32 Lecture 2 
nection with wide-band signals of continuous spectrum (i.e., noise emissions); 
in this case its use is natural since it provides the process of cross-correlation 
which is used so extensively in the mathematical consideration of such signals. 
But the use of the process in narrow-band (and coherent) echo ranging seems 
to be a new development and we are not aware that anyone else has been inves - 
tigating its practical problems. 
2.2.2. Superdirective Arrays 
There is copious literature on superdirective arrays [5], to which we have 
contributed [6], but practical results are not conspicuous. There is no doubt that 
the academic interest of superdirectivity rather exceeds its practical utility. 
Naturally, the idea of obtaining a greatly narrowed beamwidth is very attractive, 
but there are numerous disadvantages and difficulties in this method of attaining 
the desired result: 
a. The signals must be restricted to a very narrow band. 
b. The sensitivities (or excitations, since the method is applicable also to 
radiators) of the various elements of the array must be set to extremely 
close limits. 
c. On reception, the signal-noise performance (in relationto receiver noise) 
is very poor; on transmission, the efficiency is very low. 
d. The calculation of a practical array is made very difficult—at present, 
perhaps impracticable in most cases—by interelement coupling, which 
is inevitably a major effect in an array where adjacent elements must 
be less than halfa wavelength apart and must usually operate in antiphase. 
It thus seems unlikely that there will be much serious development of super- 
directive arrays of the normally considered single-frequency multielement type. 
But it will be shown in the next section that multifrequency two-element arrays 
may encourage trials of superdirectivity for limited applications, since they 
almost eliminate the difficulty of interelement coupling. 
It seems fairly certain that multiplicative arrays are superior in most re- 
spects to superdirective arrays, since they are quite easily realized and give 
a much better signal-noise performance. Figure 2.2 shows how a particular 
directional improvement can be obtained in both ways, but while the multiplicative 
method costs only 3 db in worsening of the noise factor, the superdirective 
method costs 20 db. 
2.2.3, Multifrequency Two-Element Arrays 
The idea of "space-frequency equivalence," where the effect of a multielement 
array is obtained from two elements used simultaneously at a number of fre- 
quencies, appears to be very recent [7,8]. The basic reasoning is simple; the | 
spacing of the two elements is a different number of wavelengths at each fre- 
quency, so that the use of a harmonic series of frequencies can give the effect 
of a uniformly spaced multielement array. Althoughthis kind of array is in some 
respects analogous to an ordinary multielement array, it has been shown that 
it is certainly not properly analogous [9]. Even to make equivalent transmitting 
and receiving arrays separately stretches the analogy somewhat, and it fails 
completely when one tries to associate a multifrequency two-element transmitter 
