Question

1. Consider a signal of the form (t) = 2 cos(100nt) cos(1507) This signal is first sampled at the rate of 80 samples per second and the result was processed with an ideal reconstruction filter, again assuming that sampling rate was 80 samples per second. What is the signal that results after the reconstruction? Show enough details in your answer to demonstrate that you understand the theory of sampling and reconstruction from samples. Hint: Write (t) as a sum of two sinusoids using trigonometric identities. Sketch the Fourier transform of the impulse train sampled signal and identify the signal within the bandwidth of the reconstruction (lowpass) filter. What is the bandwidth of this lowpass filter? [25 points)

Answer 1

22 (+) = 267 (luont) (soat) - Cos Cloont nisontst cos sont-100at) z Cos (250nt) + Cos(soat). = $[25_sort of sort] { [essorte ejet] - } [esar nasty a j2a(16)+] + 4f2540374 estabesse) The Fourier trawnform of x<t) will be x(bas= f[act)) - SxCty ejado dt = ? [8(6+2507) + 6(-2500) + 8(6+507) + 8(W-son) ..() == [8 (f +25)+ $(8-25)+ 8 (f+125)+ $(f=125)] The spectrum of x (f) is shown in fig-2 $ (Hz) ____- -5 P2 25 figoli Spectrum of X (f) X(t) is sampled at fs = 80 samples/ second if the sampled signal is x* (L), then **() - x6f4fs) [where , T ] The spectrum of f) is shown in fig. 2 -

1 XG) ideal Tecomten f (Hz) 50 205 E -205 -105_ go -55-4540 To 40 45 55 80 105 Efs - 2-fs fig. 2: spectrum of truy x*(F) a The sampled signal is reconstructed using ideal reconstruction filter, assuming the sampling rate 80 samples sec. fs. = 802, Bw of the ideal reconstruction filter will be 80 ltz ie how fs / to foi re from - 40Hz to 40 Hz. output, Xo (f) = 0.2 a output signal [ro (t) = 0) (Aus)