1 by the quantization noise enPl - A. I

en-1.

rate, let T = NT, so that we have N samples per received bit since T is assumed to be unity. We assume the follow- ing two conditions in the sequel.

1) The sampling rate 1/T, is equal to the Nyquist rate of the filtered signal, i.e., T, = 1/2B. It follows that B = N/2 T.

2) The filter bandwidth (as determined by the number of samples per bit N ) is large enough so that the sampled signal { s, } can be assumed to be a sequence of + 1’s and - 1’s determined by the value y,, i.e., g( t ) passes through the filter with negligible change. At the output of the low-pass filter the noise has the spectral density

and the corresponding autocorrelation 111. PERFORMANCE ANALYSIS

equiprobable and normalized to a bit length of 1 s and an amplitude of f 1 V. Specifically, we let

s ( t ) = c Y k d t - k),

s i n 2 ~ B r The transmitted NRZ signal s( t ) (see Fig. 1) is assumed R w ( r ) = NoB- (10)

~ T B T . As is well-known from the sampling theorem, this function

00 has nulls at r = n / 2 B , n = f l , f 2 ; . . , so that from condition 1) the sampled noise process n, is a sequence of i.i.d. Gaussian random variables with variances Rw(0) = NoB. It follows from the form of s ( t ) and condition 2) that the sampled signal plus noise process { x , } is independent (given s ( t ) ) and Gaussian with variance N,B and mean

k = O

O < t < l ( 6 )

where { y , } is a sequence of independent identically dis-

consider the accumulator output process { 3, } over a single Pr[y,=-1]=1/2.

Gaussian noise with two-sided spectral density N0/2 and (including the initial accumulator value 30) which takes then filtered by an ideal low-pass filter with bandwidth B values (limited by the accumulator length without over- Hz before reception at the delta detector. Define the flow) in the set s= { - M , - M+l, . . .,o,l,. . .,

Hence the accumulator range is R A 2 M + 1. To set the corrupted and filtered signal as

~ ( t ) A s f ( t ) + w ( t ) (7) input overload point, let MA = A =1 so that 2,= mA, m E S. Substitution of (2), (l), and (4) into (3) yields

where s f ( t ) and w ( t ) represent the filtered signal and noise, respectively. In the delta detector, x ( t ) is sampled at x , = ~ , - ~ + s g n ( x , - , - A.3n-l).

= s ( t = nT,). tributed (i.i.d.1 Etndom variables such that Pr [ Y k = ‘1 = To investigate the performance of the delta detector,

is corrupted by additive white bit duration. This process is a sequence of length N + 1 In the channel,

(11) - I

FLIKKEMA AND DAVISSON: DETECTION OF BASEBAND SIGNALS 539

Hence 2, is formed from the previous accumulator output Zfl-l and the input x,-,, which is independent of x,-*,. . ., xo for a given bit value. Therefore, the accumu-

Pr [ 3?,1x"?,-,, x",-z,. . . , a,] = Pr [ x"n~x"fl- l ] ,

and the row vector

z k A [ P k ( - - I ) , P k ( - M + 1, - I ) , ' . ' , lator output process { x",} is Markov, i.e., P k ( M , - l) , P k ( - M,l),' * ' > P k ( M , l)] 9 (18)

then we can construct the stochastic matrix for { z k } as n = l ; . . , N . (12)

Furthermore, by condition 2) the transition probabilities (12) are independent of n and the Markov chain is homo- geneous. Note that (2) , (3), and the finite accumulator range imply that { f,} is a finite-state birth-death process. This result is exploited in Section IV for the computation of the joint probabilities for {KO, 2,; . ., ZN}.

Recall that we are considering { x",} over a single bit duration. To find a likelihood ratio test for the observed signal, we first need to find the steady-state probability distribution for the initial accumulator value, whch is given by

lim P r [ f N k = m ] , ~ E S . (13) k + m

Here X I N k denotes the accumulator output following the final sample of bit k (the bit which takes on the value determined by y k ) . Although { Z,,} is Markov, its transi- tion probabilities are determined by the data sequence { y k } , itself a random process. Thus { x " N k } is not homoge- neous, and we are denied useful results on steady-state distributions. However, we can construct a vector-valued process including x"Nk as a component that is homoge- neous. Consider the process { z k } whose state is de- termined by both the accumulator output and the signal. More precisely, let

z k = ( x " N k , y k ) .

Thus { z k } takes values at the end of each data bit. The first component of z k is the Nth (scaled) accumulator output for the received bit generated by the random vari- able y k , and the second component carries the data se- quence { y k } explicitly. Since { x",, } is Markov and { y, } is i.i.d., { z k } is also Markov. It follows that the transition probabilities for { z , } are

p r [ z k l z k - l l = ['Nk? Y k l x " N ( k - 1 ) 3 y k - l ]

*(-l) \ k ( + l ) . q - 1 ) * (+ l ) 1 \k= [

Here, + ( - l ) and \k (+ l ) are R X R matrices, where 'P(-l) has elements given by (16) and \k(+l) has ele- ments given by (17). First-order probabilities are then given by Zk = Zo\kk, where Z, denotes any initial probabil- ity distribution. The matrices \k( - 1) and *( + 1) can be found from the stochastic matrices for 2,. Call these matrices @( - 1) and @( + l) , consisting of the elements C + ~ ~ ( U ) =Pr[x",=rIx",-,=m, y , = a ] ,

E { -1 , + 1 } , n E { N k +1;. * , Nk + N } . (19) Comparing (15) and (19), the transition probabilities for { z k } are the N-step transition probabilities for 2,. In matrix form,

q - 1 ) =+(@(- l ) )N * (+ l ) = : ( @ ( + l ) ) N .

For a stationary distribution to exist it is sufficient that { z k } be irreducible with all ergodic states [6, ch. XV]. To show that it is irreducible, relabel the states of { z k } as E,, . . , E*,. (The states of { z k } correspond to the indices of the 4M + 2 = 2R elements of the vector z k in (18).) Then from the @ matrices for { x",}, all states of { x",} are reachable (from all states), so it follows from E, through E , can be reached from one another, and similarly for E,+, through El,. However, by the form of 'k, state E , + , can be reached from E, and vice versa, so { z k } is irreducible. The chain is aperiodic because \k has a posi- tive diagonal element. Then, since it is finite, the states must all be ergodic. Therefore, a unique stationary prob- ability distribution exists for the states of { z , } which can be found by the equations

M

z \ k = z , 1 [ P ( r n , - l ) + P ( m , + l ) ] =1 (20) m = - M

= Pr [ Y k l x " N ( k - l ) t y k - 1 1 where the row vector Z and its elements are defined as Z = lim z k P ( m , a ) = lim P k ( r n , a )

k + m k + m

= [ x"NkIx"N(k- 1) 3 l ) k ] (14) a E { -1 , +1}.

where the third equality follows from the mutual indepen- dence and equiprobability of the signal sequence. If we let

Finally, the distribution for Z0 (the accumulator output prior to the first sample of a bit in steady state) is

~ r [ i , = m ] A lim ~ r [ ~ , , = r n ] k + m + I , ( b l a ) A Pr [ z k = ( j , b ) Izk - 1 = ( 2 , a ) ] 9 ( l 5 )

then using (14), = P ( m , - 1) + P ( m , + 1). (21) +J-lI+l) = k , ( - 1 l - Q (16)

(17)

To formulate a detection theory problem we view the accumulator outputs as a sequence of observations of the noise-corrupted signal x( t ) . A binary hypothesis test can be posed as follows. For a bit k in steady state let

4Jl/(+1l+1) = + J + 1 I - O If we define

pk ( rn , a ) 6 Pr [ z , = ( m , a ) ] Ho: y k = - 1 HI: y k = + 1 .

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO. 3, MAY 1988 540

The prior probabilities Po and P, and (corresponding to H , and H,, respectively) are each 1/2 by the assumptions on yk. For convenience, define the vector of observations of bit k as

f 4 ( J , , 2,; . . , JN). For the Bayes test we must find the observation probabili- ties under the two hypotheses. Since { f,} is Markov, they are

N

P ~ [ ~ I H , ] = P ~ [ z , ] n Pr[Jn1Jn-,, HI], i E {O,I} n = l

(23) where Pr[Z,] is taken from (21). Let the decision spaces corresponding to H, and Hl to be denoted by E, and E,. We assign each observation J to E, or E, according to likelihood ratio test

the two delta detector design parameters of quantization step size A and accumulator range R.

The program performs a three-step algorithm. First, the initial distribution Pr[l,] is found using (20) and (21). In the second step the conditional probabilities for f under each of the hypotheses are computed for every possible observation. Finally, (24) and (25) are used to find the probability of bit error. When finding the probability of error PeN for the final accumulator output case, (26) and (27) are used instead of (24) and (25).

For both cases the transition probabilities for { J , } under H, and Hl must be calculated for each value m E S for the accumulator output. Recalling (3) and (ll), the following are used:

Pr [Jn = m + llJfl-l = m, Hi] = Pr[u, > mA],

m E { - M ; - . , M - l )

Pr[z, = m - 1 [ J n - , = m, H,] = Pr[u, < m a ] , (24)

m E { M + l ; - - , M } That is, if L ( f ) >1, then 1 is assigned to E, and if L ( f ) <1, then f is assigned to E,. If equality is satisfied, then f is assigned arbitrarily to either Zo or E,. Once the assignment of observations is made, the probability of error is

Pr [ 2, = MlJ , - , = M , H,] = Pr [ u, > MA]

Pr[J, = - M l J , - , = - M , HI] = Pr[u, < MA] P,= P, P ~ [ J ~ H , ] + P, P r [ f \ ~ , ]

f € q f € q =’( 2 f€q Pr[fIHl]+ Pr[flH,]). (25)

Although this approach is optimal for the delta detector since all the information available (given the sampling rate T, and the quantization step size A) is used for the decision, it does not yield a simple implementation. A simpler approach would be to operate the delta detector as described previously, but make the decision by observing the accumulator output only subsequent to the final sam- ple. The analysis for this method is a special case of that presented earlier, where the last component J , of f is used for the observation. The probability of error is

‘e,%’=’( Pr [ JNIHl l+ Pr[PNIHO]), (26) 2 P , E q , 2, E El&,

and the assignment of observations to EON and E,, is made according to the likelihood ratio test

IV. NUMERICAL RESULTS

A computer program was written to determine the per- formance of the delta detector in the system. The program was designed to allow user input of the parameters that govern performance, i.e., the incoming signal E,,”, and

Pr [zn = m + jla,-,= m, Hi] =0 ,

m E S , j = 0, f 2 , + 3 , . ’ . .

The random variables u, and u1 (corresponding to H, and H , ) are Gaussian with variance N,B and means of - 1 and + 1, respectively.

The primary difference between the two cases is in the computation of the observation probabilities. In the first case, the fact that { Jn} is a birth-death process allows a binary tree structure to be used to compute Pr[fIH,]. For the case when only the final output is observed, the sto- chastic matrices @( - 1) and @( + 1) for {a , } can be used, giving Pr[J,lH,] after N matrix multiplications.

The program was executed for integer values of E,/No ranging from 2 to 18 dB. For each value of E,,”, the accumulator range and the sampling rate were varied to determine the sensitivity of the delta detection scheme to these parameters. The results are presented graphcally in Figs. 2-8.

The performance in terms of the probability of error is shown in Figs. 2 through 5. Fig. 2 shows the performance sensitivity to the sampling rate for the two observation cases. Worthy of note is the increase in the slope of the curve for N = 4 at higher E b / N o values. Fig. 3 shows that this does not occur when bit timing is used (resetting the accumulator to zero prior to sampling each bit). The additional degradation in the former case results from the nonzero probabilities for extreme initial accumulator val- ues. This has been verified by examining results with the initial accumulator distribution Pr[Z, = f MI = 1/2, which

FLIKKEMA AND DAVISSON: DETECTION OF BASEBAND SIGNALS 541

F i n a l output observed

Matched N=16 F i 1 t e r

a5

F i n a l output observed .

Matched F i 1 t e r

-6 06'-2.8 3.8 4.0 5 . 0 6.8 7.8 8 . 0 3.8 10.011.812.813.814.8 06'w2.8 3.8 4 . 0 5 . 0 6 . 0 7.8 8 . 0 9.8 10.811.812.013.814.8

Eb/Na ( d B ) \. Eb/No ( d B 1 \.

Fig. 2. Performance for R = 15. Fig. 3. Performance for R =15 and known bit transition times.

Hatched F i 1 t e r

A* Matched F i 1 t e r

9' a ' ' ' ' ' ' ' Q6'vz .~ 3 . 8 4.8 5 . 0 6 . 0 7 . 8 8 . 0 9.0 10.011.012.013.014.0

Fig. 4.

Q,'c./z.a 3 . 0 4.0 5 . 0 6 . 0 7 .0 e.a 9 .0 ia.0i1.012.0i~.0i4.0

Eb/No ( d B ) \ * Eb/Nn ( d B ) \-

Performance for observation of all outputs when N = 8. Fig. 5 . Performance for observation of final output.

542

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO. 3, MAY 1988

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were worse than those for the initial steady-state distribu- tion. From another point of view, the small number of samples limits the delta detector’s ability to react to each data bit since this phenomenon is not evident for N = 8 or N=16. In fact, the probability distribution for { z k } is markedly similar for N = 4 across the range of &/No values, while at N = 8 the distribution shows a definite response to individual data bits as Eb/No increases. Fig. 3 also reveals the behavior of observation of the final output with known bit timing. For small values of Eb/No the results are as expected, but when &/No is increased the performance (relative to the matched filter) deteriorates with greater sampling rates. This can be attributed to two factors. First, as the sampling rate increases, so does the filter bandwidth (to maintain the independence of the noise samples) and hence the noise power. This effect, coupled with the fact that much less is observed about x ( t ) (than when 5 is processed), overpowers the advantage of sampling at a higher rate.

Figs. 4 and 5 show the performance sensitivity to the accumulator range. Most noticeable is the performance for N = 8 in Fig. 5. The poor performance for R = 15 relative to the case when R = 7 indicates that a mismatch between accumulator range and sampling rate can occur when only the final output is observed since the curves for N = 16 are as expected. The slopes for the N = 8 curves indicate that the mismatch can be overcome with very high E,,/No values; indeed, at = 18 dB, the performance for R = 15 is better than for R = 7 .

The performance degradation in dB relative to the matched filter is depicted in Figs. 6-8. When all outputs are observed, increasing either N or R independently yields

,. P Y

C

Y

U

L

0

.-

m n

8.0 4 6 8 ia 12 14 16

Number o f t a m p l e a / b t t . N

Fig. 6. Degradation versus sampling rate for observation of all outputs. Eh/Na = 6 dB.

4 6 B 10 12 14 16

Number of a a m p l e a / b t t . N

Fig. 7. Degradation versus sampling rate for observation of final out- put. E,/Na = 6 dB.

A l l outputs observed Final output observed 1.0 - -.__

e .O 3 5 7 9 1 1 13 15

Rccumula tor range . R

Fig. 8. Degradation versus accumulator range. E,,/N, = 6 dB.

improved performance. Notice that improving complexity beyond R = 7 and N = 10 yields only a small improve- ment. When only the final output is observed, the impor- tance of matching N and R is made clear by Figs. 7 and 8. For example, if N is constrained to 8 or less, then increas- ing the accumulator range beyond R = 7 provides no performance improvement.

FLIKKEMA AND DAVISSON: DETECTION OF BASEBAND SIGNALS 543

REFERENCES [31

[41 [l] J. B. O’Neal, “Delta modulation quantizing noise analytical and

computer simulation results for Gaussian and television signals,” Bell .Qst. Tech. J., vol. 45, pp. 117-141, Jan. 1966. T. Fine, “Performance of an optimum digital system and applica- tions,” IEEE Trans. Inform. Theoty, vol. IT-10, pp. 287-296, Oct. 1964.

[5]

[2] [6]

S. A. Kassam, “Optimum quantization for signal detection,” IEEE Trans. Commun., vol. COM-25, no. 5, pp. 479-484, May 1977. G. L. Turin, “An introduction to digital matched filters,” Proc.

C. M. Chie, “Performance analysis of digital integrate-and-dump filters,” IEEE Trans. Commun., vol. COM-30. pp. 1979-1983, Aug. 1982. W. Feller, A n Introduction to Probability Theoy and Its Applications. New York: Wiley, 1968.

IEEE, VOI. 64, pp. 1092-1112, July 1976.