49597

# 49597 - EE 541 Class Lecture Weeks 7 10(additional slides...

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EE 541 Class Lecture Weeks 7 - 10 (additional slides to be supplied) Prof. John Choma, Professor Department of Electrical Engineering- Electrophysics University of Southern California University Park; MC: 0271; PHE #604 Los Angeles, California 90089-0271 213-740-4692 [USC Office] 213-740-7581 [USC Fax] [email protected] Classical Filter Approximations Fall 2006 Semester

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University of Southern California EE 541: Choma 182 Overview Of Lecture Overview Of Lecture z Classical Filter Forms ± Maximally Flat Magnitude ± Butterworth Maximally Flat Magnitude ± Chebyshev Equal Ripple ± Bessel Maximally Flat Delay z Second Order Confirmation Of Results z Example Filter Realizations
University of Southern California EE 541: Choma 183 n 2 2k k k1 H( j ω )1c ω = =+ () kk k k j j j1 ca b b c = =−− 2 2 22 n n P( ω ) ω ) ω )b ω = + z Magnitude Squared Function ± nth Order Network Containing m Zeros ± Normalized To Zero Frequency Gain, H(0) ± Squared Magnitude Function Is Even Function Of Frequency ω ± Maclaurin Series ± c k = 0, k = 1, 2, 3, . .. m . .. n-1 ² Eliminates Frequency Dependence In Passband ² Equivalent To Setting First (2n-1) Derivatives w/r ω To Zero ² Implies a k =b k , k = 1, 2, 3 … m ² Implies b k = 0, k = m+1, m+2, … n-1 z Resultant Maximally Flat Magnitude Expression 246 2 m 2 123 m 2 m 2 n m n 1a ω a ω a ω a ω ω ) 1b ω b ω b ω b ω b ω ++++ + = + + + " "" Generalized Transfer Function Generalized Transfer Function

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University of Southern California EE 541: Choma 184 ( ) 2 2 2n 22 b b P( ω ) H( j ω ) ω ω )P ω ω =   +  ( ) nb b b ω P ω = z Transfer Function ± 3-dB Bandwidth Is ω b ± Implication ± Alternate Form z Butterworth ± P( ω 2 ) = 1 ± Bandwidth Remains ω b ± Normalized Frequency: y = ω / ω b ± nth Order Butterworth Squared Magnitude Transfer Function z Complex Frequency ± p = jy = j ω / ω b = s / ω b ± Resultant Transfer Function In Normalized s-Plane 2 2 n n ω ) ω ) ω )b ω = + () 2 1 H( jy) 1y = + 2 n 11 H(p) 1j p p == +− Maximally Flat Magnitude Filter Maximally Flat Magnitude Filter
University of Southern California EE 541: Choma 185 Butterworth Frequency Response Butterworth Frequency Response 0 0.2 0.4 0.6 0.8 1 1.2 0.10 0.16 0.25 0.40 0.63 1.00 1.58 2.51 3.98 6.31 10.00 Normalized Frequency, y Nor mal iz ed Squar ed Tr ansf er Ma gn it ud e Order (n) = 2 n = 4 n = 7 Ideal Lowpass Brick Wall 0 0.2 0.4 0.6 0.8 1 1.2 0.10 0.16 0.25 0.40 0.63 1.00 1.58 2.51 3.98 6.31 10.00 Normalized y (n) = 2 Ideal Lowpass Brick Wall

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University of Southern California EE 541: Choma 186 () ( ) a 2n a a 2 a a aa 11 H( jy ) A 1y A1 A n 2y y log log log log =≤ + ≥≈ z Determines Requisite Order Of Filter z Strategy ± Let Transfer Function Be Attenuated By Factor Of A a At A Normalized Frequency Of y a ± y a And A a Are Both Larger Than One z Example ± 45 dB Down, One Octave Above 3-dB Bandwidth ± 45 dB Means A a = 177.8 ± 1 Octave Above ω b Means y a = 2 ± Result Is n 7.474 ± Practical Realization Must Therefore Entails An 8 th Order Butterworth, Maximally Flat Magnitude Filter 16 1 H( jy) = + Butterworth Attenuation Butterworth Attenuation
University of Southern California EE 541: Choma 187 z Poles Are At p = p k z Observations ± Left Half Plane Poles Ascribed To H(p) ±

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49597 - EE 541 Class Lecture Weeks 7 10(additional slides...

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