delta-sigma analog-to-digital converters sigma adc... · 2016. 9. 17. · delta-sigma (ΔΣ) adc...

Department of Electrical and Computer Engineering

© Vishal Saxena -1-

Delta-Sigma Analog-to-Digital

Converters Brief Overview

Vishal Saxena, Boise State University ([email protected])


Analog to Digital Converter Architectures

Bandwidth

(Hz)

Resolution

(bits)

5

10

15

20

1k 10k 100k 1M 10M 100M 1G 10G

25

S/H with 1ps

rms jitterOversampling

Integrating

SAR,

Algorithmic

Pipelined,

Folding,Flash,

Time-Interleaved


Why larger overlap on SAR and OS?

Sometimes oversampling modulators are preferred over SAR. Why?

Nyquist rate SAR converters can’s support complete bandwidth due to stringent requirements on Anti-aliasing Filters.

Consider a data converter with 250 ksps output rate.

For oversampled modulator OSR=64, input is sampled at 16 MHz.

Nyquist rate SAR samples the input at 500 kHz.

Requirements on anti-aliasing filter in the second case is very hard compared to oversampling case.


Quantization Basics

y v

QuantizerContinuous

amplitude

Discrete amplitude represented

by a binary code

N

y v

QuantizerAnalog representation of the

discrete amplitude

N

….11000101….

DAC

Modeling

Quantization error/noise: e[n] = v[n]–y[n]

Least significant bit (LSB): Δ

Quantizer has non-linear input-output characteristics

Exact analytical modeling is difficult Use a simplified model


Quantizer Modeling

Quantization noise modeling can be simplified with the assumptions Input (y) stays within the no-overload input range

e[n] is uncorrelated with the input y

Spectrum of e[n] is white

Quantization noise is uniformly distributed

Linearized quantizer model AWUN noise

Quantization noise power:

SQNR = 6.02·N + 1.76

22

12e

y v

Quantizer

kq

e[n]

Linearized

Quantizer model

y v


Quantizer Modeling

Linear quantizer model holds true for large number of quantization levels and when the input is varying fast

Linear quantizer model breaks down when Input (y) is not varying fast

Input (y) is periodic with a frequency harmonically related to fs

Quantizer overload

With quantizer overload, the effective gain of the quantizer drops as the input amplitude inscreases

y v

Quantizer

kq

e[n]

Linearized

Quantizer model

y v


Mid-Rise Quantizer (even number of levels)

Step rising at y=0 (mid-rise).

In this figure (DSM toolbox model), LSB= Δ= 2

M = Number of steps, (M is odd here) Number of levels (nLev) = M+1, (even)

Input thresholds: 0, ±2, ……, ±(M–1).

Output levels: ±1, ±3, ……, ±M.


Mid-Tread Quantizer (odd number of levels)

Flat part of the step at y=0 (mid-tread).

Here, LSB= Δ= 2

M = Number of steps, (M is even here)

• Number of levels (nLev) = M+1, (odd)

Input thresholds: 0, ±2, ……, ±(M–1).

Output levels: 0, ±2, ±4, ……, ±M.


Quantizer Characteristics : Slow ramp input

0 100 200 300 400 500 600 700 800-40

-20

0

20

40

samples

Quantization

y[n]

v[n]

0 100 200 300 400 500 600 700 800-10

-5

0

5

10

samples

e[n]


Quantizer Characteristics : Sine input

0 100 200 300 400 500 600 700 800-10

-5

0

5

10

samples

Quantization

y[n]

v[n]

0 100 200 300 400 500 600 700 800-1

-0.5

0

0.5

1

samples

e[n]


Quantization Noise : Example 1

5 10 15 20 25 30-20

-10

0

10

20

samples

Quantization

y[n]

v[n]

5 10 15 20 25 30-1

-0.5

0

0.5

1

E(e2) = 0.304

samples

e[n]

50 100 150 200 2500

10

20

30

40

50

60

70

80

FFT bins (k)

20*l

og

10|V

(k)|,

dB

Quantized Signal Spectrum

V[k]

nLev=17, Δ=2, fin/fs = 9/256 :

•E(e2) = 0.304 ≈ Δ2/12


Quantization Noise : Example 1 contd.

nLev=17, Δ=2, fin/fs = 9.1/256 :

•E(e2) = 0.295 ≈ Δ2/12

•Notice the FFT leakage.

5 10 15 20 25 30-20

-10

0

10

20

samples

Quantization

y[n]

v[n]

5 10 15 20 25 30-1

-0.5

0

0.5

1

E(e2) = 0.295

samples

e[n]

50 100 150 200 2500

10

20

30

40

50

60

70

80

FFT bins (k)

20*l

og

10|V

(k)|,

dB


V[k]


Quantization Noise : Example 1 contd.

nLev=17, Δ=2, fin/fs = 32/256 = 1/8 :

•E(e2) = 0.235 < Δ2/12

•Quantization noise approximation not valid

50 100 150 200 2500

10

20

30

40

50

60

70

80

FFT bins (k)

20*l

og

10|V

(k)|,

dB


V[k]

5 10 15 20 25 30-20

-10

0

10

20

samples

Quantization

y[n]

v[n]

5 10 15 20 25 30-1

-0.5

0

0.5

1

E(e2) = 0.235

samples

e[n]


Oversampling and Noise-Shaping


Oversampling

yv

fs

2

sf

OSR

LPF

OSR

Decimation

Dout

@fs @fs/OSR

0 50 100 150 200 250 300-8

-6

-4

-2

0

2

4

6

8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-80

-70

-60

-50

-40

-30

-20

-10

0

10

e2=0.29136

V(f)

Oversampling ratio (OSR)

Conversion bandwidth: fB = fs/2·OSR

SQNR = 6.02·N + 1.76 + 10·log10(OSR)

0.5 bits increase in resolution per doubling in OSR


Oversampling with Feedback

Can we use feedback with high loop-gain (A·kq) to reduce the error

e=|u-v| ?

Since quantizer output can not be equal to the input

The loop will be unstable as the error gets unbounded

Ay v

u

fs

High-gain

error

qA k e u v


Oversampling with Feedback

Use large loop-gain in the signal band and small loop-gain at higher frequencies

At low frequencies

At high frequencies, low loop-gain stabilizes the loop

L(z) is the loop-filter

This feedback arrangement is called a (noise) modulator

0e u v

vu

fs

error2

sf

OSR

L(z)

y


First-order Modulator

Differencing (Δ) followed by an accumulator (Σ) ΔΣ modulator

At low frequencies 0e u v

y vu

fs1

1-z-1

z-1

Δ

Σ


First-order Noise Shaping

Linearized model for the modulator

Noise transfer function (NTF) (1-z-1) : first-order differentiator

High-pass shaping of quantization noise

Signal transfer function (STF) Unit delay

y vu

fs1

1-z-1

z-1

e[n]

1 1( ) 1 ( )V z z U z z E z

STF NTF


First-order ΔΣ Modulator

y vu

fs1

1-z-1

z-1

e[n]

Sv(f)v[n]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-140

-120

-100

-80

-60

-40

-20

0DSM Output Spectrum

0 100 200 300 400 500 600 700 800 900 1000-8

-6

-4

-2

0

2

4

6

8DSM time-domain Simulation


First-order ΔΣ Modulator SQNR

In-band quantization noise (IBN)

SQNR = 6.02·N –3.4 + 30·log10(OSR)


Out-of-band noise is filtered out using a digital filter

0

2

0

22

0

22

0

2 3

0

2 2

3

( )

( )12

4sin / 212

12

12 3

36

OSRj

v

OSRj

OSR

OSR

OSR

IBN S e d

NTF e d

d

d

OSR

2 23

12 3OSR

ππ

OSR

|NTF(ejω

)|

0

2


Delta-Sigma (ΔΣ) ADC

Use oversampling (fs=2·OSR·BW) to shape the quantization noise out of the signal band

Use low-resolution ADC and DAC to higher much higher resolution

Digitally filter out the out-of band shaped (modulated) noise

Trades-off SQNR with oversampling ratio (OSR)

+

–

vinL(z)

DAC

ADCvDSM Decimation

Filter

vout

ΔΣ Modulator

E(z)

f

STF

NTF•E

E

fs/2•OSR fs/2

|VDSM(f)|vin

ffs/2

|Vout(f)|vin

fs/2•OSR


Second-Order Noise Shaping

2

1 1( ) 1 ( )V z z U z z E z

STF NTF

Linearized model for the modulator

Second-order noise-shaping

In-band quantization noise (IBN):


Can be extended to higher orders

2 45

12 5OSR

v1

1-z-1

z-1

u 1

1-z-1

y

e[n]


Second-order ΔΣ Modulator

v1

1-z-1

z-1

u 1

1-z-1

y

e[n]

Sv(f)v[n]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

0 100 200 300 400 500 600 700 800 900 1000-8

-6

-4

-2

0

2

4

6

8


-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X2

NTF(z)

NTF

zeroes

Second-order ΔΣ Modulator

10-4

10-3

10-2

10-1

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

SNDR = 73.8 dB

ENOB = 11.97 bits

@OSR = 32

Frequency

dB

Second-Order KD1S Output Spectrum

40 dB/dec

Sv(f)

NTF(z) = (1-z-1)2

Two zeroes at DC

Out-of-Band Gain (OBG) (i.e gain at ω≈π) = 4


Higher-Order ΔΣ Modulators


Higher-Order NTFs

Higher order noise shaping Reduced in-band noise, higher SQNR

For NTF=(1-z-1)-N, in-band noise (IBN):

Ideally (N+1/2) bits increase in resolution per doubling in OSR

y vu L(z)

( ) 1

( ) ( )1 ( ) 1 ( )

L zV z U z E z

L z L z

STF NTF

2 2(2 1)

12 2 1

NNOSR

N


Higher-Order NTFs

NTF gain increases at high frequencies (around ω≈π)

Can we go on increasing the order?

0.5 1 1.5 2 2.5 3-200

-150

-100

-50

0

50

|NT

F(e

j )|

Bode Diagram

Frequency (rad/s)


Third-order ΔΣ Modulator Example

NTF(z) = (1-z-1)3

OBG = 8, Full-scale input.

Unstable after few samples (look at quantize input (y) blowing up!). Signature for ΔΣ instability

Worst case for a single-bit quantizer.

0 100 200 300 400 500 600 700 800 900 1000-10

-5

0

5


u

v

0 100 200 300 400 500 600 700 800 900 1000-4

-2

0

2

4x 10

7

y


Third-order ΔΣ Modulator Example

Stable for 50% of full-Scale amplitude

Signal dependent stability Need to develop intuition for modulator stability

Reference: Stability theory from the Yellow Bible of delta-sigma

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

DSM Output Spectrum

0 100 200 300 400 500 600 700 800 900 1000-10

-5

0

5


u

v

0 100 200 300 400 500 600 700 800 900 1000-10

-5

0

5

10

y


Systematic NTF Design

NTFs of the form (1-z-1)N have stability issues The OBG (2N) are too high

A larger OBG causes more wiggling at the quantizer input This saturates the quantizer for even smaller inputs

Irrecoverable quantizer saturation causes loop instability

For high-OBG the maximum stable (input) amplitude (MSA) is small

The stability is worse for low quantizer resolutions

Thus we need to reduce OBG while maintaining high in-band noise shaping


Systematic NTF Design Procedure

Introduce poles into the NTF

NTF realizability criterion No delay-free loops in the modulator

• First sample of the NTF impulse response (i.e. h[0])=1

NTF()=1

D(z=)=1

Commonly used pole positions: Butterworth, Inverse Chebyshev and maximally flat poles (maxflat)

1(1 )( )

( )

NzNTF z

D z

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Pole-Zero Map

Real Axis

Imagin

ary

Axis

X3


NTF Response with Poles

Select appropriate OBG for the NTF to assure stability

Trade-off between stability and increased in-band noise MSA vs SQNR for a given order and quantizer resolution

0 50 100 150 200 250 300 350 400 450 500-10

-5

0

5

10OBG=8

u

v

0 50 100 150 200 250 300 350 400 450 500-10

-5

0

5

10OBG=3

u

v

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4

5

6

7

8

9

/

|NT

F(e

j )|

NTF response


Systematic NTF Design Example

Specifications SQNR > 120 dB

A signal bandwidth which results in an OSR = 64

• Study optimal clock rate for the given process and quantizer design.

Designer’s Choice Order = 3

Quantizer levels (nLev) = 16

Butterworth high-pass response for the NTF

Use MATLAB for finding coefficients of the HPF response. [b,a] = butter(order, ω3dB, ‘high’) The cutoff frequency ω3dB specifies the transfer function.


Systematic NTF Design contd.

Start with cutoff frequency ω3dB=π/8, for the butterworth HPF H(z).

Derive a realizable NTF using NTF(z)=H(z)/b0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

/

Magnitude

H(z)

NTF(z)



Map the NTF response to a loop-filter architecture (details later)

Simulate the modulator for all possible amplitudes and input tone frequencies.

Compute the peak SQNR and MSA. simulateDSM function in the toolbox.

Can use Risbo’s method shown later



Peak SNR = 107 dB

MSA = 0.9

-100 -80 -60 -40 -20 00

10

20

30

40

50

60

70

80

90

100

110

120

130

140

Input Level, dB

SN

R d

B

peak SNR = 107.2dB



If SNR is not enough, repeat the entire procedure with a higher cutoff frequency for the Butterworth HPF IBN ↓, SQNR ↑

OBG ↑ and MSA ↓

If SNR is too high, repeat the entire procedure with a lower cutoff frequency for the Butterworth HPF IBN ↑, SQNR ↓

OBG ↓ and MSA ↑



ω3dB=π/4.

Peak SNR = 119 dB, OBG = 2.25, MSA = 0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

/

Magnitude

H(z)

NTF(z)

-100 -80 -60 -40 -20 00

10

20

30

40

50

60

70

80

90

100

110

120

130

140

Input Level, dB

SN

R d

B

peak SNR = 119.7dB



ω3dB=2π/7.

Peak SNR = 121 dB, OBG = 2.54, MSA = 0.8. Design closed !

-100 -80 -60 -40 -20 00

10

20

30

40

50

60

70

80

90

100

110

120

130

140

Input Level, dB

SN

R d

B

peak SNR = 121.1dB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

/

Magnitude

H(z)

NTF(z)



An advanced version of this iterative process is implemented as the function synthesizeNTF in the delta-sigma Toolbox. Several ‘opt’ params for NTF zero (and pole) optimization

Use synthesizeChebyshevNTF for low OSR and low OBG designs.


NTF-Zero Optimization

Spread zeros in the signal band to minimize in-band noise Complex zeros on the unit circle

8dB increase in SQNR for 3rd order modulator

Bandwidth normalized NTF-zero locations obtained by toolbox function ds_optzeros(order, 1)

Already implemented in synthesizeNTF function for opt=1

10-3

10-2

10-1

-120

-100

-80

-60

-40

-20

0

20

/

|NT

F(e

j )|

NTF response

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Pole-Zero Map

Real Axis

Imagin

ary

Axis


Estimating MSA (Maximum Stable Amplitude)

MSA is found through extensive simulation

Simulate for input sinusoids of varying amplitudes for all possible signal frequencies in the signal band. For every input amplitude compute in-band SNR.

Beyond the MSA, the NTF poles move out of the unit circle.

Noise shaping is disrupted and the in-band SNR drops.

At this point the quantizer input (y[n]) blows up.

simulateSNR function in the toolbox does exactly the same

Time consuming and often impractical for iterative design


Estimating MSA using Risbo’s Method

Use a slow ramp input from 0 to FS value. Plot log10|y[n]|. Observe where this plot blows up.

Take 90% of the input amplitude where log10|y[n]| blows up as a conservative estimate for MSA.

Estimated MSA is close to that predicted by the sinewave input method.

Much quicker than the sinewave technique (simulateSNR function)

y v

u

L

t0

FS

Slow ramp input


Estimating MSA using Risbo’s Method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6

-4

-2

0

2

4

6

8

10

12

14

log |y|

Input, u

MSA = 0.843

MSA Estimation by Simulation


Loop-Filter Architectures

Several loop-filter discrete-time architectures possible

Toolbox function realizeNTF maps the synthesized NTF to loop-filter co-efficients [a,g,b,c] = realizeNTF(H, form);

Dynamic Range Scaling (DRS) performed to scale loop-filter states to a bounded value Scaling performed using ABCD matrix representation of the loop-

filter

See any introductory text on Linear Systems ABCD = stuffABCD(a,g,b,c,form);

[ABCDs umax] = scaleABCD(ABCD, nLev, f0, xLim);

[a,g,b,c] = mapABCD(ABCDs, form);


CIFB (Cascade of Integrators with Distributed Feedback)

Cascade of delaying integrators: Feedback coefficients a’s realize the zeros of L1 and thus the

NTF and STF poles.

Feed-in coefficients b’s determine zeros of L0 and thus the STF zeros.

State scaling coefficients c’s are used for dynamic range scaling.

-g1

c1

-a1 -a2

x1(n) x2(n) y(n)c2

z-1

1-z-1

z-1

1-z-1

b2b1

DAC

v(n)

-a3

x3(n)c2

z-1

1-z-1

b3

-a4

b4

u(n)


CRFB (Cascade of Resonators with Distributed Feedback)

Combine a non-delaying and a delaying integrator with local feedback around them, to form a stable resonator Local feedback coefficients g’s realize the complex zeros in the

NTF.

Implements NTF with complex zeros.

For odd-order, use an integrator in the front to avoid noise coupling due to g

-g1

c1

-a1 -a2

x1(n) x2(n+1) y(n)c2

z-1

1-z-1

1

1-z-1

b2b1

DAC

v(n)

-a3

x3(n)c2

z-1

1-z-1

b3

-a4

b4

u(n)


CIFF (Cascade of Integrators with Feed-Forward Summation)

Feedforward summation of states

For b1=bN+1=1 and b2 to bN=0, STF=1 Loop-filter only processes quantization noise, low power and

distortion

Feedforward loop-filters typically result in lower-power implementation

DAC

-g1

c2

-c1

x1(n) x2(n) y(n)c3

v(n)

x3(n)a3

a2

a1

z-1

1-z-1

z-1

1-z-1

z-1

1-z-1

b2b1 b3 b4

u(n)


CRFF (Cascade of Resonators with Feed-Forward Summation)

Use resonators with feedforward summation Implements NTF with complex zeros

For odd-order, use an integrator in the front to avoid noise coupling due to g

-g1

c1

-a1 -a2

x1(n) x2(n) y(n)c2

z-1

1-z-1

1

1-z-1

-b2-b1

DAC

v(n)

-a3

x3(n)c2

z-1

1-z-1

-b3

-a4

-b4

u(n)


ΔΣ Modulator Architectures

Cascade/ MASH architecture:

•Eg. Two first order modulators are used to implement second order

modulator.

•Stability concerns are relaxed but mismatch in the two forward paths

should be properly monitored.


ΔΣ Modulator Architectures

Feedforward modulators

•Most popular architecture.

•Input signal is summed at Nth stage integrator output.

•Summation block may be required at higher order modulators.

•Multibit quantizer is necessary.


Key Terminologies :

SQNR – Signal to quantization noise ratio Thermal/electrical noise are not included.

SNR - Signal to noise ratio Distortion is not included.

SDNR - Signal to noise and distortion ratio All noise sources are included.

ENOB – Effective number of bits (resolution) This is very important than actual number of output bits

Dynamic Range (DR) Measured with input of the modulator shorted.

Harmonic Distortion THD is usually total harmonic distortion. Or Third??

Spur Free Dynamic Range (SFDR) Very key parameter in communication systems


Frequency Domain Measurements

Dynamic range

Peak SNR

Peak SNDR


Spurious (tone) Free Dynamic Range (SFDR)

SFDR


References

Delta-Sigma Data Converters B.1 R. Schreier, G. C. Temes, Understanding Delta-Sigma Data Converters,

Wiley-IEEE Press, 2005 (the Green Bible of Delta-Sigma Converters). B.2 S. R. Norsworthy, R. Schreier, G. C. Temes, Delta-Sigma Data Converters:

Theory, Design, and Simulation, Wiley-IEEE Press, 1996 (the Yellow Bible of Delta-Sigma Converters).

B.3 S. Pavan, N. Krishnapura, “Oversampling Analog to Digital Converters Tutorial,” 21st International Conference on VLSI Design, Hyderabad, Jan, 2008.

B.4 S. H. Ardalan, J. J. Paulos, “An Analysis of Nonlinear behavior in Delta-Sigma Modulators,” IEEE TCAS, vol. 34, no. 6, June 1987.

B.5 R. Schreier, “An Empirical Study of Higher-Order Single-Bit Delta-Sigma Modulators,” IEEE TCAS-II, vol. 40, no. 8, pp. 461-466, Aug. 1993.

B.6 J. G. Kenney and L. R. Carley, “Design of multibit noise-shaping data converters,” Analog Integrated Circuits Signal Processing Journal, vol. 3, pp. 259-272, 1993.

B.7 L. Risbo, “Delta-Sigma Modulators: Stability Analysis and Optimization,” Doctoral Dissertation, Technical University of Denmark, 1994 [Online].

B.8 R. Schreier, J. Silva, J. Steensgaard, G. C. Temes,“Design-Oriented Estimation of Thermal Noise in Switched-Capacitor Circuits,” IEEE TCAS-I, vol. 52, no. 11, pp. 2358-2368, Nov. 2005.

http://eivind.imm.dtu.dk/publications/PhD_thesis/LarsRisbo_PhD_Part1.ps.gz

delta-sigma analog-to-digital converters sigma adc... · 2016. 9. 17. · delta-sigma (ΔΣ) adc...

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