# Frequencies

If you are a current student, please Log In for full access to the web site.

Note that this link will take you to an external site (

`https://shimmer.csail.mit.edu`) to authenticate, and then you will be redirected back to this page.

## 1) Part 1

Consider the following CT signal:

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].

## 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.