Please make sure your answer is 100% correct. I do not have the solutions to thi

Question

Please make sure your answer is 100% correct.

I do not have the solutions to this problem, and I rely on your answer to be correct.

Pease don't submit your answer unless you know 100% it is correct.

Music produced by an orchestra (or a rock band, for that matter) is a continuous-time signal that typically contains frequencies up to 30 KHz. Thus if we let x(t) denote this signal, then the Fourier transform X(j omega) of x(t) satisfies: On the other hand, the human ear typically cannot perceive frequencies above 10 KHz. In fact, one can model the human ear as an ideal low-pass filter with a cut-off at tvc = 10 KHz. What is the lowest frequency omega s at which music can be sampled to ensure that its digital recording does not sound distorted to human listeners? Equivalently, what is the smallest possible value of ws = 2 pi/Ts in the system below: Design a system that allows sampling music without perceivable distortion at a lower frequency than what you found in part (a), by pre-processing the signal x(t) prior to sampling it. Equivalently, it is required that at the output of the system below:

Explanation / Answer

Read 'S' as omega s.

R (sampling rate) = S KS/s
fs (signal being sampled = 30kHz
fN (the Nyquist frequency) = (S/2) KHz
fa (aliased frequency) >= 10kHz

The frequency of the aliased signal can be found from the following simple equation:

fa = R*n - fs
where n is the closest integer multiple of the sampling rate (R) to the signal being aliased (fs).

Human ear can percieve upto 10Khz.

=> Signal upto 10Khz should not be distorted.

=> fa>=10KHz

=> S*n-30Khz >= 10 Khz

Put n=1.

=>S>=40 Khz

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