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Widft12 - ECE 421 Sum 2010 Detailed Solutions Set 12...

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ECE 421 - Sum 2010 Detailed Solutions Set 12 : Inverse Discrete Fourier Transform 1 12-1. Let a time domain signal x m be the M 4 sample set given by the time-domain vector x , where x x 0 x 1 x 2 x 3 T 1 1 1 1 T Use the matrix interpretation of the DFT to compute the frequency-domain vector X , where X X 0 X 1 X 2 X 3 T DETAILED SOLUTION: Note that if we drew the x m by “connecting the dots”, we would see a DC waveform, a “low” frequency. Therefore, the DFT should produce a math result which is consistent with this observation. Substituting the x m from the problem statement into equation (4) from the Notes then gives X 0 X 1 X 2 X 3 1 1 1 1 1 e j π 2 e j π e j 3 π 2 1 e j π e j 2 π e j 3 π 1 e j 3 π 2 e j 3 π e j 9 π 2 1 1 1 1 a Evaluating the complex exponential terms in the matrix, you will see that equation a can be simplified to X 0 X 1 X 2 X 3 1 1 1 1 1 j 1 j 1 1 1 1 1 j 1 j 1 1 1 1 b Carrying out the matrix multiplication in b then gives the entire collection of frequency domain compo- nents: X 0
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