lecture_16

# lecture_16 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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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 Low-Pass Filter Design by Windowing In Lecture 15 we examined the creation of a causal FIR filter based upon an ideal low-pass filter with cut-off frequency ω c , and found that the impulse response was ω c sin( ω c n h ( n ) = π Ω c n − ∞ < n < ∞ . The resulting filter is therefore both infinite in extent and non-causal. 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 ω ) from the convolution 1 π H (e j ω ) = H (e j ν ) R (e j( ω − ν ) )d ν 2 π − π where R (e j ω ) = sin(( M + 1) ω/ 2) sin( ω/ 2) The truncation generates a Gibb’s phenomenon associated with the band edges of H (e j ω ) where, as demonstrated in the figure below: (a) Both the pass-band and the stop-band 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 16–1 1 w (b) The amplitude of the first side-lobe in the stop-band 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 right-shift of M/ 2 to the impulse response to form { h ˆ n } where h ˆ ( n ) = h ( n − M/ 2) ≤ n ≤ M + 1 . The shift was seen to have no effect on H ( e j ω ) , 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 ω ), which when convolved with the ideal low-pass filter function H ( e j ω ) 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 ω , and for comparison the spectrum of the same length rectangular window R ( e j ω ....
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lecture_16 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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