EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 1EE247Lecture 9• Switched-Capacitor Filters– “Analog” sampled-data filters:• Continuous amplitude• Quantized time– Applications:• First commercial product: Intel 2912 voice-band CODEC chip, 1979• Oversampled A/D and D/A converters• Stand-alone filtersE.g. National Semiconductor LMF100EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 2Switched-Capacitor FiltersToday• Emulating resistor via switched-capacitor network• 1storder switched-capacitor filter• Switch-capacitor filter considerations:– Issue of aliasing and how to avoid it– Tradeoffs in choosing sampling rate– Effect of sample and hold – Switched-capacitor filter electronic noise – Switched-capacitor integrator topologiesEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 3Switched-Capacitor Resistor• Capacitor C is the “switched capacitor”• Non-overlapping clocks φ1and φ2control switches S1 and S2, respectively• vINis sampled at the falling edge of φ1– Sampling frequency fS• Next, φ2rises and the voltage across C is transferred to vOUT• Why does this behave as a resistor?vINvOUTCS1 S2φ1φ2φ1φ2T=1/fsEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 4Switched-Capacitor ResistorsvINvOUTCS1 S2φ1φ2φ1φ2T=1/fs• Charge transferred from vINto vOUTduring each clock cycle is:• Average current flowing from vINto vOUTis:Q = C(vIN – vOUT)i=Q/t = Q . fsSubstituting for Q:i =fSC(vIN– vOUT)EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 5Switched-Capacitor ResistorsWith the current through the switched-capacitor resistor proportional to the voltage across it, the equivalent “switched capacitor resistance” is:vINvOUTCS1 S2φ1φ2φ1φ2T=1/fsi = fS C(vIN – vOUT)1ReqfCsExample:f1MHz,C1pFsR1Megaeq===→=ΩEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 6Switched-Capacitor Filter• Let’s build a “switched- capacitor ” filter …• Start with a simple RC LPF• Replace the physical resistor by an equivalent switched-capacitor resistor• 3-dB bandwidth:vINvOUTC1S1 S2φ1φ2C2vOUTC2REQvINC11fs3dBRCCeq22C11ffs3dB2C2ωπ==×−=×−EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 7Switched-Capacitor Filters Advantage versus Continuous-Time FiltersVinVoutC1S1 S2φ1φ2C2VoutC2ReqVin3dB1s2C1ff2Cπ−=×2eqCR121fdB3×=−π• Corner freq. proportional to:System clock (accurate to few ppm)C ratio accurate à < 0.1%• Corner freq. proportional to:Absolute value of Rs & CsPoor accuracy à 20 to 50%Ñ Main advantage of SC filtersà inherent corner frequency accuracyEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 8Typical Sampling ProcessContinuous-Time(CT) ⇒ Sampled Data (SD)Continuous-Time SignalSampled Data+ ZOH ClocktimeSampled DataEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 9Uniform SamplingNomenclature:Continuous time signal x(t)Sampling interval TSampling frequency fs= 1/TSampled signal x(kT) = x(k)• Problem: Multiple continuous time signals can yield exactly the same discrete time signal• Let’s look at samples taken at 1µs intervals of several sinusoidal waveforms …timex(kT) ≡ x(k)Tx(t)AmplitudeEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 10Sampling Sine Wavestimevoltagev(t) = sin [2π(101000)t]T = 1µsfs= 1/T = 1MHzfin= 101kHzy(nT)EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 11Sampling Sine Wavestimevoltagev(t) = - sin [2π(899000)t]T = 1µsfs= 1MHzfin= 899kHzEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 12Sampling Sine Wavestimevoltagev(t) = sin [2π(1101000)t]T = 1µsfs= 1MHzfin= 1101kHzEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 13Sampling Sine WavesProblem:Identical samples for:v(t) = sin [2π fint ]v(t) = sin [2π( fin+fs )t ]v(t) = sin [2π( fin-fs )t ]àMultiple continuous time signals can yield exactly the same discrete time signalEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 14Sampling Sine WavesFrequency Spectrumfs= 1/Ty(nT)timefs1MHzTime domain…fAmplitudefin101kHz2fsFrequency domainfs1MHz…fAmplitudefin101kHz2fsfs - fin899kHzfs + fin1101kHzVoltageBefore SamplingAfter SamplingEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 15Frequency Domain Interpretationfs……..fAmplitudefin2fsFrequency domainfsfAmplitudefin2fsFrequency domainSignal scenariobefore samplingSignal scenarioafter sampling & filteringfs /2fs /2Key point: Signals @ nfS± fmax__signalfold back into band of interestàAliasingEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 16Aliasing• Multiple continuous time signals can produce identical series of samples• The folding back of signals from nfS±fsigdown to ffinis called aliasing– Sampling theorem: fs> 2fmax_Signal• If aliasing occurs, no signal processing operation downstream of the sampling process can recover the original continuous time signalEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 17How to Avoid Aliasing?• Must obey sampling theorem:fmax_Signal< fs/2• Two possibilities:1. Sample fast enough to cover all spectral components, including "parasitic" ones outside band of interest2. Limit fmax_Signalthrough filteringEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 18How to Avoid Aliasing?fs_old……..fAmplitudefin2fs_oldFrequency domainfsfAmplitudefin2fsFrequency domain1- Push sampling frequency to x2 of the highest freq. à In most cases not practical2- Pre-filter signal to eliminate signals above fs/2 then samplefs /2fs_newEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 19Anti-Aliasing Filter ConsiderationsCase1- B= fmax _Signal= fs /2• Non-practical since an extremely high order anti-aliasing filter (close to an ideal brickwall filter) is required• Practical anti-aliasing filter àNonzero filter "transition band"• In order to make this work, we need to sample much faster than 2x the signal bandwidthà"Oversampling" 0 fs2fs...fAmplitudeBrickwallAnti-AliasingPre-Filterfs/2Anti-Aliasing FilterSwitched-CapacitorFilterRealisticAnti-AliasingPre-FilterDesiredSignalBandEECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 20Practical Anti-Aliasing Filter0 fs... fDesiredSignalBandfs/2B fs-BParasiticToneAttenuation0 ...fBAnti-Aliasing
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