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EELE 4310: Digital Signal Processing (DSP) Chapter # 8 : Efficient Computation of the DFT: Fast Fourier Transform Algorithms Spring, 2012/2013 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 1 / 15 Outline Efficient

  1. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 8 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 5 / 15

  2. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 8 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 5 / 15

  3. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 8 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 5 / 15

  4. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 8 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 5 / 15

  5. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 8 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 5 / 15

  6. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 8 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 5 / 15

  7. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 8 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 5 / 15

  8. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 8 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 5 / 15

  9. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 8 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 5 / 15

  10. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  11. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  12. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  13. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  14. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  15. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  16. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  17. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  18. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  19. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  20. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  21. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  22. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 9 Ex. Using the decimation in time FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } Check that DFT of x ( n ) = { 2 , 1 , 2 , 1 } is X ( k ) = { 6 , 0 , 2 , 0 } . EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 6 / 15

  23. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 10 Consider the case where N = 8, the first decimation yields the sequence x (0) , x (2) , x (4) , x (6) , x (1) , x (3) , x (5) , x (7). The second decimation results in the sequence x (0) , x (4) , x (2) , x (6) , x (1) , x (5) , x (3) , x (7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x ( n ) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order. Ex. the data point x (3) ≡ x (011) is placed in the position m = 110 or m = 6 in the decimated array. Therefore, the data x ( n ) after decimation is stored in bit-reversed order. If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order. EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 7 / 15

  24. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 10 Consider the case where N = 8, the first decimation yields the sequence x (0) , x (2) , x (4) , x (6) , x (1) , x (3) , x (5) , x (7). The second decimation results in the sequence x (0) , x (4) , x (2) , x (6) , x (1) , x (5) , x (3) , x (7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x ( n ) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order. Ex. the data point x (3) ≡ x (011) is placed in the position m = 110 or m = 6 in the decimated array. Therefore, the data x ( n ) after decimation is stored in bit-reversed order. If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order. EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 7 / 15

  25. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 10 Consider the case where N = 8, the first decimation yields the sequence x (0) , x (2) , x (4) , x (6) , x (1) , x (3) , x (5) , x (7). The second decimation results in the sequence x (0) , x (4) , x (2) , x (6) , x (1) , x (5) , x (3) , x (7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x ( n ) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order. Ex. the data point x (3) ≡ x (011) is placed in the position m = 110 or m = 6 in the decimated array. Therefore, the data x ( n ) after decimation is stored in bit-reversed order. If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order. EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 7 / 15

  26. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 10 Consider the case where N = 8, the first decimation yields the sequence x (0) , x (2) , x (4) , x (6) , x (1) , x (3) , x (5) , x (7). The second decimation results in the sequence x (0) , x (4) , x (2) , x (6) , x (1) , x (5) , x (3) , x (7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x ( n ) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order. Ex. the data point x (3) ≡ x (011) is placed in the position m = 110 or m = 6 in the decimated array. Therefore, the data x ( n ) after decimation is stored in bit-reversed order. If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order. EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 7 / 15

  27. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 10 Consider the case where N = 8, the first decimation yields the sequence x (0) , x (2) , x (4) , x (6) , x (1) , x (3) , x (5) , x (7). The second decimation results in the sequence x (0) , x (4) , x (2) , x (6) , x (1) , x (5) , x (3) , x (7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x ( n ) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order. Ex. the data point x (3) ≡ x (011) is placed in the position m = 110 or m = 6 in the decimated array. Therefore, the data x ( n ) after decimation is stored in bit-reversed order. If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order. EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 7 / 15

  28. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 10 Consider the case where N = 8, the first decimation yields the sequence x (0) , x (2) , x (4) , x (6) , x (1) , x (3) , x (5) , x (7). The second decimation results in the sequence x (0) , x (4) , x (2) , x (6) , x (1) , x (5) , x (3) , x (7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x ( n ) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order. Ex. the data point x (3) ≡ x (011) is placed in the position m = 110 or m = 6 in the decimated array. Therefore, the data x ( n ) after decimation is stored in bit-reversed order. If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order. EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 7 / 15

  29. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 10 Consider the case where N = 8, the first decimation yields the sequence x (0) , x (2) , x (4) , x (6) , x (1) , x (3) , x (5) , x (7). The second decimation results in the sequence x (0) , x (4) , x (2) , x (6) , x (1) , x (5) , x (3) , x (7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x ( n ) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order. Ex. the data point x (3) ≡ x (011) is placed in the position m = 110 or m = 6 in the decimated array. Therefore, the data x ( n ) after decimation is stored in bit-reversed order. If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order. EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 7 / 15

  30. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... 11 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 8 / 15

  31. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  32. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  33. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  34. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  35. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N = � ( N / 2) − 1 N + W Nk / 2 � ( N / 2) − 1 x ( n + N x ( n ) W kn 2 ) W kn n =0 n =0 N N EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  36. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N = � ( N / 2) − 1 N + W Nk / 2 � ( N / 2) − 1 x ( n + N x ( n ) W kn 2 ) W kn n =0 n =0 N N e − j π � k = ( − 1) k , then = e − j 2 π Nk / 2 N = e − j π k = Since W Nk / 2 � N EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  37. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N = � ( N / 2) − 1 N + W Nk / 2 � ( N / 2) − 1 x ( n + N x ( n ) W kn 2 ) W kn n =0 n =0 N N e − j π � k = ( − 1) k , then = e − j 2 π Nk / 2 N = e − j π k = Since W Nk / 2 � N X ( k ) = � ( N / 2) − 1 � x ( n ) + ( − 1) k x ( n + N � W kn 2 ) n =0 N EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  38. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N = � ( N / 2) − 1 N + W Nk / 2 � ( N / 2) − 1 x ( n + N x ( n ) W kn 2 ) W kn n =0 n =0 N N e − j π � k = ( − 1) k , then = e − j 2 π Nk / 2 N = e − j π k = Since W Nk / 2 � N X ( k ) = � ( N / 2) − 1 � x ( n ) + ( − 1) k x ( n + N � W kn 2 ) n =0 N By splitting (decimate) X ( k ) into even and odd numbered samples, we obtain, EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  39. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N = � ( N / 2) − 1 N + W Nk / 2 � ( N / 2) − 1 x ( n + N x ( n ) W kn 2 ) W kn n =0 n =0 N N e − j π � k = ( − 1) k , then = e − j 2 π Nk / 2 N = e − j π k = Since W Nk / 2 � N X ( k ) = � ( N / 2) − 1 � x ( n ) + ( − 1) k x ( n + N � W kn 2 ) n =0 N By splitting (decimate) X ( k ) into even and odd numbered samples, we obtain, X (2 k ) = � ( N / 2) − 1 � x ( n ) + x ( n + N � W kn 2 ) n =0 N / 2 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  40. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N = � ( N / 2) − 1 N + W Nk / 2 � ( N / 2) − 1 x ( n + N x ( n ) W kn 2 ) W kn n =0 n =0 N N e − j π � k = ( − 1) k , then = e − j 2 π Nk / 2 N = e − j π k = Since W Nk / 2 � N X ( k ) = � ( N / 2) − 1 � x ( n ) + ( − 1) k x ( n + N � W kn 2 ) n =0 N By splitting (decimate) X ( k ) into even and odd numbered samples, we obtain, X (2 k ) = � ( N / 2) − 1 � x ( n ) + x ( n + N � W kn 2 ) n =0 N / 2 = � ( N / 2) − 1 g 1 ( n ) W kn n =0 N / 2 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15

  41. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N = � ( N / 2) − 1 N + W Nk / 2 � ( N / 2) − 1 x ( n + N x ( n ) W kn 2 ) W kn n =0 n =0 N N e − j π � k = ( − 1) k , then = e − j 2 π Nk / 2 N = e − j π k = Since W Nk / 2 � N X ( k ) = � ( N / 2) − 1 � x ( n ) + ( − 1) k x ( n + N � W kn 2 ) n =0 N By splitting (decimate) X ( k ) into even and odd numbered samples, we obtain, X (2 k ) = � ( N / 2) − 1 � x ( n ) + x ( n + N � W kn 2 ) n =0 N / 2 = � ( N / 2) − 1 g 1 ( n ) W kn n =0 N / 2 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15 and,

  42. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N = � ( N / 2) − 1 N + W Nk / 2 � ( N / 2) − 1 x ( n + N x ( n ) W kn 2 ) W kn n =0 n =0 N N e − j π � k = ( − 1) k , then = e − j 2 π Nk / 2 N = e − j π k = Since W Nk / 2 � N X ( k ) = � ( N / 2) − 1 � x ( n ) + ( − 1) k x ( n + N � W kn 2 ) n =0 N By splitting (decimate) X ( k ) into even and odd numbered samples, we obtain, X (2 k ) = � ( N / 2) − 1 � x ( n ) + x ( n + N � W kn 2 ) n =0 N / 2 = � ( N / 2) − 1 g 1 ( n ) W kn n =0 N / 2 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15 x ( n ) − x ( n + N ) and, X (2 k + 1) = � ( N / 2) − 1 �� W n � W kn �

  43. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 1 The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N / 2. The N-point DFT of the sequence x ( n ) is X ( k ) = � N − 1 n =0 x ( n ) W nk N , , k = 0 , 1 , · · · , N − 1 = � ( N / 2) − 1 N + � N − 1 x ( n ) W kn n = N / 2 x ( n ) W kn n =0 N = � ( N / 2) − 1 N + W Nk / 2 � ( N / 2) − 1 x ( n + N x ( n ) W kn 2 ) W kn n =0 n =0 N N e − j π � k = ( − 1) k , then = e − j 2 π Nk / 2 N = e − j π k = Since W Nk / 2 � N X ( k ) = � ( N / 2) − 1 � x ( n ) + ( − 1) k x ( n + N � W kn 2 ) n =0 N By splitting (decimate) X ( k ) into even and odd numbered samples, we obtain, X (2 k ) = � ( N / 2) − 1 � x ( n ) + x ( n + N � W kn 2 ) n =0 N / 2 = � ( N / 2) − 1 g 1 ( n ) W kn n =0 N / 2 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 9 / 15 x ( n ) − x ( n + N ) and, X (2 k + 1) = � ( N / 2) − 1 �� W n � W kn �

  44. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 2 and, The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 10 / 15

  45. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 2 and, X (2 k + 1) = � ( N / 2) − 1 x ( n ) − x ( n + N W n W kn �� � � 2 ) n =0 N N / 2 The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 10 / 15

  46. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 2 and, X (2 k + 1) = � ( N / 2) − 1 x ( n ) − x ( n + N W n W kn �� � � 2 ) n =0 N N / 2 = � ( N / 2) − 1 g 2 ( n ) W kn n =0 N / 2 The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 10 / 15

  47. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 2 and, X (2 k + 1) = � ( N / 2) − 1 x ( n ) − x ( n + N W n W kn �� � � 2 ) n =0 N N / 2 = � ( N / 2) − 1 g 2 ( n ) W kn n =0 N / 2 The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 10 / 15

  48. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 2 and, X (2 k + 1) = � ( N / 2) − 1 x ( n ) − x ( n + N W n W kn �� � � 2 ) n =0 N N / 2 = � ( N / 2) − 1 g 2 ( n ) W kn n =0 N / 2 The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 10 / 15

  49. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  50. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  51. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  52. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  53. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  54. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  55. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  56. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  57. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  58. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  59. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  60. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  61. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  62. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  63. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  64. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  65. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  66. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 3 EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 11 / 15

  67. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 4 Ex. Using the decimation in frequency FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 12 / 15

  68. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 4 Ex. Using the decimation in frequency FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 12 / 15

  69. Efficient Computation of the DFT: FFT Algorithms Radix-2 FFT Algorithms ... Decimation in Frequency ... 4 Ex. Using the decimation in frequency FFT algorithm, compute the 4-point DFT of the sequence x ( n ) = { 5 , 0 , − 3 , 4 } EELE 4310: Digital Signal Processing (DSP) - Ch.8 Dr. Musbah Shaat 12 / 15

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