4.28. (a) Let x(t) have the Fourier transform X(jw), and let p(t) be periodic with funda-...

 

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4.28. (a) Let x(t) have the Fourier transform X(jw), and let p(t) be periodic with funda- mental frequency wo and Fourier series representation p(t)=aef 8118 Determine an expression for the Fourier transform of y(t) = x(t)p(t). (P4.28-1) (b) Suppose that X(jw) is as depicted in Figure P4.28(a). Sketch the spectrum of y(r) in eq. (P4.28-1) for each of the following choices of p(t): (i) p(t) = cos(1/2) (ii) p(t) = cost (iii) p(t) = cos 2r (iv) p(t) = (sint)(sin 2r) (v) p(t) = cos 2t-cost 5+% (vi) p(t) -8(-n) = (vii) p(t)=-8(1-2n) (viii) p(t) ==-8(-4) (ix) p(t)=6(t-2n) - -6(t - n) = (x) p(t) the periodic square wave shown in Figure P4.28(b). p (t) X (joo) (a) 1 -3 -2T ** 2TT 4T 5T t (b) Figure P4.28 4.28. (a) Let x(t) have the Fourier transform X(jw), and let p(t) be periodic with funda- mental frequency wo and Fourier series representation p(t)=aef 8118 Determine an expression for the Fourier transform of y(t) = x(t)p(t). (P4.28-1) (b) Suppose that X(jw) is as depicted in Figure P4.28(a). Sketch the spectrum of y(r) in eq. (P4.28-1) for each of the following choices of p(t): (i) p(t) = cos(1/2) (ii) p(t) = cost (iii) p(t) = cos 2r (iv) p(t) = (sint)(sin 2r) (v) p(t) = cos 2t-cost 5+% (vi) p(t) -8(-n) = (vii) p(t)=-8(1-2n) (viii) p(t) ==-8(-4) (ix) p(t)=6(t-2n) - -6(t - n) = (x) p(t) the periodic square wave shown in Figure P4.28(b). p (t) X (joo) (a) 1 -3 -2T ** 2TT 4T 5T t (b) Figure P4.28

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