EE247 Lecture 25. Oversampled ADCs (continued)

EE247 Lecture 25 Oversampled ADCs (continued) Higher order ΣΔ modulators Last lecture Cascaded ΣΔ modulators (MASH) (continued) Single-loop single-quantizer modulators with multi-order filtering in the forward path Example: 5 th order Lowpass ΣΔ Modeling Noise shaping Effect of various nonidealities on the ΣΔ performance Bandpass ΣΔ modulators EECS 247- Lecture 25 Oversampled ADCs 29 Page 1 Administrative EE247 Lecture 25 Final exam: Date: Mon. Dec. 14 th Time: 1:3pm-4:3pm (note change of time) Location: 299 Cory (change of location) Closed book/course notes No calculators/cell phones/pdas/computers You can bring two 8x11 paper with your own notes Final exam covers the entire course material unless specified EECS 247- Lecture 25 Oversampled ADCs 29 Page 2

EE247 Lecture 25 Project: Project reports due Dec. 4 th (Dec. 2 nd if you are presenting on Dec. 3 rd ) Please make an appointment with the instructor for a 2 minute meeting per team for Frid. Dec. 4 th (for early presenters Dec. 2 nd ) Prepare to give a 5 to 1 minute presentation regarding the project during the class period on Dec. 8 th (or Dec. 3 rd ) Highlight the important aspects of your approach towards the implementation of the ADC If the project is joint effort, both team members should present Email your PowerPoint presentation files to H.K. two hours prior to class to conserve class time EECS 247- Lecture 25 Oversampled ADCs 29 Page 3 EE247 Lecture 25 Homework for oversampled data converters Due to the time consuming nature of the project, homework covering oversampled converters will not be given. Please review relevant previous year homeworks & solutions e.g. http://wwwinst.eecs.berkeley.edu/~ee247/fa7/files7/homew ork/hw9_2_7.pdf http://wwwinst.eecs.berkeley.edu/~ee247/fa7/files7/homew ork/hw9_sol_lynn_wang.pdf EECS 247- Lecture 25 Oversampled ADCs 29 Page 4

Example: 2 Cascaded ΣΔ Modulators Accuracy of < + 3% 2dB loss in DR Ref: L. A. Williams III and B. A. Wooley, "A third-order sigma-delta modulator with extended dynamic range," IEEE Journal of Solid-State Circuits, vol. 29, pp. 193-22, March 1994. EECS 247- Lecture 25 Oversampled ADCs 29 Page 5 2 Cascaded ΣΔ Modulators Effect of gain parameters on signal-to-noise ratio Ref: L. A. Williams III and B. A. Wooley, "A third-order sigma-delta modulator with extended dynamic range," IEEE Journal of Solid-State Circuits, vol. 29, pp. 193-22, March 1994. EECS 247- Lecture 25 Oversampled ADCs 29 Page 6

2 Cascaded ΣΔ Modulators Measured Dynamic Range Versus Oversampling Ratio Theoretical SQNR 21dB/Octave 3dB/Octave Ref: L. A. Williams III and B. A. Wooley, "A third-order sigma-delta modulator with extended dynamic range," IEEE Journal of Solid-State Circuits, vol. 29, pp. 193-22, March 1994. EECS 247- Lecture 25 Oversampled ADCs 29 Page 7 Comparison of 2 nd order & Cascaded (2) ΣΔ Modulator Test Results Reference Architecture Dynamic Range Peak SNDR Oversampling rate Differential input range Power Dissipation Active Area Digital Audio Application, f N =44.1kHz (Does not include Decimator) Brandt,JSSC 4/91 2 nd order 98dB (16-bits) 94dB 256 (theoretical SQNR=19dB, 18bit) 4Vppd 5V supply 13.8mW.39mm 2 ( 1μ tech.) Williams, JSSC 3/94 (2+1) Order 14dB (17-bits) 98dB 128 (theoretical SQNR=128dB, 21bit!) 8Vppd 5V supply 47.2mW 5.2mm 2 ( 1μ tech.) EECS 247- Lecture 25 Oversampled ADCs 29 Page 8

Higher Order ΣΔ Modulators (1) Cascaded Modulators Summary Cascade two or more stable ΣΔ stages Quantization error of each stage is quantized by the succeeding stage/s and subtracted digitally Order of noise shaping equals sum of the orders of the stages Quantization noise cancellation depends on the precision of analog/digital signal paths Quantization noise further randomized less limit cycle oscillation problems Typically, no potential instability EECS 247- Lecture 25 Oversampled ADCs 29 Page 9 Higher Order Lowpass ΣΔ Modulators Forward Path Multi-Order Filter E(z) X(z) Σ N() z H() z = Σ D() z Y(z) H( z) 1 Y( z) = X( z) + E( z) 1 + H( z) 1 + H( z) Y( z) 1 D( z) NTF = = = E(z) 1 + H(z) D(z) + N(z) Zeros of NTF (poles of H(z)) can be positioned to minimize baseband noise spectrum Main issue Ensuring stability for 3 rd and higher orders EECS 247- Lecture 25 Oversampled ADCs 29 Page 1

Overview Building behavioral models in stages A 5 th -order, 1-Bit ΣΔ modulator Noise shaping Complex loop filters Stability Voltage scaling Effect of component non-idealities EECS 247- Lecture 25 Oversampled ADCs 29 Page 11 Building Models in Stages When modeling a complex system like a 5 th -order ΣΔ modulator, model development proceeds in stages Each stage builds on its predecessor Design goal detect and eliminate problems at the highest possible level of abstraction Each successive stage consumes progressively more engineering time Our ΣΔ model development proceeds in stages: Stage gets to the starting line: Collect references, talk to veterans Stage 1 develops a practical system built with ideal sub-circuits & simulation Stage 2 models key sub-circuit non-idealities and translates the results into real-world sub-circuit performance specifications Real-world model development includes a critical stage 3: Adding elements to earlier stages to model significant surprises found in silicon EECS 247- Lecture 25 Oversampled ADCs 29 Page 12

Procedure ΣΔ Modulator Design Establish requirements Design noise-transfer function, NTF Determine loop-filter, H Synthesize filter Evaluate performance, Establish stability criteria Ref: R. W. Adams and R. Schreier, Stability Theory for ΔΣ Modulators, in Delta-Sigma Data Converters- S. Norsworthy et al. (eds), IEEE Press, 1997 EECS 247- Lecture 25 Oversampled ADCs 29 Page 13 Example: Modulator Specification Example: Audio ADC Dynamic range DR 18 Bits Signal bandwidth B 2 khz Nyquist frequency f N 44.1 khz Modulator order L 5 Oversampling ratio M = f s /f N 64 Sampling frequency f s 2.822 MHz The order L and oversampling ratio M are chosen based on SQNR > 12dB EECS 247- Lecture 25 Oversampled ADCs 29 Page 14

Noise Transfer Function, NTF(z) % stop-band attenuation Rstop=8dB, L=5... L=5; Rstop = 8; B=2; [b,a] = cheby2(l, Rstop, B, 'high'); % normalize b = b/b(1); NTF = filt(b, a,...); Chebychev II filter chosen zeros in stop-band NTF [db] 2-2 -4-6 -8 1 4 1 6 Frequency [Hz] EECS 247- Lecture 25 Oversampled ADCs 29 Page 15 Loop-Filter Characteristics H(z) Y( z) 1 NTF = = Ez ( ) 1 + Hz ( ) 1 H( z) = 1 NTF Note: For 1 st order ΣΔ an integrator is used instead of the high order filter shown Loopfilter H [db] 1 8 6 4 2-2 1 4 1 6 Frequency [Hz] EECS 247- Lecture 25 Oversampled ADCs 29 Page 16

Filter Modulator Topology Simulation Model b1 b2 X I1 K1 z I2 K2 z I3 K3 z I4 I5 K4 z K5 z a1 I_1 a2 I_2 a3 I_3 a4 I_4 a5 I_5 DAC Gain g Comparator Q Y +1 EECS 247- Lecture 25 Oversampled ADCs 29 Page 17 Filter Coefficients a1=1; a2=1/2; a3=1/4; a4=1/8; a5=1/8; k1=1; k2=1; k3=1/2; k4=1/4; k5=1/8; b1=1/124; b2=1/16/64; g =1; Ref: Nav Sooch, Don Kerth, Eric Swanson, and Tetsuro Sugimoto, Phase Equalization System for a Digital-to-Analog Converter Using Separate Digital and Analog Sections, U.S. Patent 561925, 199, figure 3 and table 1 EECS 247- Lecture 25 Oversampled ADCs 29 Page 18

Output Spectrum [dbwn] / Int. Noise [dbfs] 4 2-2 -4-6 -8 Signal 5 th Order Noise Shaping AFE Simulation Results Notice tones around f s /2 2 4 Output Spectrum 6 Integrated Noise (2 averages).1.2.3.4.5 Frequency [ f / f s ] Mostly quantization noise, except at low frequencies Let s zoom into the baseband portion EECS 247- Lecture 25 Oversampled ADCs 29 Page 19 5 th Order Noise Shaping Output Spectrum [dbwn] / Int. Noise [dbfs] 4 2-2 -4-6 -8 Output Spectrum Integrated Noise (2 averaged) 2 Quantization noise 3dBFS 4 Signal @ band edge! 6 Band-Edge.2.4.6.8 1 Frequency [ f / f N ] f N =44.1kHz SQNR > 12dB Sigma-delta modulators are usually designed for negligible quantization noise Other error sources dominate, e.g. thermal noise are allowed to dominate & thus provide dithering to eliminate limit cycle oscillations EECS 247- Lecture 25 Oversampled ADCs 29 Page 2

In-Band Noise Shaping Magnitude [db] Output Phase Spectrum [degrees] 14 12 1 8 6 Loop Filter 4.2.4.6.8 1 4 4 3 2 1-4 -8 H(z) maxima align up with noise minima Output Spectrum Integrated Noise (2 averages) 2.2.4.6.8 1 Frequency [f/fn] 6.2.4.6 Frequency [f/f N ].8 1 Lot s of gain in the loop filter pass-band Forward path filter not necessarily stable! Remember that: NTF ~ 1/H small within passband since H is large STF=H/(1+H) ~1 within passband EECS 247- Lecture 25 Oversampled ADCs 29 Page 21 Stability Analysis e(kt) x(kt) Σ H(z) q(kt) G eff Σ y(kt) Quantizer Model Approach: linearize quantizer and use linear system theory! One way of performing stability analysis use RLocus in Matlab with H(z) as argument and Geff as variable Effective quantizer gain 2 G 2 = y eff q 2 Can obtain G eff from simulation Ref: R. W. Adams and R. Schreier, Stability Theory for ΔΣ Modulators, in Delta-Sigma Data Converters- S. Norsworthy et al. (eds), IEEE Press, 1997 EECS 247- Lecture 25 Oversampled ADCs 29 Page 22

Quantizer Gain (G eff ) ε Vin G eff Σ Vout Vout Quantizer Model G eff (small signal) dvout/dvin G eff (large signal) Vout/Vin 1 Vin Vin +1 Vin EECS 247- Lecture 25 Oversampled ADCs 29 Page 23 Stability Analysis G H ( z) STF = 1 + G H ( z) ( ) N ( z) H z = D( z) G N ( z) STF = D( z) + G N ( z) Zeros of STF same as zeros of H(z) Poles of STF vary with G For G=small (no feedback) poles of the STF same as poles of H(z) For G=large, poles of STF move towards zeros of H(z) Draw root-locus: for G values for which poles move to LHP (s-plane) or inside unit circle (z-plane) system is stable EECS 247- Lecture 25 Oversampled ADCs 29 Page 24

Modulator z-plane Root-Locus z-plane Root Locus.4 Increasing G eff.3.2.1 G eff =.45 As G eff increases, poles of STF move from poles of H(z) (G eff = ) to zeros of H(z) (G eff = ) -.1 -.2 -.3 -.4 Unit Circle.6.7.8.9 1 1.1 Note: Final exam does NOT include Root Locus Pole-locations inside unit-circle correspond to stable STF and NTF Need G eff >.45 for stability EECS 247- Lecture 25 Oversampled ADCs 29 Page 25 Effective Quantizer Gain, Geff Effective Quantizer Gain.8.6.4.2 G eff =.45 stable Large inputs comparator input grows Output is fixed (±1) G eff drops modulator unstable for large inputs Solution: Limit input amplitude unstable Detect instability (long sequence of +1 or ) and reset integrators -4-35 -3-25 -2 5-5 5 Input [dbv] Be ware that signals grow slowly for nearly stable systems use long simulations EECS 247- Lecture 25 Oversampled ADCs 29 Page 26

Internal Node Voltages Loop filter peak node voltages [V] 1 5-5 5-2 i1 i2 i3 i4 i5 q Integrator outputs Quantizer input -4-35 -3-25 -2 5-5 Input [dbv] Internal signal peak amplitudes are weak function of input level (except near overload) Maximum peak-to-peak voltage swing approach +V! Exceed supply voltage! Solution: Node scaling based on max. signal handling capability of integrators EECS 247- Lecture 25 Oversampled ADCs 29 Page 27 Node Scaling Example: 3 rd Integrator Output Voltage Scaled by α K3 * α, b1 /α, a3 / α, K4 / α, b2 * α b1 V new =V old * α b2 X K1 z I1 I_1 K2 z I2 K3 z I3 I_2 I_3 K4 z K5 z I4 I5 I_4 I_5 a1 a2 a3 a4 a5 Q DAC Gain g Comparator Y EECS 247- Lecture 25 Oversampled ADCs 29 Page 28

Node Voltage Scaling Loop filter peak voltages [V] 1.5 1.5 -.5.5-4 -35-3 -25-2 5-5 Input [dbv] Integrator output range reasonable for new parameters But: maximum input signal limited to -5dB (-7dB with safety) fix? α=1/1 k1=1/1; k2=1; k3=1/4; k4=1/4; k5=1/8; a1= 1; a2=1/2; a3=1/2; a4=1/4; a5=1/4; b1=1/512; b2=1/16/64; g =1; EECS 247- Lecture 25 Oversampled ADCs 29 Page 29 Input Range Scaling Increasing the DAC levels by using higher value for g reduces the analog to digital conversion gain: DOUT ( z) H ( z) 1 = V ( z) 1+ gh ( z) g IN V IN Σ Loop Filter H(z) Comparator D OUT +1 or g Increasing V IN & g by the same factor leaves 1-Bit data unchanged EECS 247- Lecture 25 Oversampled ADCs 29 Page 3

Scaled Stage 1 Model Loop filter peak voltages [V] 1.5 1.5 -.5 g modified: From 1 to 2.5; Overload input level shifted up by 8dB.5-4 -35-3 -25-2 5-5 Input [dbv] +2dB EECS 247- Lecture 25 Oversampled ADCs 29 Page 31 Scaled Stage 1 Model Loop filter peak voltages [V] 1.5 1.5 -.5 g = 2.5;.5-4 -35-3 -25-2 5-5 Input [dbv] EECS 247- Lecture 25 Oversampled ADCs 29 Page 32

Stability Verification Post Scaling Effective Quantizer Gain 8 7 6 5 4 3 2 G eff =4.5 stable unstable Note: Operating the AFE at signals <dbv ensures system stability 1-4 -35-3 -25-2 5-5 5 Input [dbv] EECS 247- Lecture 25 Oversampled ADCs 29 Page 33 5 th Order Modulator Final Parameter Values 1/512 1/16/64 b1 b2 X Input range ~ ±1V 1/1 1 1/4 1/4 1/8 K1 z K2 z K3 z K4 z K5 z I1 I2 I3 I4 I5 I_1 I_2 I_3 I_4 I_5 a11 1 a2 12 a3 1/2 a4 1/4 a5 1/4 Q ±2.5V DAC Gain g Comparator Y Stable input range with margin ~ ±1V EECS 247- Lecture 25 Oversampled ADCs 29 Page 34

Summary Stage 1 model verified stable and meets SQNR specification Stage 2 issues in 5 th order ΣΔ modulator DC inputs Spurious tones Dither kt/c noise EECS 247- Lecture 25 Oversampled ADCs 29 Page 35 Output Spectrum [dbwn] / Int. Noise [dbv] 5 Tones in the vicinity of f s /2 exceed input level -5 5 th Order Noise Shaping Output Spectrum Integrated Noise (3 averages) 5.5 1 1.5 Frequency [MHz] Input:.1V, sinusoid 2 15 point DFT 3 averages Note: Large spurious tones in the vicinity of f s /2 Let us check whether tones appear inband? EECS 247- Lecture 25 Oversampled ADCs 29 Page 36

Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 In-Band Noise Output Spectrum Integrated Noise (3 averages) Note: No in-band tones! While Large spurious tones appear in the vicinity of f s /2 In-Band quantization noise: 12dB! 5 1 2 3 4 5 Frequency [khz] EECS 247- Lecture 25 Oversampled ADCs 29 Page 37 Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 5 5 th Order Noise Shaping 15dB stopband attenuation needed to attenuate unwanted f s /2 components down to the in-band quantization noise level Output Spectrum Integrated Noise (3 averages).5 1 1.5 Frequency [MHz] Input:.1V, sinusoid 2 15 point DFT 3 averages Note: Digital filter required attenuation function of tones in the vicinity of fs/2 & in-band quantization noise EECS 247- Lecture 25 Oversampled ADCs 29 Page 38

Out-of-Band vs In-Band Signals A digital (low-pass) filter with suitable coefficient precision can eliminate out-of-band quantization noise No filter can attenuate unwanted in-band components without attenuating the signal We ll spend some time making sure the components at f s /2-f in will not mix down to the signal band But first, let s look at the modulator response to small DC inputs (or offset) EECS 247- Lecture 25 Oversampled ADCs 29 Page 39 Output Spectrum [dbwn] / Int. Noise [dbv] ΣΔ Tones Generated by Small DC Input Signals 5-5 6kHz 12kHz 5 1 2 3 4 5 Frequency [khz] 5mV DC input (V DAC 2.5V) Simulation technique: A random 1 st sample randomizes the noise from DC input and enables averaging. Otherwise the small tones will not become visible. EECS 247- Lecture 25 Oversampled ADCs 29 Page 4

Limit Cycles Representing a DC term with a 1/+1 pattern e.g. 1 123 1 + 1 123 1 + 1 123 1 + 1 123 1 + 1 123 1 + 1 + 1 11 1 2 3 4 5 144444444 244444444 3 1444444444 2444444444 3 1 11 Spectrum: fs 11 fs 2 11 fs 3 11 K EECS 247- Lecture 25 Oversampled ADCs 29 Page 41 Limit Cycles The frequency of the tones are indeed quite predictable Fundamental Tone velocity (useful for debugging) DC fδ = fs V DAC 5mV Note: For digital audio in this case DC signal>2mv generates tone with f δ >24kHz out-of-band no problem V = 3M Hz 2.5V = 6kHz df dv df dv δ DC δ D C f = V s DAC = 1.2kHz/mV EECS 247- Lecture 25 Oversampled ADCs 29 Page 42

ΣΔ Spurious Tones Effect of Small DC Input @ Vicinity of f s /2 Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 6kHz Output Spectrum Integrated Noise (3 averages) 5 1.47 1.48 1.49 1.5 Frequency [MHz] EECS 247- Lecture 25 Oversampled ADCs 29 Page 43 ΣΔ Spurious Tones In-band spurious tones look like signal Can be a major problem in some applications E.g. audio even tones with power below the quantization noise floor can be audible Spurious tones near f s /2 can be aliased down into the signal band of interest Since they are often strong, even a small amount of aliasing can create a major problem We will look at mechanisms that alias tones later First let s look at dither as a means to reduce or eliminate in-band spurious tones EECS 247- Lecture 25 Oversampled ADCs 29 Page 44

Dither DC inputs can be represented by many possible bit patterns Including some that are random (non-periodic) but still average to the desired DC input The spectrum of such a sequence has no spurious tones How can we get a ΣΔ modulator to produce such randomized sequences? EECS 247- Lecture 25 Oversampled ADCs 29 Page 45 Dither The target DR for our audio ΣΔ is 18 Bits, or 113dB Designed SQNR~2dB allows thermal noise to dominate at 15dB level Let s choose the sampling capacitor such that it limits the dynamic range: 2 ( V ) 1 2 FS 2 n DR = VFS = 1Vp v 2 1 n 2DR ( V ) v = = 1μV FS EECS 247- Lecture 25 Oversampled ADCs 29 Page 46

Effect of Dither on In-Band Spurious Tones Output Spectrum [dbwn] 5-5 No dither With dither (thermal noise) 5mV DC input Thermal noise added at the input of the 1 st integrator In-band spurious tones disappear Note: they are not just buried 5 1 2 3 4 5 Frequency [khz] How can we tell? EECS 247- Lecture 25 Oversampled ADCs 29 Page 47 Effect of Dither on Spurious Tones Near f s /2 Output Spectrum [dbwn] 5-5 No dither With dither 5 1.47 1.48 1.49 1.5 Frequency [MHz] Key point: Dither at an amplitude which eliminate the inband tones has virtually no effect on tones near f s /2 EECS 247- Lecture 25 Oversampled ADCs 29 Page 48

kt/c Noise So far we ve looked at noise added to the input of the ΣΔ modulator, which is also the input of the first integrator Now let s add noise also to the input of the second integrator Let s assume a 1/16 sampling capacitor value for the 2 nd integrator wrt the 1 st integrator This gives 4μV rms noise EECS 247- Lecture 25 Oversampled ADCs 29 Page 49 kt/c Noise Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 No noise 1st Integrator 2nd Integrator 5 1 2 3 4 5 Frequency [Hz] x1 4 5mV DC input Noise from 2 nd integrator smaller than 1 st integrator noise shaped Why? EECS 247- Lecture 25 Oversampled ADCs 29 Page 5

Effect of Integrator kt/c Noise b1 b2 X I1 K1 z I2 K2 z I3 K3 z I4 I5 K4 z K5 z I_1 I_2 I_3 I_4 I_5 a1 a2 a3 a4 a5 DAC Gain g Comparator Noise from 1 st integrator is referred directly to the input Noise from 2 nd integrator is first-order noise shaped Noise from subsequent integrators attenuated even further Especially for high oversampling ratios, only the first 1 or 2 integrators add significant thermal noise. This is true also for other imperfections. Q Y EECS 247- Lecture 25 Oversampled ADCs 29 Page 51 Dither Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 No noise 1st Integrator 2nd Integrator 5 1.47 1.48 1.49 1.5 Frequency [MHz] No practical amount of dither eliminates the tones near f s /2 EECS 247- Lecture 25 Oversampled ADCs 29 Page 52

Full-Scale Inputs With practical levels of thermal noise added, let s try a 5kHz sinusoidal input near full-scale No distortion is visible in the spectrum 1-Bit modulators are intrinsically linear But tones exist at high frequencies To the oversampled modulator, a sinusoidal input looks like two slowly alternating DCs hence giving rise to limit cycles EECS 247- Lecture 25 Oversampled ADCs 29 Page 53 5 Full-Scale Inputs Output Spectrum [dbwn] -5 Output Spectrum [dbwn] 5-5 5 1 2 3 4 5 Frequency [khz] Output Spectrum Integrated Noise (3 averages) 5.5 1 1.5 Frequency [MHz] EECS 247- Lecture 25 Oversampled ADCs 29 Page 54

Recap Dither successfully removes in-band tones that would corrupt the signal The high-frequency tones in the quantization noise spectrum will be removed by the digital filter following the modulator What if some of these strong tones are demodulated to the base-band prior to digital filtering? Why would this happen? Vref Interference EECS 247- Lecture 25 Oversampled ADCs 29 Page 55 V ref Interference via Modulation x 2 (t) x 1 (t) y(t) x 1 x x 1 2 () t = X1 cos( ω1t ) () t = X cos( ω t) X X 2 1 2 () t x () t = [ cos( ω t + ω t) + cos( ω t ω t) ] 2 2 2 1 2 1 2 EECS 247- Lecture 25 Oversampled ADCs 29 Page 56

Modulation via DAC V ref y(t) DAC v(t) ( ) yt= D =± 1 ref out V = 2.5V + 1mV f /2 square wave ( ) ( ) vt = yt V ref s EECS 247- Lecture 25 Oversampled ADCs 29 Page 57 Modulation via DAC D OUT spectrum V ref spectrum interferer convolution yields sum of red and green, mirrored tones and noise appear in band f s /2 f s EECS 247- Lecture 25 Oversampled ADCs 29 Page 58

V ref Interference via Modulation Output Spectrum [dbwn] 5-5 6dB (1 db/db) V 1μV 1mV Key Point: In high resolution ΣΔ modulators Vref interference via modulation can significantly limit the maximum dynamic range 5 1 2 3 4 5 Frequency [khz] EECS 247- Lecture 25 Oversampled ADCs 29 Page 59 V ref Interference via Modulation Output Spectrum [dbwn] 5-5 V 1e-6V.1V 5 1 2 3 4 5 Frequency [Hz] x 1 4 Output Spectrum [dbwn] 5-5 V 1e-6V.1V 5 1.47 1.48 1.49 1.5 Frequency [Hz] x 1 6 Symmetry of the spectra at f s /2 and DC confirm that this is modulation EECS 247- Lecture 25 Oversampled ADCs 29 Page 6

V ref Spurious Tone Velocity vs Native Tone Velocity Output Spectrum [dbwn] 5-5 V in V ref.6khz/mv = 6mV / 12mV.6V DC = 2.5V DC.12V & 1mV f s /2 4dB shift for readability Aliased Native 5 tone tone 1 2 3 4 5 Frequency [khz] Native tone velocity 1.2kHz/mV Aliased tone velocity.6khz/mv EECS 247- Lecture 25 Oversampled ADCs 29 Page 61 V ref Interference via Modulation Simulations performed to verify the effect of the DAC reference contamination via output signal interference particularly in the vicinity of f s /2 Interference modulates the high-frequency tones Since the high frequency tones are strong, a small amount (1μV) of interference suffices to create audible base-band tones Stronger interference (1mV) not only aliases spurious tones but elevated raises noise floor by aliasing high frequency quantization noise Amplitude of modulated tones is proportional to interference The velocity of modulated tones is half that of the native tones Such differences could help debugging of silicon How clean does the reference have to be? EECS 247- Lecture 25 Oversampled ADCs 29 Page 62

V ref Interference Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 Output Spectrum (1μV interference on V ref ) Integrated Noise (3 averages) Tone dominates noise floor w/o thermal noise 5 1 2 3 4 5 Frequency [khz] EECS 247- Lecture 25 Oversampled ADCs 29 Page 63 Summary The model can drive almost all capacitor sizing decisions based on: Gain scaling kt/c noise Dither Dither quite effective in the elimination of native in-band tones Extremely clean & well-isolated V ref is required for high-dynamic range applications e.g. digital audio EECS 247- Lecture 25 Oversampled ADCs 29 Page 64

Bandpass ΔΣ Modulator v IN + _ Resonator dout DAC Replace the integrator in 1 st order lowpass ΣΔ with a resonator 2 nd order bandpass ΣΔ EECS 247- Lecture 25 Oversampled ADCs 29 Page 65 Measured output for a bandpass ΣΔ (prior to digital filtering) Key Point: Bandpass ΔΣ Modulator Example: 6 th Order Quantization Noise Input Sinusoid NTF notch type shape STF bandpass shape Ref: Paolo Cusinato, et. al, A 3.3-V CMOS 1.7-MHz Sixth-Order Bandpass Modulator with 74-dB Dynamic Range, ΙΕΕΕ JSSCC, VOL. 36, NO. 4, APRIL 21 EECS 247- Lecture 25 Oversampled ADCs 29 Page 66

Bandpass ΣΔ Characteristics Oversampling ratio defined as f s /2B where B = signal bandwidth Typically, sampling frequency is chosen to be f s =4xf center where f center bandpass filter center frequency STF has a bandpass shape while NTF has a notch or band-reject shape To achieve same resolution as lowpass, need twice as many integrators EECS 247- Lecture 25 Oversampled ADCs 29 Page 67 Bandpass ΣΔ Modulator Dynamic Range As a Function of Modulator Order (K) K=6 21dB/Octave K=4 15dB/Octave K=2 9dB/Octave Bandpass ΣΔ resolution for order K is the same as lowpass ΣΔ resolution with order L= K/2 EECS 247- Lecture 25 Oversampled ADCs 29 Page 68

Example: Sixth-Order Bandpass ΣΔ Modulator Simulated noise transfer function Simulated signal transfer function Ref: Paolo Cusinato, et. al, A 3.3-V CMOS 1.7-MHz Sixth-Order Bandpass Modulator with 74-dB Dynamic Range, ΙΕΕΕ JSSCC, VOL. 36, NO. 4, APRIL 21 EECS 247- Lecture 25 Oversampled ADCs 29 Page 69 Example: Sixth-Order Bandpass ΣΔ Modulator Features & Measured Performance Summary f s =4xf center B OSR=f s /2B Ref: Paolo Cusinato, et. al, A 3.3-V CMOS 1.7-MHz Sixth-Order Bandpass Modulator with 74-dB Dynamic Range, ΙΕΕΕ JSSCC, VOL. 36, NO. 4, APRIL 21 EECS 247- Lecture 25 Oversampled ADCs 29 Page 7

Modulator Front-End Testing Should make provisions for testing the modulator (AFE) separate from the decimator (digital back-end) Data acquisition board used to collect 1-bit digital output at f s rate Analyze data in a PC environment or dedicated test equipment in manufacturing environments can be used Need to run DFT on the collected data and also make provisions to perform the function of digital decimation filter in software Typically, at this stage, parts of the design phase behavioral modeling effort can be utilized Good testing strategy vital for debugging/improving challenging designs f s Filtered Sinwave AFE Data Acq. PC Matlab EECS 247- Lecture 25 Oversampled ADCs 29 Page 71 Summary Oversampled ADCs Noise shaping utilized to reduce baseband quantization noise power Reduced precision requirement for analog building blocks compared to Nyquist rate converters Relaxed transition band requirements for analog anti-aliasing filters due to oversampling Takes advantage of low cost, low power digital filtering Speed is traded for resolution Typically used for lower frequency applications compared to Nyquist rate ADCs EECS 247- Lecture 25 Oversampled ADCs 29 Page 72