The Effect of Frequency Selective F;Ading on the -Binary Error Probabilities of Incoherent and Differentially Coherent Matched. Filter Receivers

170 IEEE TRANSACTIONS ON COlUlMUNICS TIONS SYSTEMS June

The Effect of Frequency Selective F;ading on the -Binary Error Probabilities of Incoherent and

Differentially Coherent Matched. Filter Receivers* d

P. A. BELLOt, SENIOR MEMBER, IEEE AND B. D. NELINt, MEMBER, IEEE

Summary-The presence of frequency-selective fading in a communication channel limits the maximum data rate capability of conventional communication systems. Surprisingly little analysis has been carried out to determine the effect of frequency-selective fading on digital (or analog) communication systems. This paper considers the effect of frequency-selective fading on the binary error probabilities of incoherent and differentially coherent matched filter receivers employing postdetection diversity combining. In the analysis it is assumed that the fading is slow enough so that over a’few bits at least the channel transfer function remains constant. In addition, it is assumed that the amplitude and phase fluctuations on a received carrier have the same statistical character as those of narrow-band Gaussian noise.

The general analytical results are specialized to the cases of frequency-shift keying using incoherent detection, and phase-shift keying using differentially coherent detection for the case of a Gaussian frequency autocorrelation function. For these special cases, signal-to-noise degradation curves are given as a function of the ratio of the binary data rate to the correlation (or coherence) bandwidth. Two types of FSK are considered, phase-continuous and phase-discontinuous. In phase-continuous FSK there is no discontinuity in the phase of the transmitted waveform at the mark- space or space-mark epochs. Such an FSK system results when the mark and space frequencies are obtained by frequency modulat- ing an oscillator. In phase-discontinuous FSK phase discontinuities exist at the transition epochs. Such an FSK system results, for example, when the mark and space frequencies are derived by switching between two independent oscillators. An interesting result of the analysis is that the differentially coherent PSK system and the phase-discontinuous FSK system degrade considerably more rapidly with increasing (normalized) data rate than the phase- continuous FSK system.

The existence of an irreducible error probability is demonstrated for the incoherent and differentially coherent matched filter receivers. Thus, in general, an increase in transmitted signal power cannot reduce the error probability below a certain value depending upon the ratio of data rate to correlation bandwidth and order of diversity. Theoretical curves of irreducible error probability are given for the incoherent FSK and differentially coherent PSK systems.

I. INT~ODUCTION

OST RADIO CHANNELS exhibit a phenomenon known as frequency-selective fading. This type of fading has been characterized experimentally

by transmitting sinusoids of two different frequencies and determining the correlation coefficient between the envelopes of the corresponding two received waveforms as a function of the frequency separation between the sinusoids. When the frequency separation is sufficiently small it is found that the two received envelopes fade

* Received January 5, 1963. t ADCOM, Inc., Cambridge, Mass.

essentially in step, i.e., the correlation coefficient is negligibly different from unity. However, as the frequency separation increases the two envelopes no longer fade in step and the correlation Coefficient drops below unity, eventually a:pproaching zero for scatter type radio channels. The resulting curve is called a frequency correla- Pion curve. In order to delimit the frequency range over which receiver waveforms resulting from transmitted sinusoids fade approximately in step, a quantity called the correlation. bandwidth has been defined as the “width” (in some appropriate sense) of the frequency correlation curve.

Fading rates observed on most radio channels seem to be composed of a fast fading component (e.g., up to a few cps for troposkatter) superimposed on a very much slower fading component (such as hourly or diurnal variations). This paper is concerned with the evaluation of error probabilities for a mathematical model of a linear channel! which is simply described by stating that .the channel response to a sinusoid is a narrowband Gaussian process whose parameters may change with a change in tlhe frequency of the sinusoid. Such a model appears to refect more accurately the type of behavior observed for the fast fading part of the channel variations than for the s:low fading part. Thus the error probabilities computed in this paper are probably most useful in describing the quasi-stationary behavior of the binary error probabilities as a function of the slow variations in signal strength and shape of the frequency correlation curve. In order to predict error probabilities over a 1.onger time interval it would be necessary to average the quasi-stationary error probabilities derived here over probability distributions for the long-term variations in signal strengt:h and in shape of the frequency correlation curve. While data has been compiled on the statistics of long-term signal strength variations, virtually nothing is available 011 long-term variations in the shape of the frequency correlation curve of radio channels.

In these days of an ever increasing military and com- mercial demand for data rate capability every avenue o f approach i;s being studied to increase the capacity of existing channels. The importance of knowing the effect of frequency-sdective fading on digital error probabilities arises in this context, since the data rate capability of conventional radio channels is adversely affected by frequency-selective fading which introduces symbol distortion and. intersymbol interference.

1963 Bello and Nelin: Frequency-Selective Fading and Binary Error Probabilities 171

Previous evaluations of error probabilities for fading channels and conventional receivers have, in the main, ignored the time- and frequency-selective behavior of the channel and have concentrated on what may be called the “time-flat frequency-flat” or just “flat-flat” fading channel model. The flat-flat fading model is one in which the received digital pulse differs from the transmitted pulse only in an amplitude change and a phase shift. Such a model can closely app.roximate the behavior of a radio channel with regard to digital data transmission if the bandwidth of the digital pulse is much iess than the correlation bandwidth of the channel and the digital pulsewidth is very much less than the correlation time of the fading. Unfortunately these two requirements may sometimes be incompatible since the time-bandwidth product of a digital pulse has a lower bound. Thus for the longer distance troposcatter links and for the iono- scatter links it is doubtful that calculations made for a flat-flat model are applicable for any digital pulse.

Some results have appeared on the evaluation of binary error probabilities for an idealized channel which may appropriately be called either the frequency-flat fading channel or the pure time-selective fading channel. Such a channel has the property that the amplitude and phase fluctuations observed on the channel response to a sinusoid are the same for any excitation frequency. Since any transmitted digital pulse may be resolved into a continuum of sinusoids (via the Fourier Integral) it follows that the received waveform corresponding to a transmitted digital pulse differs from the transmitted pulse only in having a superimposed random amplitude and phase fluctuation due to the channel. The frequency- flat fading channel may be used as an approximation to the behaviour of a radio channel with regard to digital data transmission when the bandwidth occupied by the

’ digital pulses is very much smaller than the correlation bandwidth of the channel.

It is perhaps of interest to present a brief summary of previous work on the evaluation of digital error probabilities for frequency-flat fading channels. In all cases the authors have assumed that the amplitude and phase fluctuations of the channel have the same statistical character as those of narrow-band Gaussian noise. Bello and Nelin [I] have determined exact expressions for binary error probabilities for a frequency-flat fading channel and incoherent and differentially coherent receivers employing postdetection diversity combining. Voelker [2] has obtained some approximate results for the case of a matched filter receiver employing phase- reversal keying and differentially coherent detection. Price [3] has obtained some analytical results for the optimum detection of binary signals over frequency-flat (and frequency-selective) fading channels. In an un- published Lincoln Laboratory report Price [4] also derives an expression for the binary error probabilities in on-off transmission and incoherent matched filter detection with a frequency-flat fading channel. Turin [5] , [6] has determined analytical expressions for binary

error probabilities in optimal diversity reception over diversity channels which individually have the frequency- flat fading property.

The type of channel analyzed in this paper is an idealized channel model appropriately called the time-flat or pure frequency-selective fading channel since it may be represented as a random but time-invariant linear filter. The time-flat fading channel may be used as an approximation to the behavior of a radio channel with regard to digital data transmission if the duration of a digital pulse is sufficiently small compared to the fading correlation time of the channel.

It appears that prior to this paper negligible work has been published’ on the evaluation of binary error probabilities for the time-flat fading channel. The present analysis will be concerned with the determination of binary error probabilities for a time-flat fading channel and incoherent and differentially coherent receivers employing postdetection diversity combining. Each diversity channel will be represented as a random time- invariant linear filter whose transfer function has appropriately defined Gaussian statistics. The individual diversity channels will be assumed to fade independently, to have identical statistics and to provide no specular component of received signal.

The method of evaluation of the required error probabilities is exactly the same as that used by the authors in a previous paper [I] wherein error probabilities were evaluated for the same types of receivers but for the frequency-flat rather than time-flat fading Channel. In fact the general error probability expressions are identical when expressed in terms of correlations between matched filter outputs. The reason for the identity of these expressions is simply that as long as the channel is Gaussian,2 whether time-flat or frequency-flat (or neither), the sampled matched filter outputs will be Gaussian random variables. Then, since the statistics of a set of Gaussian random variables are determined by their correlation matrix, the general expression for binary error probabilities in terms of the correlations between sampled matched filter outputs must be invariant to the precise time- and frequency-selective behavior of the transmission medium as long as the same decision mechanism (in terms of sampled matched filter outputs) is used in the receiver. In the case of the present paper and the previous one [l] this decision mechanism involves the comparison of a quadratic form in sampled matched filter outputs with a threshold. The general expressions for error probabilities in terms of the correlation matrix of the sampled matched filter outputs were evaluated in [l] by recognizing that the quadratic forms appropriate to the decision mechanisms in the incoherent and differentially coherent matched filter receivers were special

1 R. Freudberg and S. Stein, “Performance of Correlation Sys- tems with Bandwidths Exceeding the Coherent Bandwidth,” presented a t 1962 Nat’l Comm. Symp., Utica, N. y.; October, 1962.

a We define a Gaussian channel as one whose output may be separated into a deterministic signal plus a random Gaussian signal when the input is deterministic.

172 IEEE TRANSACTIONS ON COMMUNICATIONS SI‘STEMS Jw1.e

cases of a more general quadratic form. Error probabilities were evaluated for this more general (quadratic form) decision mechanism by applying results of Turin [7] on the characteristic function for a quadratic form in zero- mean Gaussian variable^.^ Specialization of these results yielded the required error probability expressions for the incoherent and differentially coherent matched filter receiver.

In view of the above it is clear that (8a) and (8b) of Bello and Nelin [I] ‘may be used to determine the binary error probabilities of the incoherent and. differentially coherent matched filter receiver“ for the most general randomly time-variant linear Gaussian channel in which no specular component is r e~e ived .~ Detailed results for particular types of selective-fading channels are obtained by specializing the correlation matrix (ie., ,the moments defined in (11) of [l]) to the form appropriate for that type of fading.

Before leaving this section it is perhaps worthwhile to point out that the general error probability expressions in terms of the moments (11) of [I] are sufficiently general to include any type of additive interference with the sole proviso that it be characterized as a zero-mean Gaussian process. Thus, since the presence of intersymbol and interchannel interference may be accounted for as an additive zero-mean Gaussian process (assuming a Gaussian channel with no specular component), the general error probability expressions in [I] may be specialized to account for this type of disturbance. Finally, it should be mentioned that since mismatching of matched filters, constant frequency conversion errors and constant synchronization errors do not change the Gaussian character of the sampled matched filter outputs, the general error probability expressions in [l] may be used to investigate the effects of such nonidealities on system performance.

111. COMPLEX NOTATION

Complex envelope notation will be used extensively throughout this paper. A process z ( t ) whose spectrum covers a band, of frequencies which is small compared to any frequency in the band may be expressed as Re{ z ( t ) exp [j27rfOt]} where Re{ 1 is the usual real part notation, f o is some frequency within the band and z( t ) is the complex envelope of x( t ) . This name for x ( t ) derives from the fact that the magnitude of x ( t ) is the conventional envelope of x( t ) while the angle of z ( t ) is the conventional phase of z(t) measured with respect to carrier phase wot. The non-narrow-band case may be

3 The zero-mean assumption restricts the use of the resulting error probability expressions to the subcase of a Gaussian channel in which the channel response to a detcrministic signal consists of a Gaussian process only. In more familiar terms this means that the received signal contains no specular component.

4 The results may be used for any receiver in which the diversity

involves the comparison of a quadratic form in zero-mean Gaussian channels have independent fading, and whose decision mechanism

variables of the form of (7) of [l]. 6 Bello [8] has recently estended these results to include the case

in which a cspecular component appears in the received signal.

handled with complex notation also by the use of Hilbert transforms [9]--[la]. However, the complex envelope will then no longer have the simple interpretation described a.bove.

The reception of narrow-band signals is usually accompanied by an additive “white” noise. For purposes of analysis, however, it is not necessary to consider this white noise to be flat over infinite bandwidth. It is sufficient for all practical problems to use an “equivalent” noise which has a flat power density spectrum only over the range of frequencies that will be processed by the receiver. This range of frequencies is invariably narrow e,nough so that an equivalent narrow-band noise may be used in place ‘of the white noise. On this basis we justify the use- of complex envelope notation to represent the additive noise. It is readily demonstrated that if the real additive noise has a (two-sided) ,spectral intensity of N o , then the complex envelope of the “equivalent” low-pass noise has (two-sided) spectral intensity of 4N0.

When dealing with problems in which there are wide- band filters (time-variant included) whose inputs and outputs are narrowband (when expressed with reference t’o the same (center frequency) it is possible to replace these filters with equivalent narrow-band filters which leave the input-output relations invariant. This fact becomes obvious when it is realized that by preceding and following a wide-band filter with narrow-band filters which have fiat transfer functions over the range of input and output frequencies of interest one produces a composite filter which is narrow-band and of course cannot change the input-output relations for the properly restricted class of input and output narrow-band signals. I t is readily demonstrated that (except for an unimportant constant of one half) the complex envelope of a narrow- band signal at the output of a narrow-band filter due 00 a narrow-band input may be obtained by passing tQe complex envelope of the input through an “equivalent” low-pass filter whose impulse response is just equal to the complex envelope of the narrowband filter impulse response.

111. CHARACTERIZATION OF CHANNEL Using complex envelope notation, the input-output

relationship for a time-flat fading diversity channel (temporarily ignoring additive noise) may be expressed as

Wdf) = Y ’ k ( f ) S ( I ) (1)

where Tk( f ) is the random transfer function of an “equivalent” low-pass kth diversity channel, S(f) is the spectrum of the complex envelope of the channel input and W,(f) is the spectrum of the complex envelope of the channel output.

Because of the assumed Gaussian character of the channel, Tk(f) is a complex valued Gaussian process as is W,(f) when S(f) is specified. It is perhaps worth reminding the reader that (1) can be used to characterize a randonlly time-variant linear channel for the purpose of evaluating bit error probabilities only if the fading

196s Bello and Nelin: Freqwncy-Selective Fading and Binary Error Probabilities 173

correlation time is very much longer than the time occupied by all transmitted signals which influence the error probability of a particular bit. Thus, for example, in the transmission of a single binary data stream, the fading correlation time must be much longer than a time interval equal to the sum of a bit duration plus the duration of all bits which cause intersymbol interference.

It is interesting to compare the input-output relationship (1) with the corresponding one for the frequency-flat fading channel studied in [l], namely,

W d t ) = gk(t)s(t) (2)

where s(t) , wk(t) are the complex envelopes of the input and output and g k ( t ) is the time varying (complex) Gauss- ian distributed gain function of the 16th diversity channel. Note that although (1) is a frequency domain relationship and (2) is a time domain relationship they are of identical form not only from an analytical point of view but also from a statistical point of view since Tk(f) and g k ( t ) are both complex valued Gaussian processes. It is clear that the time-flat and frequency-flat fading channels have a time-frequency dual relationship. This dual relationship is a particular example of a general body of time-frequency dual relationships developed by Bello [13], [14] which embraces the study of (time-frequency) dual stochastic processes, communication processing networks and randomly time variant linear channels.

Beyond the trivial flat-flat fading channel, the time- selective and frequency-selective fading channels are the simplest fading channels to characterize. The simplest fading channel to characterize which contains both time- and frequency-selective fading is one representable as a continuum of statistically stationary independently fluctuating scatterers. Such a model has been used by Ilagfors [15], Price and Green 1161 and Green [17] in radar astronomy to characterize reflections from planetary objects. This channel has also been studied independently by Bello [14] as a special case in the characterization of more general time- and frequency-selective fading channels with the aid of time-frequency duality concepts.

In the particular problem at hand duality considerations are useful in understanding the statistical behavior of Tk(f) by direct analogy with the statistical behavior of gk(t) which is most likely more familiar to the reader. Thus the reader is undoubtedly familiar with the fact that if gk(t) is a stationary complex-valued6 Gaussian stochastic process, it is completely determined statistically by specifying its autocorrelation function or its power density spectrum and that the latter two functions are Fourier transform pairs according to the Wiener- Iiintchine Theorem. It is clear that from a statistical point of view Tk(f) must be characterizable in exactly the same manner as gk(t) if Tk(f) is a complex-valued

be asymmetrical with respect to dc and requires a slightly altered 6 The complex character of g k ( t ) allows the power spectrum to

definition of autocorrelation function, viz., g k * ( t ) g k ( t + T ) where the bar denotes an ensemble average and the asterlsk denotes the the comples conjugate operation.

stationary Gaussian process, in the frequency variable. However, now the “autocorrelation function” and “power spectrum” of Tk(f) have a quite different physical interpretation from the corresponding functions associated with gl;(t).

Let us assume temporarily that Tk(f) is stationary in f and consider first the autocorrelation function of T k ( f ) which we shall call the frequency correlation function. This correlation function R(O) is defined by

which may be contrasted with the dual equation

where T ( T ) is the autocorrelation function of gh( t ) . The statistical averages indicated in (3) and (4) are

ensemble averages. Since single member functions rather than complete ensembles will be available in any physical situation, an experimental measurement of the correlation functions (3) and (4) on physical channels will involve time averages rather than ensemble averages.

Thus, to determine the frequency correlation function directly an appropriate experimental procedure would be to transmit two sine waves of different frequencies and measure the correlation between the complex envelopes of the two received processes as a function of the frequency separation of the sine waves.7 Of course, such a procedure can be meaningful for a practical radio channel only if the total measurement time is small enough compared to the duration of any nonstationary effects in the channel. However, even in the absence of nonstationary effects such a measurement procedure is potentially capable of measuring R(Q) only for values of O larger than the channel fading bandwidth since smaller values of Q do not allow the received processes corresponding to the two sinusoids to be separated. Since in most scatter channels the fading bandwidth is much smaller than the correlation bandwidth this re- striction is not too serious.

To make a direct determination of the fading correlation function experimentally is simpler than in the case of the frequency correlation function. One sine wave is transmitted and the autocorrelation function of the received process is evaluated.’

The Fourier transform of r ( ~ ) given by

P(f) = / m r(r)e-iz”” d r (5 )

is the familiar power spectral density of the process gk(t) and thus requires little discussion. This power spectum

* -m

measured in practice involves envelope detecting the received proc- 7 As discussed in Section I the frequency correlation function

esses before correlating. Thus what is measured is l T k ( f ) l l T k ( f + n)l rather than T k * ( f ) T k ( f + a). The former correlation function is considerably easier t,o *measure but does not contain information due to decorrelation in the phase of the transfer function.

Again, in practice an autocorrelation function of the envelope of the received process is extracted with a concomitant loss in phase- fluctuation information.

174 IEEE TRANSACTIONS ON COMMUNICATIONS SYSTEMS June

could be measured directly if a spectrum analyzer of sufficiently fine resolution were available by analyzing the response of the channel to a sinusoidal excitation. On the other hand the Fourier transform of R(Q) given by

p(T) = l: R ( Q ) e - ’ 2 T r n d~ (6)

is not so familiar, a t least on first inspection. It is clear that from an abstract point of view p ( ~ ) must be called the “spectral density” of Tk( f ) , but what its physical significance is, requires some elaboration. To aid in this elaboration let us formally define hk(t) the impulse response of the time-flat fading channel as the inverse Fourier transform of the transfer function of the channel, %.e.,

Now if u e evaluate the correlation function of h,(t) we find that I

number of scatterers with some local dependence between scattering statistics. However, if the digital pulses used for communica,tion have reciprocal bandwidths which are long compared with any path delay separations over which scatterers may be correlated, the channel will still “look” to the digital pulses as if it had the general form indicated in (9) because of the h i t e time resolution capabilities implied by such digital pulse bandwidths.

The positive function p ( t ) may be approximately determined experimentally by transmitting a periodic train of pulses, each pulse being much narrower than t.he width of th.e digital pulses it is desired to communicate with. If the received pulse train is square law envelope detected and the pulses are averaged via a recirculating delay line which stacks one pulse on top of the other, the resulting average pulse shape will be a suitable approximation to p ( t ) a t least for a digital communication with a specified class of digital pulses.

An interesting consequence of our tiiscussiou concerning the nature of bak(t) is found by considering the corresponding function in the dual channel. This function is the spectrum (not to be confused with the power spectrum) of the multiplier g k ( t ) formally defined as

= p(s)6(s - t ) = p(t)6(s - t ) By direct anallogy with our discussion concerning hk(t)

where 6 ( x : ) is a unit impulse at = o. It is clear that we conclude that for wide-sense stationary g k ( t ) , Gdf) is for hk(t) to have such an autocorrelation function it a random process which may be represented in the general must be of a rather singular character. One may quickly form verify by direct calculation that hk(t) will have an auto- G~U) = b k i S ( f - QJ + 4% vk(f) (14) correlation function of the form (8) if it is representable i

as the sum of a finite set of impulses plus a burst of wide- V,(f) i,s a white noise in the frequency variable sense stationary white noise, i e . , with autocorrelation function

where v k ( l ) is a wide-sense stationary white noise with P(f) is a positive function containing no impulses and autocorrelation function b k i , f l j are the gain and frequency shift associated with

the jth discrete moving scatterer of the kth diversity (lo) channel. The autocorrelation function of G k ( f ) is given by

G*kif)Gk(O = f‘(f)6(1 - I> (1 6)

Y*,( t )Vk(S) = 6(t - s) ,

$(t) is a positive function containing no impulses and ski, are the complex scatter gain and delay produced by the jth discrete scatter path in the kth diversity when channel. ‘The autocorrelation of the right-hand side of (9) will equal the right-hand side of (8) if we set ~’(f) = C Pi S ( f - Qi) + -fW (17)

__-

p i = I ak i 1’. (12) We have arrived a t a result which will no doubt be surprising to most readers of this paper, namely, that

It is not difficult to see that such a singular character the spectrum of a wide-sense stationary noise may be for h k ( t ) can arise when the channel consists of a finite meaningfully defined and that this noise behaves in the set of uncorrelated discrete scatter paths [producing frequency do:main as a nonstationary white noise plus the impulses in hk(t)] plus a continuum of uncorrelated a possible set of impulses with random complex areas scatter paths [producing the burst of white noise in hk( t ) ] . due to the presence of sinusoids in the original process. In practice the channel will contain a large but finite A more detailed discussion of the nature of the spectrum


of both stationary and nonstationary noises may be found in a paper by Bello [13] on time-frequency duality. Before leaving the discussion of spectra for random processes it is interesting to note that the spectrum of wide-sense stationary white noise is also a wide-sense stationary white noise with precisely the same LfspectraI’l density.

In view of the above interpretation of p ( t ) as the dual of P( f ) and in view of the physical interpretation of p ( t ) discussed above it appears appropriate to call p ( t ) the Delay Power Density Spectrum.’ The “width” of p ( t ) is a measure of the multipath spread of the time-flat fading channel while the “width” of P( j ) is a measure of the Doppler spread of the frequency-flat fading channel.”

IV. ERROR PROBABILITIES FOR INCOHERENT MATCHED FILTER RECEIVER;$

A simplified block diagram of an incoherent matched filter receiver is shown in Fig. 1. Two matched filters are indicated, each matched (apart from a phase shift) to one of the two possible waveforms that are successively repeated to form the transmitted binary communication signal. The outputs of the matched filters are envelope detected and subtracted. Diversity combining is per- formed by summing the outputs from all the diversity receivers. It is assumed that a fixed bit synchronization exists such that in the absence of selective fading the output would be sampled optimally. We shall assume that the fading statistics are stationary in time so that the bit error probability is independent of bit location. It is then sufficient to evaluate the error probabilities for a transmitted binary waveform which in the absence of selective fading would be received in the interval 0 < t < T where 1’ is the bit duration.

D

Fig. I-Block diagram of incoherent matched filter receiver.

Let the spectrum of the complex envelope of the additive Gaussian noise in the kth diversity channel be denoted by Nk(j ) . Then the spectrum of the complex envelope of the total received signal in the kth diversity channel Yk(f) is given by

Yk(f) = W ) T k ( f > + Ndl’i. (1 9)

Let y,(t) denote the complex envelope of the total received signal and let s,(t) and s,(t) denote the complex

Green [17] and the Delay Spectrum by Hagfors [15]. 9 This function has been called the Power Impulse Response by

10 A general discussion of gross channel parameters such as

tbe general randomly time-variant channel. “multipath spread’’ and “Doppler spread” is presented in [14] for

envelopes and’s,(f), s,(f) the corresponding spectra of the mark and space waveforms, respectively, as they would appear if received (noise free) in the time interval 0 < t < T . Then the combined operations of matched filtering and sampling at t = T may be expressed by the integrals

Uk = 1’ yk(ljsT(t) dt = ll Y k ( f ) S T ( f ) df

v, = lT Yk( t )S*O( t ) at = l; Y k ( f ) X * O ( f ) dl’ (20)

where, in obtaining the integrals over the frequency variable, we have used Parseval’s Theorem. When X(f) is specified Yk(f) is a stationary complex valued Gaussian process. Thus U, and V , are complex Gaussian variables since a linear operation on a Gaussian process always produces a Gaussian process.

I n Fig. 2 we have outlined the mathematical operations that take place in the incoherent matched filter receiver. The quantities I U , 1’ and I V k 1’ are the squared envelopes at the outputs of filters 1 and 0, respectively at t = T. We use square law rather than linear envelope detectors to simplify the analysis. However, for the zero threshold case (which we assume), the square law detector assumption does not change the error probabilities. I t is interesting to note that the frequency domain operations in Fig. 2 are identical in analytical form to the time domain operations in Fig. 2 of [l]. This identity is a result of the fact that the channels considered in this paper and in [I] are (time-frequency) duals and the fact that the dual of the incoherent matched filter receiver is also an incoherent matched filter receiver [13].

Fig. 2-Fnnctional diagram of incoherent matched filter receiver.

As in [I] Section I1 the input to the threshold device is a random variable q given by

where L is the order of diversity and U,, V , are complex- valued Gaussian variables. To obtain the desired error probabilities we need the probabilities P,(q > 0), Pr(g < 0) for general complex Gaussian distributed U,, V k . The probabilities are given by (Sa) and (8b) of [I] and are repeated below for convenient reference”

l1 Binary error probabilities have been evaluated by J. N. Pierce [IS] for the same type of receiver discussed in this section for the case of flat-flat fading. His expression for error probability is given by G L ( ~ ) where p is a SNR given by SoTlno in his notation. Fig. 1 of his paper shows plots of G L ( ~ ) for p from 0 to 20 db and for L = 1 - 5, 10, 20.

176 IEEE TRANSACTIONS ON COMfiIUNICATIONX SYSTEMS June

where C,“ is the usual binomial coefficient n!/m!(n - m)!. For the incoherent matched filter receiver y is given by

Y= - 1 - moo) (23)

~ 1 1 - ~ 0 0 ) 2 + ~ ( ~ 1 1 ~ 0 0 - ~ ~ 1 0 ~ 2 ) ~ ( ~ l l ~ ~ o o )

where

Noting that the additive noise Nk( f ) and the random filter Tk(f) are independent and have zero means we find that the moments (24) are given by the double integral

m,,= j’’m 1; ~ * k ( f l ) ~ k ( f z ) s * ( ~ l j ~ ( ~ z ~ ~ ~ ~ ~ l ) ~ ~ ~ ~ 2 ) df, df2

The autocorrelation function of the spectrum of the additive .noise1‘ Nk(f) is given by

where S(fP - fz) is a unit impulse a t f l = fz and N o is the spectral intensity of the real additive noise. Use of (3) and (26) in (25) leads to the following expression for the moments

mr8 = 1: n(fi)xT,(a) dfi + 4fi-oEr8 (27)

where

and

In the subsequent calculations it is convenient to express (27), (28) and (29) in alternate time-domain forms. Thus if we define

D,(T) = s*(t)s,(t + T ) dt = 1; S*(f)S7(f)e’2“f‘ df (30)

and use I’arseval’s Theorem in (28) we find that

a - m

X , ~ ( Q = Irn D r ( 7 ) D $ ( T ) e ’ z r *r dT. (31) - m

Section 111:. l2 See comment on the spectrum of white noise at the end of

A further application of Parseval’s Theorem to (27) then results in [see definition of p(.j, (6)]

m;, = 1; p ( 7 ) ~ , ( 7 ) ~ * , ( 7 ) d7 + ~ N ~ E , , . (32)

Another obvious application of Parseval’s Theorem yields

E,, = s,( t )st( t ) d l . s,’ (33)

The two error probabilities of concern are given by

p , = Pr[q > 0 I s( t ) = so ( t ) ; 0 < t < 7’3

p1 = Pr[q < 0 I s(t) = s1(t); 0 < t < TI. (34)

To use (34) properly, one must recognize that due t o the frequenqy-selective behavior of the channel, intersymbol interference will be produced. Thus the waveform received in the time interval 0 < t < T can, in theory, depend upon the form of s( t ) not only for time instant$ in the interval 0 < t < T but for all time instants. I n practice, as frequency-selective fading is gradually introduced i:nto an otherwise flat-fading channel, only values of s(t:) for time instants adjacent to the interval 0 < t < T will be effective in producing intersymbol interference. Then, in computing bit error probabilities, one must consider the intersymbol interference caused by adjacent pulses. Thus, for example, in order to find the probability p o that a “1” is printed when a “0” is transmitted one has to find the probabilities that a “I” is printed when 000, 101, 100 and 001 are transmitted. It follows that

P, = a ( P o 0 0 + PlOl + P I 0 0 + Po01 I

pt = %{P111 + polo + POI, + P l l O }

(35)

where

p a o c = Pr[q > 0 I s( t ) = s,oc(t>l

Pal, = Prlq < 0 I s ( 0 = sa1c ( t ) J

a , c = 0 , 1

(36)

and the time function sabc(t) is equal to that portior of the received signal (unity gain channel) corresponding to the transmission of the bit sequence abc where a, Z, c = 0, 1, and where it is assumed that the bit b resulk in a correspclnding binary signal received in the interval 0 < t < T . Note that in (35) we have assumed 1 and 0 equally likely a priori .

Further increase in the degree of frequency selectivity in the channel will eventually necessitate the inclusion of the effects’ of two adjacent bits before and after a bit in order to compute bit error probability, and so on for higher degreles of selectivity. However, it appears that when more than the adjacent bits have to be included: system degradation will be so high as to render such a communication system uninteresting from a practical viewpoint. I n this paper we will carry out detailed


calculations only for the case in which the intersymbol interference from adjacent bits is important. Other cases, although more tedious from a computational point of view, may be treated in an obvious analogous fashion.

In order to evaluate the error probabilities p o and p , it is necessary to evaluate y and thus the moments mrs for the eight cases of transmitted pulse combinations considered in (35). Let these values be denoted by yobc(a , 6, c, = 0, 1) where the subscript shows which pulse combination is considered. Then

Po = 4 c c F L t Y a O J 1 '

a=O e = O (37)

In general p o # p,. Thus, although binary symmetric operation may take place for nonselective fading, the introduction of frequency selectivity may bring nonsymmetric operation. A study of the moment expressions reveals an interesting case in which binary symmetric operation occurs for all degrees of selective fading. To describe this case most simply we define an operation - which when applied to a received signal converts each mark signal to a space signal and vice versa. Thus the time function $( t ) is the received signal which results from a binary stream which is the complement of the binary stream that produces s(t) . The condition for binary symmetric operation independent of selective fading then becomes

r = 0, 1, all s( t ) . (38)

When the transmitted signal is generated by pulsing conjugate mark and space filters, (38) simplifies to

A study of the expressions for the y 0 b c in terms of the relevant moments shows that as the additive noise vanishes (No 4 0) and thus the SNR goes to infinity, the Y a b c in general approach finite values when selective fading is present. Thus, there exists an irreducible error probability caused by the selective fading. This is to be expected since the intersymbol interference and waveform distortion introduced by the selective fading will cause errors even in the absence of additive noise.

V. ERROR PROBABILITIES FOR FSIZ In this section we will specialize the general expressions

derived above for the case of FSIZ transmission and a Gaussian shaped frequency autocorrelation function. As mentioned in the Summary we shall consider two types of FSK systems, phase-continuous and phase-

discontinuous, depending upon whether phase discontinuities do not or do appear, respectively, a t baud transition epochs. These two cases may be handled analytically by defining complex envelopes for mark and space oscillators, sM(t) and ss ( t ) , respectively, as follows:

where $M and #s are phase shifts associated with the oscillators, the integer n is equal to the frequency separation between the mark and space frequencies normalized with respect to the data rate 1/T and E is the received energy in one bit for a unity gain channel. To generate the mark and space waveforms let us assume that we switch between the oscillators with a timing such that baud transitions as observed at the receiver occur at the time instants t = kT where k is an integer. It is readily seen that a phase discontinuity of value

will occur at a baud transition. When snr(t) and ss( t ) are generated by independent oscillators, 9 may be expected to vary randomly over all angles with a rapidity depending upon the oscillator stability. Even when snr(t) and ss( t ) are generated from a common master oscillator, 9 will vary randomly over all angles.

If one could frequency modulate an oscillator with step changes in frequency for mark and space, it is readily seen that no phase discontinuities will result. From an analytical point of view such a frequency modulated output is identical to switching between two oscillators as described above but with 9 = 0. It is realized that abrupt step changes in oscillator frequency are impractical. However, if the transition time is small compared to baud duration T , a situation often attainable, little error will result in our subsequent analysis by assuming step changes in oscillator frequency.

In qur subsequent analysis we carry through analytical computations for a fixed value of y5(#Af and $s enter the computations only through $s + $M because we are considering an incoherent detection case) and give numerical results for the best case, 9 = 0 and the worst case 9 = T. As mentioned above, the case 9 = 0 cor- responds to phase-continuous FSII. By comparing results for the extreme cases 9 = 0 and 9 = ?r one may obtain bounds on the performance of randomly phase-discontinuous FSII (random 9).

I n the case of FSH it may be shown that the symmetry conditions (38) are satisfied. Thus binary symmetric operation will result for all degrees of frequency-selective fading. We need then evaluate p , only. The y's involved in the calculation of p , are yolo, ylll, yell, yo',. To compute these values of y we need to evaluate the moments mr. for s(t) equal to s,,,(t); a, c = 0, 1. Examination of (32) shows that s ( t ) enters in the determination of the moments

178 IEEE TRANSACTIONS ON COMMUNICATIONS SYSTEMS June

only through the functions D,(T) ; r = 0, 1. Thus we define

DZ"(7) == szbbe(t)sr(t + 7) d t ; a , b , c , T = 0 , 1

as the function D,(T) when s ( t ) = sabc(t) .

s It is readily found that

D:"(T) = 2Ee-in*r/T; I 7 I < T D y ( 7 ) = 0; b I < T

u:'o(7)

Note that we have computed the relevant D functions only for values of 7 in the interval (- T , T ) since, as may be seen from (32)) only such values of T are of relevance when the intersymbol interference is caused essentially only by adjacent pulses.

Before we may proceed further it is necessary to specify some form of frequency correlation function. Due to the lack of experimental data on frequency correlation functions, i t is difficult to make a choice of frequency correlation function for a specific scatter channel. We arbitrarily select a Gaussian frequency correlation function

R( a) = 2u2 exp --- [ z21 where B,, the correlation bandwidth, is defined here as the frequency spacing between the l /e points on the frequency correlation curve. The quantity u2 is just equal to the average power that would be received when a sinusoid of unity peak value is transmitted. The p ( ~ ) function corresponding to R(g) in (49) is

P(T) = u2&BE exp [-(?TI. TB, r (50)

In order for intersymbol interference effects on a given baud to be caused by adjacent bauds only, p ( r ) must have negligible values for 1 T I > T. For a given medium, this means that the data rate 1/T must be sufficiently small compared to the correlation bandwidth B,. If the normalized data rate d = l /TBc equals 7~/4 the function p ( ~ ) in (50) will have 99.5 per cent

from our calculations that satisfaction of the inequality

d < z 7T

is more than sufficient to ensure that intersymbol interference effects on a given baud :ire caused essentially only by adja'cent bauds.

To compulbe -yalc we need the moments m:tc defined by

r , s , a , c = 0 , 1. (52)

We have computed these moments only approximatelJ although very closely by using the D functions listed ir (43)-(48) in the integrand of (52) even though thest equations define only those parts of the necessary L functions lyi.ng in the range 1 7 I < T . The justificatior for this approximation may be appreciated by recalling our assumption that for the calculations of this papel p ( ~ ) will be essentially confined to the interval I 7 I < T Thus even t'hough the functions in (43)-(48) are used for 1 r I > T rind thus outside of their domain of definition p ( 7 ) will be so small for I 7 I > T that essentially al. the significant contribution to the integral in (52) wil come from integration over the interval -T < 7 < T

The moments evaluated according to the approximatior discussed above are exceedingly involved algebraicallJ and have b'een relegated to the Appendix. For verJ large mark-space frequency separation (Le., n = a) thesc expressions E:implify considerably and are shown below

my:' = 8E2,,'(1 - - 4d + 2d2 + ')I any + 7TG P

= 8 E 2 , r 2 ( 7 - 2d2 4- L, P

= o I J

(55)

in which

a2R Signal Power in Data Channel N O Noise Power in Data Channel p = - - = (56) of its area concentrated in the interval 1 7 I < T . It appears . _~._ ~ . ~ ~ ~~ ~ . . ~ ~ ~ ~ ~-


where we define the noise power in the data channel as the noise power in a rectangular band-pass filter of unity gain and bandwidth l/T.

For most practical applications the evaluation of system degradation for small amounts of selective fading is of more importance than the evaluation of degradation for large amounts of selective fading. Consequently we have computed Taylor series expansions of the moments mztc in the relative data rate parameter d. Upon using these expansions in the relevant expression for yal.we find that for small d the yalc are given by

Y l l l = P 7

for phase continuous FSK with mark-space frequency separation of 1/T cps, and by

Y l l l = P

Yo10 = -

d2 << 1

l u l l I I U U

1 + P - ( p + 2)lr >J for phase-discontinuous FSK with maximum phase discontinuity and with mark-space frequency separation l / T cps.

For the case of very large mark-space frequency separation (n = a) we find from (53)-(55) that

Y l l l = P 1

An examination of (57)-(59) reveals some interesting facts. First note that the phase-continuous FSK for

than either the phase-discontinuous FSK for n = 1, or general FSK for n = a. This lack of sensitivity is evidenced by the appearance of d in the third and fourth powers in (57) in corresponding places where d appears in the first and second powers, respectively, in (58) and (59).

Note also that as p ---f a only ylll + a, the other 7's approaching finite values dependent upon d. Since GL(y) decreases monotonically with y , approaching zero only as y -+ a, the existence of any finite values of yale for p -+ a means that the error probabilities approach finite nonzero values for large SNR. Thus there always exists an irreducible error probability when FSK (of the type described in this paper) is used over a frequency- selective fading medium.

As an illustrative calculation, in Figs. 3 and 4 we present curves of signal-to-noise degradation as a function of relative data rate d for an error probability of lo-* and for second- and fourth-order diversity ( L = 2, 4).

3.10.' 10-2 3.10.' Io.' 3.10" RELATIVE'OATA RATE d = l I E c T

Fig. 3-SNR degradation as a function of relative data rate for

width. FSK with mark-space frequency separation of reciprocal pulse-

n = I is less sensitive to frequency-selective fading Fig. 4-SNR degradation aa a function of relative data rate for

FSK with large mark-space frequency separation.

180

10.'

t

J

6 0 LL

c- m m

a

-

LL p 10-9 W

W 1 m a V 3

W CK n -

lo-6

IO-'

IEEE TRANSACTIONS ON COMiWUNICA3"IONS SYSTEMS June

I I T :PULSE WIDTH

E,:= CORRELATION BANDWIDTH BETWEEN e - ' POINTS ON I

1 1 FREQ. CORRELATION CURVE / / I / f Y L = <

I " 7 FREQ. SEPARATION BETWEEN MARK AND SPACE SIGNALS I I I/ I I1

L =ORDER OF DIVERSITY PHASE DISCONTINUITY AT MARK-SPACE TRANSITION

-1 l o - 2 3.10-' lo- ' 3.10.' I

RELATIVE DATA RATE d = I / B , T

1

Fig. 5-Irreducible error probability as a function of relative data rate for FSK.

Fig. 3 deals with phase-continuous and phase-discontinuous FSK for a mark-space frequency separation of a reciprocal pulsewidth while Fig. 4 deals with very large mark-space frequency separation.

The better performance of phase-continuous FSK against frequency-selective fading is clearly evidenced by these curves. It is interesting to note that increasing the order of diversity has a beneficial effect in reducing signal-to-noise degradation caused by frequency-selective fading. A similar observation has been made for the case of pure time-selective fading [l]. Fig. 5 presents curves of irreducible error probability as a function of relative data rate.

VI. ERROR PROBABILITIES FOR DIFFERENTIALLY COHERENT RECEIVERS

A simplified block diagram of the differentially coherent matched. filter receiver is shown in Fig. 6. It is assumed that only two possible pulses sl(t) , so(t) are transmitted as with the incoherent matched filter receiver. However, the transmitted bit is encoded into the change or lack of change of successive adjacent pulses. Thus a transmission of the pairs sl(t) + so(t + T ) or so(t) + sl(t + T ) denotes a "zero') say, while the pairs sl(t) + sl(t + T )

Fig. 6-Block diagram of differentially coherent matched filter receiver.

or so(t) + so(!: + T ) denotes a "one." The receiver consists in part of two separate coherent matched filter receivers with filters matched to waveforms sl(t) and so(t) but differing by .an input delay of T seconds. Thus the two receivers are operating upon adjacent bauds. The actual detector output is obtained by using the output of one coherent recI3iver as a reference for the other. This is done, as shown in Fig. 6, by a multiplication followed by extraction of the baseband component. In the special case when the pulses so(t), s l ( t ) are f l over the baud duration, the differentially coherent system specializer to the differentially coherent phase reversal Kineplea System.

As mentioned in Section IV of this paper we shal assume that the fading statistics are stationary in time It is then sufficient for us to evaluate the error prob. abilities for a pair of transmitted binary waveform: which in the absence of selective fading would be receivec in the time interval - T < t < T. If, for purposes o analysis, the operations of matched filtering and samplinl of t = T are combined, the complex envelopes of thc sampled output of the undelayed matched filter receivc may be expressed as

Vk = s,' Y k ( t ) M t ) - s%(t)l dt (60

while the complex envelope at the sampled output o the delayed matched filter receiver, may be expressed a

U k = 1' yk( t - T)[sT(t) - s*,(t)] dt (61

where y k ( t ) is the complex envelope of the receivec signal. Applying Parseval's Theorem to (60) and (61 and noting that Y,(f), the spectrum of y k ( t ) , nlay b expressed a& shown in (19), we find that U , and V k ma; be expressed. as the following frequency domain integrals

0

"m

i 96s Bello and Nelin: Frequency-Selective Fading and Binary Error Probabilities 181

As in [l] Section I11 the input to the threshold device is a random variable q given by

L

q = c [U*,V, + v,v*,1 (63) b=l

where Uk, V k are complex-valued Gaussian random variables. Fig. 7 shows in block diagram form the detailed mathematical operations involved in determining the diversity receiver output at the time t = T . Unlike the noncoherent detection case, the frequency domain operations in Fig. 7 are not identical in analytical form to the time domain operations shown for the same type of receiver in Fig. 7 of [l]. The reason for this dissymmetry is that the dual of the differentially coherent receiver is not a differentially coherent receiver of the type described here.13

To evaluate the desired error probabilities we need the probabilities Pr(q > 0) , Pr(q < 0) where q is given by (63). These probabilities are given by the same general expressions derived in [l] for the differentially coherent receiver, namely, (22a) and (22b) with y given by

2(%o + mT0) . \ / ( ~ l ~ + ~ ~ n ~ ~ + ~ ( ~ i ~ ~ n o - /m1n12)-(m10+mTo)

y= ____ + (64)

The required moments in (64) are determined from (62) [with the aid of the definitions (24)] as

e iz*( fr- -2r)T [sl(f> - so(f)l[S:(l) - ~ X O I df d l . (65)

Recalling that the additive noise and channel transfer function are independent zero-mean complex-valued Gaussian processes and using (3) and (26) in (65) one readily finds that Inr8 may be expressed as

in [13] to involve the comparison of a pulse with another pulse ad- 13 The dual of the PSK differentially coherent receiver is shown

jacent in frequency rather than time.

and

b = [ I Sl(f) - So(f) 1' d f = 1 I sl(t) - s d t ) I' at

a,, = r = s {:: r # s.

(68)

In the particular computations carried out in this paper it is convenient to express (66)-(68) in alternate time domain forms. Thus, if we use the definition

B,(T) = j" S*(f)e 'z*rfTISl( f ) - ~ ~ ( f ) ] e j 2 * " df

= [ S*(t - L!')[Sl(t f T ) - S o ( t T ) ] dt (69)

(where the second integral may be obtained from the first by an application of Parseval's Theorem), we find by an application of ParseVal's Theorem to (67) that

= [ B7(7)BZ(r)eizr Rr d r . (70)

A further application of Parseval's Theorem to (66) then results in [see definition of p ( ~ ) , (6 ) ]

mrs = 1; p ( r ) ~ , ( T ) ~ f ( r ) dT + 4i\iob6,,. (71)

The two error probabilities of concern are given by

p o = P r [ p > 0 I s(t) = s l ( t + T ) + so(t);

- T < t < T ]

+ $Pr[p > 0 I ~ ( t ) = so(l + T ) + sl ( t ) ;

- T < t < l ' ] (72)

pl = V r [ q > 0 I s(t) = so(t + T ) + so(t); - T < t < T ]

+ W - [ q > 0 1 S\t ) = Sl(t + T ) + a ( t ) ;

-T < t < l ' ] .

In the correct use of (72) the effects of intersymbol interference must be accounted for. Thus the waveform received in the time interval - T < t < T can, in theory, depend upon the form of s( t ) not only for time instants in the interval - T < t < T but for all time instants. If the degree of frequency-selective fading is small enough intersymbol interference effects need be considered only from pulses adjacent to the interval - T < t < T . Thus, for purposes of discussion, let us suppose this is the case and that 1's and 0's are equally likely to be transmitted. Then the probability p , that a "one" is printed when a transition is transmitted can be expressed as

(73)

182 IEEE TRANSACTIONS ON C0MMUNICAI”IONS SYSTEMS June

where 6 is the binary complement of b ( i e . , i = 0,6 = l),

P o b b c = pr[p > 0 I s(t) = s a b s c ( t ) ] (74)

and S a & ( t ) is equal to that portion of s(t) corresponding to the transmission of the bit sequence ab&, assuming that in the absence of selective fading the bit sequence b6 is received in the interval - T < t < T.

In an entirely analogous fashion, the probability that a “zero” is printed when no transition is transmitted is given by

p l = P a b b c 1 ’

(75) o = O b - 0 c - 0

where

p a b b c = p r [ q < 0 I s(t) = S a b b c ( t ) l (76)

and S,,bbc(t) has the obvious interpretation analogous to

A further increase in the degree of frequency-selective fading will eventually necessitate an inclusion of the effects of intersymbol interference produced by pulses further removed. However, as mentioned in the incoherent detection case, it appears that when more than adjacent pulses have to be considered, system degradation will be so high as to render such a communication system uninteresting from a practical point of view.

In order to evaluate the error probabilities po and p , it is necessary to evaluate y , (64), and thus the moments m,,, (71), for the 16 cases of transmitted pulse combinations considered in (73) and (75). To identify these values of y and m,., let -f&d and nzZfcd ; a, b, c, dl r , s = 0, 1, denote the value of y, and the. corresponding set of moments, computed on the assumption that s ( t ) = S a b c d ( t ) .

Our desired error probabilities are then

s a b x c ( t ) *

P O = ’ 2 2 k F L ( ’ Y a b x c ) (77) 8 o=-O b - 0 c - 0

P I = ’ 2 2 2 G,(yaaac>. (78) 8 a=@ b=O c = O

In general p , # p,. Thus, just as with the incoherent receiver, although binary symmetric operation may take place for nonselective fading, the introduction of frequency selectivity may be expected to bring nonsymmetric operation. However, in the incoherent detection case it was found that for certain practically interesting types of transmitted signals binary symmetric operation could still result irrespective of the degree of selective fading. We have not yet established the existence of such a class of . transmitted signals for the differentially- coherent receiver. In the case of phase-reversal keying studied in detail in the following section, binary nonsymmetric operation always results when frequency selectivity is introduced.

One may readily verify the existence of irreducible error probab:ilities for the differentially-coherent receiver by studying the behavior of y a b c d as the additive noise vanishes.

VII. ERROR PROBABILITIES FOR DPSK In this sec1;ion we will specialize the general expressions

derived above for the case of phase-reversal keying and the Gaussian shaped frequency correlation function already defin’ed in (49).

The two possible received binary signals in the absence of fading (unity gain channel) are assumed given by

for the case where they are received in the time interval 0 < t < T . Because binary symmetric operation does not exist fclr DPSK with frequency-selective fading it is necessary to use both (77) and (78) which together involve the determination of 16 values of y. Since each value of y requires the determination of three moments, the numerical determination of error probabilities appears quite involved algebraically. However, due to certain symmetry co’nditions many of the y’s can be shown t o be equal with the result that only 6 7’s need be computed.

To describe the symmetry conditions let abcd denote a sequence of transmitted bits and s & d ( t ) the corresponding received waveform (unity gain channel). The symmetry conditions are

p o b c d = pib;a (80)

p d c b a = P a b e d . (81)

Eq. (80) ma,y be established by noting that s z ; ~ ( t ) i z just the negakive of s a b e d ( t ) for the case of phase-reversal keying (79). .An examination of the differentially coherent receiver operation readily reveals that a change in polarity of the transmitted signal does not change the (signal part of) the detector output. Eq. (81) may be established by noting that S d e b a ( t ) = S & d ( - t ) , and then realizing from (62) that replacing s( t ) by s ( - t ) is statistically equivalent14 t o interchanging u k and v, which cannot change the detector output statistics.

In the subsequent discussion it will be convenienl to define “eq,uivalent” SNR’s as follows:

Pabbc = Y o b b o (82:

14 This assumes that the statistics of T k ( t ) and N k ( j ) are the samt

is valid if R(Q:) is an even function and N k ( f ) is white as assumec as those of T k ( -f) and N k ( - j ) , respectively. Such an assumptior

here.

i 963 Bello and Nelin: Frequency-Selective Fading and Binary Error Proba.bilities 183

With these definitions one may express the general conditional error probability as

P o b c d = G L ( P o b c d ) . (84)

The reason for defining equivalent SNR's here is the same as the reason for defining equivalent SNR's in the previous analysis [l] of the effects of pure time- selective fading, namely, by using the equivalent SNR in Pierce's expression" for error probability in the flat-flat fading channel with incoherent matched filter detection one may obtain the desired error probabilities for the more general fading channel with differentially coherent

Just as for the D functions in the incoherent detection case, we have computed the B functions only for values of r in the interval (- T , T ) since only such values of r are of relevance when the intersymbol interference is caused essentially only by adjacent pulses, as is assumed in the numerical calculations in this paper.

To compute y a b c d we need the moments [see (71)J r n ; p c d defined by

rn;".' = [ ~ ( T ) B , " * " ~ ( T ) [ B ~ ~ ~ ~ ( T ) ] * dT + 32EN06,,

a , b , c , d , r , s = 0 , l . (94) detection. When the selective fading dissappears one As in the analogous situation in the incoherent detection may verify that 2P as it since case, we have computed the moments only approximately has a 3-db SNR advantage Over FSK for a flabflat although very closely by using the B functions listed fading channel. in (88)-(92) in the integrand of (94) even though these

equations are valid only for I r I < T. The justification ties may be expressed as for this approximation here is the same as for the corres-

Due to the symmetry relations (88), (81) our probabili-

PO = t[GL(~olol) + G L ( ~ 1 l o o ) + 2G~(~moo)I (85)

p1 = t [ G L ( ~ o 1 1 0 ) + GL(~1111) + 2GL(~o111)1. (86)

The B function (69) appropriate to a specific received signal sequence S o b c d ( t ) is conveniently designated

B:bcd((7) = [ S z b c d ( t - rT)[sl(t + 7) - s0(t + r )] d t . (87)

It is readily determined that the B functions necessary to evaluate the required error probabilities are as listed below.

u:'"O(T) = I-4E; - T < r < O

ponding approximation in the incoherent detection case [see discussion following (52)] namely, that for d < 7r/4, the p ( r ) function of (50) may be regarded as essentially confined to values of 1 r I < T , at least for purposes of calculation of error probabilities.

The integrals defining the moments in (94) are quite simple and readily evaluated. If these moments are used to determine the required values of y, it is found from (82) and (83) that the equivalent SNR's needed in (85) and (86) are given by

Po101 = 2 4 1 - 7r4 - 4d + g

4d 1 - 7

[-4&(1 - $); 0 < T < I' 4d 4d2 (89)

Po100 = 2P (97) 1 -- 7r6 + 7

(,,(, + G ) ; -I' < 7 < 0 B:lO"(T) =

m; o < r < / 1 4d 1 - -

4d' 7r6

Po110 = 2P 1 + P ? r "

B:"o(T) = B;'""() Pllll = 2P (99) 2d

B,0"0(T) = -B;'"(T) 1 - - 7r4 B:"'(T) = 4E

Po111 = 2P -7- 2d2 (100) 1 7 1 < T (92) 1 + P ? r "

B:"'(T) = 4E

B;"'(T) = 4E l r l < T . (93) are only valid for d < 7r/4. In (97), (98) and (100) addi- B:"'(T) = B:'"(T) tional approximations were used to simplify the expres-

~ T / < T (91)

1 where it should be understood that these expressions

184 IEEE TRANSACTIONS ON COMII//UNICATIONS SYSTEMS June TABLE I

FSK-DPSK BREAK-EVEN RELATIVE DATA RATES !?OR A ERROR PROBABILITY

- -

Type of Relative Data Rate FSK d = 1/B-2' Order of Diversity

- -

n = l

2 . 7 . lo-' 6.4. lop2 2

4

2.i.10-1 4 b = ? r 6 . 5 . lo-? 2 n = l

2.5.10-' 4 + = o 5.8.10-2 2

- -

- - n = m

RELATIVE DATA RATE d i l / B C T

Fig. 8-SNR degradation as a function of relative data rate for: DPSK.

sions. Ilowever, for d < ?r/4 there is negligible error in these further approximations.

Examination of (95)-(100) reveal that as the input SNR p --+ 00 only pl l l l + m , all other equivalent SNR's approaching finite values.. Thus, as for the FSK case, an irreducible error probability is present.

It is interesting to note that for small d polo,, plloo and poloo decrease linearly with d while pollo, pl l l l and polll decrease only as d2. As a result p,, the probability of incorrectly detecting a transition, will increase more rapidly with normalized data rate than will p,, the probability of detecting incorrectly the absence of a transition. A possible heuristic explanation for this behavior is that a transition occupies more bandwidth than a lack of transition and thus is more adversely affected by the frequency-selective fading.

As an illustrative calculation we have plotted curves, shown in Fig. 8, of SNR degradation for second- and fourth-order diversity as a function of the ratio of the data rate to the correlation bandwidth and for a total error probability

p = 3bl + pol (101)

of The general behavior of these curves is the same as that for FSK. However, it is interesting to note that FSK appears generally less sensitive to the effects of frequency-selective fading than DPSII. Thus although DPSII is better than FSK by 3 db for a flat-flat fading channel, it eventually becomes worse than FSIC for a frequency-selective fading channel as the degree of frequency selectivity is increased. Table I gives a few values of relative data rate at which the DPSII and FSK systems break even for an error probability of For higher relative data rates more transmitter power is required with DPSK than with FSK to achieve the specified error probability of

Fig. 9 presents curves of irreducible error probability for the DPSK system for second- and fourth-order diversity as a function of relative data rate.

> c i m a m a. a 0

K 0

i d S W

W -I

0 L3 O W

m

a E

Io-(

=PULSE WIDTH

3,=.CORRELATION BANDWIDTH BETWEEN e - ' POINTS ON FREQUENCY CORRELATION CURVE

. = ORDER OF DIVERSITY

I

RELATIVE DATA R A T E d = I / B , T

Fig. 9-Irreducible error probability as a function of relative data rate for DPSK.

VIII. CONCLUDING REMARKS This paper has derived exact expressions for the binary

error probabilities of incoherent and differentially coherent matched filter receivers using postdetection dj.versity combining of independently time-flat (or pure frequency-selective fading) diversity channels containing no specular components. To illustrate these results, error probabilities were evaluated and SNR degradation curves presented for FSK and DPSK systems assuming a Gaussian shaped frequency correlation curve.

As discussed in the introduction the results of this paper will be 'of use in the estimation of the effects of

196s Bello and Nelin: Frequency-Selective Fading and Binary Error Probabilities 185

frequency-selective fading on binary error probabilities for a channel whose fading bandwidth is sufficiently small so that its transfer function changes negligibly for the duration of a binary signaling element and whose fading statistics may be characterized by stating that the transmission of a sinusoid results in the reception of a narrow-band Gaussian process. Thus, the results of this paper complement previous error probability results [I] which are useful in evaluating error probabilities for a channel with Gaussian statistics but with a correlation bandwidth large enough compared to the bandwidth of the binary signaling element so that purely time-selective fading results.

In a physical channel both time-selective and frequency- selective effects occur simultaneously. Thus when the duration of a binary signaling element is anywhere near the fading "correlation-time," SNR degradation will result due to the time-selective fading of the channel. A decrease in the duration of the signaling element will decrease the SNR degradation caused by the time- selective fading. However, since the time-bandwidth product of a pulse has a lower bound, the reduction in duration of the signaling element is accompanied by an increase in its bandwidth. Thus decreasing the duration of the signaling element will decrease the SNR degradation only up to the point at which the bandwidth of the signaling element begins to approach the correlation bandwidth of the medium. Then SNR degradation will begin to increase with further reduction in signaling element duration.

It follows that a curve of SNR degradation (or binary error probability) as a function of binary data rate will exhibit a minimum for physical channels. If the time-bandwidth product of the signaling elements are not too large and if the correlation bandwidth is several orders of magnitude bigger than the fading bandwidth this minimum will be very broad and, in fact, the channel will appear like a flat-flat fading channel in the vicinity of the minimum, like a frequency-flat fading channel for lower data rates, and like a time-flat fading channel for larger data rates.

Unfortunately, for large time-bandwidth product signaling elements or for values of the ratio of correlation bandwidth to fading bandwidth which are not very large, the minimum occurs in a region wherein both time- and frequency-selective effects affect the SNR degradation. The authors have studied this combined time frequency-selective fading case and are presently readying a paper [20] on this subject.

APPENDIX

In this Appendix we list values of the moments m::c obtained by using the D functions (43)-(48) and the p(7 ) function (50) in (52). As discussed in the body of the paper this procedure results in an approximation which is quite accurate for d < a/4. These moments are

(103)

186 IEEE TRANSACTIONS ON COM.MUNICATIONS SYSTEMS June

where the function O(z), given by the integral [lO] J. Dugundji, “Envelopes and pre-envelopes of real waveform,” I R E TRANS. ON INFOEMATION THEORY, vol. IT-4, pp. 53-57; March, 1958.

[ll.] R. Arens, “Complex processes for envelopes of normal noise,” O(x) =- 1’ et ’ dt 0 IRE TRANS. ON INFORMATION THEORY, vol. IT-3, pp. 204-207;

has been ta,bulated by Jahnke and Emde [19]. [12] D. Gabor, “Theory of communications,” J. IEE, vol. 93,

[13] P. A. Bello, ‘“Time-Frequency Duality,” Sect. 1 of vol. I1 of [l] P. A. Bello and B. D. Nelin, “The influence of fading spectrum Final Report to ITT Communication Systems, Inc., Paramus,

on the binary error probabilities of incoherent and differentially- N. J., Subcontract No. 480.116D, Task No. 6003-0326; coherent matched filter receivers,” IRE TRANS. ON COMMUNI- ADCOM, Inc., Cambridge, Mass., 1962. (To be published in CATION SYSTEMS, vol. CS-10, pp: 160-168; June, 1962.,,

[2] H. B. Voelcker, “Phase-shift keying in fading channels, Proc. [14] -, “Characterization of Randomly Time-Variant Linear the IEEE TRANS. ON INFORMATION THEORY.)

IEE, vol. 107, pt. B, pp. 31-38; January, 1960. 131 R. Price, “Optimum detection of random signals in noise, with munication Systems Inc., Paramus, N. J., Subcontract No.

Channels,” Sect. 1 of vol. I1 of Final Report to ITT Com-

application to scatter multipath communication, I,” I R E 480.117D, Task No. 6003-0326; ADCOM, Inc., Cambridge, TRANS. ON INFORMATION THEORY, pp. 125-135; December, 1956.

Mass., 1962. (To be published in IEEE TRANS. ON COMMUNI-

[4] R. Price, “Error Probabilities for the Ideal Detection of Signals [15] T. Hagfors, “‘Some properties of radio waves reflected from CATIONS SYSTEMS.)

Perturbed by Scatter and Noise,” M.I.T. Lincoln Lab., Lexing- ton, Mass., Group Rept. No. 3440; 1955.

the moon and their relation to the lunar surface,” J . Geophs.

[5] G. L. Turin, “On optimal diversity reception,” I R E TRANS. [16] R. Price and P. E. Green, Jr., “Signal Processing in Radar Res., vol. 66, pp. 777-785; March, 1961.

[6] G. L. Turin, “On optimal diversity reception 11,” IRE TRANS. ON INFORMATION THEORY, vol. IT-7, pp. 154-166; July, 1961. Astronomy,” M.I.T. Lincoln Lab., Lexington, Mass., Rept.

No. 234; 1960. ON COMMUNICATION SYSTEMS, vol. CS-10, pp. 22-31; March, [l7] P. E. Green, Jr., “Radar measurement of target characteristics,” 1962.

[7] G. L. Turin, “The characteristic function of hermitian quad- in “Radar Astronomy,” by J. V . Harrington and J. V. Evans. (To be publiihed.)

ratic forms in complex normal variables,” Biometrika, vol. 47, [18] J . N. Pierce, “Theoretical diversity improvement in frequency- pp. 199-201; June, 1960.

[8] P. A. Bello, “Binary Error Probabilities for Matched Filter [19] E. Jahnke and -F. Emde, Tables of Functions, Dover Publica- shift keying,” PROC. IRE, vol. 46, pp. 903-910; May, 1958.

Rece tion Over Selectively Fading Channels Containing Specu- lar &mponents,” ADCOM, Inc., Cambridge, Mass., Research [20] P. A. Bello and B. D. Nelin, “Binary Error Probabilities for

tions, Inc., Nmew York, N. Y., p. 32; 1945.

Rept. No. 6; 1963. [9] P. M. Woodward, “Probability and Information Theory,”

Non-Cohered; and Differentially-Coherent Matched Filter Re- ception Over ‘Time and Frequency Selective Fading Channels,”

McGraw-Hill Book Company, Inc., New York, N. Y.; 1953. ’ Research Rept. No. 7, ADCOM, Inc., Cambridge, Mass.; 1962.

September, 1957.

pt. 111, pp. 4:!9-457; November, 1946. REFERENCES

Retransmission Error Control*

Summary-This paper discusses a simple, easily implemented form of retransmission error control; namely, the use of vertical and horizontal block parity checks. The method discussed provides a level of error rate performance which is far better than that which can be expected from the equipment into which it must operate. For instance, for a random bit error probability of a 22 X 22 bit block yields an output error rate of 4.4(10)-14, with an efficiency of 87 per cent. This form of retransmission error control, therefore, represents the basic method against which other methods can be compared and evaluated. An analysis is presented which determines the parameters for optimum performance for both random and bunched errors.

INTRODUCTION T MUST CERTAINLY ‘be agreed that in order to transmit digital information over existing transmission channels for which the average bit error prob-

ability is greater than the required output bit error rate, some form of error control is required. This control may be some very simple procedure, such as decreasing

* Received April 30, 1963. t ITT Communication Systems, Inc., Paramus, N. J.

or increasing transmission rate, increasing transmitter power, increasing receiver sensitivity, repeating all messages, or, indeed, ceasing transmission during known periods of high noise intensity. These and countless ot8her precauticlnary procedures must be included in the term “error control.” However, for the purposes of this discussion, it will be assumed that most of the above- mentioned LLcontroIsJ’ are in effect and that the task of the communicator is to provide additional control so that a sufficient number of the remaining errors (the proportion to be dictated by the user) may be detected and corrected. However, it should be emphasized that sometimes a great deal of error control may take place with ab’solutely no automatic error correction. It is often the case, because of contextual redundancy or the fact that unimportant portions of the message are affected, that simple identification of the characters in. error is sufticient. In times of emergency it should be the prerogative of the message recipient, as well as the originator, to cancel the correction properties of any system in order to make more prudent use of

Documents

The Effect of Frequency Selective F;Ading on the -Binary Error Probabilities of Incoherent and Differentially Coherent Matched. Filter Receivers