# DFT

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## 1) Part 1

Consider a signal x_1[n] = (-j)^n + e^{j\pi n/4}

Enter your answer as a python list of 8 literal values.

For example, {1\over 2}\delta[k-1] would be

`[0, 0.5, 0, 0, 0, 0, 0, 0]`

.

Enter a python list of 16 literal values for the DFT coefficients.

## 2) Part 2

Consider a different signal x_2[n] that is zero outside the range 0\leq n < 6, whose corresponding DFT (computed with N=6) is given by:

Try to arrive at your solution without explicitly determining x_2[n].

Enter your solution as a python list of 6 literal values. For example,

`[0, 0, 1j, 0, 0, 0]`

.

Try to arrive at your solution without explicitly determining x_2[n].

## 3) Part 3

Consider a signal x_3[n], which is nonzero only for n=0,1,2,3.

You know the values of the **DTFT** at two points only. In particular, you
know the values X_3(\frac{\pi}{2}) and X_3(\frac{3\pi}{2}). You also know
the value of \sum_m x_3[m] and \sum_m e^{j\pi m}x_3[m].

**Check Yourself:**Explain how you could determine x_3[n] from this information.

This question isn't graded, but it's really good quiz prep! So you should
**solve the problem on your own** before looking at someone else's solution.
Once you've done the problem, you can see the staff solution if you'd like by
pressing Show/Hide below.

## Show/Hide

x_3[n] is non-zero for n=0,1,2,3. The DTFT of x_3[n] is:

Let X_3[K] be the DFT of x_3[n] with analysis window N=4. Then,

The right hand side of each of the equations are given as known in the problem. We can determine x_3[n] by applying the DFT synthesis equation using these four values.