Signal Changes

Let x[n] represent a periodic, discrete-time signal with a period of 6, a portion of which is shown below:

In this problem, we will explore alternative signals that we can make by changing a small number of samples per period in the signal above. Consider the related signals below (where x_1[\cdot] through x_5[\cdot] represent time-domain signals and X_1[\cdot] through X_5[\cdot] represent the associated Fourier series).

For each, we want to know whether it is possible to create a signal with the given properties by modifying at most two samples per period of x[n]. If it is possible, specify which value(s) you wish to change, and what you wish to change them to. If it is not possible, put an X in each box.

There may be multiple solutions to some of the following parts. You need only find one solution for full credit.

Part a. x_1

We would like to create x_1[\cdot] by modifying at most two samples in x[\cdot] so that X_1[\cdot] is a symmetric function of k.

Part b. x_2

We would like to create x_2[\cdot] by modifying at most two samples in x[\cdot] so that e^{j5\pi k/ 3}X_2[\cdot] is purely imaginary.

Part c. x_3

We would like to create x_3[\cdot] by modifying at most two samples in x[\cdot] so that \sum_{m=0}^{17} X_3[m] = 0.

Part d. x_4

We would like to create x_4[\cdot] by modifying at most two samples in x[\cdot] so that X_4[0] = 0.

Part e. x_5

We would like to create x_5[\cdot] by modifying at most two samples in x[\cdot] so that X_5[k] = -X_5[k+1] for all k.

Please upload a single pdf file that contains your answers to all parts of this problem:

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