INTRODUCTION TO DELTA-SIGMA ADCS
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1 ECE37 Advanced Analog Circuits Lecture INTRODUCTION TO DELTA-SIGMA ADCS Richard Schreier Trevor Caldwell Course Goals Deepen understanding of CMOS analog circuit design through a top-down study of a modern analog system The lectures will focus on Delta-Sigma ADCs, but you may do your project on another analog system. Develop circuit insight through brief peeks at some nifty little circuits The circuit world is filled with many little gems that every competent designer ought to recognize. ECE37 - Logistics Format: Meet Mondays 3:-5: PM except Feb 4 and Feb 8 -hr lectures plus proj. presentation Grading: 4% homework 6% project References: Schreier & Temes, Understanding Σ Johns & Martin, Analog IC Design Razavi, Design of Analog CMOS ICs Lecture Plan: ECE37-3 Date Lecture Ref Homework 8--7 RS Introduction: MOD & MOD S&T -3, A Matlab MOD 8--4 RS Example Design: Part S&T 9., J&M Switch-level sim 8-- RS 3 Example Design: Part J&M 4 -level sim 8--8 TC 4 Pipeline and SAR ADCs Arch. Comp ISSCC No Lecture 8-- RS 5 Advanced Σ S&T 4, 6.6, 9.4, B CTMOD; Proj Reading Week No Lecture 8--5 RS 6 Comparator & Flash ADC J&M TC 7 SC Circuits J&M 8-3- TC 8 Amplifier Design TC 9 Amplifier Design TC Noise in SC Circuits S&T C Project Presentation TC Matching & MM-Shaping Project Report RS Switching Regulator -level sim ECE37-4 NLCOTD: Level Translator DD > DD, e.g. 3- logic? - logic DD < DD, e.g. - logic? 3- logic Constraints: CMOS - and 3- devices no static current ECE37-5 What is Σ? Σ is NOT a fraternity It is more like a way of life Simplified Σ ADC structure: Analog In Loop Filter DAC Coarse ADC Digital Out (to digital filter) Key features: coarse quantization, filtering, feedback and oversampling uantization is often quite coarse: bit! ECE37-6
2 What is Oversampling? Oversampling is sampling faster than required by the Nyquist criterion For a lowpass signal containing energy in the frequency range (, f B ), the minimum sample rate required for perfect reconstruction is f s = f B The oversampling ratio is OSR f s ( f B ) For a regular ADC, OSR 3 To make the anti-alias filter (AAF) feasible For a Σ ADC, OSR 3 To get adequate quantization noise suppression. All signals above are removed digitally. f B ECE37-7 Oversampling Simplifies AAF OSR ~ : OSR = 3: Desired Signal f s Undesired Signals f First alias band is very close Wide transition band f s f Alias far away ECE37-8 How Does A Σ ADC Work? Coarse quantization lots of quantization error. So how can a Σ ADC achieve -bit resolution? A Σ ADC spectrally separates the quantization error from the signal through noise-shaping analog input desired signal t u Σ v Decimation w digital ADC Filter output s n@f B undesired signals f B f s f s f B shaped noise Nyquist-rate PCM Data ECE37-9 t f B digital input (interpolated) A Σ DAC System u Σ v Reconstruction w analog Modulator Filter output s signal shaped noise analog output f B f s f B f s 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. ECE37 - t Why Do It The Σ Way? ADC: Simplified Anti-Alias Filter Since the input is oversampled, only very high frequencies alias to the passband. These can often be removed with a simple RC section. 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. ECE37 - Highlights (i.e. What you will learn today) st - and nd -order modulator structures and theory of operation Inherent linearity of binary modulators 3 Inherent anti-aliasing of continuous-time modulators 4 Spectrum estimation with FFTs ECE37 -
3 Background (Stuff you already know) The SNR * of an ideal n-bit ADC with a full-scale sine-wave input is (6.n +.76) db 6 db = 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. X H(z) The power in any frequency band is the integral of the PSD over that band *. Signal-to-uantization-Noise Ratio Y S yy () f = He ( j πf ) S xx () f ECE37-3 Poor Man s Σ DAC Suppose you have low-speed 6-bit data and a high-speed 8-bit DAC How can you get good analog performance? 6-bit data 6 5 khz? DAC 5 MHz Good uality Audio ECE37-4 Simple (-Minded) Solution Only connect the MSBs; leave the LSBs hanging 6 MSBs khz LSBs 8 DAC 5 MHz (or 5 khz) 6-b Input Data DAC Output us Time ECE37-5 Spectral Implications Desired Signal 5 khz Unwanted Images sin( x ) DAC frequency response x Frequency uantization 8-bit level SNR = 5 db ECE37-6 Better Solution Exploit oversampling: Clock fast and add dither 6 khz 5 MHz dither spanning one 8-bit LSB 6-b Input Data DAC ECE MHz DAC Output Time Spectral Implications uantization noise is now spread over a broad frequency range Oversampling reduces quantization noise density.5 MHz OSR = = db 5 khz Frequency 5 khz.5 MHz In-band quantization noise power = % of total quantization noise power SNR = 7 db ECE37-8
4 Even More Clever Method Add LSBs back into the input data 6 5 MHz 8 z 6-b Input Data DAC 5 MHz DAC Output Time ECE37-9 Mathematical Model Assume the DAC is ideal, model truncation as the addition of error: E = LSBs U z - E = U + ( z )E Hmm Oversampling, coarse quantization and feedback. Noise-Shaping! Truncation noise is shaped by a z transfer function, which provides ~35 db of attenuation in the -5 khz frequency range ECE37 - Spectral Implications uantization noise is heavily attenuated at low frequencies Shaped uantization Noise Frequency 5 khz.5 MHz In-band quantization noise power is very small, 55 db below total power SNR = 5 db! ECE37 - U MOD: st -Order Σ Modulator Standard Block Diagram z - Σ z - Y DAC ECE37 - uantizer (-bit) v Feedback DAC v Since two points define a line, a binary DAC is inherently linear. y v MOD Analysis Exact analysis is intractable for all but the simplest inputs, so treat the quantizer as an additive noise source: U z - Y (z) = Y(z) + E(z) (z) = U(z) + ( z )E(z) ECE37-3 z - Y(z) = ( U(z) z - (z) ) / ( z - ) ( z - ) (z) = U(z) z - (z) + ( z - )E(z) Y E The Noise Transfer Function In general, (z) = STF(z) U(z) + NTF(z) E(z) For MOD, NTF(z) = z The quantization noise has spectral shape! 4 3 NTF e j πf ( ) ω for ω « Normalized Frequency (f /f s ) The total noise power increases, but the noise power at low frequencies is reduced ECE37-4
5 In-band Noise Power Assume that e is white with power i.e. S ee ( ω) = σ e π The in-band noise power is N = ω B He ( jω ) S ee ( ω)dω π π σ Since OSR , N e ω = ( OSR) B 3 3 For MOD, an octave increase in OSR increases SNR by 9 db.5-bit/octave SNR-OSR trade-off. ECE37-5 ω B σ e σ e ω π dω dbfs/nbw A Simulation of MOD Full-scale test tone SNR = 55 OSR = 8 4 Shaped Noise 6 8 db/decade 3 Normalized Frequency NBW = 5.7x 6 ECE37-6 CT Implementation of MOD R i /R f sets the full-scale; C is arbitrary Also observe that an input at f s is rejected by the integrator inherent anti-aliasing Integrator R f C Latched Comparator v y MOD-CT Waveforms u = u =.6 v y u R i y clock D DFF CK B v 5 5 Time 5 5 Time With u=, v alternates between + and With u>, y drifts upwards; v contains consecutive +s to counteract this drift ECE37-7 ECE37-8 Summary So Far Σ works by spectrally separating the quantization noise from the signal Noise-shaping is achieved by the use of filtering and feedback A binary DAC is inherently linear, and thus a binary modulator is too MOD has NTF(z) = z Arbitrary accuracy for DC inputs..5 bit/octave SNR-OSR trade-off. MOD-CT has inherent anti-aliasing ECE37-9 MOD: nd -Order Σ Modulator Replace the quantizer in MOD with another copy of MOD: E U z - (z) = U(z) + ( z )E (z), E (z) = ( z )E(z) (z) = U(z) + ( z ) E(z) E ECE37-3 z - z - z -
6 U U Simplified Block Diagrams z z z z z - - NTF z ECE37-3 E E NTF( z) = ( z ) STF( z) = z ( ) = ( z ) STF( z) = z NTF ( e jπf ) (db) NTF Comparison MOD MOD MOD has twice as much attenuation at all frequencies 3 Normalized Frequency ECE37-3 For MOD, In-band Noise Power He ( jω ) ω 4 ω B As before, N = He ( jω ) S ee ( ω)dω and S ee ( ω) = σ e π Simulation Example Input at 75% of FullScale Time Domain So now N π 4 σ e = ( OSR) 5 5 With binary quantization to ±, = and thus σ e = = 3. An octave increase in OSR increases MOD s SNR by 5 db (.5 bits) ECE Simulated Noise Density Frequency Domain Predicted Noise Density 8 Agreement is fair 4-point FFT.5.5 ECE37-34 dbfs/nbw Simulated MOD PSD Input at 5% of FullScale SNR = 86 OSR = 8 Simulated spectrum (smoothed) 4 db/decade Theoretical PSD (k = ) NBW = Normalized Frequency ECE37-35 SNR (db) SNR vs. Input Amplitude MOD & OSR = 56 Predicted SNR MOD Simulated SNR Input Amplitude (dbfs) MOD ECE37-36
7 SNR vs. OSR Audio Demo: MOD vs. MOD SNR (db) MOD (Theoretical curve assumes -3 dbfs input) MOD (Theoretical curve assumes dbfs input) Sine Wave Slow Ramp Speech MOD MOD Predictions for MOD are optimistic. Behavior of MOD is erratic ECE37-37 ECE37-38 MOD + MOD 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 and ADC; image filter in a DAC Binary quantization yields inherent linearity CT loop filter provides inherent anti-aliasing MOD is better than MOD 5 db/octave vs. 9 db/octave SNR-OSR trade-off. uantization noise more white. Higher-order modulators are even better. ECE : 3: NLCOTD ECE37-4 Homework # Create a Matlab function that computes MOD s output sequence given a vector of input samples and exercise your function in the following ways. erify that the average of the output equals the input for DC inputs in [,]. Produce a spectral plot like that on Slide a) Construct a SNR vs. input amplitude curve for OSR = 8 for amplitudes from to dbfs. b) Determine approximately how much the interstage gain and feedback coefficients need to shift in order to have a significant (~3-dB) impact. 4 Compare the in-band quantization noise of your system with a half-scale sine-wave input against the relation given on Slide 33 for OSR in [ 3, ]. ECE37-4 un ( ) U x ( n) MOD Expanded z - z z z x ( n + ) x ( n + ) ECE37-4 E x z - ( n) vn ( ) Difference Equations: v( n) = x ( ( n) ) x ( n + ) = x ( n) vn ( ) + un ( ) x ( n + ) = x ( n) vn ( ) + x ( n + )
8 Example Matlab Code function [v,x] = simulatemod(u) x = ; x = ; for i = :length(u) v(i) = quantize( x ); x = x + u(i) - v(i); x = x + x - v(i); end return function v = quantize( y ) if y>= v = ; else v = -; end return ECE37-43 ~ 4 -Point Simulations ECE37-44 u v (v u) x Example Spectrum Nfft = ^; ftest = ; t = :Nfft-; u =.5*sin(*pi*ftest/Nfft*t); % Has ftest cycles in Nfft points v = simulatemod(u); U = fft(u); = fft(v); f = linspace(,,nfft+); f=f(:nfft); semilogx(f,dbv(u),'m', f,dbv(),'b'); figuremagic([e-4.5],[],[],[-8 8],,); 8 6 Peak at +5dB? Normalized Frequency ECE37-45 db (?) FFT Considerations (Partial) The FFT implemented in MATLAB is X M k + N ( ) = x M ( n + )e j n = If xn ( ) = Asin( πfn N), then Xk ( ) = πkn N AN , k = f or N f, otherwise Need to divide FFT by ( N ) to get A.. f is an integer in (, N ). I ve defined Xk ( ) X M ( k + ), xn ( ) x M ( n + ) since Matlab indexes from rather than ECE37-46 x The Need For Smoothing The FFT can be interpreted as taking sample from the outputs of N complex FIR filters: h ( n) h ( n) h k ( n) y ( N) = X ( ) y ( N) = X( ) h k ( n) e j n πk = N, n < N, otherwise y k ( N) = X( k) How To Do Smoothing Average multiple FFTs Implemented by MATLAB s psd() function Take one big FFT and filter the spectrum Implemented by the Σ Toolbox s logsmooth() function logsmooth() averages an exponentiallyincreasing number of bins in order to reduce the density of points in the high-frequency regime and make a nice log-frequency plot h N ( n) y N ( N) = X( N ) an FFT yields a high-variance spectral estimate ECE37-47 ECE37-48
9 dbfs Smoothed Spectrum 4 db / decade Normalized Frequency ECE37-49 uantization Noise Spectrum? Assume that the quantization error e is uniformly distributed in [,+] e = ( y) y ρ e σ e = ρ e ( e)e de.5 y e = e3 3 Assume e is white -sided PSD: S ee () f Multiply S ee () f by NTF( e j πf ) to get the PSD of the shaped error ECE f σ e.5 = S ee () f df S ee () f = σ e = -- 3 dbfs Simulation vs. Theory Simulated Spectrum Theoretical. Noise? Slight Discrepancy (~4 db) Normalized Frequency ECE37-5 What Went Wrong? We normalized the spectrum so that a full-scale sine wave (which has a power of.5) comes out at db (whence the dbfs units) We need to do the same for the error signal. i.e. use S ee () f = 4 3. But this makes the discrepancy 3 db worse. We tried to plot a power spectral density together with something that we want to interpret as a power spectrum Sine-wave components are located in individual FFT bins, but broadband signals like noise have their power spread over all FFT bins! The noise floor depends on the length of the FFT. ECE37-5 Spectrum of a Sine Wave + Noise ( dbfs ) Ŝ x () f 3 N = 6 N = 8 N = N = dbfs Sine Wave dbfs Noise SNR = db 3 db/ octave Observations The power of the sine wave is given by the height of its spectral peak The power of the noise is spread over all bins The greater the number of bins, the less power there is in any one bin. Doubling N reduces the power per bin by a factor of (i.e. 3 db) But the total integrated noise power does not change Normalized Frequency, f ECE37-53 ECE37-54
10 So How Do We Handle Noise? Recall that an FFT is like a filter bank The longer the FFT, the narrower the bandwidth of each filter and thus the lower the power at each output We need to know the noise bandwidth (NBW) of the filters in order to convert the power in each bin (filter output) to a power density For a filter with frequency response Hf (), NBW Hf () df = Hf ( ) NBW Hf () f ECE37-55 f FFT Noise Bandwidth hn ( ) = exp j πk n N N Hf () = hn ( ) exp( j πfn) f Hf () NBW n = N k = ----, Hf ( N ) = = N = hn ( ) = n = N [Parseval] Hf () df N = = = Hf ( ) N ECE N dbfs/nbw Better Spectral Plot passband for OSR = 8 Simulated Spectrum Theoretical. Noise NBW = / N =.5 5 N = Normalized Frequency 4 -- NTF() f 3 NBW ECE37-57 SNR Calculation S = power in the signal bin N = sum of the powers in the non-signal inband noise bins Using MATLAB to perform these calculations for the preceding simulation yields SNR = 84. db at OSR = 8 Can also eyeball SNR from the plot: S = 6 db N = 3 + dbp (BW/NBW) = 89 db SNR = 83 db. dbp(x ) = log (x ). ECE37-58 SNR (db) SNR vs. Amplitude 5A ( OSR) π 4 Simulation Theory Signal Amplitude (dbfs) ECE37-59 Tolerable Coefficient Errors? un ( ) z z z - c a a z a & a are the feedback coefficients; nominally c is the interstage coefficient; nominally You should find that the SNR stays high even if these coefficients individually vary over a : range ECE37-6 z - vn ( )
11 SNR (db) SNR vs. OSR for MOD Half-Scale Input (A =.5) 5A ( OSR) π 4 4 Theory Simulation OSR ECE37-6 Windowing Σ data is usually not periodic Just because the input repeats does not mean that the output does too! A finite-length data record = an infinite record multiplied by a rectangular window: wn ( ) =, n < N Windowing is unavoidable. Multiplication in time is convolution in frequency db Frequency response of a 3-point rectangular window: Slow roll-off out-of-band. noise may appear in-band ECE37-6 db 4 4 Example Spectral Disaster Rectangular window, N = 56 Out-of-band quantization noise obscures the notch! Normalized Frequency, f Actual Σ spectrum W( f) w Windowed spectrum ECE37-63 (db) Wf () W ( ) Window Comparison (N = 6) Rectangular Hann Hann Normalized Frequency, f ECE37-64 Window Properties Window Rectangular Hann wn ( ), n =,,, N ( wn ( ) = otherwise) Number of non-zero FFT bins w = wn ( ) W ( ) = wn ( ) NBW = w W ( ) πn cos N Hann πn cos N N 3N/8 35N/8 N N/ 3N/8 /N.5/N 35/8N. MATLAB s hann function causes spectral leakage of tones located in FFT bins unless you add the optional argument periodic. ECE37-65 Window Length, N Need to have enough in-band noise bins to Make the number of signal bins a small fraction of the total number of in-band bins <% signal bins >5 in-band bins > 3 OS Make the SNR repeatable N = 3 OSR yields std. dev. ~.4 db. N = 64 OSR yields std. dev. ~. db. N = 56 OSR yields std. dev. ~.5 db. N = 64 OSR is recommended ECE37-66
12 Good FFT Practice [Appendix A of Schreier & Temes] Use coherent sampling Need an integer number of cycles in the record. Use windowing A Hann window works well. Use enough points N = 64 OSR Scale the spectrum A full-scale sine wave should yield a -dbfs peak. State the noise bandwidth For a Hann window, NBW =.5 N. Smooth the spectrum if you want a pretty plot ECE37-67 What You Learned Today And what the homework should solidify MOD and MOD structure and linear theory SNR-OSR trade-offs: 9 db/octave for MOD 5 db/octave for MOD Inherent linearity of binary modulators 3 Inherent anti-aliasing of continuous-time modulators 4 Proper use of FFTs for spectral analysis 5 (Hwk) MOD and MOD are tolerant of large coefficient errors ECE37-68
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