1965 Viterbi: Optimum Detection and Signal Selection ability density function of x1 (and x,) is equal to the input power spectrum P,(X) and the probability density func- AT, = s f”Pz(f) df = x;” = XT tion of y, Pr(z), is equal to the Fourier transform of T(T). Moments of the spectra P,(f), P#(f) are then identical to we readily hnd that averages of random variables. Tyo) = (27rj)m(x, - xJrn From (8) we note that 239 033 T(%)(O) = (27rj)” 1 f”P&) df = (27rj)m 2 cy Mm-,M,( - 1)” (611 (59) k=0 where CT is the familiar binomial coefficient. It should Thus, upon computing the average of y’” in (58) and be noted that the above calculations are meaningful only defining when the various moments exist. Optimum Detection and Signal Selection for Partially Coherent Binary Communication ANDREW VITERBI, SEXIOR MEMBER, IEEE Absirncl-The optimum detectors for coherent and noncoherent reception of binary signals in additive Gaussian noise and the re- sulting error probabilities were obtained by Helstrom [l]. In many practical communication systems a reasonable estimate of the phase of the received signal is available as the result of an auxiliary track- ing operation of the carrier signal by a coherent tracking device such as a phase-locked loop. It is shown that the optimum detector for this case, which we refer to as partially coherent reception, is a linear combination of the correlation detector and the squared envelope correlation detector, which are optimum for the coherent and noncoherent cases, respectively. The error probabilities are also obtained as a function of the energy-to-noise ratio of the channel and the variance of the error in the phase estimate, which is a function of the signal-to-noise (SNR) in the tracking loop. The signal selection problem is con- sidered in terms of these parameters. 7r RANSMISSION of a sequence of binary digits over a physical channel perturbed by additive Gaussian noise is achieved by selecting the signal waveform so(t) corresponding to a “0” and sl(t) .corre- sponding to a ‘I 1”. The waveforms are amplitude and phase-modulated sinusoids, which may be denoted as Sk(t) = A,(t) cos [wd + e,(t) + 41 k = 0, 1 Olt<T (1) where Ak(t) and t?,<(t)represent the envelope and phase modulating waveform, w0 and 4 are the carrier frequency Manuscript received May 12, 1964; revised December 16? 1964. This paper presents the results of one phase of research carried out at the Jet Propulsion Lab., California Institute of Technology, Pasadena! under Contract NAS 7-100, sponsored by the National Aeronautics and Space Administration. The author is with the University of California, Los Angeles, Calif. and is a consultant to the Jet Propulsion Lab., California Institute of Technology, Pasadena, Calif. and phase, and T is the transmission time per digit or the inverse data rate. It is frequently convenient to rewrite the expression as sk(t) = Re [M,c(t)e~‘““t+4’] (k = 0, 1) (2) where AT,(L) = ilk(t)ez8hct) is referred to as the complex envelope. It is generally assumed that w0 is known exactly to the receiver; if 4 is also known exactly the reception is said to be coherent. If 4 is unknown at the receiver it is generally taken to be a random variable with the uniform probability density function and the reception is referred to as noncoherent. The opti- mum detectors and the resulting performances are well known for both cases and for any choice of signals Sk(t). [I] Quite often, however, the receiver has obtained some information relative to the carrier phase, typically by means of a coherent carrier tracking device known as a phase-locked loop. In such cases the receiver has some a priori probability density function other than the uniform one on the unknown phase parameter 4. It has recently been shown [2], [3] that when the simplest (first- order) tracking loop is employed to produce an esti- mate of the ca,rrier phase,l the stationary probability 1 This is obtained by tracking an unmodulated carrier. If the carrier phase varies randomly with time so that +(t) is a stationary random process the model is still reasonable provided the band- width of 4(t) and of the loop are very small compared to l/T. When a filter is employed in the phase-locked loop, (3) is still a good ap- proximation to t’he phase error probability density for sufficiently large 01.