EE247 Lecture 2. Butterworth Chebyshev I Chebyshev II Elliptic Bessel Group delay comparison example. EECS 247 Lecture 2: Filters

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1 EE247 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 example 24 H.K. Page Nomenclature Filter Types Lowpass Highpass Bandpass Band-reject (Notch) H( jω) H( jω) H( jω) H( jω) All-pass H( jω) ω ω ω ω ω Provide frequency selectivity Phase shaping or equalization EECS 247 Lecture 2: Filters 24 H.K. Page2

2 Filter 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 outputreferred third order intercept point) Power/pole & Area/pole 24 H.K. Page3 Lowpass Filter Frequency Characteristics H ( ) jω Passband Ripple (Rpass) f 3dB H( ) Passband Gain H( jω) 3dB Transition Band Stopband Rejection H( jω) Passband f c f stop Frequency (Hz) Stopband Frequency x f 24 H.K. Page4

3 Quality 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 each 24 H.K. Page5 Component Quality Factor (Q) For any component with a transfer function: ( ω) H j = R ( ω) + jx ( ω) Quality factor is defined as: X Q = R ( ω) ( ω ) EnergyStored AveragePower Dissipation perunittime 24 H.K. Page6

4 Inductor & Capacitor Quality Factor Inductor Q : YL= Q ω Rs+ jωl L= Rs L Rs L Capacitor Q : Z C = QC= ωcrp Rp + jωc Rp C 24 H.K. Page7 Pole Quality Factor jω s-plane ω x ω P σ x σ Q Pole = ω 2σ x x 24 H.K. Page8

5 Bandpass Filter Quality Factor (Q) H( jf ) Q= f center /Df Magnitude (db) -3dB Df f center Frequency. 24 H.K. Page9 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 two sinewaves at slightly different frequencies ( ω<<ω): The filter output is: What is Group Delay? G(jw) G(jw) e jq(w) v IN (t) = A sin(wt) + A 2 sin[(w+dw) t] v OUT (t) = A G(jw) sin[wt+q(w)] + A 2 G[ j(w+dw)] sin[(w+dw)t+ q(w+dw)] 24 H.K. Page

6 What is Group Delay? v OUT (t) = A G(jw) sin w t + Since q(w) { [ ]} w + { [ ]} + A 2 G[ j(w+dw)] sin (w+dw) t + q(w+dw) w+dw Dw w << [ ] 2 then Dw w q(w+dw) q(w)+ [ dq(w) dw Dw ][ - Dw w w q(w) w + ( ] dq(w) q(w) ( dw - w ) Dw w 24 H.K. Page What is Group Delay? Signal Magnitude and Phase Impairment { [ ]} q(w) v OUT (t) = A G(jw) sin w t + w + + A 2 G[ j(w+dw)] sin (w+dw) t + q(w) dq(w) { [ w + q(w) ( dw - w ) Dw w ]} If the second term in the phase of the 2 nd sinwave is non-zero, then the filter s output at frequency ω+ ωis time-shifted differently than the filter s output at frequency ω Phase distortion If the second term is zero, then the filter s output at frequency ω+ ω and the output at frequency ω are each delayed in time by -θ(ω)/ω τ PD -θ(ω)/ω is called the phase delay and has units of time 24 H.K. Page2

7 What is Group Delay? Signal Magnitude and Phase Impairment Phase distortion is avoided only if: dq(w) dw q(w) - w = 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 GR -dθ(ω)/dω is called the group delay and also has units of time. For a linear phase filter τ GR τ PD =k τ GR = τ 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 24 H.K. Page3 What is Group Delay? Signal Magnitude and Phase Impairment If τ GR = τ PD No phase distortion [ ( )] + A 2 G[ j(w+dw)] sin (w+dw) [ ( t - t GR)] v OUT (t) = A G(jw) sin w t - t GR + If also G( jω) = G[ j(ω+ ω)] for all input frequencies within the signal-band, v OUT is a scaled, time-shifted replica of the input, with no signal magnitude distortion : In most cases neither of these conditions are realizable exactly 24 H.K. Page4

8 Summary Group Delay Phase delay is defined as: τ PD -θ(ω)/ω [ time] Group delay is defined as : τ GR -dθ(ω)/dω [time] If θ(ω)=kω, k a constant, no phase distortion For a linear phase filter τ GR τ PD =k 24 H.K. Page5 Maximally flat amplitude within the filter passband N d H(j ω ) dω ω= Moderate phase distortion Filter Types Butterworth Lowpass Filter = Magnitude (db) Phase (degrees) Normalized Frequency 5 3 Normalized Group Delay Example: 5th Order Butterworth filter 24 H.K. Page6

9 Butterworth Lowpass Filter All poles Poles located on the unit circle with equal angles jω s-plane σ Example: 5th Order Butterworth filter 24 H.K. Page7 Filter Types Chebyshev I Lowpass Filter Chebyshev I filter Equal-ripple passband Sharper transition band compared to Butterworth Poorer group delay Magnitude (db) Phase (degrees) Normalized Frequency Example: 5th Order Chebyshev filter 35 Normalized Group Delay 24 H.K. Page8

10 Chebyshev 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 Q jω s-plane σ Chebyshev I LPF 3dB passband ripple Chebyshev I LPF.dB passband ripple Example: 5th Order Chebyshev I Filter 24 H.K. Page9 Filter Types Cheybshev II Lowpass Chebyshev II filter Ripple in stopband Sharper transition band compared to Butterworth Passband group delay superior to Chebyshev I Phase (deg) Magnitude (db) Bode Diagram Frequency [Hz] Example: 5th Order Chebyshev II filter 24 H.K. Page2

11 Both poles & zeros No. of poles n No. of zeros n- Poles located both inside & outside of the unit circle Zeros located on jω axis Ripple in the stopband only Filter Types Cheybshev II Lowpass jω s-plane σ Example: 5th Order Chebyshev II Filter poles zeros 24 H.K. Page2 Filter Types Elliptic Lowpass Filter Elliptic filter Ripple in passband Ripple in the stopband Sharper transition band compared to Butterworth & both Chebyshevs Poorer group delay Phase (degrees) Magnitude (db) Normalized Frequency Example: 5th Order Elliptic filter 24 H.K. Page22

12 Both poles & zeros No. of poles n No. of zeros n- Zeros located on jω axis Sharp cut-off Narrower transition band Pole Q higher compared to the previous filters Filter Types Elliptic Lowpass Filter jω s-plane σ Pole Zero Example: 5th Order Elliptic Filter 24 H.K. Page23 Bessel All poles Maximally flat group delay Poor amplitude attenuation Poles outside unit circle (s-plane) Relatively low Q poles Filter Types Bessel Lowpass Filter jω s-plane σ Pole Example: 5th Order Bessel filter 24 H.K. Page24

13 Filter Types Comparison of Various LPF Magnitude Response Magnitude (db) Normalized Frequency Magnitude (db) All 5th order filters with same corner freq. Bessel Butterworth Chebyshev I Chebyshev II Elliptic 24 H.K. Page25 Filter Types Comparison of Various LPF Singularities Poles Bessel Poles Butterworth Poles Elliptic Zeros Elliptic Poles Chebyshev I.dB jω s-plane σ 24 H.K. Page26

14 Comparison of Various LPF Groupdelay 5 28 Bessel Chebyshev I.5dB Passband Ripple 2 Butterworth 4 Ref: A. Zverev, Handbook of filter synthesis, Wiley, H.K. Page 27 Group Delay Comparison Example Lowpass filter with khz corner frequency Chebyshev I versus Bessel Both filters 4th order- same -3dB point Passband ripple of db allowed for Chebyshev I 24 H.K. Page 28

15 Magnitude Response Bode Magnitude Diagram - Magnitude (db) th Order Chebychev 4th Order Bessel Frequency [Hz] 24 H.K. Page29 Phase Response -5 - Phase [degrees] th Order Chebychev 4th Order Bessel Frequency [Hz].5 2 x 5 24 H.K. Page3

16 Group Delay 4 2 4th Ord. Chebychev 4th Ord. Bessel Group Delay [µ s] Frequency [Hz] 24 H.K. Page3 Normalized Group Delay 3 4th Ord. Chebychev 4th Ord. Bessel 2.5 Group Delay [normalized] Frequency [Hz] 24 H.K. Page32

17 Step Response.4.2 4th Order Chebychev 4th Order Bessel Amplitude Time (sec) x H.K. Page33 Intersymbol Interference (ISI) ISI Broadening of pulses resulting in interference between successive transmitted pulses Example: Simple RC filter 24 H.K. Page34

18 Pulse Broadening Bessel versus Chebyshev.5 Input Output x -4 8th order Bessel x -4 4th order Chebyshev I Chebyshev has more pulse broadening compared to Bessel More ISI 24 H.K. Page35 Response to Random Data Chebyshev versus Bessel.5 Input Signal: 3kHz max. signal spectral density x x -4 4th order Bessel x -4 4th order Chebyshev I 24 H.K. Page36

19 Measure of Signal Degradation Eye Diagram Eye diagram is a useful graphical illustration for signal degradation Consists of many overlaid traces of a signal using an oscilloscope where the symbol timing serves as the scope trigger It is a visual summary of all possible intersymbol interference waveforms The vertical opening immunity to noise Horizontal opening timing jitter 24 H.K. Page37 Measure of Signal Degradation Eye Diagram Magnitude (db) Bessel Chebychev Random data with max. power spectral density of: 5kHz khz 3kHz x 4 Frequency [Hz] Group Delay [normalized] th Ord. Chebychev 4th Ord. Bessel x 4 Frequency [Hz] 24 H.K. Page38

20 Eye Diagram Chebyshev versus Bessel Input Signal Input Signal Random data maximum power spectral density 3kHz Time x -5 4th Order Bessel 4th Order Chebychev Time x -5 4th order Bessel Time x -5 4th order Chebyshev I 24 H.K. Page39 Eye Diagrams 4th Order Bessel 4th Order Chebychev % Eye opening % Eye opening Time x Time x Random data maximum power spectral density 5kHz 24 H.K. Page4

21 Eye Diagrams 4th Order Bessel 4th Order Chebychev % Eye opening % Eye opening Time x Time x -5 Random data maximum power spectral density khz Filter with constant group delay More open eye Lower BER (bit-error-rate) 24 H.K. Page4 Summary Filter Types Filters with high signal attenuation per pole poor phase response For a given signal attenuation requirement of preserving constant groupdelay Higher order filter In the case of passive filters higher component count Case of integrated active filters higher chip area & power dissipation In cases where filter is followed by ADC and DSP Possible to digitally correct for phase non-linearities incurred by the analog circuitry by using phase equalizers 24 H.K. Page42

22 Summary Filter Types Filters with high signal attenuation per pole poor phase response For a given signal attenuation requirement of preserving constant groupdelay Higher order filter In the case of passive filters higher component count Case of integrated active filters higher chip area & power dissipation In cases where filter is followed by ADC and DSP possible to digitally correct for phase non-linearities incurred by the analog circuitry by using digital phase equalizers 24 H.K. Page43 RLC Filters Bandpass filter: R V o s Vo = RC Vin s2+ ωo 2 Q s+ ωo V in L C ωo = LC Q= ωorc = R L ωo 24 H.K. Page44

23 RLC Filters Design a bandpass filter with: Center frequency of khz Q of 2 V in R L C V o Assume that the inductor has series R resulting in an inductor Q of 4 What is the effect of finite inductor Q on the overall Q? 24 H.K. Page45 RLC Filters Effect of Component Finite Q = + Qfilt Q ideal Qind. filt Q=2 (ideal L) Q=3.3 (QL=4) Component Q must be much higher compared to desired filter Q 24 H.K. Page46

24 RLC Filters R V o V in L C Question: Can RLC filters be integrated on-chip? 24 H.K. Page47 Monolithic Inductors Feasible Quality Factor & Value Feasible monolithic inductor in CMOS tech. <nh with Q <7 Ref: Radio Frequency Filters, Lawrence Larson; Mead workshop presentation H.K. Page48

25 Monolithic LC Filters Monolithic inductor in CMOS tech. L<nH with Q<7 Max. capacitor C< pf LC filters in the monolithic form feasible: - freq >5MHz - Only low quality factor filters Learn more in EE H.K. Page49 Monolithic Filters Desirable to integrate filters with critical frequencies << 5MHz Per previous slide LC filters not a practical option in the integrated form Good alternative: Integrator based filters 24 H.K. Page5

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