EE247 Lecture 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 filters E.g. National Semiconductor LMF100 EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 1 Switched-Capacitor Filters Today • Emulating resistor via switched-capacitor network • 1st order 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 topologies EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 2 Switched-Capacitor Resistor f1 f2 • Capacitor C is the “switched capacitor” vIN vOUT S1 S2 • Non-overlapping clocks f1 and f2 control switches S1 and S2, C respectively • vIN is sampled at the falling edge of f1 – Sampling frequency fS f1 • Next, f2 rises and the voltage across C is transferred to vOUT f2 • Why does this behave as a resistor? T=1/fs EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 3 Switched-Capacitor Resistors • Charge transferred from v to IN f1 f2 vOUT during each clock cycle is: vIN vOUT S1 S2 Q = C(v – v ) IN OUT C • Average current flowing from vIN to vOUT is: i=Q/t = Q . fs f1 Substituting for Q: f2 i =fS C(vIN – vOUT) T=1/fs EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 4 Switched-Capacitor Resistors i = fS C(vIN – vOUT) f1 f2 With the current through the switched- vIN vOUT capacitor resistor proportional to the S1 S2 voltage across it, the equivalent “switched capacitor resistance” is: C 1 Req = fCs Example: f1 fs ==1MHz,C1pF f2 T=1/f ®Req=W1Mega s EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 5 Switched-Capacitor Filter REQ • Let’s build a “switched- capacitor vIN vOUT ” filter … • Start with a simple RC LPF C2 • Replace the physical resistor by an equivalent switched- f capacitor resistor 1 f2 • 3-dB bandwidth: vIN vOUT C S1 S2 w =1 =´f 1 -3dB RCCs eq22 C1 C2 1 C1 ff-3dB =´s 2Cp 2 EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 6 Switched-Capacitor Filters Advantage versus Continuous-Time Filters f1 f2 Req V Vout Vin Vout in S1 S2 C2 C1 C2 1 1 1 C1 f-3dB = ´ ff-3dB =´s 2p R C 2Cp 2 eq 2 • Corner freq. proportional to: • Corner freq. proportional to: System clock (accurate to few ppm) Absolute value of Rs & Cs C ratio accurate à < 0.1% Poor accuracy à 20 to 50% Ñ Main advantage of SC filtersà inherent corner frequency accuracy EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 7 Typical Sampling Process Continuous-Time(CT) Þ Sampled Data (SD) Continuous- time Time Signal Sampled Data Sampled Data + ZOH Clock EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 8 Uniform Sampling Nomenclature: x(kT) º x(k) Continuous time signal x(t) x(t) Sampling interval T Sampling frequency fs = 1/T Amplitude Sampled signal x(kT) = x(k) • Problem: Multiple continuous time signals can yield exactly the same discrete time signal T • Let’s look at samples taken at 1ms intervals of several sinusoidal waveforms … time EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 9 Sampling Sine Waves T = 1ms fs = 1/T = 1MHz fin = 101kHz time voltage y(nT) v(t) = sin [2p(101000)t] EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 10 Sampling Sine Waves T = 1ms fs = 1MHz fin = 899kHz time voltage v(t) = - sin [2p(899000)t] EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 11 Sampling Sine Waves T = 1ms fs = 1MHz fin = 1101kHz time voltage v(t) = sin [2p(1101000)t] EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 12 Sampling Sine Waves Problem: Identical samples for: v(t) = sin [2p fint ] v(t) = sin [2p( fin+fs )t ] v(t) = sin [2p( fin-fs )t ] àMultiple continuous time signals can yield exactly the same discrete time signal EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 13 Sampling Sine Waves Frequency Spectrum Time domain fs = 1/T Voltage time y(nT) Frequency domain Before Sampling After Sampling + fs - fin fs fin 899kHz 1101kHz Amplitude Amplitude f f f f fin s 2fs … in fs 2fs … 101kHz 1MHz 101kHz 1MHz EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 14 Frequency Domain Interpretation Frequency domain Signal scenario before sampling Amplitude f in fs /2 fs 2fs …….. f Frequency domain Signal scenario after sampling & filtering Amplitude f in fs /2 fs 2fs f Key point: Signals @ nfS ± fmax__signal fold back into band of interest àAliasing EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 15 Aliasing • Multiple continuous time signals can produce identical series of samples • The folding back of signals from nfS±fsig down to ffin is 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 signal EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 16 How 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 interest 2. Limit fmax_Signal through filtering EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 17 How to Avoid Aliasing? 1- Push Frequency domain sampling frequency to x2 of the highest freq. Amplitude fin f à In most cases fs_old 2fs_old …….. s_new f not practical 2- Pre-filter signal to Frequency domain eliminate signals above then sample Amplitude fs/2 f in fs /2 fs 2fs f EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 18 Anti-Aliasing Filter Considerations Desired Signal Brickwall Band Anti-Aliasing Realistic Pre-Filter Anti-Aliasing Pre-Filter Anti-Aliasing Switched-Capacitor Filter Filter Amplitude 0 fs/2 fs 2fs ... f Case1- 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 bandwidth à"Oversampling" EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 19 Practical Anti-Aliasing Filter Desired Signal Parasitic Band Tone Anti-Aliasing Switched-Capacitor Filter Filter Attenuation Case2 - B= f _ << f /2 max Signal s 0 B fs/2 fs-B fs ... f • More practical anti-aliasing filter • Preferable to have an anti- aliasing filter with: àThe lowest order possible 0 B ... f àNo frequency tuning required (if frequency tuning is required then why use switched-capacitor filter, just use the prefilter!?) EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 20 Tradeoff Oversampling Ratio versus Anti-Aliasing Filter Order Maximum Aliasing Dynamic Range Filter Order fs /fin-max * Assumptionà anti-aliasing filter is Butterworth type (not a necessary requirement) àTradeoff: Sampling speed versus anti-aliasing filter order Ref: R. v. d. Plassche, CMOS Integrated Analog-to-Digital and Digital-to-Analog Converters, 2nd ed., Kluwer publishing, 2003, p.41] EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 21 Effect of Sample & Hold ... ... ... ... Sample & Hold Tp Ts Ts •Using the Fourier transform of a rectangular impulse: T sin(pfT ) H ( f ) = p p Ts pfTp EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 22 Effect of Sample & Hold on Frequency Response 1 0.9 Tp sin(pfTp ) 0.8 | H ( f ) |= T pfT 0.7 Tp=Ts s p More practical 0.6 0.5 abs(H(f)) T =0.5T 0.4 p s 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 f / fs EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 23 Sample & Hold Effect (Reconstruction of Analog Signals) Time domain Tp=Ts e T =T g p s a t l o time v ZOH Magnitude sin()p fT Frequency domain Hf()= s droop due p fTs to sinx/x Amplitude effect f in fs 2fs …….. f EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 24 Sample & Hold Effect (Reconstruction of Analog Signals) Magnitude droop due to sinx/x effect: Time domain Case 1) f =f /4 sig s Voltage time -1dB Frequency domain Droop= -1dB Amplitude f in fs f EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 25 Sample & Hold Effect (Reconstruction of Analog Signals) Time domain Magnitude droop due 1 to sinx/x effect: 0.8 0.6 0.4 Case 2) 0.2 0 Amplitude fsig=fs /32 -0.2 -0.4 -0.6 -0.8 sampled data after ZOH -1 0 0.5 1 1.5 2 2.5 3 3.5 Time -5 x 10 -0.0035dB Droop= -0.0035dB Frequency domain à High Amplitude oversampling ratio fin f f desirable s EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K. Page 26 Sampling Process Including S/H H(Z) Sampler S/H e.g. (S.C.F) Vi Time Domain t Freq. Domain f f f 2f fs 2fs fs 2fs Freq. in s s Domain General Signal f f f f 2f B s 2fs s 2fs s s EECS 247 Lecture 9: Switched-Capacitor Filters © 2005 H.K.