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Unformatted text preview: 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 . Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing Continuous and Discrete Fall Term 2008 Lecture 16 1 Reading: Proakis & Manolakis, Sec. 10.2 Oppenheim, Schafer & Buck, Chap. 7. Cartinhour, Chap. 9. 1 FIR LowPass Filter Design by Windowing In Lecture 15 we examined the creation of a causal FIR filter based upon an ideal lowpass filter with cutoff frequency c , and found that the impulse response was c sin( c n h ( n ) = c n < n < . The resulting filter is therefore both infinite in extent and noncausal. To create a finite length filter we truncated the impulse response by multiplying { h ( n ) } with an even rectangular window function { r ( n ) } of length M + 1, where r ( n ) = 1  n  M/ 2 otherwise. The result was to create a modified filter { h } with a real frequency response function n H (e j ) from the convolution 1 H (e j ) = H (e j ) R (e j( ) )d 2 where R (e j ) = sin(( M + 1) / 2) sin( / 2) The truncation generates a Gibbs phenomenon associated with the band edges of H (e j ) where, as demonstrated in the figure below: (a) Both the passband and the stopband exhibit significant ripple, and the maxima of the ripple is relatively independent of the chosen filter length M + 1. 1 copyright c D.Rowell 2008 161 1 w (b) The amplitude of the first sidelobe in the stopband ia approximately 0.091, corre sponding to an attenuation of 21 dB, at that frequency. (c) The width of the transition region decreases with M + 1, the filter length. M = 1 M = 2 M = 3 M = 4 . 4 p p  H ' ( e j w )  . 0 9 1 A causal filter was then formed by applying a rightshift of M/ 2 to the impulse response to form { h n } where h ( n ) = h ( n M/ 2) n M + 1 . The shift was seen to have no effect on H ( e j ) , but created a linear phase taper (lag). The windowing method of FIR seeks to improve the filter characteristic by selecting an alternate length M + 1 window function { w ( n ) } with improved spectral characteristics W ( e j ), which when convolved with the ideal lowpass filter function H ( e j ) will produce a better filter. There are many window functions available. We first look at three common fixed param eter windows: The Bartlett Window: The length M + 1 Bartlett window is a even triangular window 1 + 2 n/M M/ 2 n w ( n ) = 1 2 n/M n M/ 2 otherwise, as shown for M + 1 = 40 in the figure below. Also plotted is the spectrum W ( e j , and for comparison the spectrum of the same length rectangular window R ( e j ....
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 Fall '08
 DerekRowell

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