Cascaded Noise Shaping for Oversampling A/D and D/A Conversion Bruce A. Wooley Stanford University Bruce A. Wooley - 1 - Copyright 2005, Stanford University Outline . Oversampling modulators for A-to-D conversion . Cascaded ΣΔ modulators . Low-voltage ΣΔ modulator design for MHz-bandwidth signals . Cascaded noise shaping for bandpass oversampling D-to-A conversion Bruce A. Wooley - 2 - Stanford University, 2005 Analog-to-Digital Conversion 00001011 11000011 01001000 Digital Processor Filtering Sampling Quantization Processing Quantizer model: p(eQ) eQ[n] 1/Δ Σ -Δ/2 Δ/2 Bruce A. Wooley - 3 - Stanford University, 2005 Quantization Noise Shaping S (f) NB Q Quantizer resolution increased by 3 dB per octave of OVERSAMPLING -fS/2 -fB fB fS/2 SQ(f) Quantizer resolution NB increased through NOISE SHAPING -fS/2 -fB fB fS/2 Bruce A. Wooley - 4 - Stanford University, 2005 Oversampling Modulators . Embed quantizer in a feedback loop to achieve larger improvement in resolution with increased oversampling . Feedback can be used for PREDICTION (Δ modulation) or NOISE SHAPING (ΣΔ modulation) . Noise shaping modulators are more robust and easier to implement than predictive modulators Bruce A. Wooley - 5 - Stanford University, 2005 Sigma-Delta (or Delta-Sigma) Modulation EQ(z) Integrator z-1 X(z) A/D Y(z) Σ 1− z -1 D/A "1 "1 Y (z) = z X(z) + (1" z )EQ(z) 2 N (f) 2sin f / f N (f) E = [ ( " S )] Q ! Bruce A. Wooley - 6 - Stanford University, 2005 ! ΣΔ Modulator Response 0.6 0.4 0.2 0 -0.2 -0.4 Modulator Input, Quantizer Output -0.6 0 50 100 150 200 250 Time (t/T) Bruce A. Wooley - 7 - Stanford University, 2005 Noise Shaping Ideal Digital Lowpass Filter First-Order Noise Shaping f /2 fB fN S Frequency Bruce A. Wooley - 8 - Stanford University, 2005 Higher-Order Noise-Shaping Modulators The order of the noise shaping can be increased using either . Single quantizer modulators − Multi-loop noise differencing − Single loop with multi-order filtering or . Cascaded (multistage) modulators Bruce A. Wooley - 9 - Stanford University, 2005 Noise-Differencing ΣΔ Modulators Y(z) = z–1X(z) + (1– z–1)LE(z) L=3 L=2 LP Filter L=1 Noise Shaping fB fN fS/2 Frequency Bruce A. Wooley - 10 - Stanford University, 2005 ΣΔ Modulator Dynamic Range (with 1-bit quantization) 140 120 100 L=3 L=2 80 L=1 60 40 Dynamic Range (dB) 20 0 4 8 16 32 64 128 256 512 Oversampling Ratio Bruce A. Wooley - 11 - Stanford University, 2005 Single-Quantizer ΣΔ Modulator E(z) + X(z) – A(z) + Y(z) – F(z) Y (z) = HX(z)X(z) + HE(z)E(z) where A(z) 1 HX(z) = , HE(z) = ! 1+ A(z)F(z) 1+ A(z)F(z) and H (z) 1" H (z) A(z) = X , F(z) = E H (z) H (z) ! E X Bruce A. Wooley - 12 - Stanford University, 2005 ! Noise Differencing Modulators . Y(z) = z–1X(z) + (1– z–1)LE(z) . Can implement with a single quantizer and L nested loops . Limit cycle instability for L > 2 . For L = 2 z"1 A(z) = and F(z) = 2 " z"1 "1 2 (1" z ) Integrator 1 Integrator 2 Integrator L E ! 1 + + 1 + 1 + z! + X " " " !1 " Y 1!z!1 1!z!1 1!z ! ! ! Bruce A. Wooley - 13 - Stanford University, 2005 2nd-Order Modulator Implementation E A(z) + + + + + X " " " z!1 " Y ! + + z!1 ! F(z) " z!1 + + + " ! E + + + + + + X " " " " z!1 " Y ! + ! + z!1 Unity + + Transfer " " + ! Function 1 z!1 z! Bruce A. Wooley - 14 - Stanford University, 2005 2nd-Order ΣΔ Modulator E + + + + + + X " " " " z!1 " Y ! + ! + z!1 ! E + + + + + + + X " " z!1 " " " z!1 " Y + ! + ! + Bruce A. Wooley - 15 - Stanford University, 2005 2nd-Order ΣΔ Modulator E + + + + + + X " " z!1 " " z!1 " Y ! + ! + 2 ! E + + 1 + + + + !1 !1 X " 2 " z " 2 " z " Y ! + ! + + Bruce A. Wooley - 16 - Stanford University, 2005 2nd-Order Noise-Differencing ΣΔ Modulator * (with 1-bit quantization) Can scale only w/ 1-bit quantizer INTEGRATOR 1 INTEGRATOR 2 QUANTIZER x(nT) + + + + y(nT) 1 1 1-bit A/D ! 2 ! DELAY ! 2 ! DELAY – + – + D/A q(nT) * B. Boser, JSSC, Dec .1988 Bruce A. Wooley - 17 - Stanford University, 2005 Cascaded ΣΔ Modulators . Quantizer error quantized by subsequent stage and then digitally filtered and subtracted from output of preceding stage . Cancellation of lower-order noise-shaping terms depends on matching of analog and digital paths . No potential instability if use first- and second-order stages Bruce A. Wooley - 18 - Stanford University, 2005 Cascaded Noise-Shaping Modulator Analog + Digital x !" y Delay ! In – Out e Digital ADC Difference Matches noise shaping of quantization error in first-stage Bruce A. Wooley - 19 - Stanford University, 2005 Maximum Improvement in Dynamic Range 80.0 Simulation Closed form equation 60.0 40.0 20.0 Independent of OSR 0.0 0.01 0.1 1 10 Coefficient Mismatch (%) Bruce A. Wooley - 20 - Stanford University, 2005 Third-Order (2-1) Cascaded Modulator nd y x 2 order !" 1 Error y Cancellation e1 y2 1st order !" "2 "1 2 Y1(z) = z X(z) + (1" z ) E1(z) "1 "1 Y2(z) = z E1(z) + (1" z )E2(z) Y(z) z"1Y (z) (1 z"1)2Y (z) z"3X(z) (1 z"1)3E (z) = 1 " " 2 = + " 2 Bruce A. Wooley - 21 - Stanford University, 2005 ! 2-1 Cascaded ΣΔ Modulator + + x $ # $ # y1 - - b " + ! $ - + $ # y2 - Bruce A. Wooley - 22 - Stanford University, 2005 Matching Error in 2-1 Cascade 10 8 6 4 2 Calculated Simulated 0 Loss in Dynamic Range (dB) -2 -10 -5 0 5 10 Matching Error (%) Bruce A. Wooley - 23 - Stanford University, 2005 Spectrum of 2-1 Cascade w/ Mismatch 0 -50 -100 Spectral Power (dB) -150 -200 0 5 10 15 20 25 Frequency (kHz) Bruce A. Wooley - 24 - Stanford University, 2005 Matching Error in 1-1-1 Cascade 35 30 25 20 15 Calculated Simulated 10 Loss in Dynamic Range (dB) 5 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Matching Error (%) Bruce A. Wooley - 25 - Stanford University, 2005 Spectrum of 1-1-1 Cascade w/ Mismatch 0 -50 -100 Spectral Power (dB) -150 -200 0 5 10 15 20 25 Frequency (kHz) Bruce A. Wooley - 26 - Stanford University, 2005 Advantages of 2-1 Cascade . Low sensitivity to precision of analog circuits . Suppression of spurious noise tones resulting for correlation of quantization noise with input . Considerable design flexibility . No potential instability Bruce A. Wooley - 27 - Stanford University, 2005 Analog Integration in CMOS . Continuous Time (tune gm or MOS-R) – gm-C – MOSFET-C . Sampled Data – Switched current – Switched capacitor Bruce A. Wooley - 28 - Stanford University, 2005 Analog Integration in CMOS . Continuous-time (gm-C or MOSFET) – Tune gm or MOS resistor – Performance limited by timing jitter, waveform asymmetry and integrator linearity . Switched-current – Limitations: • current sources must be cascoded to increase output resistance high supply voltage • large VGS – VT needed to reduce sensitivity to VT mismatch high power dissipation • sensitive to switch parasitics and charge injection • noise introduced into “compressed” signal Switched-capacitor approach preferable for obtaining high resolution at low supply voltage and low power dissipation Bruce A. Wooley - 29 - Stanford University, 2005 Low-Power ΣΔ Modulator Design * Ci 1 Cs 2 x ! H(z) Q y 2 1 + . High-resolution converters should by limited by thermal noise – kT/C noise in switched-capacitor circuits . First stage limit performance and therefore dissipates a large fraction of power . Minimize power by minimizing capacitor size in switched-C implementations * S. Rabii, JSSC, June 1997 Bruce A. Wooley - 30 - Stanford University, 2005 ΣΔ Modulator Implementation 1/5 1/6 y ! ! 1 x 1/5 " 1/3 " H (z) + # + # 1 1/20 ! y2 + 3/4 " H2(z) " + ! # ! 1/5 y Bruce A. Wooley - 31 - Stanford University, 2005 First Stage of Modulator 1.8V 0V 1.8V 0V S2 S5 S6 0.3V S7 0.3V 20pF 0.9V S3 S6 4pF S1 S4 S5 S8 Vi( ) + + ! + ! Vi(!) ! + ! + S1 4pF S4 S5 S8 S3 S6 0.3V 20pF 0.9V S7 S2 S5 S6 0.3V 1.8V 0V 1.8V 0V S1, S5: !1delayed, CMOS S3, S7: !1, NMOS S2, S6: !2delayed, CMOS S4, S8: !2, NMOS Bruce A. Wooley - 32 - Stanford University, 2005 2-Stage Class A/AB Amplifier 1.8 V M9 M7 Vcmfb1 M8 M10 M13 Vbz Vbz Mz Vin(+) M1 M2 Vin(!) Mz Vout(+) Vout(!) CC CC Vbias Vo1(!) Vo1(+) M11 M5 M3 M4 M6 M12 Bruce A. Wooley - 33 - Stanford University, 2005 Die Photo of 1.8-V CMOS ΣΔ Modulator Bruce A. Wooley - 34 - Stanford University, 2005 Measured SNR and SNDR 100 SNR SNDR 80 60 40 20 0 -100 -80 -60 -40 -20 0 Input Level (dB) Bruce A. Wooley - 35 - Stanford University, 2005 Measured Baseband Output Spectrum 0 -20 -40 !20 dB, 2 kHz Input -60 -80 -100 -120 -140 -160 0 5 10 15 20 25 Frequency (kHz) Bruce A. Wooley - 36 - Stanford University, 2005 Dynamic Range vs. Oversampling Ratio 110 100 90 80 70 60 20 50 100 200 300 Oversampling Ratio Bruce A. Wooley - 37 - Stanford University, 2005 Dynamic Range & Power vs. Supply Voltage 100 4.0 3.5 99 3.0 98 Power 2.5 Dynamic range 97 2.0 1.5 1.7 1.9 2.1 2.3 2.5 Supply Voltage (V) Bruce A. Wooley - 38 - Stanford University, 2005 1.8-V CMOS ΣΔ Modulator Performance Dynamic range 99 dB Peak SNR 99 dB Peak SNDR 95 dB Bandwidth 25 kHz Oversampling ratio 80 Power dissipation 2.5 mW Active area 1.5 mm2 Technology 0.8-µm CMOS Threshold voltages +0.65 V, –0.75 V Bruce A.