The input signal for the sampling and reconstruction system in Fig. P-1 isfor – ∞<t ∞....

The input signal for the sampling and reconstruction system in Fig. P-1 is

for – ∞<t ∞. The frequency response of the lowpass reconstruction filter is

where Ts is the sampling period.

(a) Determine the Fourier transform X(jω)and plot the Fourier transform Xs(jω)for – 2π/Ts<ω 2π/Ts for the case where 2π/Ts = 1000π. Carefully label your sketch. What is the output xr(t)in this case?

(b) Now assume that ωs.= 2π/T, = 500π. Determine an equation for the output xr(t).

(c) Is it possible to choose the sampling rate so that

where A is a constant? If so, what is the value of Ts and what is the numerical value of A?