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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 . 1 . 9 . 5 1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing Continuous and Discrete Fall Term 2008 Lecture 7 1 Reading: Class handout: Introduction to Continuous Time Filter Design . Butterworth Filter Design Example (Same problem as in the Class Handout). Design a Butterworth lowpass filter to meet the power gain specifications shown below:  H ( j 9 )  2 1 2 9 p a s s b a n d s t o p b a n d t r a n s i t i o n b a n d At the two critical frequencies 1 = 0 . 9 = 0 . 3333 1 + 2 1 = 0 . 05 = 4 . 358 1 + 2 Then log( / ) N = 3 . 70 log( r / c ) 1 copyright c D.Rowell 2008 71 we therefore select N=4. The 4 poles ( p 1 . . . p 4 ) lie on a circle of radius r = c 1 /N = 13 . 16 and are given by  p n  = 13 . 16 p n = (2 n + 3) / 8 for n = 1 . . . 4, giving a pair of complex conjugate pole pairs p 1 , 4 = 5 . 04 j 12 . 16 p 2 , 3 = 12 . 16 j 5 . 04 The transfer function, normalized to unity gain, is 29993 H ( s ) = ( s 2 + 10 . 07 s + 173 . 2)( s 2 + 24 . 32 s + 173 . 2) and the filter Bode plots are shown below. Bode Diagram 1 1 2 3 10 10 10 10 10 Frequency (rad/sec) 2 Chebyshev Filters The order of a filter required to met a lowpass specification may often be reduced by relaxing the requirement of a monotonically decreasing power gain with frequency, and allowing 150 100 50 0 50 Magnitude (dB) 360 270 180 90 0 Phase (deg) 72 ripple to occur in either the passband or the stopband. The Chebyshev filters allow these conditions: Type 1  H ( j )  2 = 1 + 2 T N 1 2 ( / c ) (1) 1 Type 2  H ( j )  2 = 1 + 2 ( T N 2 ( r / c ) /T N 2 ( r / )) (2) Where T N ( x ) is the Chebyshev polynomial of degree N . Note the similarity of the form of the Type 1 power gain (Eq. (1)) to that of the Butterworth filter, where the function T N ( / c ) has replaced ( / c ) N . The Type 1 filter produces an allpole design with slightly different pole placement from the Butterworth filters, allowing resonant peaks in the pass band to introduce ripple, while the Type 2 filter introduces a set of zeros on the imaginary axis above r , causing a ripple in the stopband. The Chebyshev polynomials are defined recursively as follows T ( x ) = 1 T 1 ( x ) = x T 2 ( x ) = 2 x 2 1 T 3 ( x ) = 4 x 3 3 x ....
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 Fall '08
 DerekRowell

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