lecture_12 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 12 1 Reading: Class Handout: The Fast Fourier Transform Proakis and Manolakis (4th Ed.): Secs. 8.1 8.3 Oppemheim Schafer & Buck (2nd Ed.): Secs. 9.0 9.3 The Fast Fourier Transform (contd.) In Lecture 11 we saw that we could write the DFT of a length N sequence as N 1 N 1 j 2 πmn F m = f n e N = f n W N mn , m =0 ,...,N 1 n =0 n =0 where W N =e j2 π/N . We noted that the number of complex multiplication operations to compute the DFT is N 2 , but if we divided the original sequence into two length N/ 2 sequences (based on even and odd samples) and computed the DFT of each shorter sequence, they could be combined F m = A m + W N m B m for m ... ( 2 1) , and F m = A m N/ 2 W N m N/ 2 B m N/ 2 for m = 2 ( N 1) where { A m } is the DFT of the even-numbered samples, and { B m } is the DFT of the odd- numbered samples. 0 1 2 3 4 5 6 7 A A A A B B B B 0 0 1 1 2 2 3 3 4 - p o i n t D F T f r o m e v e n s a m p l e s 4 - p o i n t D F T f r o m o d d s a m p l e s { f , f , f , f } 0 2 4 6 { f , f , f , f } 1 3 5 7 F 0 = A 0 + W 8 0 B 0 F 1 = A 1 + W 8 1 B 1 F 2 = A 2 + W 8 2 B 2 F 3 = A 3 + W 8 3 B 3 F 4 = A 4 + W 8 4 B 4 = A 0 - W 8 0 B 0 F 5 = A 5 + W 8 5 B 5 = A 1 8 1 B 1 F 6 = A 6 + W 8 6 B 6 = A 2 8 2 B 2 F 7 = A 7 + W 8 7 B 7 = A 3 8 3 B 3 1 copyright ± c D.Rowell 2008 0–1
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F F F F 3 F 4 F F 6 F 0 1 2 5 7 The total number of required complex multiplications is ( N/ 2) 2 for each shorter DFT, and 2 to combine the two, giving a total of N ( N +1) / 2, which is less than N 2 . If N is divisible by 4, the process may be repeated, and each length 2 DFT may be formed by decimating the two 2 sequences into even and odd components, forming the length 4 DFTs, and combining these back into a length 2 DFT, as is shown for N =8 below: 4 2 - p o i n t D F T s 2 4 - p o i n t D F T s 8 - p o i n t D F T 0 1 2 3 4 5 6 7 2 - p o i n t D F T f r o m s a m p l e s { f , f } 0 4 0 0 2 2 4 4 6 6 2 - p o i n t D F T f r o m s a m p l e s { f , f } 2 6 2 - p o i n t D F T f r o m s a m p l e s { f , f } 1 5 2 - p o i n t D F T f r o m s a m p l e s { f , f } 3 7 Notice that all weights in the figure are expressed by convention as exponents of W 8 . In general, if the length of the data sequence is an integer power of 2, that is N =2 q for integer q , the DFT sequence { F m } may be formed by adding additional columns to the left and halving the length of the DFT at each step, until the length is two. For example if N =256=2 8 a total of seven column operations would be required. The final step is to evaluate the N/2 length-2 DFTs. Each one may be written F 0 = f 0 + W 2 0 f 1 = f 0 + f 1 F 1 = f 0 + W 2 1 f 1 = f 0 f 1 , which is simply the sum and difference of the two sample points. No complex multiplications are necessary. The 2-point DFT is shown in signal-flow graph form below, and is known as the FFT butterfly .
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.161 taught by Professor Derekrowell during the Fall '08 term at MIT.

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lecture_12 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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