TABLE OF CONTENTS The Fourier transforms of time functions, f( t), we have discussed to this point are valid only for those for which the integral of its magnitude is finite so that they satisfy Eq. (5.9). However, the Doppler radar echo is not such a waveform. For its analysis, we consider it to be a waveform that has existed for a very long time, in fact an infinite time so that its existence is for ? ? < t < ?. Such waveforms are called continuing waveforms because they continue forever. Our objective is to determine the frequency spectrum of such a waveform. Now, the mean-square value of a waveform f a( t) is [5]
Clearly, the mean-square value of normal continuing waveforms is greater than zero. It can be shown that the Fourier transform of any waveform with a nonzero mean-square value does not exist. [6] This places a difficulty in our determination of the waveform frequency spectrum because Fourier transform theory cannot be used. We can, however, determine the waveform power density spectrum by determining its autocorrelation function.
The autocorrelation function of a real continuing time function f a( t) is
For ? = 0, we observe that 
. There are some important properties of the autocorrelation function. One property is that it is an even function of ?. To show this, we have from our definition of the autocorrelation...
