1.34 What is the result of the following MATLAB command? Do not use direct computation of the DFT. Show your derivation or explain your answer. >> fft([1 0 -1 0 1 0 -1 0 1 0 -1 0])
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First, we need to understand what the fft function does in MATLAB. The fft function stands for Fast Fourier Transform, which is a mathematical algorithm used to compute the discrete Fourier transform (DFT) of a sequence or signal. The input to the fft function Show more…
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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 specified by the DFT equation X(k) = Σ_{n=0}^{N-1} x[n]e^{frac{-j2πkn}{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] = δ[n] for N = 10. b. x[n] = 1 for N = 10. c. x[n] = e^{frac{j2πn}{N}} for N = 10. d. x[n] = cos(frac{2πn}{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
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The known sequence x(n)={1, 2, 3, 4}, n=0:3, completes the following calculations: (1) x(n) ⊗ x(n) [take the circumferential convolution length N=5] (2) x(n) ⊗ x(n) [take the circumferential convolution length N=7] (3) x(n) * x(n) (4) Write the step of calculating the circular convolution using the DFT method.
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