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EE247 - Lecture 2 Filters EE247 - Lecture 2 Filters • Filters: – Nomenclature – Specifications • Quality factor • Frequency characteristics • Group delay – Filter types • Butterworth • Chebyshev I & II • Elliptic • Bessel – Group delay comparison example – Biquads EECS 247 Lecture 2: Filters © 2009 H.K. Page 1 Nomenclature Filter Types wrt Frequency Range Selectivity Lowpass Highpass Bandpass Band-reject All-pass (Notch) H ()jω H ()jω H ()jω H ()jω H ()jω ω ω ω ω ω Provide frequency selectivity Phase shaping or equalization EECS 247 Lecture 2: Filters © 2009 H.K. Page 2 Filter Specifications • Frequency characteristics: – 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) • Area/pole & Power/pole EECS 247 Lecture 2: Filters © 2009 H.K. Page 3 Filter Frequency Characteristics Example: Lowpass H ()j[dB]ω Passband Ripple (Rpass) f−3dB H ()0 Passband 3dB Transition Gain H ()jω Band Stopband Rejection 0 H ()jω Frequency (Hz) f fc f x 10 stop Stopband Passband Frequency EECS 247 Lecture 2: Filters © 2009 H.K. Page 4 Filters • Filters: – Nomenclature – Specifications • Frequency characteristics • Quality factor • Group delay – Filter types • Butterworth • Chebyshev I & II • Elliptic • Bessel – Group delay comparison example – Biquads EECS 247 Lecture 2: Filters © 2009 H.K. Page 5 Quality Factor (Q) • The term quality factor (Q) has different definitions in different contexts: –Component quality factor (inductor & capacitor Q) –Pole quality factor –Bandpass filter quality factor • Next 3 slides clarifies each EECS 247 Lecture 2: Filters © 2009 H.K. Page 6 Component Quality Factor (Q) • For any component with a transfer function: Hj()ω = 1 RjX()ωω+ () • Quality factor is defined as: X ()ω Energy Stored Q = → per unit time R()ω Average Power Dissipation EECS 247 Lecture 2: Filters © 2009 H.K. Page 7 Component Quality Factor (Q) Inductor & Capacitor Quality Factor • Inductor Q : ==1Lω YQLL+ ω L Rs jL Rs Rs • Capacitor Q : Rp 1 ZCC==QCRω p 1 + jωC Rp C EECS 247 Lecture 2: Filters © 2009 H.K. Page 8 Pole Quality Factor jω • Typically filter s-Plane singularities include pairs of complex conjugate poles. ω ωx P • Quality factor of complex conjugate σ poles are defined as: σ x ω p Q = Pole σ 2 x EECS 247 Lecture 2: Filters © 2009 H.K. Page 9 Bandpass Filter Quality Factor (Q) Hjf() = /Δ Q fcenter f 0 -3dB Δ f = f2 -f1 Magnitude [dB] Magnitude 0.1f1 fcenter 1f2 Frequency 10 EECS 247 Lecture 2: Filters © 2009 H.K. Page 10 Filters • Filters: – Nomenclature – Specifications • Frequency characteristics • Quality factor • Group delay – Filter types • Butterworth • Chebyshev I & II • Elliptic • Bessel – Group delay comparison example – Biquads EECS 247 Lecture 2: Filters © 2009 H.K. Page 11 What is Group Delay? • Consider a continuous time filter with s-domain transfer function G(s): jθ(ω) G(jω) ≡⏐G(jω)⏐e • Let us apply a signal to the filter input composed of sum of two sinewaves at slightly different frequencies (Δω<<ω): ω ω Δω vIN(t) = A1sin( t) + A2sin[( + ) t] • The filter output is: ⏐ ω ⏐ ω θ ω vOUT(t) = A1 G(j ) sin[ t+ ( )] + ⏐ ω Δω ⏐ ω Δω θ ω Δω A2 G[ j( + )] sin[( + )t+ ( + )] EECS 247 Lecture 2: Filters © 2009 H.K. Page 12 What is Group Delay? θ(ω) v (t) = A ⏐G(jω)⏐ sin ω OUT 1 {]}[ t + ω + ⏐ ω Δω ⏐ θ(ω+Δω) + A2 G[ j( + )] sin (ω+Δω) t + {]}[ ω+Δω Δω Since then Δω 2 ω <<1 []ω 0 θ(ω+Δω) dθ(ω) Δω ≅ θ(ω)+ Δω 1 ω+Δω [][dω ω (]1 - ω ) ≅ θ(ω) dθ(ω) θ(ω) Δω ω + ( dω - ω ) ω EECS 247 Lecture 2: Filters © 2009 H.K. Page 13 What is Group Delay? Signal Magnitude and Phase Impairment θ(ω) v (t) = A ⏐G(jω)⏐ sin ω OUT 1 {]}[t + ω + θ(ω) dθ(ω) θ(ω) Δω + A ⏐G[ j(ω+Δω)]⏐sin (ω+Δω) t + 2 {]}[ ω +( dω - ω ) ω τ ≡ θ ω ω • PD - ( )/ is called the “phase delay” and has units of time • If the second delay term is zero, then the filter’s output at frequency ω+Δω and the output at frequency ω are each delayed in time by -θ(ω)/ω • If the second term in the phase of the 2nd sin wave is non-zero, then the filter’s output at frequency ω+Δω is time-shifted differently than the filter’s output at frequency ω “Phase distortion” EECS 247 Lecture 2: Filters © 2009 H.K. Page 14 What is Group Delay? Signal Magnitude and Phase Impairment • Phase distortion is avoided only if: dθ(ω) θ(ω) dω - ω = 0 • Clearly, if θ(ω)=kω, k a constant, no phase distortion • This type of filter phase response is called “linear phase” Phase shift varies linearly with frequency τ ≡ θ ω ω • GR -d ( )/d is called the “group delay” and also has units of τ ≡τ time. For a linear phase filter GR PD =k τ τ GR= PD implies linear phase • Note: Filters with θ(ω)=kω+c are also called linear phase filters, but they’re not free of phase distortion EECS 247 Lecture 2: Filters © 2009 H.K. Page 15 What is Group Delay? Signal Magnitude and Phase Impairment τ τ •If GR= PD No phase distortion ⏐ ω ⏐ ω t - τ vOUT(t) = A1 G(j ) sin [)]( GR + + A ⏐G[ j(ω+Δω)]⏐ sin (ω+Δω) t - τ 2 [ ( GR )] • If also⏐G( jω)⏐=⏐G[ j(ω+Δω)]⏐ for all input frequencies within the signal-band, vOUT is a scaled, time-shifted replica of the input, with no “signal magnitude distortion” : • In most cases neither of these conditions are exactly realizable EECS 247 Lecture 2: Filters © 2009 H.K. Page 16 Summary Group Delay • Phase delay is defined as: τ ≡ θ ω ω PD - ( )/ [ time] • Group delay is defined as : τ ≡ θ ω ω GR -d ( )/d [time] •If θ(ω)=kω, k a constant, no phase distortion τ ≡τ • For a linear phase filter GR PD =k EECS 247 Lecture 2: Filters © 2009 H.K. Page 17 Filters • Filters: – Nomenclature – Specifications • Frequency characteristics • Quality factor • Group delay – Filter types • Butterworth • Chebyshev I & II • Elliptic • Bessel – Group delay comparison example – Biquads EECS 247 Lecture 2: Filters © 2009 H.K. Page 18 Filter Types wrt Frequency Response Lowpass Butterworth Filter 0 • Maximally flat amplitude within -20 the filter passband -40 N dH(j)ω (dB) Magnitude = 0 -60 dω 0 5 ω=0 -200 3 Delay Group • Moderate phase distortion Phase (degrees) -400 1 0 1 2 Normalized Normalized Frequency Example: 5th Order Butterworth filter EECS 247 Lecture 2: Filters © 2009 H.K. Page 19 Lowpass Butterworth Filter • All poles jω s-plane • Number of poles equal to filter order • Poles located on the unit σ circle with equal angles pole Example: 5th Order Butterworth Filter EECS 247 Lecture 2: Filters © 2009 H.K. Page 20 Filter Types Chebyshev I Lowpass Filter 0 • Chebyshev I filter – Ripple in the passband -20 – Sharper transition band compared to Butterworth (for [dB] Magnitude -40 the same number of poles) 0 35 – Poorer group delay compared to Butterworth -200 – More ripple in passband [degrees] Phase poorer phase response -400 0 Group Delay Normalized 0 1 2 Normalized Frequency Example: 5th Order Chebyshev filter EECS 247 Lecture 2: Filters © 2009 H.K. Page 21 Chebyshev I Lowpass Filter Characteristics jω s-plane • All poles • Poles located on an ellipse inside the unit circle σ • Allowing more ripple in the passband: Narrower transition band Sharper cut-off Higher pole Q Chebyshev I LPF 3dB passband ripple Poorer phase response Chebyshev I LPF 0.1dB passband ripple Example: 5th Order Chebyshev I Filter EECS 247 Lecture 2: Filters © 2009 H.K. Page 22 Filter Types Chebyshev II Lowpass • Chebyshev II filter 0 – No ripple in passband -20 – Nulls or notches in -40 stopband -60 Magnitude (dB) – Sharper transition band 0 compared to -90 Butterworth -180 – Passband phase more -270 linear compared to (deg) Phase -360 Chebyshev I 0 0.5 1 1.5 2 Normalized Frequency Example: 5th Order Chebyshev II filter EECS 247 Lecture 2: Filters © 2009 H.K. Page 23 Filter Types Chebyshev II Lowpass jω • Poles & finite zeros – No. of poles n (n −> filter order) s-plane – No. of finite zeros: n-1 • Poles located both inside & outside of the unit circle σ • Complex conjugate zeros located on jω axis • Zeros create nulls in stopband Example: 5th Order pole Chebyshev II Filter zero EECS 247 Lecture 2: Filters © 2009 H.K. Page 24 Filter Types Elliptic Lowpass Filter 0 • Elliptic filter -20 – Ripple in passband -40 Magnitude (dB) Magnitude -60 – Nulls in the stopband 0 – Sharper transition band compared to Butterworth & -200 both Chebyshevs Phase (degrees) Phase – Poorest phase response -400 0 1 2 Normalized Frequency Example: 5th Order Elliptic filter EECS 247 Lecture 2: Filters © 2009 H.K. Page 25 Filter Types Elliptic Lowpass Filter ω • Poles & finite zeros j – No. of poles: n s-plane – No. of finite zeros: n-1 • Zeros located on jω axis σ • Sharp cut-off Narrower transition band Pole Q higher compared to the previous filter types Pole Zero Example: 5th Order Elliptic Filter EECS 247 Lecture 2: Filters © 2009 H.K. Page 26 Filter Types Bessel Lowpass Filter • Bessel jω s-plane –All poles –Poles outside unit circle σ –Relatively low Q poles –Maximally flat group delay –Poor out-of-band attenuation Pole Example: 5th Order Bessel filter EECS 247 Lecture 2: Filters © 2009 H.K.
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