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By Paul A. Lynn (auth.)

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3(a). ". ifnl n't:~. 3 -I (a) A repetitive pulse waveform, and its real exponential RJurier coefficients for (b) k = 3; and (c) k = 5 practical interest because similar waveforms occur widely in such devices as digital computers and radar and communication systems; and secondly it is of analytical interest because it provides a good starting point for a discussion of the Fourier transform. For convenience we will assume (as in the diagram) that the period of the wavefonn is K times as large as the pulse duration.

To take a straightforward example, suppose we have (s+ a) G(s) =(s + ~)(s + 1) 48 ANALYSIS AND PROCESSING OF SIGNALS We write the identity A B (s + a) - - + - - = --'-----''--(s + (3) (s + r) (s + (3)(s + r) and multiply both sides by (s + (3)(s + r), giving A(s + r) + B(s + (3) = (s + a) Equating coefficients of equal powers of s yields A +B= 1 and Ar +B(3= a from which B= (r -a) (r - (3) and A = (a - (3) (r - (3) We now have G(s) expressed in a much simpler form; both A/(s + (3) and B/(s + r) have an inverse transform of simple exponential form, so the time function corresponding to G(s) is equal to the sum of two exponentials.

7. The imaginary part ofG(w) peaks strongly in the region of W = wo, which is not surprising since the waveform f(t) = e -crt sin wot must be expected to contain large sinusoidal components at frequencies close to woo On the other hand we would not anticipate strong cosine components in this frequency region and the form of the real part of G(w) confirms this view. 8 (a) The real and (b) imaginary parts of the spectrum of the waveform shown in figure 3. 7 part of G( w) is an even function of w, whereas the imaginary part is odd.

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