MODULE-10 - MODULE 10 DISCRETE FOURIER TRANSFORM Inversion...

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Unformatted text preview: MODULE 10 DISCRETE FOURIER TRANSFORM Inversion and Properties Cyclic Convolution Uniqueness of DFT for Convolution Linear Convolution via DFT Sectioned Convolutions Page 10.1 INDEX Discrete Fourier Transform Proof of DFT Inversion Discrete Fourier Series DFS Interpretation of the DFT Relationship of DFT to Other Transforms Properties of the DFT Cyclic Convolution Example: Cyclic Convolution Uniqueness of DFT for Convolution DFT Matrix Linear Convolution by DFT Length of Linear Convolution Example - Linear Convolution Via Cyclic Convolution Sectioned Convolutions Problem with DFT-Based Convolution Sectioned Convolution via "Overlap-Add" Selection of Section Length L Example - Sectioned Convolution Sectioned Convolution via "Overlap-Save" Direct vs. DFT Computation of Convolution MAIN INDEX Page 10.2 10. DISCRETE FOURIER TRANSFORM index READ : Chapter 8 of Oppenheim & Schafer. Work as many related problems as possible. Definition : The Discrete Fourier Transform ( DFT ) of the finite- length sequence { x ( n ); n = 0 ,..., N-1} is: X ( m ) = DFT N { x ( n ) } = - = 1 N n x ( n ) e-j 2 nm / N ( DFT ) for m {0 ,..., N-1}. Definition : The Inverse Discrete Fourier Transform ( IDFT ) of { X ( m ); m = 0 ,..., N-1} is given by: x ( n ) = IDFT N { X ( m ) } = 1 N - = 1 N m X ( m ) e j 2 nm / N ( IDFT ) for n {0 ,..., N-1}. The following notation will sometimes be used: x ( n ) DFT X ( m ) Page 10.3 Proof of DFT Inv ersion index For any n, the IDFT is: 1 N - = 1 N m X ( m ) e j 2 nm / N = 1 N - = 1 N m [ - = 1 N k x ( k ) e-j 2 km / N ] e j 2 nm / N X ( m ) = 1 N - = 1 N k x ( k ) - = 1 N m e j 2 m ( n-k )/ N = -- =-- n k e e n k N N k n j N N k n j ; 1 1 ; / ) ( 2 / ) ( 2 ] [ = N ( n- k ) = 1 N x ( n ) N = x ( n ) QED Page 10.4 Discrete Fourier Series index Periodic Extension : Given a finite-length sequence { x ( n ); n = 0 ,..., N-1} define the periodic sequence { x ( n ); - < n < } by x ( n ) = x ( n ) N = x ( n mod N ). The sequence x ( n ) with period N is called the periodic extension of x ( n ). It has a fundamental frequency 2 N . The sequence x ( n ) does not have a Z-transform or a convergent Fourier sum (why?). But it does have ... Page 10.5 Discrete Fourier Series : Any period- N sequence has a Discrete Fourier Series ( DFS ) representation - a finite sum of complex exponentials e j (2 / N ) kn with frequencies 2 N k : x ( n ) = 1 N - = 1 N m X m e j 2 nm / N with DFS coefficients : X m = - = 1 N n x ( n ) e-j 2 nm / N The proof is essentially identical to the DFT inversion proof....
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MODULE-10 - MODULE 10 DISCRETE FOURIER TRANSFORM Inversion...

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