S-6] 



NONLINEAR AND TIME-DEPENDENT OPERATIONS 



257 



A Clamping Circuit. Clamping circuits, sometimes called pulse 

 stretchers or boxcar detectors, are another common component of radar 

 systems. They also provide an example of an operation with a periodic 

 time dependency. Such a circuit clamps the output to a sampled value of 

 the input for a fixed period of time; at the end of this period, the output is 

 clamped to a new value of the input. The operation of such a circuit is 

 shown in Fig. 5-8. Symbolically, the output of this device can be repre- 



Clamped Output 



nput Signal 



-7 H TIME 



Fig. 5-8 Operation of a Clamping Circuit. 



sented by 



y{t) = x{tk), t, < t < tk + i = tk-\- T. (5-64) 



Clearly the autocorrelation of_y(/) is dependent upon time. As with the case 

 of the product demodulator, however, the time average of the autocorre- 

 lation function and power density spectrum yield results which can be used 

 for almost all applications. To determine the average autocorrelation of 

 y{t), consider that when the delay ti — t\ = r, used in computing the 

 autocorrelation functions, is a multiple of the sampling interval T, the 

 average value of the productjyiV2 of the sampled and stretched process must 

 be the same as the average value of the product x-^Xi because at the sample 

 points ^1 = xi and jy2 = Xi- Thus, for t = kT, 



<Py{kT) = ^{kT). (5-65) 



When the time delay is intermediate between these isolated points, say 

 kT < t < (k -\- l)T, the autocorrelation function of jy will sometimes be 

 (p(kT) and sometimes (p(kT -\- T) depending upon the value of /. The 

 fraction of the time during which <py{T) takes one of the other of these values 

 is proportional to the relative values of r — kT and {k -\- \)T — r. Thus, 

 the average value of ^2,(t) should vary linearly between its values at the 

 discrete points where r = kT, and it will be composed of these points 

 connected by straight lines. 



A limiting case of special interest occurs when the sampling frequency is 

 much smaller than the width of the input spectrum. In this case, the 

 autocorrelation function of the input is narrow compared with the sampling 

 period. That is, values of the process which are separated by more than 

 the sampling period are very nearly independent. Since in this case 



