CONCISE PAPERS 237

in (46) gives finally n

REFERENCES [ 11 R. W. Lucky, J . Salz. and E. J . Weldon, Principles of Data Commu-

[ 21 R. W. Lucky, “Automatic equalization for digital communication,” nicafion. New York: McGraw-Hill, 1968, ch. 5 and 6.

BellSyst. Tech. J . , vol. 44, pp. 547-588, Apr. 1965.

A New Digital Coherent Demodulator

D. A . SPAULDING

Abstract-A coherent demodulator is described which is particularly well suited to digital implementation. Its important features are that no low-pass filter is required to eliminate double-frequency terms and that the passband signal need be sampled at only twice the highest frequency of the baseband signal, rather than at twice the highest frequency of the passband signal. A demodulator of this type will be useful in those systems that do not require passband digital signal processing for other purposes.

I. INTRODUCTION The development of inexpensive highly integrated digital cir-

cuits has stimulated the use of digital signal processing tech- niques in many areas of communication and control. Digital circuits now are performing many of the functions previously performed by analog circuits. One such function is demodula- tion of a passband signal t o baseband. A straightforward ap- proach for “digitizing” the usual analog multiplier low-pass filter demodulator requires that the passband signal be sampled at least at twice the highest frequency occurring in the passband signal.’ However, one might expect intuitively that the sampling rate required might in fact be only twice the highest frequency of the baseband signal.2 This paper presents a method for performing digital demodulation that requires sampling passband signals at only twice the highest frequency of the baseband signal. Included is a brief analysis of the errors introduced by this demodulator.

11. DIGITAL DEMODULATOR Fig. 1 shows a standard analog demodulator. The passband

signal r ( t ) is multiplied by the carrier signal and low-pass filtered to produce the baseband signal y ( t ) . The carrier frequency is denoted by w, and an arbitrary phase angle is denoted by 6. The passband signal is assumed to be of the form of (1) where the spectra of a ( t ) and b ( t ) are zero for frequencies above w,.

r ( t ) = a ( t ) cos wc t + b ( t ) sin wc t . (1)

What might be called a standard digital implementation of the analog demodulator of Fig. 1 would place sampling switches before the multiplier. The sampling rate required at these points in order to avoid aliasing effects is at least twice the highest frequency of passband signal. These samples are then low-pass filtered digitally to produce the sampled output.

Paper approved by the Data Communications Committee of the IEEE Communications Society for publication without oral presentation. Manuscript received August 2 0 , 1972.

The author is with the Bell Laboratories, Holmdel, N.J. 07733.

band signal occurs with a double sideband signal. ‘The lowest sampling rate of twice the highest frequency of the pass-

a passband signal of bandwidth w hertz in terms of complex time 2A related result is given by Rowe [ 1 ] where he gives an expansion of

samples spaced at l / w .

LOW PASS FILTER

6 cos(w,t+8)

Fig. 1 . Standard analog demodulator.

- y \ ”

Fig. 2. Phase-shift-type demodulator.

Another analog approach for implementing a demodulator is shown in Fig. 2. Here the signal Y( t ) is multiplied by the carrier signal and added to the Hilbert transform of the passband signal multiplied by the carrier signal shifted by 90’. The addition of these two signals again produces the baseband sig- nal. An important point in this particular circuit is that no low-pass filter is needed to eliminate the double-frequency terms generated by the multiplication of the passband signals with the carrier signal.

In constructing a digital equivalent of the demodulator shown in Fig. 2, one would sample the output of the demodu- lator y ( t ) a t a rate that was twice the highest frequency of the components in the baseband signal. I t is clear from Fig. 2 that the sampling operation can be moved through the demodu- lator, which has zero memory, to the passband. At this point the signals, r ( t ) and ?(t) , are sampled at the baseband sample rate and not at twice the highest frequency of the passband signal.3

ERROR ANALYSIS

In order to determine the performance of this digital demod- ulator, an error model is constructed as is shown in Fig. 3. The signal ?(t) is the Hilbert transform of the received pass- band signal. The signal ~ ( t ) is a signal that represents the error introduced by the filter performing Hilbert transform operation. The two passband signals are now sampled at the baseband sampling rate to produce the signals shown after the sampling switch. The parameter tk indicates the particular sampling instant. The quantities ni and nq represent the quantizing noise associated with the analog-to-digital conver- sion of the sampled signals. Sampled values of the sine and cosine of the carrier are then multiplied by the sampled values of the passband signal and its Hilbert transform. The numbers n, and ns indicate the quantizing noise associated with the quantization of the sine and cosine of the carrier. The angle 6 may take on the values of 0’ and 90°, respectively, to gener- ate the in-phase and quadrature baseband signals. The output of the demodulator is indicated as y ( t k ) and consists of three components. The component y l ( t k ) represents the sampled value of an ideal demodulated baseband signal. The numbers n l and n 2 represent, respectively, first-and second-order errors

would be analog-to-digital conversion associated with the sampling of 31n a physical digital implementation of this demodulation there

r ( t ) and p ( t ) ; digital values of the sine and cosine would be read from read-only memories.

238 IEEE TRANSACTIONS ON COMMUNICATIONS, MARCH 1973

f [ tk)+c(tk~+nq

Fig. 3. Sampled version of phase-shift-type demodulator.

introduced by the Hilbert transform filtering operation and the sampling operation.

Equation ( 2 ) indicates the expression for the Hilbert trans- form of the passband signal; (3a)-(3d) give

?(t) = a ( t ) sin a, t - b( t ) cos a, t (2) Y ( t k ) = Y l ( t k ) + n l + n Z (3a)

Y1 ( t k ) = a ( t k ) cos 6 - b ( t k ) sin 6 (3b) n 1 = [ a ( t k ) cos tk + b ( t k ) sin t k ] nc

+ [ a ( t k ) sin tk - b ( t k ) COS 0, t k ] nc + n i cos (0, tk + 6 ) + nq sin (a, t k + 6 )

+ ~ ( t k ) sin (ac tk + 6) (3c) nz = ni nc + nq n, + E(tk) n, (3d)

the expressions for the various quantities appearing at the out- put of the demodulator. In any reasonable system the second- order error terms can be neglected. One can compute the mean-square error associated with the various errors introduced in the demodulator. Taking the expected value of the quantity n l , the following expression can be obtained for the mean- square error Z at the output of t h e d e m ~ d u l a t o r . ~

(?2:)/Ps = 2 Nq + M q / P s + PF/Ps (4a) Nq = (nz) = (n:) (4b) M~ = ( n f ) = (4c) PF = k 2 ( t k ) Sin2 (0, tk + 6 ) ) = ( E z ( t k ) ) / 2 (4d) ps = (r2 ( f k ) ) . (4e)

In (4a)-(4e) the quantity Ps represents the passband signal power, Nq the quantizing noise associated with the sine and the cosine of the carrier, Mq the quantizing noise associated with sampling the passband signals, and PF is related to the error introduced by the filter generating the Hilbert transform and is proportional to the signal power. Evaluating the mean- square error from (4a)-(4e) for some typical numbers shows that the number of bits carried in the digital signal processing can be selected in such a manner so as to keep the mean-square error introduced by the digital demodulator to acceptable levels. I t is also possible to design the filter generating the Hilbert transform of the passband signal accurately enough so as to keep this error reasonable. For many applications a low-order analog filter will be adequate [ 1 ] .

As an example consider a signal with a flat spectrum from 300 to 3000 Hz. The sampling rate of the signals r ( t ) and F ( t ) is 5400. An eighth-order phasesplitter (four second-order

signals, an arbitrary angle uniformly distributed between [0, 2n] was ‘(.) indicates expected value. In taking expected values of passband

assumed. Also a ( t k ) and b( tk) were assumed to have equal power.

active-filter sections) realizes a Hilbert transformer with a phase error of 0.0 1 rad. Using 10 bits to represent the sampled signals and assuming that the passband signal power is 10 dB below the maximum power that can be represented by 10 bits (this gives the analog-to-digital converter some margin against saturation), the mean-square error is -40 dB. This error is due primarily to the Hilbert transformer errors; the digital quanti- zation errors are about two orders of magnitude smaller.

IMPLEMENTATION COMPLEXITY

One can compare the complexity of the digital demodula- tor of Fig. 3 with a digital equivalent of Fig. 1. An analog filter is required to generate the Hilbert transform of the pass- band signal. However, simple analog filters are required in any case to eliminate aliasing noise when sampling occurs. If we assume that the bandwidth of the passband signal is less than twice the carrier frequency (this is usually the case) the samp- ling rate required in Fig. 3 is less than the sampling rate re- quired in a digital equivalent of Fig. 1. In addition, the circuit of Fig. 3 does not require the large number of signal processing steps associated with the digital implementation of the low- pass filter. It should be clear then that in many cases of practi- cal importance the circuit of Fig. 3 is simpler.

V. CONCLUSIONS A new digital demodulator suitable for coherent demodula-

tion has been described. The circuit uses the passband signal and its Hilbert transform sampled at the baseband sampling rate to generate the sampled version of the baseband signal. The advantage of this implementation is that the sampling rate is that determined by the baseband signal,and not the twice the highest frequency of the passband signal. For the special case of synchronous PAM data transmission systems [ 21 where the baseband bandwidth is greater than the Nyquist frequency (one-half the symbol rate), a further reduction in sampling rate is possible because only samples at the symbol rate are re- quired. The demodulator described in this paper will be useful in those systems where passband digital signal processing, and thereby a higher sampling rate, is not required for other pur- poses. Such might be the case when the only passband filtering is simple noise limiting or when the frequencies coupled with the filter complexity do not make digital processing at pass- band attractive.

REFERENCES [ 11 H.E. Rowe, SignalsandNoise in Communication Systems. Prince-

[ 2 ] S. D. Bedrosian, “Normalized design of 90° phase-difference net- ton, N.J.: Van Nostrand, 1965, p. 53.

works,” IRE Trans. Circuit Theory, vol. CT-7, pp. 128-136, June 1960.

[ 3 ] R. W. Lucky, J . Salz, and E. J . Weldon, J r . , Principles of Data Communzcation. New York: McGraw-Hill, 1968, ch. 4 and 6.

Intersymbol Interference in Binary Communication Systems with Single-Pole Band-Limiting Filters

I. KORN

Abstruct-The exact value of the probability of error is computed for binary communication systems with sampling detector or integrating- and-dumping detector, with nonreturn-to-zero or split-phase signals

IEEE Communications Society for publication without oral presenta- Paper approved by the Data Communications Committee of the

tion. Manuscript received July 30, 1972.

Tex. He is now with the Technion-Israel Institute of Technology, The author was with the NASA Manned Spacecraft Center, Houston,

Haifa, Israel.