340 Lecture 18 
Let us consider the sonar equation as a possible starting point or gauge for 
research in signal processing. Now, the central point concerning the sonar equa- 
tion is that itis a power signal-to-noise relationship. It epitomizes what 
may be called the "signal-to-noise" approach to system performance in general 
and to signal processing in particular. In such a relationship, propagation phe- 
nomena and signal processing are linked only in terms of power and intensity 
loss. Several comments may be made concerning this signal-to-noise approach. 
Since power is the parameter used, the processing techniques implied in 
this approach are those directly related to or of a lower order than power. 
Speaking statistically, for most cases of interest power is a second central 
moment of the basic physical data. Power can be used to measure directly two 
averages of the second-order joint probability density function of the basic 
physical data: the power spectrum and the cross- or autocorrelation function. 
Thus we find that such average processing can readily be incorporated into the 
sonar equation. Whether other processing principles bear a relation to power 
simple enough to permit such incorporation is another question. 
A second comment is that variability and fluctuations due to the medium 
and other acoustic elements of the problem are taken into account only on an 
ad hoc basis. For many situations of interest, both waveform and wavefront 
variability and fluctuation may reasonably be expected to be of the same order 
of magnitude as, or much greater than, the deterministic elements of the prob- 
lem, and so must certainly enter our thinking other than as an error estimate 
or a first-order correction. 
Closely related to this point is the fact that since there are a number of 
single terms, each connected with a specific item of acoustic generation or 
propagation or processing in the sonar equation, there is no explicit treatment 
of interactions. Interaction between different sources of interference, between 
our instrumentation and the acoustic field, and generally between the acoustic 
and processing elements of the problem, cannot be treated on other than an 
ad hoc basis. In addition, the separability implied by these single terms leads 
directly to a point of view in which we assume we can take propagation loss 
from one set of measurements, source level from another, and so forth. In fact, 
however, the question of just what interactions are involved, whether they are 
negligible, and what degree of separability is possible and useful must be con- 
sidered carefully from the very start of a processing study. 
As to research in signal processing, however, the most serious comment 
I would like to present is that the signal-to-noise approach, as embodied in the 
sonar equation, does not guide us very effectively in improving our processing. 
What Eq. (1) tells us, for example, is that to improve the signal differential, 
and hence our decision-making ability, we must look for a noisier target, or a 
stretch of ocean with less transmission loss, or build a bigger array, or find 
an area with lower noise background, or somehow find a processor which gives 
us the same decision-making ability with a lower signal differential, but such 
comments are inevitably phrased in terms of what we have done in proc- 
essing design—in the signal-to-noise approach, the interesting and heuristic 
physics has already been smoothed over. 
