The following sampling system will be used for the next few problems. An input signal r(t)...

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The following sampling system will be used for the next few problems. An input signal r(t) is passed 2π through an anti-aliasing filter Ha (w) before it is sampled at T = seconds. The resulting sampled signal 5 2π xg(t) is then passed through a reconstruction filter with frequency spectrum H,(w) = rect (5). 5 x(t) Anti-Aliasing Filter: Ha(w) Ta(t) sampler: ST, (t) x, (t) Reconstruction Filter: H, (w) r(t) = sinc² xr(t) Problem 6 Determine and sketch the CTFT of anti-aliasing filter Ha(w) -12 ≤ w≤ 12 with maximum cutoff frequency to prevent aliasing. Hint: It should have magnitude equal to one and a cutoff frequency in rad/sec we wg /2 (half the Nyquist rate). = Problem 7 We now want to take the same signal of an anti-aliasing filter you designed in Problem 6. (a) Determine and sketch X(w) for -12 <w ≤ 12 (b) Determine and sketch Xa(w) for -12 ≤ ≤ 12 (c) Determine and sketch X, (w) for -12 ≤w ≤ 12 (d) Determine and sketch X, (w) for -12 ≤ w≤ 12 (e) Does aliasing occur for this signal in this system? Explain your answer. (f) Does x, (t) = xa(t)? (g) What changes to the system (not the signal) would be needed to get x,(t) = x(t)? Explain in detail! Hint: Each of the 3 blocks will require changes. 3 2 t and pass it through this system with The following sampling system will be used for the next few problems. An input signal r(t) is passed 2π through an anti-aliasing filter Ha (w) before it is sampled at T = seconds. The resulting sampled signal 5 2π xg(t) is then passed through a reconstruction filter with frequency spectrum H,(w) = rect (5). 5 x(t) Anti-Aliasing Filter: Ha(w) Ta(t) sampler: ST, (t) x, (t) Reconstruction Filter: H, (w) r(t) = sinc² xr(t) Problem 6 Determine and sketch the CTFT of anti-aliasing filter Ha(w) -12 ≤ w≤ 12 with maximum cutoff frequency to prevent aliasing. Hint: It should have magnitude equal to one and a cutoff frequency in rad/sec we wg /2 (half the Nyquist rate). = Problem 7 We now want to take the same signal of an anti-aliasing filter you designed in Problem 6. (a) Determine and sketch X(w) for -12 <w ≤ 12 (b) Determine and sketch Xa(w) for -12 ≤ ≤ 12 (c) Determine and sketch X, (w) for -12 ≤w ≤ 12 (d) Determine and sketch X, (w) for -12 ≤ w≤ 12 (e) Does aliasing occur for this signal in this system? Explain your answer. (f) Does x, (t) = xa(t)? (g) What changes to the system (not the signal) would be needed to get x,(t) = x(t)? Explain in detail! Hint: Each of the 3 blocks will require changes. 3 2 t and pass it through this system with

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