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Problem3 - Problem3(a Explain in detail the differences...

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Problem3(a): Explain in detail the differences between the DTFT and DFT for an N-point sequence. Fourier Transform pair 1: Discrete Time fourier Transform(DTFT) ( ) ( ) j j n n H e h n e ϖ ϖ - = -∞ = IDFT: 1 ( ) ( ) 2 j j n h n H e e d π ϖ ϖ π ϖ π - - = Fourier Transform Pair 2: Special case of DTFT is DFT: Discrete Fourier Transform(DFT) 2 1 [ ] 0 ( ) [ ] ( ) k N j k n j N n X e x n e X k π ϖ - - = = = IDFT: 2 1 [ ] 0 1 [ ] ( ) 0 1 N j k n N n x n X k e k N N π - = = - ATTRIBUTES OF DTFT are Continuous ,Periodic with period 2π and Complex Whereas ATTRIBUTES OF DFT are Discrete ,Periodic with period 2π and Complex In DTFT Sampling is performed only in time domain whereas DFT is obtained by performing sampling operation in both the time and frequency domains. The sufficient condition for the existense of DTFT is it should be absolutely summable.i.e;the series should converge whereas DFT exists for every finite length sequence. ( ) k S h k = - ∞ = < ∞
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DFT is the solution of DTFT at discrete frequencies. Example: Let 0 [ ] ( ) 1 1 n n n h n u n β β β = = = < ∞ - β<1 Absolutely summable DTFT 0 0 1 ( ) ( ) 1 j n j n n j n j n n H e e e e ϖ ϖ ϖ ϖ β β β - - - = = = = = - For n=N-1 i.e;for finite number of frequency samples we can compute DFT Problem3(b): Explain in detail what is scalloping in using DFT.
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