Delta-Sigma Analog-to-Digital Converters Notes/Delta...© Vishal Saxena-3- Oversampling y v f s 2 f s OSR L PF OSR Dec i m at i o n D o u t @ f s @ f s /OSR 0 50 100 150 200 250 300-8-6-4-2

© Vishal Saxena -1-

Delta-Sigma Analog-to-Digital

Converters

Oversampling Data Converters Basics

Vishal Saxena


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

© Vishal Saxena -4-© Vishal Saxena

Oversampling

0 50 100 150 200 250 300 350 400 450 500-20

-10

0

10

20Oversampling: Time-domain waveforms

u[n]

v[n]

vf[n]

0 50 100 150 200 250 300 350 400 450 500-1

-0.5

0

0.5

1

e[n]

ef[n]

File: oversampling1.m


Oversampling

0 0.1 0.2 0.3 0.4-80

-70

-60

-50

-40

-30

-20

-10

0

10Quantization Noise Spectrum

e2=0.30483

V(f)

0 0.1 0.2 0.3 0.4-80

-70

-60

-50

-40

-30

-20

-10

0

10Filtered Quantization Noise Spectrum

q2=0.0082827

Vf(f)

File: oversampling1.m


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

File: First_Order_DSM.m


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

File: Second_Order_DSM.m


-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


2nd order DSM

-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

0 50 100 150 200 250 300 350 400 450 500-8

-6

-4

-2

0

2

4

6

8

v

u

File: Second_Order_DSM_Zero_Opt.m

Set variable opt=0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.12

0.304

0.728

1.152

1.576

2

2.424

2.848

3.272

3.696

4.12

Normalized Frequency ( rad/sample)

Magnitu

de

Magnitude Response


2nd order DSM: contd.

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



Comparison: 1st and 2nd order modulator waveforms

60 80 100 120 140 160 180 200 220 240

-6

-4

-2

0

2

4

6

DSM time-domain Simulation

860 880 900 920 940 960 980

-6

-4

-2

0

2

4

6

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

OBG = 2

Max LSB jump = 1

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

OBG = 4

Max LSB jump = 3


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

File: Third_Order_DSM.m


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

File: Third_Order_DSM.m


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


2nd order DSM: NTF Zero Optimization

File: Second_Order_DSM_Zero_Opt.m

Set variable opt=1.

-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

0 50 100 150 200 250 300 350 400 450 500-8

-6

-4

-2

0

2

4

6

8

v

u


2nd order DSM: NTF Zero Optimization contd.

10-4

10-3

10-2

10-1

-160

-140

-120

-100

-80

-60

-40

-20

0

SNDR = 79.3 dB

ENOB = 12.89 bits

@OSR = 32

Frequency

dB


•5.5 dB increase in SQNR.

NTF pole (if any) optimization to be discussed later.


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

File: MSA_Risbo_Method.m


Simulation with input with MSA

0 100 200 300 400 500 600 700 800-20

-10

0

10


u

v

0 100 200 300 400 500 600 700 800-20

-10

0

10

20Loop-filter States

x1

x2

y=x3



Simulated SNR with input with MSA

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-250

-200

-150

-100

-50

0

SNDR = 119.3 dB

ENOB = 19.52 bits

@OSR = 64

/

dB

FS

Modulator Output Spectrum



Simulation with input with 1.2*MSA

0 100 200 300 400 500 600 700 800-20

-10

0

10


u

v

0 100 200 300 400 500 600 700 800-50

0

50Loop-filter States

x1

x2

y=x3



Simulation with input with 1.2*MSA

0 100 200 300 400 500 600 700 800-20

-10

0

10


u

v

0 100 200 300 400 500 600 700 800-2

-1

0

1

2x 10

6 Loop-filter States

x1

x2

y=x3



Bode Sensitivity Integral

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

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

/

log|H

|

C1=0.083472, C

2=0.083569

File: BodeSensitivity1.m

Single pole/zero transfer function with pole/zero inside the unit circle.

Area above and below the 0-dB axis are equal.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-160

-140

-120

-100

-80

-60

-40

-20

0

20

/

10 log|N

TF

|

C1=3.27, C

2=3.367


Butterworth NTF.




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70

-60

-50

-40

-30

-20

-10

0

10

20

/

10 log|H

|

C1=3.949, C

2=4.0776


Inverse Chebyshev NTF.




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70

-60

-50

-40

-30

-20

-10

0

10

20

/

10*l

og|H

|


Better in-band performance results in worse out-of-band performance.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70

-60

-50

-40

-30

-20

-10

0

10

20

/

10*l

og|H

|


Complex NTF zeros result in better in-band performance for the same OBG.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70

-60

-50

-40

-30

-20

-10

0

10

20

/

10*l

og|H

|


Higher-order NTF results in better in-band performance for the same OBG.


Loop Filter Architectures


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 SystemsABCD = 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)


CIFB Example 1

CIFB, order = 4

All NTF zeros at z=1, i.e. opt =0.

OBG = 3, OSR = 16, nLev = 15.

Only single input coupling is used

b(2:end) = 0

Maxflat poles in STF

a = [0.16 0.86 1.9 2.1]

b = [0.16 0 0 0]

c = [1 1 1 1]

g = [0 0]

File: CIFB_4th_Order_1.m

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1NTF(z)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1STF(z)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

L0(z)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

L1(z)


CIFB Example 1 contd. : NTF and STF


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

/

Mag

NTF

STF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-150

-100

-50

0

50

/

Mag,

dB

NTF

STF


CIFB Example 1 contd. : Loop-Filter

States


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

-8

-6

-4

-2

0

2

4

6

8

10Transient Simulation

samples, n

u

v

0 100 200 300 400 500 600 700 800 900 1000-20

-15

-10

-5

0

5

10

15

20Integrator States

samples, n

x1

x2

x3

x4


CIFB Example 1 contd. : Simulated

Spectrum


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

SNDR = 80.7 dB

ENOB = 13.11 bits

@OSR = 16

/

dB

FS



Other Examples of Feedback

Topologies


CRFB with single feed-in CRFB_4th_Order_1.m

Low-distortion CRFB topology CRFB_4th_Order_2.m

CIFB with single feed-in and optimized NTF zeros CIFB_Opt_4th_Order_1.m

Low-distortion CIFB topology with optimized NTF zeros CIFB_Opt_4th_Order_2.m


CIFF Example 1

CIFF, order = 4


OBG = 3, OSR = 16, nLev = 15.

Low-distortion topology

b(1) =b(5)= 1

b(2:4)=0

a = [2.1 1.9 0.86 0.16]

b = [1 0 0 0 1]

c = [1 1 1 1]

g = [0 0]


-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1NTF(z)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1STF(z)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

L0(z)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

L1(z)


CIFF Example 1 contd. : NTF and STF

File: CIFF_4th_Order_1.m

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

/

Mag

NTF

STF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-150

-100

-50

0

50

/

Mag,

dB

NTF

STF


CIFF Example 1 contd. : Loop-Filter

States


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

-8

-6

-4

-2

0

2

4

6

8


samples, n

u

v

0 100 200 300 400 500 600 700 800 900 1000-6

-4

-2

0

2

4

6Integrator States

samples, n

x1

x2

x3

x4


CIFF Example 1 contd. : Simulated

Spectrum


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

SNDR = 80.4 dB

ENOB = 13.06 bits

@OSR = 16

/

dB

FS



CIFF Example 2

CIFF, order = 4


OBG = 3, OSR = 16, nLev = 15.

Only single input feed-in used

b(2:end)=0

a = [2.1 1.9 0.86 0.16]

b = [1 0 0 0 0]

c = [1 1 1 1]

g = [0 0]


-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1NTF(z)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1STF(z)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

L0(z)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

L1(z)


CIFF Example 2 contd. : NTF and STF


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

3.5

/

Mag

NTF

STF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-150

-100

-50

0

50

/

Mag,

dB

NTF

STF

Notice the significant STF peaking !


CIFF Example 2 contd. : Loop-Filter

States


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

-8

-6

-4

-2

0

2

4

6

8


samples, n

u

v

0 100 200 300 400 500 600 700 800 900 1000-50

-40

-30

-20

-10

0

10

20

30

40

50Integrator States

samples, n

x1

x2

x3

x4

Last integrator output has significant signal content Use dynamic range scaling.

Last integrator will burn more power in this case.


CIFF Example 2 contd. : Simulated

Spectrum


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

SNDR = 82.1 dB

ENOB = 13.34 bits

@OSR = 16

/

dB

FS



Other Examples of Feed-forward

Topologies

Low-distortion CRFF topology CRFF_4th_Order_1.m

CRFF with single feed-in CRFF_4th_Order_2.m

Low-distortion CIFF topology with optimized NTF zeros CIFF_Opt_4th_Order_1.m

CIFF with single feed-in and optimized NTF zeros CIFF_Opt_4th_Order_2.m

STF peaking in FF topologies with single feed-in is an issue CT FF DSM will have STF peaking as full-feedforward branch can’t be

used.

The feed-in coefficients b’s can be strategically used to realize CIFF/CRFB topology with better out-of-band STF attenuation.


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

Data Conversion Fundamentals

A.1 M. Gustavsson, J. Wikner, N. Tan, CMOS Data Converters for Communications, Kluwer Academic Publishers, 2000.

A.2 B. Razavi, Principles of Data Conversion System Design, Wiley-IEEE Press, 1994.

A.3 ADC and DAC Glossary by Maxim.

A.4 B. Murmann, "ADC Performance Survey 1997-2009," [Online].

A.5 S. Pavan, N. Krishnapura, EE658: Data Conversion Circuits Course at IIT Madras [Online].

A.6 The Fundamentals of FFT-Based Signal Analysis and Measurement, T.I. App Note here.

http://pdfserv.maxim-ic.com/en/an/AN641.pdfhttp://www.stanford.edu/~murmann/adcsurvey.htmlhttp://www.ee.iitm.ac.in/vlsi/courses/ee658_2009/starthttp://www.lumerink.com/courses/ECE697A/docs/The Fundamentals of FFT-Based Signal Analysis and Measurements.pdf


References

Delta-Sigma Data ConvertersB.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


References

CAD for Mixed-Signal Design

D.1 K. Kundert, “Principles of Top-Down Mixed-Signal Design,” Designer’s Guide Community [Online].

D.2 R. Schreier, Matlab Delta-Sigma Toolbox, 2009 [Online], [Manual], [One page summary].

D.3 Jose de la Rosa, “SIMSIDES Toolbox: An Interactive Tool for the Behavioral Simulation of Discrete-and Continuous-time SD Modulators in the MATLAB,” University of Sevilla, Spain, [Contact the authors for the software].

D.4 P. Malcovati, Simulink Delta-Sigma Toolbox 2, 2009. Available [Online].

Example Datasheets

E.1 A 16-bit, 2.5MHz/5 MHz/10 MHz, 30 MSPS to 160 MSPS Dual Continuous Time Sigma-Delta ADC – AD9262, Analog Devices, 2008.

E.2 A 24-bit, 192 kHz Multi-bit Audio ADC – CS5340, Cirrus Logic, 2008.
http://www.designers-guide.org/Design/tdd-principles.pdfhttp://www.mathworks.com/matlabcentral/fileexchange/19http://www.lumerink.com/courses/ECE697A/docs/DSToolbox.pdfhttp://www.lumerink.com/courses/ECE697A/docs/OnePageStory.pdfhttp://www.mathworks.de/matlabcentral/fileexchange/25811-sdtoolbox-2http://www.analog.com/en/analog-to-digital-converters/ad-converters/ad9262/products/product.htmlhttp://www.cirrus.com/en/pubs/proDatasheet/CS5340_F2.pdf

Documents

Delta-Sigma Analog-to-Digital Converters Notes/Delta...© Vishal Saxena-3- Oversampling y v f s 2 f s OSR L PF OSR Dec i m at i o n D o u t @ f s @ f s /OSR 0 50 100 150 200 250 300-8-6-4-2