dfs1.pdf - Dr M Venu Gopala Rao Professor Dept of ECE KL...

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Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University. 1 Course Notes on Digital Signal Processing Discrete Fourier Series (DFS) Consider a periodic sequence [ ] p x n that is periodic with period ‘ N ’., so that [ ] [ ] p p x n rN x n for any value of ‘ r ’. Such a sequence [ ] p x n can be represented by sum of sine and cosine sequences or equivalently complex exponential sequences with frequencies that are integer multiples of fundamental frequency 2 N associated with periodic sequence. The Discrete Fourier Series ‘Analysis’ and ‘Synthesis’ equations are expressed as 1 0 [ ] [ ] , 0,1, 2,..., 1 N Kn p p N n X K x n W K N Analysis equation 1 0 1 [ ] [ ] , 0,1, 2,..., 1 N Kn p p N K x n X K W n N N Synthesis equation where 2 j N N W e , and both [ ] p x n and [ ] p X K are periodic sequences with period N . Example 1 : Determine the Fourier series coefficients of a periodic signal shown in Figure. Solution: From the given figure, the periodic signal can be expressed as [ ] {0, 1, 2, 3} x n with period N = 4. Therefore 1 0 3 4 0 0 3 2 4 4 4 4 3 2 4 4 4 [ ] [ ] , 0,1, 2, ..., 1 [ ] , 0,1, 2, 3 [0] [1] [2] [3] , 0,1, 2, 3 0 2 3 , 0,1, 2, 3 N Kn p p N n Kn p n K K K p p p p K K K X K x n W K N x n W K x W x W x W x W K W W W K
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Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University. 2 , 2 0 0 0 0 4 4 4 4 2 1 2 3 1 4 2 4 4 4 4 2 4 6 2 1 1 4 4 4 4 4 4 3 6 9 3 4 4 4 4 [0] 0 2 3 0 1 2 3 6 1 [1] 2 3 2( 1) 3 2 2 cos sin 2 2 [2] 2 3 1 2 3( 1) 2 1 [3] 2 3 2 2) K j K N p N j j p p p X W W W W e W X W W W j j j W e e j j X W W W W W W j j X W W W j W W                     X o o o o 0 0 0 2 1 4 4 1 [ ] {6, ( 2 2), 2, ( 2 2)} [ ] {6 0 , 2.8284 135 , 2 180 , 2.8284 135 } | [ ]| {6, 2.8284, 2, 2.8284} [ ] {0,135 ,180 , 135 } p p p p W j j X K j j X K X K X K         X The Discrete Fourier Series can be written as 1 3 0 0 0 4 4 4 4 3 2 2 3 2 2 1 1 [ ] [ ] , 0,1, 2, ..., 1 [ ] , 0,1, 2,3 4 1 6 ( 2 2) 2 ( 2 2) 4 1 6 ( 2 2) 2 ( 2 2) 4 3 1 1 2 2 2 2 N Kn Kn p p p N N K K n n n j n j n j n j n j n j n x n X K W n N X K W n N W j W W j W j e e j e j j e e e                
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