10-IIR_Filters_2009

10-IIR_Filters_2009 - SYSC5602: Digital Signal Processing...

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SYSC5602: Digital Signal Processing IIR_Filters_2009.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 1 / 32 INFINITE IMPULSE RESPONSE (IIR) FILTERS Analog Filter Theory Analog Filters may be described in terms of their transfer functions, which take the form: The impulse response of an analog filter is given by the inverse Laplace Transform of its transfer function H(s). For example, consider a second order filter with: where d 1 and d 2 are the poles of the transfer function. The impulse response is given by: for Hs () Ys Xs ---------- K b k s Lk k 0 = L a k s Nk k 0 = N ------------------------------ == Ks s 2 a 1 sa 0 ++ ------------------------------- C 1 sd 1 ------------- C 2 2 + ht C 1 e d 1 t C 2 e d 2 t + = t 0
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SYSC5602: Digital Signal Processing IIR_Filters_2009.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 2 / 32 For the system to be stable, h(t) must approach 0 as t approaches infinity. Therefore, stability requires that all poles of H(s) must have negative real parts. In other words, all poles must be in the left half of the s-plane. How to find H(s), assuming |H(j ω )| 2 is known: We know that Therefore real imag S-PLANE poles of a stable system Hj ω () Hs sj ω = =
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SYSC5602: Digital Signal Processing IIR_Filters_2009.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 3 / 32 It follows that The procedure to find H(s) can now be summarized as follows: 1) Find an analytic form for H(s)H(-s) from: 2) Find the 2N poles of H(s)H(-s), where N is the order of H(s) 3) Facts: If H(s) has poles at d 1 , d 2 , . ....d N then H(-s) has poles at -d 1 , -d 2 , . ..... - d N H(s) is required to be stable; i.e. the poles of H(s) must be in the left half of the s-plane. Aided by the two facts above, H(s) is chosen to have as poles the subset of the poles (found in step 2) that fall in the left half of the s-plane. For example if H(s)H(-s) had 4 poles: (.5+j0.5), (.5-j0.5), (-.5-j0.5) and (-.5+j0.5) then H(s) will have two poles: (-.5-j0.5) and (-.5+j0.5). Hs () Hj ω ω js = = ω ω ω = ω 2 ω = == ω 2 ω = =
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SYSC5602: Digital Signal Processing IIR_Filters_2009.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 4 / 32 Analog Filter Approximations Butterworth Low Pass Prototype (LPP) The magnitude of the frequency response is described by: Hj ω () 2 1 1 ω 2 N + ------------------- = 0 0.5 1 1.5 2 2.5 3 3.5 4 -25 -20 -15 -10 -5 0 5 Butterworth LPP, orders 1 to 6 Filter gain in dB frequency in rad/sec N=6 N=1
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SYSC5602: Digital Signal Processing IIR_Filters_2009.fm MOHAMMED EL-TANANY, PROFESSOR SYSTEMS & COMPUTER ENGINEERING, CARLETON UNIVERSITY, OTTAWA, ONTARIO 5 / 32 Main Characteristics N is the filter order. The 3 dB frequency is ω =1 regardless of the filter order The frequency response of this filter is maximally flat, since the first 2N-1 derivatives are zero at ω =0.
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This note was uploaded on 10/24/2009 for the course SCE sysc5602 taught by Professor El-tanany during the Winter '09 term at Carleton CA.

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10-IIR_Filters_2009 - SYSC5602: Digital Signal Processing...

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