1 consider the sinusoidal signalxt 8 sin6pit

Question

1.  Consider the sinusoidal signal

x(t) = 8 sin(6Πt + φ₀).

Assume φ₀ = Π/4 for this question and φ₀ = 0 for the following two questions. Compute analytically the spectrum. Sketch the magnitude and phase spectrum.

What is the meaning of "Nyquist rate", and what is the Nyquist rate for this signal? (Consider the signal to be a low-pass signal.)

2. The sampling function

T_s = 1/4, is used to obtain the sampled signal y(t) = x(t) • s(t).

Sketch y(t) alongside x(t) (at least 3 periods). Sketch the magnitude spectrum of y(t). Hint: Explicit computation of the spectrum Y(f) is not required.

3. Assume now T_s = 1/16. The signal x(t) is reconstructed from the sampled signal y(t) using flat-top sampling, giving the reconstructed signal z(t).

Sketch z(t) alongside x(t) (at least three periods). Discuss if the original signal x(t) can be perfectly reconstructed from z(t).

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