DFTs of Note

The questions below are due on Thursday April 10, 2025; 02:00:00 PM.

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DFT

In this problem, we will consider a CT signal consisting of two distinct notes, one an "C" at 523Hz, and another an "A" at 1760Hz, each represented by a pure cosine.

Imagine sampling this signal at a rate f_s = 2000 samples per second to produce a DT signal, and then computing the DFT of that signal with some value N, where N is a power of 2. Plotting the DFT magnitudes gives us the following graph for some range of k values (not necessarily the whole range of k values). Note also that we have plotted this as a line graph for clarity even though X[k] is actually a function of discrete k):

Under this analysis, the "C" note (523Hz) corresponds to peaks centered at k=\pm 268. Given this information, answer the following questions:

What value of N was used in this calculation?
N=~

The "A" note (1760Hz) will manifest as peaks at \pm some k value, where k is in the range [0, N/2]. At what value of k will this peak occur?
Enter your answer as a Python expression (not necessarily simplified); and note that you can use Python's round function to round to the nearest integer; you do not need to calculate the exact result by hand.

Which of the following statements best describes the relationship of these signals to the graph above?

The "C" note caused the peaks at \pm k_1, and the "A" note caused the peaks at \pm k_2.

The "C" note caused the peaks at \pm k_2, and the "A" note caused the peaks at \pm k_1.