EE247 Lecture 26. EE247 Lecture 26

EE247 Lecture 26 Administrative Final exam: Date: Tues. Dec. 13 th Time: 12:3pm-3:3pm Location: 285 Cory Office hours this week: Tues: 2:3p to 3:3p Wed: 1:3p to 2:3p (extra) Thurs: 2:3p to 3:3p 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 EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 1 EE247 Lecture 26 Higher order ΣΔ modulators Last lecture Cascaded ΣΔ modulators (MASH) This lecture Bandpass ΣΔ modulators This lecture Forward path multi-order filter Example: 5 th order Lowpass ΣΔ Modeling Noise shaping Effect of various nonidealities on the ΣΔ performance EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 2

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 26: Oversampling Data Converters 25 H. K. Page 3 Bandpass ΔΣ Modulator Measured output for a bandpass ΣΔ (prior to digital filtering) Quantization Noise Input Sinusoid Key Point: 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 26: Oversampling Data Converters 25 H. K. Page 4

Bandpass ΣΔ Characteristics Oversampling ratio defined as f s /2B where B = STF -3dB 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 shape To achieve same resolution as lowpass, need twice as many integrators EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 5 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 26: Oversampling Data Converters 25 H. K. Page 6

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 26: Oversampling Data Converters 25 H. K. Page 7 Example: Sixth-Order Bandpass ΣΔ Modulator Features & Measured Performance Summary 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 26: Oversampling Data Converters 25 H. K. Page 8

Higher Order Lowpass ΣΔ Modulators Forward Path Multi-Order Filter E(z) X(z) Σ H() z Σ Y(z) H( z) 1 Y( z) = X( z) + E( z) 1 + H( z) 1 + H( z) Y( z) 1 NTF = = E( z ) 1 + H( z ) Zeros of NTF (poles of H(z)) can be positioned to flatten baseband noise spectrum Main issue Ensuring stability for 3 rd and higher orders EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 9 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 26: Oversampling Data Converters 25 H. K. Page 1

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 & simulated 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 26: Oversampling Data Converters 25 H. K. Page 11 Stage 1 In stage 1, we ll study a model for a practical ΣΔ modulator topology built with ideal blocks Stage 1 model focus Signal amplitudes Stability Identifying worst-case inputs Unstable systems can t graduate to stage 2 Quantization noise shaping Verify performance and functionality for all regions of operation, find and test worst-case inputs Determine appropriate performance metrics and build the software infrastructure EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 12

ΣΔ Modulator Design Procedure 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 26: Oversampling Data Converters 25 H. K. 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 26: Oversampling Data Converters 25 H. K. Page 14

Modulator Block Diagram x(kt) Σ Loop Filter H(z) Comparator y(kt) Y( z) H( z) STF = = X( z) 1 + H( z) NTF Y( z) 1 E( z) 1 H( z) = = + Design NTF and solve for H(z) Approach: EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 15 Noise Transfer Function, NTF(z) % stop-band attenuation Rstop... Rstop = 8; [b,a] = cheby2(l, Rstop, 1/M, 'high'); % normalize b = b/b(1); NTF = filt(b, a, 1/fs); NTF [db] 2-2 -4-6 -8 1 4 1 6 Frequency [Hz] EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 16

Loop-Filter, H(z) Y( z) 1 NTF = = Ez ( ) 1 + Hz ( ) 1 H( z) = 1 NFT Loopfilter H [db] 1 8 6 4 2-2 1 4 1 6 Frequency [Hz] EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 17 Modulator Topology Simulation Model Filter 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 Q DAC Gain g Comparator Y +1 EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 18

Rounded 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 26: Oversampling Data Converters 25 H. K. Page 19 5 th Order Noise Shaping Output Spectrum [dbwn]/ Int. Noise [dbfs] 4 2-2 -4-6 -8 Signal 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 26: Oversampling Data Converters 25 H. K. Page 2

5 th Order Noise Shaping Output Spectrum [dbwn] / Int. Noise [dbfs] 4 2-2 -4-6 -8 Output Spectrum Integrated Noise (2 averages) 2 Quantization noise 3dBFS at 4 Signal band edge! 6 Band-Edge.2.4.6.8 1 Frequency [ f / f N ] EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 21 5 th Order Noise Shaping Output Spectrum [dbwn] / Int. Noise [dbfs] 4 2-2 -4-6 -8 2 4 6.2.4.6.8 1 Frequency [f / f N ] sigma_delta_l5.m Output Spectrum Integrated Noise (2 averages) Digital decimation filter removes out-of-band quantization noise SQNR > 12dB Sigma-delta modulators are usually designed for negligible quantization noise Other error sources dominate, e.g. thermal noise EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 22

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 passband Remember that NTF ~ 1/H STF=H/(1+H) EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 23 Stability Analysis Quantization Error e(kt) x(kt) Σ H(z) q(kt) G eff Σ y(kt) Quantizer Model Approach: linearize quantizer and use linear system theory! Effective quantizer gain 2 G 2 = y eff q 2 Obtain G eff from simulation EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 24

Modulator 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 Pole-locations inside unit-circle correspond to stable STF and NTF G eff >.45 for stability EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 25 Effective Quantizer Gain, G eff Effective Quantizer Gain 2 1.8 1.6 1.4 1.2 1 stable unstable.8.6 G eff =.45.4.2-4 -35-3 -25-2 5-5 5 Input [dbv] Large inputs comparator input grows Output is fixed (±1) G eff drops modulator unstable for large inputs Solution: Limit input amplitude Detect instability (long sequence of +1 or ) and reset integrators Beware of worst-case inputs (e.g. square waves near high-q poles attenuate with antialiasing filter) Note that signals grow slowly for nearly stable systems use long simulations EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 26

Internal Node Voltages Loop filter peak voltages [V] 1 5-5 5-2 i1 i2 i3 i4 i5 q -4-35 -3-25 -2 5-5 Input [dbv] Internal signal amplitudes are weak function of input level (except near overload) Maximum peak-topeak voltage swing approach +V! Exceed supply voltage! Solutions: Reduce V ref?? Node scaling EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 27 Internal Node Voltage Scaling If we scale filter k 1 by.1, All state variables and Q scale by.1 But since the comparator output is fixed and input is decreased by 1, G increases 1X The change in k 1 doesn t change the shape of the root locus, either The effective gain for each root position is increased 1X G new =1XG old G new > 4.5 is thus required to ensure stability EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 28

5 th Order Modulator Scaling Only the sign of Q matters, so we can make k 1 whatever we want without changing the 1-Bit data at all 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 Q DAC Gain g Comparator Y EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 29 Loop Voltage Scaling (cont.) Note that 3, 4, and 5 have substantially larger swings than 1 and 2 Just about any filter topology allows node scaling which change internal state variable amplitudes without changing the filter output (recall filter node scaling) The next slide shows an example EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 3

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 26: Oversampling Data Converters 25 H. K. Page 31 Voltage Scaling Loop filter peak voltages [V] 1.5 1.5 -.5.5-4 -35-3 -25-2 5-5 Input [dbv] 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; Integrator output range is fine now But: maximum input signal limited to -5dB (-7dB with safety) fix? EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 32

Input Range Scaling Increasing the DAC levels by g reduces the analog to digital conversion gain: D V IN ( z) H ( z) = ( z) 1+ gh ( z) g OUT 1 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 26: Oversampling Data Converters 25 H. K. Page 33 Input Range Scaling Scaling the DAC output levels adjusts the modulator input range If V IN and the DAC outputs are scaled up by the same factor g, the 1-Bit data is completely unchanged Note that increasing the range also increases the quantization noise the dynamic range and peak SQNR remain constant! If the DAC output levels are increased and the analog full scale is held constant, the stability margin improves at the expense of reduced SQNR EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 34

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 26: Oversampling Data Converters 25 H. K. Page 35 Scaled Stage 1 Model 8 Effective Quantizer Gain 7 6 5 4 3 2 G eff =4.5 stable unstable 2dB safety margin for stability 1-4 -35-3 -25-2 5-5 5 Input [dbv] EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 36

5 th Order Modulator Final Parameters 1/512 1/16/64 b1 b2 X Stable 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 ±2.5V DAC Gain g Stable input range ~ ±1V Comparator Q Y EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 37 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 26: Oversampling Data Converters 25 H. K. Page 38

Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 5 th Order Noise Shaping Tones at f s /2-Nf in exceed input level 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 fs/2 Let us check whether tones appear inband? EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 39 In-Band Noise Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 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 26: Oversampling Data Converters 25 H. K. Page 4

5 th Order Noise Shaping Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 5 15dB stopband attenuation needed to attenuate unwanted f s /2-Nf in 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 26: Oversampling Data Converters 25 H. K. Page 41 Out-of-Band vs In-Band Signals A digital (low-pass) filter with suitable coefficient precision can eliminate out-ofband 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-Nf 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 26: Oversampling Data Converters 25 H. K. Page 42

ΣΔ Tones Generated by Small DC Input Signals Output Spectrum [dbwn] / Int. Noise [dbv] 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 26: Oversampling Data Converters 25 H. K. Page 43 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 26: Oversampling Data Converters 25 H. K. Page 44

Limit Cycles The frequency of the tones are indeed quite predictable Fundamental VDC fδ = fs V DAC 5mV = 3MHz 2.5V = 6kHz Tone velocity (useful for debugging) df dv Note: For digital audio in this case DC signal>2mv generates tone with f δ >24kHz out-of-band no problem df dv δ DC δ DC f = V s DAC = 1.2kHz/mV EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 45 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 26: Oversampling Data Converters 25 H. K. Page 46

ΣΔ Spurious Tones In-band spurious tones look like signals 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 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 26: Oversampling Data Converters 25 H. K. Page 47 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 26: Oversampling Data Converters 25 H. K. Page 48

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: 1 2 ( V ) FS 2 n 2 DR = VFS = 1Vp v 2 1 n 2DR ( V ) v = = 1μV FS EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 49 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 26: Oversampling Data Converters 25 H. K. Page 5

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 26: Oversampling Data Converters 25 H. K. Page 51 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 26: Oversampling Data Converters 25 H. K. Page 52

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 26: Oversampling Data Converters 25 H. K. Page 53 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 26: Oversampling Data Converters 25 H. K. Page 54

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 26: Oversampling Data Converters 25 H. K. Page 55 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 26: Oversampling Data Converters 25 H. K. Page 56

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 26: Oversampling Data Converters 25 H. K. Page 57 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 26: Oversampling Data Converters 25 H. K. Page 58

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 26: Oversampling Data Converters 25 H. K. Page 59 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 26: Oversampling Data Converters 25 H. K. Page 6

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 26: Oversampling Data Converters 25 H. K. Page 61 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 26: Oversampling Data Converters 25 H. K. Page 62

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 26: Oversampling Data Converters 25 H. K. Page 63 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 26: Oversampling Data Converters 25 H. K. Page 64

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 help debugging of silicon How clean does the reference have to be? EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 65 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 26: Oversampling Data Converters 25 H. K. Page 66

Summary Our stage 2 model can drive almost all capacitor sizing decisions 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 Next we will add relevant component imperfections: Effect of component nonlinearities on ΣΔ performance EECS 247 Lecture 26: Oversampling Data Converters 25 H. K. Page 67