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4.1 Computing the DFT 1. We will now develop our own DFT functions to help our understanding of how the DFT comes from the DTFT. Write your own Matlab function X = DFTeqn(x) to implement the DFT of equation where x is an N point vector containing the values x [0], ···, x [N – 1] and X is the corresponding DFT. Your routine should implement the DFT exactly as specied by the DFT equation X(k) = ?_{n=0}^{N-1} x[n]e^{frac{-j2?kn}{N}}, using for-loops for n and k. 2. Test your routine DFTeqn by computing X(k) for each of the following cases: a. x[n] = ?[n] for N = 10. b. x[n] = 1 for N = 10. c. x[n] = e^{frac{j2?n}{N}} for N = 10. d. x[n] = cos(frac{2?n}{10}) for N = 10 Then, plot the magnitude of each of the DFT's. 3. Write a second Matlab function x = IDFTeqn(X) for computing the inverse DFT where X is the N point vector containing the DFT and x is the corresponding time-domain signal. Use the function IDFTeqn to invert each of the DFT's computed in the previous problem. Plot the magnitudes of the inverted DFT's, and verify that those time-domain signals match the

          4.1 Computing the DFT

1. We will now develop our own DFT functions to help our understanding of how the DFT comes from the DTFT. Write your own Matlab function X = DFTeqn(x) to implement the DFT of equation where x is an N point vector containing the values x [0], ···, x [N – 1] and X is the corresponding DFT. Your routine should implement the DFT exactly as specied by the DFT equation X(k) = ?_{n=0}^{N-1} x[n]e^{frac{-j2?kn}{N}}, using for-loops for n and k.

2. Test your routine DFTeqn by computing X(k) for each of the following cases:
a. x[n] = ?[n] for N = 10.
b. x[n] = 1 for N = 10.
c. x[n] = e^{frac{j2?n}{N}} for N = 10.
d. x[n] = cos(frac{2?n}{10}) for N = 10
Then, plot the magnitude of each of the DFT's.

3. Write a second Matlab function x = IDFTeqn(X) for computing the inverse DFT where X is the N point vector containing the DFT and x is the corresponding time-domain signal. Use the function IDFTeqn to invert each of the DFT's computed in the previous problem. Plot the magnitudes of the inverted DFT's, and verify that those time-domain signals match the

$4.1 Computing the DFT 1. We will now develop our own DFT functions to help our understanding of how the DFT comes from the DTFT. Write your own Matlab function X = DFTeqn(x) to implement the DFT of equation where x is an N point vector containing the values x [0], ···, x [N – 1] and X is the corresponding DFT. Your routine should implement the DFT exactly as specied by the DFT equation X(k) = ?n=0^N-1 x[n]e^frac-j2?knN, using for-loops for n and k. 2. Test your routine DFTeqn by computing X(k) for each of the following cases: a. x[n] = ?[n] for N = 10. b. x[n] = 1 for N = 10. c. x[n] = e^fracj2?nN for N = 10. d. x[n] = cos(frac2?n10) for N = 10 Then, plot the magnitude of each of the DFT's. 3. Write a second Matlab function x = IDFTeqn(X) for computing the inverse DFT where X is the N point vector containing the DFT and x is the corresponding time-domain signal. Use the function IDFTeqn to invert each of the DFT's computed in the previous problem. Plot the magnitudes of the inverted DFT's, and verify that those time-domain signals match the$

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