ECE1371 Advanced Analog Circuits Higher-Order ∆Σ Modulators and the ∆Σ Toolbox Richard Schreier richard.schreier@analog.com NLCOTD: Dynamic Flip-Flop • Standard CMOS version D Q CK Q • Can the circuit be simplified? Is a complementarty clock necessary? ECE1371 2
Review: A ∆Σ ADC System desired signal E U ⁄ Y V f B f s 2 U Loop ∆Σ Filter Modulator shaped noise DAC V ⁄ f B f s 2 V ( z ) = STF ( z ) U ( z ) + NTF ( z ) E ( z ) Digital Decimator STF ( z ): signal transfer function NTF ( z ): noise transfer function Nyquist-rate PCM Data E ( z ): quantization error W f B ECE1371 3 Review: MOD1 E U Q V ( ) z 1 – NTF z 1 ( ) = – z -1 STF z 1 ( ) = z -1 NTF poles & zeros: NTF e j 2 π f 4 ) 2 ( 3 2 ≅ ω 2 π 2 σ e 2 ( ) 3 1 – IQNP - OSR = - - - - - - - - - - - - 3 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Frequency ( f ) ECE1371 4
Review: MOD2 E U z 1 Q V z − 1 z − 1 ( ) 2 z 1 – NTF z 1 ( ) = – z 1 – STF z ( ) = 16 NTF e j 2 π f NTF poles & zeros: ) 2 ( 8 ≅ ω 4 π 4 σ e 2 ( ) 5 - OSR – IQNP - - - - - - - - - - - - = 5 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Frequency ( f ) ECE1371 5 Review Summary ∆Σ works by spectrally separating the • quantization noise from the signal ≡ ⁄ ( ) Requires oversampling. OSR f s 2 f B . Achieved by the use of filtering and feedback. • A binary DAC is inherently linear, and thus a binary ∆Σ modulator is too • MOD1-CT has inherent anti-aliasing MOD1 has NTF ( z ) = 1 – z –1 • ⇒ Arbitrary accuracy for DC inputs; 9 dB/octave SQNR-OSR trade-off. MOD2 has NTF ( z ) = (1 – z –1 ) 2 • ⇒ 15 dB/octave SQNR-OSR trade-off. ECE1371 6
MOD N [Ch. 4 of Schreier & Temes] N integrators ( N –1) non-delaying, 1 delaying z z 1 V U Q z − 1 z − 1 z − 1 z 1 STF z – ( ) = ( ) N z 1 NTF z 1 – = ( ) – MOD N ’s NTF is the N th power of MOD1’s NTF • ECE1371 7 NTF Comparison 40 20 (dB) 0 –20 ) 1 NTF ej2 π f D O M –40 ( –60 MOD2 MOD3 MOD4 MOD5 –80 –100 -3 -2 -1 10 10 10 Normalized Frequency ( f ) ECE1371 8
Predicted Performance • In-band quantization noise power ⁄ 0.5 OSR ∫ ) 2 S ee f NTF e j 2 π f ⋅ IQNP d f = ( ( ) 0 ⁄ 0.5 OSR ∫ ≈ ( 2 π f ) 2 N ⋅ 2 σ e 2 d f 0 π 2 N - σ e 2 = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ( ) OSR ( ) 2 N + 1 2 N 1 + Quantization noise drops as the (2 N +1) th power • of OSR— (6 N +3) dB/octave SQNR-OSR trade-off ECE1371 9 Improving NTF Performance– NTF Zero Optimization NTF 2 • Minimize the integral of over the passband Normalize passband edge to 1 for ease of calculation: Need to find the a i which minimize ( ) 2 the integral H f 1 ( ) 2 x ∫ x 2 2 a 1 d , n = 2 – 1 – 1 x 2 x 2 ) 2 x ( 2 ∫ a 1 d , n = 3 – – 1 – a –1 a 1 f / f B 1 ( ) 2 x 2 ( ) 2 x ∫ x 2 2 2 a 1 a 2 d , n = 4 – – 1 – … ECE1371 10
Solutions Up to Order = 8 Optimal Zero Placement SQNR Order Relative to f B Improvement 1 0 0 dB ± ⁄ 2 3.5 dB 1 3 ± ⁄ 3 0, 8 dB 3 5 ± ⁄ ± ( ⁄ ) 2 ⁄ 4 13 dB 3 7 3 7 3 35 – ± ⁄ ± ( ⁄ ) 2 ⁄ 5 0, 18 dB 5 9 5 9 5 21 [Y. Yang] – ± 0.23862, ± 0.66121, ± 0.93247 6 23 dB 0, ± 0.40585, ± 0.74153, ± 0.94911 7 28 dB ± 0.18343, ± 0.52553, ± 0.79667, ± 0.96029 8 34 dB ECE1371 11 Topological Implication • Feedback around pairs integrators: Non-delaying + Delaying 2 Delaying Integrators Integrators (LDI Loop) - g - g 1 1 z 1 z − 1 z − 1 z − 1 z − 1 Poles are the roots of Poles are the roots of 2 gz 1 1 0 + - - - - - - - - - - - - - - - - - - - - = 1 g - - - - - - - - - - - - 0 + = ( ) 2 z 1 z 1 – – ± e j θ ± θ ⁄ i.e. z 1 j g i.e. z cos 1 g 2 = = = – , Not quite on the unit circle, Precisely on the unit circle, but fairly close if g <<1. regardless of the value of g . ECE1371 12
Problem: A High-Order Modulator Wants a Multi-bit Quantizer E.g. MOD3 with an Infinite Quantizer and Zero Input 7 Quantizer input gets 5 large, even if the 3 input is small. 1 v 6 quantizer levels are -1 used by a small input. -3 -5 -7 0 10 20 30 40 Sample Number ECE1371 13 Simulation of MOD3-1b (MOD3 with a Binary Quantizer) Long 1 strings v of +1/–1 0 –1 0 10 20 30 40 200 HUGE! 100 y 0 –100 –200 0 10 20 30 40 Sample Number • MOD3-1b is unstable, even with zero input! ECE1371 14
Solutions to the Stability Problem Historical Order 1 Multi-bit quantization Initially considered undesirable because we lose the inherent linearity of a 1-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 ECE1371 15 Multi-bit Quantization A modulator with NTF = H and STF = 1 is < guaranteed to be stable if u u max at all times, ∞ ∑ where u max nlev 1 h 1 and h 1 h i = + – = ( ) i 0 = ( ) N z 1 • In MOD N H z 1 – , so ( ) = – { , , , , … ( ) N a N 0 … , } > h n 1 a 1 a 2 a 3 1 , a i 0 ( ) = – – – 2 N and thus h 1 H 1 = ( – ) = 2 N • nlev implies u max nlev 1 h 1 1 = = + – = MOD N is guaranteed to be stable with an N -bit quantizer if the input magnitude is less than ∆ /2 = 1. This result is quite conservative. 2 N 1 + • Similarly, nlev guarantees that MOD N is = stable for inputs up to 50% of full-scale ECE1371 16
M -Step Symmetric Quantizer ∆ = 2, ( nlev = M + 1) M odd: mid-rise M even: mid-tread v M v M e = v – y e = v – y 2 1 y y – M –1 M +1 – M –1 M +1 – M v : odd integers – M v : even integers from – M to + M from – M to + M ≤ ⇒ ≤ ∆ 2 ⁄ • No-overload range: y nlev e 1 = ECE1371 17 Inductive Proof of Criterion h 1 ( ∀ ) u n ( ≤ ) • Assume STF = 1 and n u max ( ) ≤ < • Assume e i 1 for i n .[Induction Hypothesis] ( ) ∞ ∑ y n u n h i ( ) e n i ( ) = ( ) + ( – ) i 1 = ∞ ≤ ∑ u max h i ( ) e n i + ( – ) i 1 = ∞ ≤ ∑ u max h i u max h 1 1 + ( ) = + – i 1 = Then u max nlev 1 h 1 = + – ⇒ ≤ y n nlev ( ) ⇒ ≤ e n 1 ( ) ≤ • So by induction e i 1 for all i > 0 ( ) ECE1371 18
More General NTF ⁄ z n • Instead of NTF z A z ( ) B z with B z , ( ) = ( ) ( ) = use a more general B z ( ) Roots of B are the poles of the NTF and must be inside the unit circle. Moving the poles away from z = 1 toward z = 0 makes the gain of the NTF approach unity. ECE1371 19 The Lee Criterion for Stability ≤ in a 1-bit Modulator: H ∞ 2 [Wai Lee, 1987] • The measure of the “gain” of H is the maximum magnitude of H over frequency, aka the infinity- H e j ω ≡ norm of H : H ∞ max ( ) ω ∈ [ 0 2 π , ] 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 ∞ 1.5 often works. ECE1371 20
Simulated SQNR vs. H ∞ 5 th -order NTFs; 1-b Quant.; OSR = 32 90 SQNR (dB) SQNR has a broad maximum 70 cliff! 50 H ∞ 1 1.25 1.5 1.75 2 0 (dBFS) umax –10 Stable input limit drops as H ∞ increases. –20 H ∞ 1 1.25 1.5 1.75 2 ECE1371 21 SQNR Limits— 1-bit Modulation N = 8 N = 7 N = 6 N = 5 N = 4 N = 3 140 N = 2 120 Peak SQNR (dB) 100 N = 1 80 60 40 20 0 4 8 16 32 64 128 256 512 1024 OSR ECE1371 22
SQNR Limits for 2-bit Modulators N = 8 N = 7 N = 6 N = 5 N = 4 N = 3 N = 2 140 120 Peak SQNR (dB) 100 N = 1 80 60 40 20 0 4 8 16 32 64 128 256 512 1024 OSR ECE1371 23 SQNR Limits for 3-bit Modulators N = 8 N = 7 N = 6 N = 5 N = 4 N = 3 N = 2 140 120 Peak SQNR (dB) N = 1 100 80 60 40 20 0 4 8 16 32 64 128 256 512 1024 OSR ECE1371 24
Generic Single-Loop ∆Σ ADC • Linear Loop Filter + Nonlinear Quantizer: E L 0 U Y V L 1 Y L 0 U L 1 V = + ⋅ ⋅ V STF U NTF E , where = + V Y E = + 1 ⋅ NTF & STF L 0 NTF = - - - - - - - - - - - - - - - = 1 L 1 – Inverse Relations: L 1 = 1 – 1/ NTF , L 0 = STF / NTF ECE1371 25 ∆Σ Toolbox http://www.mathworks.com/matlabcentral/fileexchange Search for “Delta Sigma Toolbox” Specify OSR, lowpass/bandpass, no. of Q. levels. scaleABCD synthesizeNTF calculateTF stuffABCD Parameters for ABCD: state- NTF (and STF) realize- a specific space description available. mapABCD NTF topology. of the modulator. predictSNR, simulateDSM, simulateSNR Time-domain Also mapCtoD simulation and simulateESL SNR measure- designHBF ments. Manual is delsig.pdf (App. B of Schreier & Temes) ECE1371 26
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