1. Determine the 4-point DFT matrix D4 and the corresponding inverse DFT matrix D^-1 4.

2. This problem investigates zero padding applied in the frequency domain. Plot each N-point DFT as a function of frequency fk = k/N.

(a) In MATLAB, create a vector x that contains one period of the sinusoid Plot the result. How "sinusoidal"

does the signal appear?

(b) Using the fft command, compute the DFT X of vector x. Plot the magnitude of the DFT coefficients. Do they make sense?

(c) Zero pad the DFT vector to a total length of 100 by inserting the appropriate number of zeros in the middle of the vector X. Call this zero-padded DFT sequence Y. Why are zeros inserted in the middle rather than at the end? Take the inverse DFT of Y and plot the result. What similarities exist between the new signal y and the original signal x? What are the differences between x and y? What is the effect of zero padding in the frequency domain? How is this type of zero padding similar to zero padding in the time domain?

(d) Derive a general modification to the procedure of zero padding in the frequency domain to ensure that the amplitude of the resulting time-domain signal is left unchanged.

(e) Consider one period of a square wave described by the length-8 vector [1, 1, 1, 1, -1, -1, -1, -1]. Zero pad the DFT of this vector to a length of 100, and call the result S. Scale S according to (d), take the inverse DFT, and plot the result. Does the new time-domain signal s[n] look like a square wave? Explain.