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Frequencies

The questions below are due on Monday March 29, 2021; 10:00:00 PM.
 
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Note that this exercise is intended to be solved by hand, without the use of computation.

1) Part 1

Consider the following CT signal:

x(t) = 6\cos(42\pi t) + 4\cos(18\pi t - 0.5\pi)

Now imagine that this signal is sampled with a sampling rate of f_s = 60Hz to obtain a discrete-time signal, which is periodic. We want to find the DFT coefficients associated with this signal when analyzed with N equal to the fundamental period of x[n].

In this analysis, which values of k in the range [0, N) are associated with non-zero X[k]? Enter your answer as a Python list of integers.

If you instead analyze with N=80, which values of k in the range [0, 80) are associated with non-zero X[k]? Enter your answer as a Python list of integers.

If you instead analyze with N=42, which values of k in the range [0, 42) are associated with non-zero X[k]? Enter your answer as a Python list of integers.

2) Part 2

Consider an abitrary CT signal x_c(\cdot), which is "band-limited" so that it does not contain any frequency content for any \omega such that |\omega| \geq (2\pi\times 1000), i.e., X(\omega)=0 for all |\omega| \geq (2\pi\times 1000). This signal is then sampled with a sampling rate f_s to produce a new DT signal x[\cdot], where x[n] = x_c(n / f_s).

Then, length-N portions of x[\cdot] are analyzed using the DFT. For reasons associated with computational efficiency, we will assume N is a power of 2. Both f_s and N can be chosen at will, subject to the constraint that aliasing must be avoided and N=2^v for some integer v.

Determine the minimum value of N, and also f_s, so that the frequency spacing between DFT coefficients is less than or equal to 2Hz. Enter your answer as a tuple of integers (N, f_s) below.

N and f_s: