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FFT Decimation in Time 1 Decimation in Time In this method we split x (n) into the even indexed x (2m) and the odd indexed x (2m + 1) each N/2 long. DFT can be expressed as: X (k ) N / 2 1 x(2m)W m 0 X (k ) 2 mk N N / 2 1 m 0 x(2m)W mk N /2 N / 2 1 W x(2m 1)W m 0 k ( 2 m 1) N N / 2 1 k N x(2m 1)W m 0 km N /2 K = 0,1,2,3,4,……………..,N-1 We have used the fact WN2 mk e j 2 mk2 / N e j 2mk /( N / 2) WNmk/ 2 (1) Decimation in Time Define new functions as G (K) is a DFT for the even samples of x (n), or G (k ) N / 2 1 x(2m)W m 0 k 0,1,2,...., N / 2 1 mk N /2 H (K) is a DFT for the odd samples of x(n), or H (k ) N / 2 1 x(2m 1)W m 0 km N /2 k 0,1,2,3,..., N / 2 1 Then equation (1) for the first N/2 points can be expressed as X (k ) G (k ) WNk H (k ) k 0,1,2,...., N / 2 1 Decimation in Time But WN( N / 2 k ) WNk G (k ) G ( N / 2 k ), k 0,1,2,..., N / 2 1 H (k ) H (k N / 2) k 0,1,2,...., N / 2 1 Then X (k N / 2) G (k ) WNk H (k ) k N / 2, N / 2 1,...., N This can be expressed in the following diagram Decimation in Time G(0) x(0) x(2) x(4) G(1) 4-point DFT x(7) G(2) H(0) x(1) x(5) X(1) X(2) G(3) x(6) x(3) X(0) 4-point DFT X(3) W80 H(1) W81 H(2) W82 H(3) W83 -1 X(4) X(5) -1 -1 -1 X(6) X(7) Decimation in Time x(0) x(4) x(2) x(6) x(1) x(5) x(3) x(7) X(0) 2-point DFT 2-point DFT X(1) W80 X(2) -1 W82 X(3) -1 W80 2-point DFT 2-point DFT -1 W81 X(5) -1 W82 W80 -1 W83 W82 -1 X(4) -1 -1 X(6) X(7) Decimation in Time X(0) x(0) x(4) W80 -1 W80 x(2) x(6) X(1) W82 W80 W80 -1 W81 W80 -1 x(3) x(7) X(3) -1 -1 x(1) x(5) X(2) -1 -1 W80 W82 -1 X(5) -1 W82 W80 -1 W83 X(4) -1 -1 X(6) X(7) Decimation in Time Example Given a sequence x(n) where x(0) = 1, x(1) = 2, x(2) = 3, x(3) = 4 and x(n) = 0 elsewhere ,find DFT for the first four points Solution 4 x(0)=1 x(2)=3 W40 1 -2 -1 6 x(1)=2 x(3)=4 X(0 )= 10 W40 1 -2 -1 X(1)= -2+2j W40 1 -1 W41 j -1 X(2)= -2 X(3)= -2-2j Inverse Fourier Transform The inverse discrete Fourier can be calculated using the same method but after changing the variable WN and multiplying the result by 1/N Example Given a sequence X(n) given in the previous example. Find the IFFT using decimation in time method Solution X(0) =10 8 4 12 8 1/4 x(1) = 2 12 1/4 x(2) = 3 16 1/4 x(3) = 4 1/4 x(0) = 1 ~ X(2) =-2 W40 1 -1 X(1) = -2+2j ~ X(3) = -2-2j ~ -4 j4 W40 1 -1 W40 1 -1 ~ W41 j -1