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Do the following questions with Matlab code

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Do the following questions with Matlab code.

The Discrete Fourier Transform & Filtering

Implement the matrix-vector form of this discrete Fourier transform.

 

To do this, write two MATLAB functions x_jw=myfft(x_n) and x_n=myifft(x_jw) which must

reside in their own MATLAB files: myfft.m and myifft.m respectively. The function x_jw=myfft(x_n)

implements the discrete fourier transform by computing the matrix Ww and multiplying this matrix times

the signal, x_n, which is assumed to be a column vector. The result is a column vector which 1s the discrete

Fourier transform of the input, x_jw. The function x_n=myifft(x_jw) implements the inverse discrete

Fourier transform by computing the matrix W 4, and multiplying this matrix times the signal, x_jw,

which is assumed to be a column vector. The result is a column vector which is the inverse discrete Fourier

transform of the input, x_n.

 

Test your DFT function using a MATLAB script (name it as myp.m)

 

1. Generate a sinusoidal sequence z[n] = cos(Zn) 0 <n < N —1 where N = 6.

 

2. Compute the DFT of x[n], X[k].

 

3. Compute the IDFT of X[k], x[n].

 

4. Compute the sum of the point-by-point errors in the reconstruction: e = ∑^N_n=0|x[n] – x[x]|.

If e >10^-15 you have a bug in your code, find it and fiz it!

 

1. Turn in your MATLAB code consisting of printouts of myfft.m, myifft.m, and myp.m.

 

2. Using subplot() to generate a 3x1 matrix of plots, show the input signal x[n] from (1), the magnitude

response of DFT X[k] from] (2) and the phase response of the DFT from (2) &{k]. Note: You may

consider rounding the DFT values to their 6th digit of precision to prevent numerical instability, 1e.,

0.12345678... rounds to 0.123457.

 

3. Explain the following: For each index k, explain the (possibly complex) observed value of the DFT,

X[k], in terms of the magnitude and phase of the complex number (there are 5 such numbers).

 

4. Change the value of N such that N = 7 and use subplot() to generate a 2x1 matrix of plots showing the

input signal x[n] from (1) the magnitude response of the DFT X[k] from (2) and the phase response

of the DFT from (2) O{k]. This DFT should look different!

 

5. Explain why the DFT from steps (2) and (4) are different.

 

6. Change the phase of x{n] by generating a new signal x[n] = cos(4(n- 1) O<n<N-1

where N = 6 and recompute the DFT.

 

7. (10pts) Using subplot() to generate a 3x1 matrix of plots, show the input signal x{n] from (1), the

magnitude response of the DFT |X[k]| from (2) and the phase response of the DFT from (2) &k).

 

8. Explain why the DFT from steps (2) and (7) are different.

 

g. For a sinusoidal sequence x[n] = cos(n) 0 <n<N

 

Determine if there is a value for N Which makes the magnitude response X[k] zero everywhere except

for two indices of k as in Task 1, step (2). Write a paragraph explaining your solution and show a plot

of your results.