lecture_07

# lecture_07 - 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 . 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 low-pass 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 −→ λ = 4 . 358 1 + λ 2 Then log( λ/ ) N ≥ = 3 . 70 log(Ω r / Ω c ) 1 copyright c D.Rowell 2008 7–1 we therefore select N=4. The 4 poles ( p 1 . . . p 4 ) lie on a circle of radius r = Ω c − 1 /N = 13 . 16 and are given by | p n | = 13 . 16 p n = π (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 low-pass 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) 7–2 “ripple” to occur in either the pass-band or the stop-band. The Chebyshev filters allow these conditions: Type 1 | H ( j Ω) | 2 = 1 + 2 T N 1 2 (Ω / Ω c ) (1) 1 Type 2 | H ( j Ω) | 2 = 1 + 2 ( T N 2 (Ω r / Ω c ) /T N 2 (Ω r / Ω)) (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 (Ω / Ω c ) has replaced (Ω / Ω c ) N . The Type 1 filter produces an all-pole 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 Ω r , causing a ripple in the stop-band. 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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lecture_07 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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