ECE1371 Advanced Analog Circuits INTRODUCTION TO DELTA-SIGMA ADCS Richard Schreier [email protected] NLCOTD: Level Translator VDD1 > VDD2, e.g. 3-V logic ? 1-V logic • VDD1 < VDD2, e.g. 1-V logic ? 3-V logic • Constraints: CMOS 1-V and 3-V devices no static current ECE1371 2 Highlights (i.e. What you will learn today) 11st-order modulator (MOD1) Structure and theory of operation 2 Inherent linearity of binary modulators 3 Inherent anti-aliasing of continuous-time modulators 42nd-order modulator (MOD2) 5 Good FFT practice ECE1371 3 0. Background (Stuff you already know) • The SQNR* of an ideal n-bit ADC with a full-scale sine-wave input is (6.02n + 1.76) dB “6 dB = 1 bit.” • The PSD at the output of a linear system is the product of the input’s PSD and the squared magnitude of the system’s frequency response i.e. XY j 2πf 2 ⋅ H(z) Syy()f = He()Sxx()f • The power in any frequency band is the integral of the PSD over that band *. SQNR = Signal-to-Quantization-Noise Ratio ECE1371 4 1. What is ∆Σ? • ∆Σ is NOT a fraternity • Simplified ∆Σ ADC structure: Analog LoopLoop CoarseCoarse Digital In FilterFilter ADCADC Out (to digital filter) DACDAC • Key features: coarse quantization, filtering, feedback and oversampling Quantization is often quite coarse (1 bit!), but the effective resolution can still be as high as 22 bits. ECE1371 5 What is Oversampling? • Oversampling is sampling faster than required by the Nyquist criterion For a lowpass signal containing energy in the (), frequency range0 f B , the minimum sample rate required for perfect reconstruction isf s = 2f B . ≡ ⁄ () • The oversampling ratio is OSR f s 2f B • For a regular ADC, OSR ∼ 23– To make the anti-alias filter (AAF) feasible • For a ∆Σ ADC, OSR ∼ 30 To get adequate quantization noise suppression. Signals betweenf B and ~f s are removed digitally. ECE1371 6 Oversampling Simplifies AAF Desired Undesired OSR ~ 1: Signal Signals f ⁄ f s 2 First alias band is very close OSR = 3: Wide transition band f ⁄ f s 2 Alias far away ECE1371 7 How Does A ∆Σ ADC Work? • Coarse quantization ⇒ lots of quantization error. So how can a ∆Σ ADC achieve 22-bit resolution? •A∆Σ ADC spectrally separates the quantization error from the signal through noise-shaping 1 t t –1 analog uv∆Σ Decimation w digital input ADC Filter output 1 bit @fs n@2fB desired signal Nyquist-rate undesired shaped signals noise PCM Data ⁄ ⁄ f B f s 2 f B f s 2 f B ECE1371 8 A ∆Σ DAC System 1 t –1 digital uv∆Σ Reconstruction w analog input Modulator Filter output (interpolated) 1 bit @fs analog signal shaped output noise ⁄ ⁄ f B f s 2 f B f s 2 f B f s • Mathematically similar to an ADC system Except that now the modulator is digital and drives a low-resolution DAC, and that the out-of-band noise is handled by an analog reconstruction filter. ECE1371 9 Why Do It The ∆Σ Way? • ADC: Simplified Anti-Alias Filter Since the input is oversampled, only very high frequencies alias to the passband. A simple RC section often suffices. If a continuous-time loop filter is used, the anti-alias filter can often be eliminated altogether. • DAC: Simplified Reconstruction Filter The nearby images present in Nyquist-rate reconstruction can be removed digitally. + Inherent Linearity Simple structures can yield very high SNR. + Robust Implementation ∆Σ tolerates sizable component errors. ECE1371 10 2. MOD1: 1st-Order ∆Σ Modulator [Ch. 2 of Schreier & Temes] Quantizer “∆”“Σ” (1-bit) v 1 Y y UVQ –1 z-1 Feedback V’ DAC z-1 DAC v’ v Since two points define a line, a binary DAC is inherently linear. ECE1371 11 MOD1 Analysis • Exact analysis is intractable for all but the simplest inputs, so treat the quantizer as an additive noise source: Y UVQ -1 z E z-1 Y V V(z) = Y(z) + E(z) Y(z) = ( U(z) – z-1V(z) ) / (1–z-1) ⇒(1–z-1) V(z) = U(z) – z-1V(z) + (1–z-1)E(z) V(z) = U(z) + (1–z–1)E(z) ECE1371 12 The Noise Transfer Function (NTF) • In general, V(z) = STF(z)•U(z) + NTF(z)•E(z) • For MOD1, NTF(z) = 1–z–1 The quantization noise has spectral shape! Poles & zeros: 4 π NTF()ej 2 f 2 3 2 ≅ ω2 for ω « 1 1 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Frequency (f /fs) • The total noise power increases, but the noise power at low frequencies is reduced ECE1371 13 In-band Quant. Noise Power σ2 • Assume that e is white with power e ω σ2 ⁄ π i.e. See()= e • The in-band quantization noise power is ω ω B B σ2 jω 2 ω ω ≅ e ω2 ω IQNP= ∫ H() e See()d ------π ∫ d 0 0 π π2σ2 ≡ e ()–3 • SinceOSR ω------- , IQNP = ------------- OSR B 3 • For MOD1, an octave increase in OSR increases SQNR by 9 dB “1.5-bit/octave SQNR-OSR trade-off.” ECE1371 14 A Simulation of MOD1 0 Full-scale test tone –20 SQNR = 55 dB @ OSR = 128 –40 Shaped “Noise” –60 dBFS/NBW –80 20 dB/decade NBW = 5.7x10–6 –100 10–3 10–2 10–1 Normalized Frequency ECE1371 15 CT Implementation of MOD1 •Ri/Rf sets the full-scale; C is arbitrary Also observe that an input at fs is rejected by the integrator— inherent anti-aliasing Integrator Latched Comparator C Rf Ri u y D Q v DFF clock CK QB ECE1371 16 MOD1-CT Waveforms u = 0 u = 0.06 1 1 v v –1 –1 y 0 y 0 0 5 10 15 20 0 5 10 15 20 Time Time • With u=0, v alternates between +1 and –1 • With u>0, y drifts upwards; v contains consecutive +1s to counteract this drift ECE1371 17 1 – z –1 MOD1-CT STF = ----------------- s Recall ze= s ω s-plane Zeros @ s = 2kπi σ Pole-zero cancellation @ s = 0 ECE1371 18 MOD1-CT Frequency Responses 10 NTF = 1–z–1 0 –10 1–z–1 STF = s –20 dB –30 Quant. Noise Inherent Notch Anti-Aliasing –40 –50 0123 Frequency (Hz) ECE1371 19 Summary • ∆Σ works by spectrally separating the quantization noise from the signal ≡ ⁄ () Requires oversampling.OSR f s 2f B . • Noise-shaping is achieved by the use of filtering and feedback • A binary DAC is inherently linear, and thus a binary ∆Σ modulator is too • MOD1 has NTF (z) = 1 – z–1 ⇒ Arbitrary accuracy for DC inputs. 1.5 bit/octave SQNR-OSR trade-off. • MOD1-CT has inherent anti-aliasing ECE1371 20 NLCOTD 1V 1V 3V → 1V: 3V 3V 3V 1V → 3V: 3V 3V 3V 3V ECE1371 21 3. MOD2: 2nd-Order ∆Σ Modulator [Ch. 3 of Schreier & Temes] • Replace the quantizer in MOD1 with another copy of MOD1 in a recursive fashion: E1 E UVQ z-1 z-1 z-1 z-1 –1 –1 V(z) = U(z) + (1–z )E1(z), E1(z) = (1–z )E(z) ⇒V(z) = U(z) + (1–z–1)2E(z) ECE1371 22 Simplified Block Diagrams E U z 1 Q z−1 z−1 V NTF() z = ()1 – z –1 2 STF() z = z –1 E 1 1 U Q z−1 z−1 V –1 2 -1 -2 NTF() z = ()1 – z STF() z = z –2 ECE1371 23 NTF Comparison 0 −20 (dB) MOD1 f −40 π j2 MOD2 () −60 MOD2 has twice as much NTF e attenuation as MOD1 −80 at all frequencies −100 10–3 10–2 10–1 Normalized Frequency ECE1371 24 In-band Quant. Noise Power • For MOD2, He()jω 2 ≈ ω4 ω B jω 2 ω ω • As before,IQNP= H() e See()d and ∫0 ω σ2 ⁄ π See()= e π4σ2 • So now IQNP = ------------e-()OSR –5 5 With binary quantization to ±1, ∆ σ2 ∆2 ⁄ ⁄ = 2 and thuse ==12 13 . • “An octave increase in OSR increases MOD2’s SQNR by 15 dB (2.5 bits)” ECE1371 25 Simulation Example Input at 75% of FullScale 1 0 –1 0 50 100 150 200 Sample number ECE1371 26 Simulated MOD2 PSD Input at 50% of FullScale 0 SQNR = 86 dB –20 @ OSR = 128 –40 –60 Simulated spectrum (smoothed) –80 Theoretical PSD (k = 1) –100dBFS/NBW 40 dB/decade –120 − NBW = 5.7×10 6 –140 10–3 10–2 10–1 Normalized Frequency ECE1371 27 SQNR vs. Input Amplitude MOD1 & MOD2 @ OSR = 256 120 100 80 MOD2 60 Predicted SQNR Simulated SQNR SQNR (dB) 40 MOD1 20 0 –100 –80 –60 –40 –20 0 Input Amplitude (dBFS) ECE1371 28 SQNR vs. OSR 120 100 MOD2 (Theoretical curve assumes -3 dBFS input) 80 60 MOD1 SQNR (dB) (Theoretical curve assumes 0 dBFS input) 40 20 Predictions for MOD2 are optimistic. Behavior of MOD1 is erratic. 0 4 8 16 32 64 128 256 512 1024 ECE1371 29 Audio Demo: MOD1 vs. MOD2 [dsdemo4] Sine Wave MOD1 Slow Ramp MOD2 Speech ECE1371 30 MOD1 + MOD2 Summary • ∆Σ ADCs rely on filtering and feedback to achieve high SNR despite coarse quantization They also rely on digital signal processing. ∆Σ ADCs need to be followed by a digital decimation filter and ∆Σ DACs need to be preceded by a digital interpolation filter. • Oversampling eases analog filtering requirements Anti-alias filter in an ADC; image filter in a DAC. • Binary quantization yields inherent linearity • MOD2 is better than MOD1 15 dB/octave vs.