EECS 247 Lecture 2: Filters © © 2004 H.K. Page1EE247 Lecture 2• Material covered today:– Nomenclature– Filter specifications • Quality factor• Frequency characteristics• Group delay– Filter types• Butterworth• Chebyshev I• Chebyshev II• Elliptic• Bessel– Group delay comparison exampleEECS 247 Lecture 21: Filters © © 2004 H.K. Page2NomenclatureFilter Types()ωjH()ωjHLowpass HighpassBandpass Band-reject(Notch)ωωωProvide frequency selectivity()ωjH( )ωjHωωAll-pass()ωjHPhase shaping or equalizationEECS 247 Lecture 2: Filters © © 2004 H.K. Page3Filter Specifications• Frequency characteristics (lowpass filter):– Passband ripple (Rpass)– Cutoff frequency or -3dB frequency – Stopband rejection– Passband gain• Phase characteristics:– Group delay• SNR (Dynamic range)• SNDR (Signal to Noise+Distortion ratio)• Linearity measures: IM3 (intermodulation distortion), HD3 (harmonic distortion), IIP3 or OIP3 (Input-referred or output-referred third order intercept point)• Power/pole & Area/poleEECS 247 Lecture 2: Filters © © 2004 H.K. Page40x 10Frequency (Hz)Lowpass Filter Frequency Characteristics()ωjH()ωjH()0HPassband Ripple (Rpass)Transition BandcfPassbandPassband GainstopfStopband FrequencyStopband Rejectionf()ωjH3dBf−dB3EECS 247 Lecture 2: Filters © © 2004 H.K. Page5Quality Factor (Q)• The term Quality Factor (Q) has different definitions:– Component quality factor (inductor & capacitor Q)– Pole quality factor– Bandpass filter quality factor• Next 3 slides clarifies eachEECS 247 Lecture 2: Filters © © 2004 H.K. Page6Component Quality Factor (Q)• For any component with a transfer function:• Quality factor is defined as:( )( ) ( )( )( )EnergyStoredperunittimeAveragePowerDissipation1HjRjXXQRωωωωω→=+=EECS 247 Lecture 2: Filters © © 2004 H.K. Page7Inductor & Capacitor Quality Factor• Inductor Q :• Capacitor Q :RsLLL1LYQRsjLRsωω==+RpCCC1ZQCRp1jRpCωω==+EECS 247 Lecture 2: Filters © © 2004 H.K. Page8Pole Quality FactorxσxωωjσPωxPolexQ 2ωσ=s-PlaneEECS 247 Lecture 2: Filters © © 2004 H.K. Page9Bandpass Filter Quality Factor (Q) -20-15-10-500.1 1 10fcenter0-3dB∆f()HjfFrequencyMagnitude (dB)Q= fcenter/∆fEECS 247 Lecture 2: Filters © © 2004 H.K. Page10• Consider a continuous time filter with s-domain transfer function G(s):• Let us apply a signal to the filter input composed of sum of twosinewaves at slightly different frequencies (∆ω<<ω):• The filter output is:What is Group Delay?vIN(t) = A1sin(ωt) + A2sin[(ω+∆ω) t]G(jω) ≡ G(jω)ejθ(ω)vOUT(t) = A1 G(jω) sin[ωt+θ(ω)] + A2 G[ j(ω+∆ω)] sin[(ω+∆ω)t+ θ(ω+∆ω)]EECS 247 Lecture 2: Filters © © 2004 H.K. Page11What is Group Delay?{ ]}[vOUT(t) = A1 G(jω) sinωt + θ(ω)ω+{ ]}[+ A2 G[ j(ω+∆ω)] sin(ω+∆ω)t +θ(ω+∆ω)ω+∆ωθ(ω+∆ω)ω+∆ω≅ θ(ω)+dθ(ω)dω∆ω[ ][1ω)( ]1 -∆ωωdθ(ω)dωθ(ω)ω+θ(ω)ω-()∆ωω≅∆ωω<<1Since then ∆ωωà0[ ]2EECS 247 Lecture 2: Filters © © 2004 H.K. Page12What is Group Delay?Signal Magnitude and Phase Impairment{ ]}[vOUT(t) = A1 G(jω) sin ωt + θ(ω)ω+{ ]}[+ A2 G[ j(ω+∆ω)]sin (ω+∆ω) t +dθ(ω)dωθ(ω)ω+θ(ω)ω-()∆ωω• If the second term in the phase of the 2ndsinwave is non-zero, then the filter’s output at frequency ω+∆ωis time-shifted differently than the filter’s output at frequency ωà “Phase distortion”• If the second term is zero, then the filter’s output at frequency ω+∆ωand the output at frequency ω are each delayed in time by -θ(ω)/ω• τPD≡ -θ(ω)/ω is called the “phase delay” and has units of timeEECS 247 Lecture 2: Filters © © 2004 H.K. Page13• Phase distortion is avoided only if:• Clearly, if θ(ω)=kω, k a constant, à no phase distortion• This type of filter phase response is called “linear phase”àPhase shift varies linearly with frequency• τGR ≡ -dθ(ω)/dω is called the “group delay” and also has units of time. For a linear phase filter τGR ≡ τPD=k à τGR=τPDimplies linear phase• Note: Filters with θ(ω)=kω+c are also called linear phase filters, but they’re not free of phase distortionWhat is Group Delay?Signal Magnitude and Phase Impairmentdθ(ω)dωθ(ω)ω-= 0EECS 247 Lecture 2: Filters © © 2004 H.K. Page14What is Group Delay?Signal Magnitude and Phase Impairment• If τGR= τPDà No phase distortion[ )](vOUT(t) = A1 G(jω) sin ωt - τGR+[+ A2 G[ j(ω+∆ω)] sin (ω+∆ω))](t - τGR• If alsoG( jω)=G[ j(ω+∆ω)] for all input frequencies within the signal-band, vOUTis a scaled, time-shifted replica of the input, with no “signal magnitude distortion” :• In most cases neither of these conditions are realizable exactlyEECS 247 Lecture 2: Filters © © 2004 H.K. Page15• Phase delay is defined as:τPD≡ -θ(ω)/ω [ time]• Group delay is defined as :τGR ≡ -dθ(ω)/dω [time]• If θ(ω)=kω, k a constant, à no phase distortion• For a linear phase filter τGR ≡ τPD=kSummaryGroup DelayEECS 247 Lecture 2: Filters © © 2004 H.K. Page16Filter Types Butterworth Lowpass Filter• Maximally flat amplitude within the filter passband• Moderate phase distortion-60-40-200Magnitude (dB)1 2-400-200Normalized Frequency Phase (degrees)531Normalized Group Delay00Example: 5th Order Butterworth filterN0dH(j)0dωωω==EECS 247 Lecture 2: Filters © © 2004 H.K. Page17Butterworth Lowpass Filter• All poles• Poles located on the unit circle with equal angless-planejωσExample: 5th Order Butterworth filterEECS 247 Lecture 2: Filters © © 2004 H.K. Page18Filter Types Chebyshev I Lowpass Filter• Chebyshev I filter– Equal-ripple passband– Sharper transition band compared to Butterworth– Poorer group delay12-40-200Normalized FrequencyMagnitude (dB)-400-2000Phase (degrees)0Example: 5th Order Chebyshev filter350Normalized Group DelayEECS 247 Lecture 2: Filters © © 2004 H.K. Page19Chebyshev I Lowpass Filter Characteristics• All poles• Poles located on an ellipse inside the unit circle• Allowing more ripple in the passband:_Narrower transition band_Sharper cut-off_Higher pole QExample: 5th Order Chebyshev I Filters-planejωσChebyshev I LPF 3dB passband rippleChebyshev I LPF 0.1dB passband rippleEECS 247 Lecture 2: Filters © © 2004 H.K. Page20Bode DiagramFrequency [Hz]Phase (deg)
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