Second and Higher-Order Delta-Sigma Modulators
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1 Second and Higher-Order Delta-Sigma Modulators MEAD March 28 Richard Schreier ANALOG DEVICES Overview MOD2: The 2 nd -Order Modulator MOD2 from MOD NTF (predicted & actual) SQNR performance Stability Deadbands, Distortion & Tones (audio demo) Topological Variants 2 Higher-Order Modulators MODN from MOD NTF Zero Optimization Stability SQNR limits for binary and multi-bit modulators Topology Overview 2
2 . MOD2 from MOD Replace the quantizer in MOD with another copy of MOD in a recursive fashion: Replace Q with MOD E U X Q E V z - z - z - 3 z - V = U + ( z )E V = X + E = X + ( z )E E = ( z )E V = U + ( z ) 2 E Simplified Diagram Combine feedback paths, absorb feedback delay into second integrator U z z z Q E V Vz () = z Uz () + ( z ) 2 Ez () NTF() z = ( z ) 2 and the STF is STF() z = z MOD2 s NTF is the square of MOD s NTF 4
3 NTF Comparison NTF ( e j2πf ) (db) MOD MOD2 Twice as much attenuation at all frequencies 3 2 Normalized Frequency 5 Predicted Performance In-band quantization noise power IQNP = ( 2 OSR) ( 2 OSR) π 4 2 σ = e 5( OSR) 5 NTF( e j2πf ) 2 S ee ( f) ( 2πf ) 4 2 2σ e d f Quantization noise drops as the 5 th power of OSR! SQNR increases at 5 db per octave increase in OSR. d f 6
4 4 2 Predicted SQNR vs. OSR SQNR (db) MOD2 MOD OSR For OSR = 28 and binary quantization, the predicted SQNR of MOD2 is 94.2 db 7 dbfs/nbw MOD2 Simulated PSD Half-scale sine-wave input with SQNR = 85 OSR = 28 Simulated spectrum (smoothed) Observed PSD similar to theory (4 db/decade slope) But 3 rd harmonic is visible and in-band PSD is slightly higher. 8 f f s 5 4 db/decade 3 2 Normalized Frequency Theoretical PSD k = NBW = 5.7 6
5 Gain of the Quantizer in MOD2 The effective quantizer gain can be computed from the simulation data using v, y E[ y ] k = = [S&T Eq. 2.5] y, y E[ y 2 ] For the preceding simulation, k =.63. k alters the NTF: NTF k () z = NTF () z k + ( k)ntf () z Revised PSD Prediction Theoretical PSD k =.63 dbfs/nbw Simulated Spectrum (smoothed) 2 NBW = Normalized Frequency Agreement is now excellent
6 Variable Quantizer Gain When the input is small (below -2 dbfs), the effective gain of the quantizer is k =.75 The small-signal NTF is thus ( z ) NTF() z = z 2.5z +.25 This NTF has 2.5 db les quantization noise suppression than the ( z ) 2 NTF derived from the assumption that k = Thus the SQNR should be about 2.5 db lower than. As the input signal increases, k decreases and the suppression of quantization noise degrades SQNR increases less quickly than the signal power, and eventually the SQNR saturates and then decreases as the signal power is increased. Simulated SQNR of MOD2 5A 2 ( OSR) π 4 SQNR (db) Input Amplitude (dbfs) High-frequency input Low-frequency input SP = A 2 2 π IQNP 4 ( 3) = ( OSR) 5 SP SQNR = IQNP Well-modeled by ideal formula; less erratic than MOD Saturation at high signal levels due to decreased quantizer gain and altered NTF. (Worse with low-frequency inputs.) 2
7 u.5 Stability of MOD2 Known to be stable with DC inputs up to full-scale, but the state bounds blow up as u Hein [ISCAS 99]: u x (output of st u + 2 integrator) x ( 5 u ) 2 (output of 2 nd 2 integrator) ( u ) However, with a hostile input (whose magnitude is less than 3% of full-scale) MOD2 can be driven unstable! As a result of this input-dependent stability, it is wise to keep the input below 7-9% of full-scale 3 u =.3 Deadbands, Distortion & Tones Audio Comparison of MOD and MOD khz NBW = 3. Hz -8 NBW =.8 Hz - khz 4
8 Observations Tones Quantization noise of MOD is distinctly non-white Audible tones when input is near zero, or near other simple rational fractions of full-scale. MOD2 is better than MOD in terms of its tendency toward tonal behavior Dead-bands MOD has dead-bands whose widths are proportional to /A, where A is the gain of the internal op-amp MOD2 has dead-bands whose widths are proportional to /A 2 Dead-band behavior is less problematic in MOD2 5 Topological Variant Delaying Integrators E U z z Q V - -2 NTF() z = ( z ) 2 STF() z = z 2 + Delaying integrators reduce the settling requirements Can decouple the integration phase from the driving phase 6
9 Topological Variant Feed-Forward E U z z z Q V NTF() z = ( z ) 2 STF() z = 2z z 2 + Output of first integrator has no DC component Dynamic range requirements of this integrator are relaxed. Although STF near ω =, STF = 3 for ω = π Instability is more likely. 7 Topological Variant Feed-Forward with Extra Input Feed-In U z z z Q E V NTF() z = ( z ) 2 STF() z = + No DC component in either integrator s output Reduced dynamic range requirements in both integrators. + Perfectly flat STF No increased risk of instability. 8
10 Topological Variant Error Feedback E U Q V z - 2 z - E NTF() z = ( z ) 2 STF() z = + Simple Very sensitive to gain errors Only suitable for digital implementations. 9 U z z 2. MODN from MOD2 z z z Q V V V = E z V () z = z Uz () + ( z ) N Ez () z z V z ( z V + U) ( z ) N V = ( z ) N E (( z ) N + ( z ) N )z V + z ( z ) N V ( z ) N ( z E ) N = z z V + z U ( z ) N V ( z ) N ( z E ) N = z V + z U z NTF of MODN is the N th power of MOD s NTF U 2
11 4 NTF Comparison NTF ( e j2πf ) (db) MOD MOD2 MOD3 MOD4 MOD Normalized Frequency 2 Predicted Performance In-band quantization noise power IQNP = ( 2 OSR) ( 2 OSR) NTF( e j2πf ) 2 S ee ( f) ( 2πf ) 2N 2 2σ e d f π 2N 2 = σ ( 2N + ) ( OSR) 2N + e d f Quantization noise drops as the (2N+) th power of OSR! SQNR increases at (6N+3) db per octave increase in OSR. 22
12 Improving NTF Performance Zero Optimization Minimize the rms in-band value of H by finding the a i which minimize the integral of H 2 over the passband. Normalize passband edge to for ease of calculation. H( f) i which minimize the integral x 2 2 ( a ) 2d x, n = 2 x 2( x 2 2 a ) 2dx, f f B n = 3 x 2 2 ( a ) 2( x 2 2 a ) 2dx 2, n = 4 - -a a 23 Solutions Up to Order = 8 Order Optimal Zero Placement Relative to f B SQNR Improvement db 2 ± db 3, ± db 4 ± ± db 5, ± ± -- [Y. Yang] db 6 ±.23862, ±.662, ± db 7, ±.4585, ±.7453, ± db 8 ±.8343, ±.52553, ±.79667, ± db 24
13 Topological Implication Apply feedback around pairs integrators: 2 Delaying Integrators -g Non-delaying + Delaying Integrators (LDI Loop) -g z z z z z Poles are the roots of + g = z i.e. z = ± j g Not quite on the unit circle, but fairly close if g<<. 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. 25 Problem: High-Order Modulators Want Multi-bit Quantizers e.g. a 3 rd -Order Modulator with an Infinite Quantizer and Zero Input Sample Number v Quantizer input gets large, even if the input is small. 6 quantizer levels are used by a small input. 26
14 v The 3 rd -Order Modulator is Unstable with a Binary Quantizer Long strings of +/- y Sample Number The quantizer input grows without bound The modulator is unstable, even with an arbitrarily small input. 27 HUGE! Solutions to the Stability Problem Historical Order Use multi-bit quantization Originally considered undesirable because the inherent linearity of a -bit DAC is lost when a multi-bit quantizer is used. Less of an issue now that mismatch-shaping is available. 2 Use a more general NTF (not pure differentiation) Lower the NTF gain so that quantization error is amplified less. A common rule of thumb is to limit the maximum NTF gain to ~.5. Unfortunately, limiting the NTF gain reduces the amount by which quantization noise is attenuated. 3 Use a multi-stage (MASH) architecture More on this later in the course. Combinations of the above are possible 28
15 Multi-bit Quantization Can show that a modulator with NTF H and unity 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, }, where a i > and thus h = H( ) = 2 N. Thus 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 extremely conservative. Similarly, nlev = 2 N + guarantees the modulator is stable for inputs up to 5% of full-scale. 29 Proof of Criterion h By Induction Assume STF = and ( n) ( un ( ) u max ). Assume ei () for i< n. [Induction Hypothesis] Then yn ( ) = un ( ) + hi ()en ( i) i = u max + hi () en ( i) i = u max + hi () = u max + h i = Thus ( u max nlev + h ) ( yn ( ) nlev) ( en ( ) ) And by induction ei () for all i >. QED 3
16 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), otherwise known as the infinity-norm of H: max( He ( jω )) H ω [ 2π, ] Q: Is the Lee criterion necessary for stability? For MOD2, Hz () = ( z ) 2 and so H = H( ) = 4. Since MOD2 is known to be stable, the Lee criterion is not necessary. Q: Is the Lee criterion sufficient to ensure stability? No. There are lots of counter-examples, but H.5 often works. Let s look at some examples 3 The NTF Family Used by the Σ Toolbox Poles chosen such that den( Hz ()) is a maximally flat transfer function. Pole Locus for 5 th -order NTFs For lowpass modulators, the pole placement is similar to a Butterworth transfer function. Yields a flat STF for both lowpass and bandpass modulators employing the CRFB topology with one feed-in
17 σ e vs. H Binary modulators of order N = 3, 4, 5 with a small input σ e Toolbox default is H =.5 N =5 N =4 N =3 Value of H resulting in instability depends on modulator order, N H Two 5 th -Order Binary Modulators NTF Magnitude -2 H =.5 H =.75-4 db Higher H has more aggressive noise shaping Normalized Frequency 34
18 σ e vs. u DC For the two 5 th -order modulators σ e 2 NTF with higher H has a lower stable input limit DC Input 35 SQNR vs. H 5 th -Order NTFs, Binary Quantization, OSR = 32 SQNR (db) 9 7 SQNR has a broad maximum u max (dbfs) - Stable input limit drops as increases. H H
19 SQNR Limits for Binary Modulators 4 N = 8 N = 7 N = 6 N = 5 N = 4 N = 3 N = 2 2 Peak SQNR (db) N = OSR 37 SQNR Limits for 2-bit Modulators 4 2 N = 8 N = 7 N = 6 N = 5 N = 4 N = 3 N = 2 Peak SQNR (db) N = OSR 38
20 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 39 Theoretical SQNR Limits for Multi-Bit Modulators SQNR OSR = 8 with LSB (peak) input H 2 = H = 4 N = 2 H = 8 N = 3 H = 6 N = 4 H = 32 N = 5 N = 6 N = 8 N = 7 2 Total RMS Noise Power (LSBs) 4
21 6 Example Waveforms 7 th -order 7-level modulator, H = 2 SQNR = 55 OSR = Time Step 4 6 Example Waveforms 7 th -order 7-level modulator, H = 8 SQNR = 5 OSR = Time Step 42
22 Spectra 2 4 SNR = OSR=8 SNR = OSR=8 dbfs/nbw NBW=.8x Normalized Frequency SNR Curves SNR (db) peak SNR = OSR = 8 peak SNR = OSR = 8 stable input limit = dbfs 2 stable input limit = 6dBFS Input Level (dbfs) 44
23 Example Topology Feedback -g U I I I Q V -a -a 2 -a 3 Adjust a i to get the desired NTF N integrators precede the quantizer Feedback from the quantizer to the input of each integrator (via a DAC) Local feedback around pairs of integrators is possible Multiple input feed-in branches are also possible 45 Example Topology Feedforward a a 2 U I I I a 3 -g Q V N integrators in a row Each integrator output is fed forward to the quantizer Local feedback around pairs of integrators is possible Multiple input feed-in branches are also possible 46
24 General Single-Quantizer Σ Modulator The input to the quantizer is some linear combination of the input to the modulator and the fed-back output E U L Y V L Y = L U + L V V = Y + E V = GU + HE, where H = & G = L L H Inverse Relations: L = /H, L = G/H 47 Summary MOD2 is better than MOD Higher SQNR Whiter quantization noise Smaller deadbands MODN is better than MOD2 Even higher SQNR Tonal behavior unlikely Deadbands virtually eliminated BUT high-order modulators must deal with instability Modify the NTF, reduce the input range, and/or use multi-bit quantization 48
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