EECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 1EE247Lecture 9• Switched-capacitor filters–Tradeoffs in choosing sampling rate–Effect of sample and hold –Switched-capacitor network electronic noise –Switched-capacitor integrators• DDI integrators• LDI integrators• Effect of parasitic capacitance• Bottom-plate integrator topology–ResonatorsEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 2Summary of last lecture• Continuous-time filters continued– Various Gm-C filter implementations– Comparison of continuous-time filter topologies• Switched-capacitor filters– Emulating resistor via switched-capacitor network– 1st order switched-capacitor filter– Switch-capacitor filter considerations:• Issue of aliasing and how to avoid it– Sample at high enough frequency so that the entire range of signals including the parasitics are at freqs < fs /2– Use of anti-aliasing prefiltersEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 3Sampling Sine WavesProblem:Identical samples for:v(t) = cos [2πfint ]v(t) = cos [2π(n.fs+fin )t ]v(t) = cos [2π(n.fs-fin)t ]• (n-integer)ÆMultiple continuous time signals can yield exactly the same discrete time signalEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 4Sampling Sine WavesFrequency Spectrumf /fsAmplitudefs1MHz…fAmplitudefin100kHz2fs600kHz1.2MHzContinuous-TimeDiscrete TimeSignal scenariobefore samplingSignal scenarioafter samplingKey point: Signals @ nfS±fmax__signalfold back into band of interestÆAliasing0.50.10.40.21.7MHz0.3EECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 5How to Avoid Aliasing?• Must obey sampling theorem:fmax-Signal< fs /2*Note: Minimum sampling rate of fs=2xfmax-Signalis called Nyquist rate• Two possibilities:1. Sample fast enough to cover all spectral components, including "parasitic" ones outside band of interest2. Limit fmax_Signalthrough filtering Æ attenuate out-of-band components prior to samplingEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 6How to Avoid Aliasing?1-Sample Fastfs_old……..fAmplitudefin2fs_oldFrequency domainPush sampling frequency to x2 of the highest frequency signal to cover all unwanted signals as well as wanted signalsÆ In vast majority of cases not practicalfs_newEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 7How to Avoid Aliasing?2-Filter Out-of-Band Signal Prior to SamplingPre-filter signal to eliminate/attenuate signals above fs/2- then samplefs……..fAmplitudefin2fsFrequency domainfs……..fAmplitudefin2fsFrequency domainfs /2FilterEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 8Anti-Aliasing Filter ConsiderationsCase1- B= fsigmax=fs /2• Non-practical since an extremely high order anti-aliasing filter (close to an ideal brickwall filter) is required• Practical anti-aliasing filter ÆNon-zero filter "transition band"• In order to make this work, we need to sample much faster than 2x the signal bandwidthÆ"Oversampling" 0fs2fs...fAmplitudeBrickwallAnti-AliasingPre-Filterfs/2Anti-Aliasing FilterSwitched-CapacitorFilterRealisticAnti-AliasingPre-FilterDesiredSignalBandEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 9Practical Anti-Aliasing Filter0fs... fDesiredSignalBandfs/2Bfs-BParasiticToneAttenuation0 ...B/fsAnti-Aliasing FilterSwitched-CapacitorFilterCase2 - B= fmax_Signal<< fs/2• More practical anti-aliasing filter• Preferable to have an anti-aliasing filter with:ÆThe lowest order possibleÆNo frequency tuning required (if frequency tuning is required then why use switched-capacitor filter, just use the prefilter!?)f /fs0.5EECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 10TradeoffOversampling Ratio versus Anti-Aliasing Filter OrderÆTradeoff: Sampling speed versus anti-aliasing filter orderMaximum Aliased SignalRejectionfs /fin-maxFilter OrderRef: R. v. d. Plassche, CMOS Integrated Analog-to-Digital and Digital-to-Analog Converters, 2nd ed., Kluwer publishing, 2003, p.41* AssumptionÆanti-aliasing filter is Butterworth type (not a necessary requirement)EECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 11Effect of Sample & Hold......TpTs......TsSample &Hold•Using the Fourier transform of a rectangular impulse:shapesin)sin()(xxfTfTTTfHppsp→=ππEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 120 0.5 1 1.5 2 2.5 300.10.20.30.40.50.60.70.80.91f / fsabs(H(f))Effect of Sample & Hold onFrequency ResponsesspsppspffTTfffTfTTTfHππππ)sin()sin(|)(|×==integer0|)(||)0(|→====nTTnffHTTfHpssspTp=TsTp=0.5TsMore practicalEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 13Sample & Hold Effect (Reconstruction of Analog Signals)Time domaintimevoltageZOHfs……..fAmplitudefin2fsFrequency domainsin( )()fTsHffTsππ=Tp=TsMagnitude droop due to sinx/xeffectTp=TsEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 14Sample & Hold Effect (Reconstruction of Analog Signals)Time domaintimeVoltagefsfAmplitudefinFrequency domainMagnitude droop due to sinx/x effect:Case 1) fsig=fs /4Droop= -1dB-1dBEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 15Sample & Hold Effect (Reconstruction of Analog Signals)Time domainMagnitude droop due to sinx/x effect:Case 2) fsig=fs /32Droop= -0.0035dB• Insignificant droop ÆHigh oversampling ratio desirablefsfAmplitudefinFrequency domain-0.0035dB0 0.5 1 1.5 2 2. 5 3 3. 5x 10-5-1-0.8-0.6-0.4-0.200.20.40.60.81TimeAmplitudesampled dataafter ZOHEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 16Sampling Process Including S/HfsTime Domain2fstViFreq. Domainfs2fsffinfs2fsfBfs2fsfs2fsfs2fsFreq. DomainGeneralSignalSamplerH(Z)e.g. (S.C.F)S/HEECS 247 Lecture 9 Switched-Capacitor Filters © 2007 H. K. Page 171stOrder
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