Question

1) Discrete Fourier Transform (DFT) and Its Properties: a) By hand (analytically), determine the DFT of the vector: x[n] = [-1 2 -5 4]. b) Consider the periodic, infinite-duration sequence: x[n] = cos($\frac{\pi}{10}$n). This sequence has a period of N=20 samples. In MATLAB, construct the vector x1, which comprises the first 20 samples of this sequence (i.e., for sample indexes 0?n<20). Use the \"fft()\" function to numerically compute the DFT of this sequence. Stem plot your results. (In general, the DFT is complex-valued and will require a separate magnitude and phase plot versus frequency index \"k\"-unless all DFT values happen to be real-valued!) In MATLAB, next construct the vector x2 using the first 25 samples of the analytic sequence (i.e., for sample indexes 0?n<25). Use the MATLAB \"fft()\" function to numerically compute the DFT and again stem plot the results versus \"k\". These two plot sets should look significantly different. Hand in the two sets of plots (you need not hand in the MATLAB code) and explain why the DFT representation of this signal is so different between the two DFTs. c) The first five points of the eight-point DFT of a real-valued sequence are: [36 -4j10 -4 -j4 -4-j2 -4]. Write the complete eight-point DFT sequence.

          1) Discrete Fourier Transform (DFT) and Its Properties:
a) By hand (analytically), determine the DFT of the vector: x[n] = [-1 2 -5 4].
b) Consider the periodic, infinite-duration sequence: x[n] = cos($\frac{\pi}{10}$n). This sequence has a period of N=20 samples.
In MATLAB, construct the vector x1, which comprises the first 20 samples of this sequence (i.e., for sample indexes 0?n<20). Use the \"fft()\" function to numerically compute the DFT of this sequence. Stem plot your results. (In general, the DFT is complex-valued and will require a separate magnitude and phase plot versus frequency index \"k\"-unless all DFT values happen to be real-valued!)
In MATLAB, next construct the vector x2 using the first 25 samples of the analytic sequence (i.e., for sample indexes 0?n<25). Use the MATLAB \"fft()\" function to numerically compute the DFT and again stem plot the results versus \"k\".
These two plot sets should look significantly different. Hand in the two sets of plots (you need not hand in the MATLAB code) and explain why the DFT representation of this signal is so different between the two DFTs.
c) The first five points of the eight-point DFT of a real-valued sequence are: [36 -4j10 -4 -j4 -4-j2 -4]. Write the complete eight-point DFT sequence.

1) Discrete Fourier Transform (DFT) and Its Properties:
a) By hand (analytically), determine the DFT of the vector: x[n] = [-1 2 -5 4].
b) Consider the periodic, infinite-duration sequence: x[n] = cos((π)/(10)n). This sequence has a period of N=20 samples.
In MATLAB, construct the vector x1, which comprises the first 20 samples of this sequence (i.e., for sample indexes 0?n<20). Use the f̈ft()f̈unction to numerically compute the DFT of this sequence. Stem plot your results. (In general, the DFT is complex-valued and will require a separate magnitude and phase plot versus frequency index k̈-̈unless all DFT values happen to be real-valued!)
In MATLAB, next construct the vector x2 using the first 25 samples of the analytic sequence (i.e., for sample indexes 0?n<25). Use the MATLAB f̈ft()f̈unction to numerically compute the DFT and again stem plot the results versus k̈.̈
These two plot sets should look significantly different. Hand in the two sets of plots (you need not hand in the MATLAB code) and explain why the DFT representation of this signal is so different between the two DFTs.
c) The first five points of the eight-point DFT of a real-valued sequence are: [36 -4j10 -4 -j4 -4-j2 -4]. Write the complete eight-point DFT sequence.

Added by Miriam L.

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