Filters. EE247 - Lecture 2 Filters. Filter. Filters
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1 EE247 - Lecture 2 Filters 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 EECS 247 Lecture 2: Filters 25 H.K. Page Filters H( jω) H( jω) ω Filter V out ω V in Filters Provide frequency selectivity and/or phase shaping EECS 247 Lecture 2: Filters 25 H.K. Page 2
2 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 25 H.K. Page 3 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 EECS 247 Lecture 2: Filters 25 H.K. Page 4
3 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 EECS 247 Lecture 2: Filters 25 H.K. Page 5 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 EECS 247 Lecture 2: Filters 25 H.K. Page 6
4 Component Quality Factor (Q) For any component with a transfer function: ( ω) H j = R Quality factor is defined as: ( ω) + jx ( ω) X Q = R ( ω) ( ω) EnergyStored AveragePowerDissipation EECS 247 Lecture 2: Filters 25 H.K. Page 7 Inductor & Capacitor Quality Factor Inductor Q : YL= Q Rs j L L= ω + ω Rs L Rs L Capacitor Q : Z C= QC= ωcrp + jωc Rp Rp C EECS 247 Lecture 2: Filters 25 H.K. Page 8
5 Pole Quality Factor jω s-plane ω x ω P σ x σ Q Pole = ω 2σ x x EECS 247 Lecture 2: Filters 25 H.K. Page 9 H( jf ) Bandpass Filter Quality Factor (Q) Q= f center /Df Magnitude (db) -3dB Df = f 2 - f f f center f 2 Frequency. EECS 247 Lecture 2: Filters 25 H.K. Page
6 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)] EECS 247 Lecture 2: Filters 25 H.K. Page What is Group Delay? v OUT (t) = A G(jw) sin w t + q(w) { [ ]} w + { [ ]} + A 2 G[ j(w+dw)] sin (w+dw) t + q(w+dw) w+dw Dw Since w << then [ Dw w ] 2 q(w+dw) q(w)+ [ dq(w) dw Dw ][ - Dw w w q(w) w + dq(w) q(w) ( dw - w ) Dw w ( ] EECS 247 Lecture 2: Filters 25 H.K. Page 2
7 What is Group Delay? Signal Magnitude and Phase Impairment v OUT (t) = A G(jw) sin w t + q(w) { [ ]} + A 2 G[ j(w+dw)] sin (w+dw) t + w + q(w) dq(w) { [ w + q(w) ( dw - w ) Dw w ]} If the second term in the phase of the 2 nd sin wave is non-zero, then the filter s output at frequency ω+ ω is 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 EECS 247 Lecture 2: Filters 25 H.K. Page 3 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 EECS 247 Lecture 2: Filters 25 H.K. Page 4
8 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( jw) = G[ j(w+dw)] 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 EECS 247 Lecture 2: Filters 25 H.K. Page 5 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 EECS 247 Lecture 2: Filters 25 H.K. Page 6
9 Maximally flat amplitude within the filter passband N d H(j ω ) dω Moderate phase distortion Filter Types Lowpass Butterworth Filter ω= = Magnitude (db) Phase (degrees) Normalized Frequency 5 3 Normalized Group Delay Example: 5th Order Butterworth filter EECS 247 Lecture 2: Filters 25 H.K. Page 7 Lowpass Butterworth Filter All poles Poles located on the unit circle with equal angles jω s-plane σ Example: 5th Order Butterworth Filter EECS 247 Lecture 2: Filters 25 H.K. Page 8
10 Filter Types Chebyshev I Lowpass Filter Chebyshev I filter Ripple in the passband Sharper transition band compared to Butterworth Poorer group delay As more ripple is allowed in the passband: Sharper transition band Poorer phase response Magnitude (db) Phase (degrees) Normalized Frequency Example: 5th Order Chebyshev filter 35 Normalized Group Delay EECS 247 Lecture 2: Filters 25 H.K. Page 9 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 Poorer phase response jω s-plane σ Chebyshev I LPF 3dB passband ripple Chebyshev I LPF.dB passband ripple Example: 5th Order Chebyshev I Filter EECS 247 Lecture 2: Filters 25 H.K. Page 2
11 Filter Types Chebyshev II Lowpass Chebyshev II filter Ripple in stopband Sharper transition band compared to Butterworth Passband phase more linear compared to Chebyshev I Phase (deg) Magnitude (db) Bode Diagram Frequency [Hz] Example: 5th Order Chebyshev II filter EECS 247 Lecture 2: Filters 25 H.K. Page 2 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 Chebyshev II Lowpass jω s-plane σ Example: 5th Order Chebyshev II Filter poles zeros EECS 247 Lecture 2: Filters 25 H.K. Page 22
12 Filter Types Elliptic Lowpass Filter Elliptic filter Ripple in passband Ripple in the stopband Sharper transition band compared to Butterworth & both Chebyshevs Poorest phase response Phase (degrees) Magnitude (db) Normalized Frequency Example: 5th Order Elliptic filter EECS 247 Lecture 2: Filters 25 H.K. Page 23 Filter Types Elliptic Lowpass Filter 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 jω s-plane σ Pole Zero Example: 5th Order Elliptic Filter EECS 247 Lecture 2: Filters 25 H.K. Page 24
13 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 EECS 247 Lecture 2: Filters 25 H.K. Page 25 Magnitude Response of a Bessel Filter as a Function of Filter Order (n) Magnitude [db] Filter Order Increased n= Normalized Frequency EECS 247 Lecture 2: Filters 25 H.K. Page 26
14 Filter Types Comparison of Various Type LPF Magnitude Response Magnitude (db) Normalized Frequency Magnitude (db) All 5th order filters with same corner freq. Bessel Butterworth Chebyshev I Chebyshev II Elliptic EECS 247 Lecture 2: Filters 25 H.K. Page 27 Filter Types Comparison of Various LPF Singularities Poles Bessel Poles Butterworth Poles Elliptic Zeros Elliptic Poles Chebyshev I.dB jω s-plane σ EECS 247 Lecture 2: Filters 25 H.K. Page 28
15 Comparison of Various LPF Groupdelay 5 28 Bessel Chebyshev I.5dB Passband Ripple 2 Butterworth 4 Ref: A. Zverev, Handbook of filter synthesis, Wiley, 967. EECS 247 Lecture 2: Filters 25 H.K. Page 29 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 EECS 247 Lecture 2: Filters 25 H.K. Page 3
16 Magnitude Response Bode Magnitude Diagram - Magnitude (db) th Order Chebychev 4th Order Bessel Frequency [Hz] EECS 247 Lecture 2: Filters 25 H.K. Page 3 Phase Response -5 - Phase [degrees] th Order Chebychev 4th Order Bessel Frequency [Hz].5 2 x 5 EECS 247 Lecture 2: Filters 25 H.K. Page 32
17 Group Delay 4 2 4th Ord. Chebychev 4th Ord. Bessel Group Delay [µ s] Frequency [Hz] EECS 247 Lecture 2: Filters 25 H.K. Page 33 Normalized Group Delay 3 4th Ord. Chebychev 4th Ord. Bessel 2.5 Group Delay [normalized] Frequency [Hz] EECS 247 Lecture 2: Filters 25 H.K. Page 34
18 Step Response.4.2 4th Order Chebychev 4th Order Bessel Amplitude Time (sec) x -5 EECS 247 Lecture 2: Filters 25 H.K. Page 35 Intersymbol Interference (ISI) ISI Broadening of pulses resulting in interference between successive transmitted pulses Example: Simple RC filter EECS 247 Lecture 2: Filters 25 H.K. Page 36
19 Pulse Broadening Bessel versus Chebyshev.5 Input Output th order Bessel x th order Chebyshev I x -4 Chebyshev has more pulse broadening compared to Bessel More ISI EECS 247 Lecture 2: Filters 25 H.K. Page 37 Response to Random Data Chebyshev versus Bessel.5 Input Signal: Symbol rate /3kHz x x -4 4th order Bessel x -4 4th order Chebyshev I EECS 247 Lecture 2: Filters 25 H.K. Page 38
20 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 EECS 247 Lecture 2: Filters 25 H.K. Page 39 Measure of Signal Degradation Eye Diagram Magnitude (db) Bessel Chebychev Random data with symbol rates: /5kHz /khz /3kHz x 4 Frequency [Hz] Group Delay [normalized] th Ord. Chebychev 4th Ord. Bessel x 4 Frequency [Hz] EECS 247 Lecture 2: Filters 25 H.K. Page 4
21 Eye Diagram Chebyshev versus Bessel Input Signal Input Signal Random data Symbol rate:/ 3kHz Time x -5 4th Order Bessel 4th Order Chebychev Time x -5 4th order Bessel Time x -5 4th order Chebyshev I EECS 247 Lecture 2: Filters 25 H.K. Page 4 Eye Diagrams 4th Order Bessel 4th Order Chebychev % Eye opening % Eye opening Time x Time x Random data maximum power symbol rate /5kHz EECS 247 Lecture 2: Filters 25 H.K. Page 42
22 Eye Diagrams 4th Order Bessel 4th Order Chebychev % Eye opening % Eye opening Time x Time x -5 Random data maximum symbol rate /khz Filter with constant group delay More open eye Lower BER (bit-error-rate) EECS 247 Lecture 2: Filters 25 H.K. Page 43 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 EECS 247 Lecture 2: Filters 25 H.K. Page 44
23 RLC Filters Bandpass filter: R V o s Vo = RC Vin s2+ ωo s+ ω2 Q o V in L C ωo = LC Q= ωorc= R L ωo Singularities: Complex conjugate poles + zeros and zero & infinity EECS 247 Lecture 2: Filters 25 H.K. Page 45 RLC Filters Design a bandpass filter with: Center frequency of khz Q of 2 V in R L V o C 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? EECS 247 Lecture 2: Filters 25 H.K. Page 46
24 RLC Filters Effect of Finite Component Q = + Qfilt ideal Q Qind. filt Q=2 (ideal L) Q=3.3 (Q ind. =4) Component Q must be much higher compared to desired filter Q EECS 247 Lecture 2: Filters 25 H.K. Page 47 RLC Filters R V o V in L C Question: Can RLC filters be integrated on-chip? EECS 247 Lecture 2: Filters 25 H.K. Page 48
25 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 999 EECS 247 Lecture 2: Filters 25 H.K. Page 49 Monolithic LC Filters Monolithic inductor in CMOS tech. L<nH with Q<7 Max. capacitor size (based on realistic chip area) C< pf LC filters in the monolithic form feasible: - Frequency >5MHz - Only low quality factor filters Learn more in EE242 EECS 247 Lecture 2: Filters 25 H.K. Page 5
26 Monolithic Filters Desirable to integrate filters with critical frequencies << 5MHz Per previous slide LC filters not a practical option in the integrated form for non-rf frequencies Good alternative: Active filters built without the need for inductors EECS 247 Lecture 2: Filters 25 H.K. Page 5
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