The maximum likelihood receiver (MLR) to detect a pulsed signal propagated through a turbulent medium is found under assumptions that approximate optical propagation over a long path through the atmosphere. Limiting cases of the MLR are the coherent heterodyne receiver (CHR) and the total-energy-measuring receiver. With an additional assumption, the probability of error of the MLR is approximated. The CHR, even with an optimally chosen aperture weighting function, can be markedly inferior to the MLR. Unlike the optimized CHR, the MLR can effectively use arbitrarily large apertures. A coherent optical system is given that, in principle, realizes the MLR.