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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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k n - k 0 0 - 3 f k n - k 0 h n - k n = 8 8 1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 18 1 Reading: Proakis and Manolakis: 7.3.1, 7.3.2, 10.3 Oppenheim, Schafer, and Buck: 8.7.3, 7.1 FFT Convolution for FIR Filters The response of an FIR filter with impulse response { h k } to an input { f k } is given by the linear convolution y n = f k h n k . k = −∞ The length of the convolution of two finite sequences of lengths P and Q is N = P + Q 1. The following figure shows a sequence { f n } of length P = 6, and a sequence { h n } of length Q = 4 reversed and shifted so as to compute the extremes of the convolved sequence y 0 and y 8 . P = 6 n = 0 h n - k 5 Q = 4 Q = 4 The convolution property of the DFT suggests that the FFT might be used to convolve two equal length sequences y n = IDFT { DFT { f n } . DFT { h n }} . 1 copyright ± c D.Rowell 2008 18–1
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However, DFT convolution is a circular convolution, involving periodic extensions of the two sequences. The following figure shows the circular convolution of length 6, on two sequences { f n } of length P = 6 and { h n } of length Q = 4. The periodic extensions cause overlap in the first Q 1 samples, generating “wrap-around” errors in the DFT convolution. P = 6 Q = 4 k n - k 0 5 0 - 3 f k h n - k n = 0 o v e r l a p o f Q - 1 s a m p l e s DFT convolution of two sequences of length P and Q ( P Q ) in DFTs of length P 1. Produces an output sequence of length P , whereas linear convolution produces an output sequence of length P + Q 1. 2. Introduces wrap-around error in the first Q 1 samples of the output sequence. The solution is to zero-pad both input sequences to a length N P + Q 1 and then to use DFT convolution with the length N sequences. For example, if { f n } is of length P = 237, and { h n } is of length Q = 125, for error-free convolution we must perform the DFTs in length N 237 + 125 1 = 461. If the available FFT routine is radix-2, we should choose N=512. The use of the FFT for Filtering Long Data Sequences: The DFT convolution method provides an attractive alternative to direct convolution when the length of the data record is very large. The general method is to break the data into manageable sections, then use the FFT to to perform the convolution and then recombine the output sections. Care must be
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lecture_18 - MIT OpenCourseWare http:/ocw.mit.edu 2.161...

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