Higher-Order Modulators: MOD2 and MODN

Transcription

1 ECE37 Advanced Analog Circuits Lecture 2 Higher-Order Modulators: MOD2 and MODN Richard Schreier richard.schreier@analog.com Trevor Caldwell trevor.caldwell@utoronto.ca Course Goals Deepen understanding of CMOS analog circuit design through a top-down study of a modern analog system a delta-sigma ADC 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 know. ECE37 2-2

2 Date Lecture Ref Homework RS Introduction: MOD ST 2, A : MOD in Matlab RS 2 MOD2 & MODN ST 3, 4, B 2: MOD2 in Matlab RS 3 Example Design: Part ST 9., CCJM 4 3: Sw.-level MOD TC 4 SC Circuits R 2, CCJM 4 4: SC Integrator TC 5 Amplifier Design TC 6 Amplifier Design 5: SC Int w/ Amp Reading Week + ISSCC No Lecture RS 7 Example Design: Part 2 CCJM 8 Start Project RS 8 Comparator & Flash ADC CCJM TC 9 Noise in SC Circuits ST C TC Matching & MM-Shaping ST , RS Advanced Σ ST 6.6, TC 2 Pipeline and SAR ADCs CCJM 5, No Lecture Project Presentation ECE NLCOTD: Dynamic Flip-Flop Standard CMOS version D CK Q Q Can the circuit be simplified? Is a complementarty clock necessary? ECE37 2-4

3 Highlights (i.e. What you will learn today) Second-order modulator (MOD2) 2 Arbitrary-order modulator (MODN) design with the Σ Toolbox ECE desired signal f B shaped noise f B f B Review: A Σ ADC System f s 2 f s 2 Nyquist-rate PCM Data U Σ Modulator V Digital Decimator W U Loop Filter DAC V (z )= STF (z )U( z )+ NTF (z )E (z ) STF (z ): signal transfer function NTF (z ): noise transfer function E (z ): quantization error ECE Y E V

4 U Review: MOD Q E Vz ( ) = NTF( z)ez ( ) + STF( z)uz ( ) NTF( z) = ( z ) z - STF( z) = z - NTF poles & zeros: 4 3 NTF e j 2πf ( ) 2 2 ω 2 IQNP = Normalized Frequency (f ) π 2 σ e ( OSR) 3 3 ECE Review 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 MOD has NTF (z) = z Arbitrary accuracy for DC inputs. 9 db/octave SQNR-OSR trade-off. MOD-CT has inherent anti-aliasing ECE37 2-8

5 MOD2: 2 nd -Order Σ Modulator [Ch. 3 of Schreier & Temes] Replace the quantizer in MOD with another copy of MOD in a recursive fashion: U E Q E V z - z - z - z - V(z) = U(z) + ( z )E (z), E (z) = ( z )E(z) V(z) = U(z) + ( z ) 2 E(z) ECE U Simplified Block Diagrams z z z Q E V NTF( z) = ( z ) 2 STF( z) = z E U z z Q V - -2 NTF z ( ) = ( z ) 2 STF( z) = z 2 ECE37 2-

6 NTF Comparison NTF ( e j2πf ) (db) MOD MOD2 MOD2 has twice as much attenuation as MOD at all frequencies 3 2 Normalized Frequency ECE37 2- In-band Quant. Noise Power For MOD2, As before, IQNP = H( e jω ) 2 S ee ( ω)dω and S ee ( ω) = σ 2 e π So now IQNP He ( jω ) 2 ω 4 = ω B π 4 σ e ( OSR) 5 5 With binary quantization to ±, = 2 and thus σ2 e = 2 2 = 3. An octave increase in OSR increases MOD2 s SQNR by 5 db (2.5 bits) ECE37 2-2

7 Simulation Example Input at 75% of FullScale Sample number ECE dbfs/nbw Simulated MOD2 PSD Input at 5% of FullScale SQNR = 86 OSR = 28 Simulated spectrum (smoothed) 4 db/decade Theoretical PSD (k = ) NBW = Normalized Frequency ECE37 2-4

8 2 SQNR vs. Input Amplitude MOD & OSR = 256 SQNR (db) Predicted SQNR MOD2 Simulated SQNR MOD Input Amplitude (dbfs) ECE SQNR vs. OSR SQNR (db) MOD2 (Theoretical curve assumes -3 dbfs input) MOD (Theoretical curve assumes dbfs input) 2 Predictions for MOD2 are optimistic. Behavior of MOD is erratic ECE37 2-6

9 Audio Demo: MOD vs. MOD2 [dsdemo4] Sine Wave Slow Ramp Speech MOD MOD2 ECE MOD + 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 MOD 5 db/octave vs. 9 db/octave SQNR-OSR trade-off. Quantization noise more white. Higher-order modulators are even better. ECE37 2-8

10 MODN [Ch. 4 of Schreier & Temes] N integrators U z z (N ) non-delaying, delaying z z z Q V STF( z) = z NTF( z) = ( z ) N MODN s NTF is the N th power of MOD s NTF ECE NTF Comparison 2 NTF ( e j2πf ) (db) MOD MOD2 MOD3 MOD4 MOD Normalized Frequency (f ) ECE37 2-2

11 Predicted Performance In-band quantization noise power IQNP = =.5 OSR.5 OSR NTF( e j 2πf ) 2 S ee () f ( 2πf ) 2N 2σ2 e df π 2N σ2 ( 2N + ) ( OSR) 2N + e Quantization noise drops as the (2N+) th power of OSR (6N+3) db/octave SQNR-OSR trade-off ECE df Improving NTF Performance NTF Zero Optimization Minimize the integral of NTF 2 over the passband Normalize passband edge to for ease of calculation: Hf a a Need to find the a i which minimize () 2 the integral f s /f B x 2 a 2 ( ) 2 dx, n = 2 x 2 x 2 a 2 ( ) 2 dx, n = 3 ( x 2 a 2 ) 2 x 2 a 2 ( 2 ) 2 dx, n = 4 ECE

12 Order Solutions Up to Order = 8 Optimal Zero Placement Relative to f B SQNR Improvement db 2 ± db 3, ± db 4 ± 3 7 ± ( 3 7) db 5, ± 5 9 ± ( 5 9) 5 2 [Y. Yang] 8 db 6 ±.23862, ±.662, ± db 7, ±.4585, ±.7453, ± db 8 ±.8343, ±.52553, ±.79667, ± db ECE Topological Implication Feedback around pairs integrators: 2 Delaying Integrators z -g z Poles are the roots of + g = z i.e. z = ± j g Not quite on the unit circle, but fairly close if g<<. Non-delaying + Delaying Integrators (LDI Loop) z z -g z Poles are the roots of gz ( z ) 2 = i.e. z = e ± jθ, cosθ = g 2 Precisely on the unit circle, regardless of the value of g. ECE

13 Problem: A High-Order Modulator Wants a Multi-bit Quantizer E.g. MOD3 with an Infinite Quantizer and Zero Input v Sample Number Quantizer input gets large, even if the input is small. 6 quantizer levels are used by a small input. ECE v Simulation of MOD3-b (MOD3 with a Binary Quantizer) Long strings of +/ y Sample Number HUGE! MOD3-b is unstable, even with zero input! ECE

14 Solutions to the Stability Problem Historical Order Multi-bit quantization Initially considered undesirable because we lose the inherent linearity of a -bit DAC. 2 More general NTF (not pure differentiation) Lower the NTF gain so that quantization error is amplified less. Unfortunately, reducing the NTF gain reduces the amount by which quantization noise is attenuated. 3 Multi-stage (MASH) architecture Combinations of the above are possible ECE Multi-bit Quantization A modulator with NTF = H and STF = is guaranteed to be stable if u < u max at all times, where u max = nlev + h and h = hi () i = In MODN Hz ( ) = ( z ) N, so hn ( ) = {, a, a 2, a 3, ( ) N a N, }, a i > and thus h = H( ) = 2 N nlev = 2 N implies u max = nlev + h = MODN is guaranteed to be stable with an N-bit quantizer if the input magnitude is less than /2=. This result is quite conservative. Similarly, nlev = 2 N + guarantees that MODN is stable for inputs up to 5% of full-scale ECE

15 M-Step Symmetric Quantizer = 2, (nlev = M + ) M odd: mid-rise M even: mid-tread e = v y M v e = v y y M M + M M + M 2 v y M v: odd integers from M to +M M v: even integers from M to +M No-overload range: y nlev e 2 = ECE Inductive Proof of Criterion h Assume STF = and ( n) ( un ( ) u max ) Assume ei () for i < n.[induction Hypothesis] i = i = i = yn ( ) = un ( ) + hi ()en ( i) u max u max + hi () en ( i) + hi () = u max + h Then u max = nlev + h yn ( ) nlev en ( ) So by induction ei () for all i > ECE37 2-3

16 More General NTF Instead of NTF( z) = Az ( ) Bz ( ) with Bz ( ) = z n, use a more general Bz ( ) Roots of B are the poles of the NTF and must be inside the unit circle. Moving the poles away from z = toward z = makes the gain of the NTF approach unity. ECE The Lee Criterion for Stability in a -bit Modulator: H 2 [Wai Lee, 987] The measure of the gain of H is the maximum magnitude of H over frequency, aka the infinitynorm of H: H max ( He ( jω )) ω [ 2π, ] Q: Is the Lee criterion necessary for stability? No. MOD2 is stable (for DC inputs less than FS) but H = 4. Q: Is the Lee criterion sufficient to ensure stability? No. There are lots of counter-examples, but H.5 often works. ECE

17 Simulated SQNR vs. H 5 th -order NTFs; -b Quant.; OSR = 32 SQNR (db) 9 7 SQNR has a broad maximum cliff! H umax (dbfs) Stable input limit drops as increases. H H ECE SQNR Limits -bit Modulation Peak SQNR (db) N = 8 N 7 N = 6 N = 5 N = 4 N = 3 N = 2 N = OSR ECE

18 SQNR Limits for 2-bit Modulators 4 N = 8 N = 7 N = 6 N = 5 N = 4 N = 3 N = 2 Peak SQNR (db) N = OSR ECE SQNR Limits for 3-bit Modulators 4 N = 8 N = 7 N = 6 N = 5 N = 4 N = 3 N = 2 Peak SQNR (db) N = OSR ECE

19 Generic Single-Loop Σ ADC Linear Loop Filter + Nonlinear Quantizer: E U L Y V L Y = L U + L V V = Y + E V = STF U + NTF E, where NTF = & STF = L L NTF Inverse Relations: L = /NTF, L = STF / NTF ECE Σ Toolbox Search for Delta Sigma Toolbox Specify OSR, lowpass/bandpass, no. of Q. levels. NTF (and STF) available. synthesizentf calculatetf Parameters for a specific topology. stuffabcd mapabcd scaleabcd ABCD: statespace description of the modulator. Time-domain simulation and SNR measurements. realize- NTF predictsnr, simulatedsm, simulatesnr Also mapctod simulateesl designhbf Manual is delsig.pdf (App. B of Schreier & Temes) ECE

20 Σ Toolbox Modulator Model Modulator: U [-M, +M ] Quantizer: L L Y M = M = 2 M = 3 3 v 2 v v y y y e 2 e e -3 ECE V [-M, +M ] = 2 Loop filter can be specified by NTF or by ABCD, a state-space representation Mid-tread quantizer; v: even integers [ M,+M ] NTF STF = = L L L Mid-rise quantizer; v: odd integers [ M,+M ] NTF Synthesis synthesizentf Not all NTFs are realizable Causality requires h( ) =, or, in the frequency domain, H( ) =. Recall Hz ( ) = h( )z + h( )z + Not all NTFs yield stable modulators Rule of thumb for single-bit modulators: H <.5 [Lee]. Can optimize NTF zeros to minimize the mean-square value of H in the passband The NTF and STF share poles, and in some modulator topologies the STF zeros are not arbitrary Restrict the NTF such that an all-pole STF is maximally flat. (Almost the same as Butterworth poles.) ECE37 2-4

21 Lowpass Example [dsdemo] 5 th -order NTF, all zeros at DC Pole/Zero diagram: OSR = 32; H = synthesizentf(5); plotpz(h); f = linspace(,.5); z = exp(2i*pi*f); H_z = evaltf(h,z); plot(f,dbv(h_z)); g = rmsgain(h,,.5/osr) ECE Lowpass NTF Out-of-band gain =.5 db 2 4 rms in-band gain = 47 db / Normalized Frequency (f /f s ) ECE

22 Improved 5 th -Order Lowpass NTF Zeros optimized for OSR=32 optimization flag OSR = 32; H = synthesizentf(5,osr,);... Zeros spread across the band-of-interest to minimize the rms value of the NTF. ECE Improved NTF db 2 6 rms gain = 66 db / Normalized Frequency (f /f s ) ECE

23 Bandpass Example OSR = 64; center frequency f = /6; H=synthesizeNTF(6,OSR,,[],f);... [] or NaN means use default value, i.e. Hinf =.5 ECE Bandpass NTF and STF db NTF all-pole STF with same poles as Σ toolbox NTF Normalized Frequency (f /f s ) ECE

24 Summary: NTF Selection If OSR is high, a single-bit modulator may work To improve SQNR, Optimize zeros, H Increase, or Increase order. If SQNR is insufficient, must use a multi-bit design Can turn all the above knobs to enhance performance. Feedback DAC assumed to be ideal ECE NTF-Based Simulation [dsdemo2] order=5; OSR=32; ntf = synthesizentf(order,osr,); N=2^7; fbin=959; A=.5; % 28K points input = A*sin(2*pi*fbin/N*[:N-]); output = simulatedsm(input,ntf); spec = fft(output.*ds_hann(n)/(n/4)); plot(dbv(spec(:n/(2*osr)))); In mex form; 28K points in <. sec dbfs/nbw 5 5 NBW=.5 bins FFT Bin Number (28K-point FFT) ECE

25 Simulation Example Cont d u dbfs/nbw v Time (sample number) 8 NBW =.8x 4 f s (8k-point FFT) 2.5 Normalized Frequency (f /f s ) ECE SNR vs. Amplitude: simulatesnr [snr amp] = simulatesnr(ntf,osr); plot(amp,snr,'b-^'); 8 OSR=32 Peak SNR is 85 db. Max. input is 3 dbfs. SQNR (db) Input Amplitude (dbfs) ECE37 2-5

26 Homework #2 Part A Create a Matlab function that computes MOD2 s output sequence given a vector of input samples and exercise your function in the following ways: Plot the error v u as a function of u using - point rectangular and triangular * averages for v. 2 Produce a spectral plot like that on Slide 4. 3 Compare the in-band quantization noise of your system with a half-scale sine-wave input against the relation given on Slide 2 for OSR in [2 3,2 ]. N *. v tri = tri( n)vn ( ), tri = rect ( N 2)*rect ( N 2) ECE37 n = 2-5 Homework #2 Part B Extract code from dsdemo & dsdemo2 and modify it to: Create a 3 rd -order NTF with zeros optimized for OSR = 32. Plot the poles/zeros and frequency response of your NTF. 2 Simulate a 3-level Σ modulator with this NTF. Plot example input and output waveforms. Plot a spectrum and the predicted noise curve. Plot the SQNR vs. input amplitude curve and note the maximum stable input.. Beware that with an M-step modulator, the full-scale is M and this must be accounted for in normalizing the FFT and in computing the expected noise curve. You ought to check your code using a large value for M, say M = 2. ECE

27 U MOD2 Expanded z z z Q E V un ( ) x ( n + ) x 2 ( n + ) x ( n) z - z - x 2 ( n) vn ( ) Q Difference Equations: v( n) = Qx ( 2 ( n) ) x ( n + ) = x ( n) vn ( ) + un ( ) x 2 ( n + ) = x 2 ( n) vn ( ) + x ( n + ) ECE Example Matlab Code function [v] = simulatemod2(u) x = ; x2 = ; for i = :length(u) v(i) = quantize( x2 ); x = x + u(i) - v(i); x2 = x2 + x - v(i); end return function v = quantize( y ) if y>= v = ; else v = -; end return ECE

28 What You Learned Today And what the homework should solidify Second-order modulator (MOD2) 2 Arbitrary-order modulator (MODN) design with the Σ Toolbox ECE NLCOTD: True Single-Phase Dynamic FF D C Q + Clock not inverted anywhere + Small + Fast ECE

29 TSPFF Operation D C C Y Z Q X C Can drop inverter. (Handy if making a divider.) D C X Y Z Q d ~d d ~d d ECE TSPFF Gotchas Leakage: Won t work if clock is too slow. Possible high current if clock is stopped. Need to add devices that hold the dynamic nodes at a safe value. No positive feedback Vulnerable to metastability. ECE