Design of an Audio Delta - Sigma A/D ConverterHari Prasath Venkatram

Contents

I Introduction 1

II Architecture 1

III Modulator Design - I CRFB - using MAT-LAB Toolbox 1

IV Modulator Design - II -CIFF using MAT-LAB Toolbox 2

V Switched-Capacitor Realization and kT/CNoise 3

V-A Switched-Capacitor Implementation . . . . 4

VI Spectre-Verification 4

VIIDecimation Filter 9VII-AHogenauer Filter Structure . . . . . . . . . 9

VIIINon-Ideal Effects 9VIII-AFinite-DC Gain . . . . . . . . . . . . . . . . 9VIII-BBandwidth and Slew-Rate . . . . . . . . . . 12VIII-CAnalog-Noise . . . . . . . . . . . . . . . . . 12VIII-DCapacitor Mismatch - D/A . . . . . . . . . 12

VIII-D.1Butterfly Randomization . . . . . . 12VIII-D.2First-Order Mismatch Shaping . . . 13

VIII-EDigital Truncation Errors . . . . . . . . . . 13

IX Conclusion 14

I. Introduction

Parameter ValueSignal BW 0-20 kHzClock Freq ≤ 5.2 MHzAccuracy ≥ 18 bit

II. Architecture

The maximum oversampling ratio for the given signalbandwidth and the clock frequency is 128. For OSR of 128,several modulators were simulated with 1-4 bit quantizers.Modulator Order Vs Peak SNR for different Quantizers isshown in the figure 1. From the figure 1, the 3-order Mod-ulator with 2 bit quantizer has a peak SNR of 122 dB. The2-nd Order Modulator with 4-bit quantizer has a peak SNRof 116 dB. 3-rd Order Modulator with 2-bit quantizer waschosen to reduce the complexity of the quantizer used in thedelta-sigma modulator. The SQNR for the chosen 3-rd Or-der Modulator is 122 dB. Thus, the design is thermal noiselimited. The complexity of the Dynamic Element Match-ing Logic is reduced by the choice of a 2-bit quantizer. The

1 2 3 4 5 640

60

80

100

120

140

160

Modulator Order

Pea

kSN

Rin

dB

Modulator Order Vs Peak SNR, OSR = 128

1 bit Quantizer 2 bit Quantizer 3 bit Quantizer 4 bit Quantizer

Fig. 1. Order Vs SNR, OSR = 128

Out-Of-Band Gain H∞ = 2 was chosen to allow maximuminput to the quantizer. The advantages of having a low-pass nature for the STF in the CIFB and CRFB modulatorcomes at the cost of large capacitors required for their real-izations. After my initial design in MATLAB for CRFB, Ifound that the capacitors required are very large and hencechanged the modulator architecture to CIFF which gives areasonable values for the capacitors. The following sectionexplains my initial CRFB design and then a CIFF designwhich was implemented in Spectre.

III. Modulator Design - I CRFB - using MATLABToolbox

Among the different architectural choices CIFB,CIFF,CRFBand CRFF, I initially chose CRFB for the following rea-sons [1].• The flicker noise corner for deep-sub micron process arein the order of MHz range. Hence, the STF was chosen tohave a maximally flat lowpass nature to filter any out-bandnoise or tones.• Further, NTF zero optimization was performed to im-prove SQNR by 8-dB when compared to NTF without zerooptimization. This allows me to use a 4 level quantizerinstead of a 9 level quantizer without zero optimizationand reduce the digital logic required for dynamic elementmatching.Further, the dynamic-range scaled coefficients were con-verted to rational fractions. The quantized coefficientscause a peaking of 0.01 dB in the in-band and the over-all peaking in the STF is less than 0.03 dB. Hence, I chosethese quantized coefficients to realize the capacitor ratiosfor the circuit-level implementation.

Z-1

1 - Z-1c1b1

u

a1 a2 a3

Z-1

1 - Z-1c2

1

1 - Z-1c3

g1

x1 x2

x3

y v

TABLE I

Unscaled - CRFB Modulator Coefficients

i ai gi

1 0.1617253168531 3.614248218646310e-042 0.5797742477013 03 0.7504538491571 0

i bi ci

1 0.1617253168531 12 0 13 0 1

TABLE II

Dynamic Range Scaled -CRFB Modulator Coefficients

i ai gi

1 0.255504864607152 9.155624560380091e-042 0.539232455621136 03 0.275531395915048 0

i bi ci

1 0.255504864607152 0.5887025816652572 0 0.3947571457098133 0 2.72366002670908

The choice of this modulator is not optimum in termsof practical realization. The capacitors required for realiz-ing this modulator coefficients are very large. Therefore,I changed my design to CIFF architecture. The follow-ing sections explain the similar design procedure for CIFFmodulator. [1]

TABLE III

Dynamic Range Scaled -CRFB Quantized Modulator

Coefficients

i ai gi bi ci

1 0.25 0.001 0.25 0.62 0.5 0 0 0.43 0.25 0 0 2.7

−120 −100 −80 −60 −40 −20 0

20

40

60

80

100

120

140

Amplitude in dB

SQ

NR

indB

SQNR Vs Amplitude , OSR = 128, H∞=2

−6 −4 −2 0

115

120

125

130

Fig. 2. Amplitude Vs SNR, OSR = 128

−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

Real Axis

Imagin

ary

Axis

Pole - Zero Plot

Fig. 3. Pole - Zero Map

IV. Modulator Design - II -CIFF using MATLABToolbox

Among the different architectural choices CIFB,CIFF,CRFBand CRFF, I chose CIFF for the following reasons.

• The Integrators process only the quantization noise,hence the linearity requirements for the integrators are lesswhen compared to CIFB .• Further, NTF zero optimization was performed to im-prove SQNR by 8-dB when compared to NTF without zero

10−3

10−2

10−1

−140

−120

−100

−80

−60

−40

−20

0

Normalized Frequency

dB

NTF w/ and w/o Zero Optimization for 3-rd Order Modulator

w/ Zero Optimization

w/o Zero Optimization

Fig. 4. NTF - Frequency Response

0 0.1 0.2 0.3 0.4 0.5−140

−120

−100

−80

−60

−40

−20

0

20

Normalized Frequency

dB

NTF and STF for 3-rd Order Modulator

NTF

STF

1 1.5 2 2.5 3 3.5

x 10−3

−140

−120

−100

−80

−60

Fig. 5. STF - Maximally Flat Response, NTF

optimization. This allows me to use a 4 level quantizerinstead of a 9 level quantizer without zero optimizationand reduce the digital logic required for dynamic elementmatching.• However, The CIFF architecture requires a active adderbefore the quantizer.

Further, the dynamic-range scaled coefficients were con-verted to rational fractions. The quantized coefficients andthe modified NTF and STF are shown. The scaling ofx3 and x2, leads to very small values of ’g’. Therefore,x3 and x2 were aggressively scaled. I played around withthe DR-scaled coefficients to arrive at the quantized ratio-nal fractions. There is negligible change in the modulatorperformance even after the quantized coefficients. The co-efficients of CIFF modulator at the summing node beforethe quantizer is realized using a switched-capacitor block.Therefore, the coefficients are normalized with common-

TABLE IV

Unscaled - CIFF Modulator Coefficients

i ai gi

1 1.330589521680359 3.613921648891427e-042 0.741499597211506 03 0.161244452225379 0

i bi ci

1 1 12 0 13 0 14 1 -

TABLE V

Dynamic Range Scaled -CIFF Modulator Coefficients

i ai gi

1 1.628625880702360 0.0013134623524472 1.758748119141032 03 1.390005133904675 0

i bi ci

1 0.817001336799664 0.8170013367996642 0 0.5160413542922583 0 0.2751446695185404 1 -

denominator.

V. Switched-Capacitor Realization and kT/CNoise

Considering the switch-ON resistance and the amplifiernoise in fig 13, the total noise power across C1 is given by,

v̄2c1 =

kT

C1

(

1 +x

1 + x+

4

3(1 + x)

)

, x = 2Rongm1

The noise power can be minimized if x ≫ 1. Under theseassumptions, the noise at the input of the first-integratoris 2kT

C1

. The modulator maximum input signal is -2dBFS.Assuming that the modulator has -3 dBFS signal and thethermal noise contribution is about 75%, the capacitancerequired to meet SNR of 110 dB with OSR of 128 is

C1 =4

3

2kT × 10110/10

0.25 × OSR= 34.4pF (1)

TABLE VI

Dynamic Range Scaled -CIFF Quantized Modulator

Coefficients

i ai gi bi ci

1 32/20 0.001 4/5 4/52 17/20 0 0 13 14/20 0 0 1/44 - - 20/20 -

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

Un-S

cale

dC

oeff

Integrator Output States

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

DR

-Sca

led

Coeff

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

Normalized Input

Quanti

zed

Sca

led

Coeff

Fig. 12. Dynamic - Range Scaled State Outputs

Z-1

1 - Z-1c2

u

c1

Z-1

1 - Z-1c3

1 - Z-1a3

g1

x1 x2x3 y v

b1 b4

Z-1

D/A

a2

a1

The noise contribution from the second and third integra-tors is very small and unit capacitors were used to realizethe ratio required for the modulator coefficients. The ca-pacitor values and the switched-capacitor implementationis shown in fig. 14 and fig. 16.

A. Switched-Capacitor Implementation

The CIFF modulator was implemented in Spec-tre [2]. The three integrators(fig(14)), active summer,quantizer(fig(16)) and timing(fig(15)) diagram are shownas follows,

VI. Spectre-Verification

The CIFF-3rd Order Modulator was simulated in Ca-dence using macro-models. The simulated model is a dif-

ferential version, whereas the single-ended version is shownin fig 14 for simplicity. Voltage scaling was performed totransform the matlab-model to spectre. The input rangewas set to ±1 Volt and the integrator swings were alsorestricted to ±1 V. The artifacts that were noticeable inthe modulator spectrum are the fifth and seventh harmon-ics. This can be attributed to the non-linear quantizermodel.The Spectre model deviates from the MATLAB PSDat regions close to fs/2. The Out-of-band Modulator spec-trum is not flat. I think this artifact was due to the finite-switch resitance and the non-linear variation of the switchresistance in the cadence model. Further, I used a mid-risequantizer instead of mid-tread quantizer. The simulatedmodulator output spectrum is shown in the fig 17. Thetime-domain simulation output is shown in fig( 18).

−

+

C8

C10

Φ1

Φ1

Φ2Φ2X2

C7

Φ1

Φ2X1

Y

C9

Φ1

Φ2X3

C10

Φ2

Φ1Vin+

Φ2

V

Int / Sum

I - 1

Capacitance(pF)

I- 2

I - 3

Active

C1 = Cd1,2= 50 pF , C2 = 62.5 pF

C3 = C4 = 5 pF

C5 = 0.3 pF, C6 = 1.2 pF

C7 = 3.2 pF, C8 = 1.7 pFC9 = 1.4 pF, C10= C11 = 2 pFSummer

Total-Capacitance - 182.3 pF

Fig. 14. CIFF- SC-Sum and Quantizer

Φ2 Φ1 Φ1 Φ1 Φ1Φ2 Φ2 Φ2 Φ2

X1(n) X1(n+1) X1(n+2) X1(n+3) X1(n+4)X1(n-1)

X2(n) X2(n+1) X2(n+2) X2(n+3) X2(n+4)X2(n-1)

X3(n) X3(n+1) X3(n+2) X3(n+3) X3(n+4)X3(n-1)

V(n) V(n+1) V(n+2) V(n+3)

X1(n) X1(n+1) X1(n+2) X1(n+3) X1(n+4)X1(n-1)

Timing Diagram

Fig. 15. CIFF- SC-Timing

− +

C1

C2

Φ1

Φ1

Φ2

Φ2

Vin

-

− +

C3

C4

Φ1

Φ1

Φ2

Φ2

− +

C5

C6

Φ1

Φ1

Φ2

Φ2

C1,

2Φ

1

Φ1

Φ2

Φ2

Vda

c1,2

x 1x 2

x 3

Φ1

Φ2

Fig.16.CIFF-SC-Integrators

−120 −100 −80 −60 −40 −20 0

20

40

60

80

100

120

140

Amplitude in dB

SQ

NR

indB

SQNR Vs Amplitude , CIFF, OSR = 128, H∞=2

−6 −4 −2 0 2 4

115

120

125

130

Fig. 6. Amplitude Vs SNR, OSR = 128

−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

Real Axis

Imagin

ary

Axis

Pole-Zero Plot of NTF

Fig. 7. Pole - Zero Map, CIFF

10−3

10−2

10−1

−140

−120

−100

−80

−60

−40

−20

0

Normalized Frequency

dB

NTF w/ and w/o Zero Optimization for 3-rd Order Modulator

Fig. 8. NTF - Frequency Response, CIFF

0 0.1 0.2 0.3 0.4 0.5−120

−100

−80

−60

−40

−20

0

20

Normalized Frequency

dB

NTF and STF for 3-rd Order Modulator

0.005 0.01 0.015

−120

−100

−80

−60

−40

Fig. 9. STF, NTF, CIFF

0 100 200 300 400 500−3

−2

−1

0

1

2

3

Normalized Input Points

Modula

tor

Outp

ut

CIFF-Modulator Output

V

U

Fig. 10. Modulator Input/Output - Time Domain Simulation

0 0.1 0.2 0.3 0.4 0.5−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

Normalized Frequency

Modula

tor

Outp

ut

Spec

trum

indB

CIFF Modulator Spectrum

0 2 4 6 8

x 10−3

−200

−150

−100

−50

0

Fig. 11. PSD - Simulated and Expected, CIFF

−

+

C1

C2

Φ1

Φ1

Φ2

Φ2

ΑVin

Vout

Φ1 Φ2 Φ1 Φ2Φ2

Φ1

CL

Fig. 13. Integrator-Noise

10−2

10−1

−180

−160

−140

−120

−100

−80

−60

−40

−20

Cadence and MATLAB Model Verification

Normalized Frequency

Mod

ulat

or O

utpu

t Spe

ctru

m in

dB

10−1

−80

−60

−40

MATLAB Model

Spectre Model

Ideal NTF

5th Harmonic7th Harmonic

Fig. 17. Spectre-Verification

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

x 10−6

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time in second(s)

Sta

teO

utp

uts

inV

Spectre-Modulator Output

X1

X2

X3

Y

Fig. 18. State Outputs

VII. Decimation Filter

The Decimation filter should satisfy the following crite-ria,

• The composite response of the NTF and the decimationfilter should be a low-pass transfer function.• The gain response should be flat and supress harmonicsat fs/OSR.

A. Hogenauer Filter Structure

As a first-cut design solution, a 4th order HogenauerSinc-Filter structure was implemented with downsamplingof 128. The droop in the pass-band edge is around 15 dB.The filter response and decimated time domain output isshown in fig VII-A and fig 20 respectively.

10−3

10−2

10−1

−160

−140

−120

−100

−80

−60

−40

−20

0

Normalized Frequency

Magnit

ude

indB

Decimation Filter and NTF Response

Hogenauer Sinc4 Filter Structure

CIFF,3rd Order Modulator NTF

Signal−Band Edge

Downsample by 128. Notches @nf

s/128, n = 1,2.....

15 dB Droop at band−edge

Fig. 19. Sinc4 Hogenauer Structure

Therefore, Decimation filter was designed in two-steps.First, A 4th order Sinc-filter was implemented with down-sampling of 4 [3]. This was followed by a 8th order Ellip-tic filter with downsampling of 32 to give the data at theNyquist rate. The Filter-response normalized to the sam-pling frequency fs is shown in the fig 22. Note that thereare two-different set of axes in the fig 22. The decimationfilter-block is shown in fig 21. The entire A/D model wassimulated in MATLAB and Cadence. The power-spectraldensity of Spectre-Model and Hogenauer decimated outputis shown in fig(24) and fig(23) are shown below.

0 0.5 1 1.5 2 2.5 3 3.5

x 10−3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Am

plitu

de

inV

Decimated A/D Time-Domain Output at Nyquist Rate

Fig. 20. Decimated time domain output

Sinc4 Filter 4 Elliptic - 8th32

Order Filter

1.28 MHz∆Σ Output 40 kHz

Decimation Filter

1 - Z-1

1

1 - Z-1

1

1 - Z-1

1

1 - Z-1

1

1 - Z-11 - Z-11 - Z-11 - Z-1

KK = (1/128)4

128

∆Σ Output

Yout

Hogenauer Filter Structure

Fig. 21. Decimation in 2-steps, Sinc4,Elliptic Filter

VIII. Non-Ideal Effects

A. Finite-DC Gain

The finite DC-gain for the non-inverting integrator isanalyzed as follows [4],

Vo(z)

Vin(z)=

C1

C2(1 + 1Adc

)

Z−1

1 + C1

C2(A+1) − Z−1

Therefore, the zero is shifted from DC to C2(A+1)C2(A+1)+C1

. If

the signal bandwidth is much higher than the above cornerfrequency, the effect of DC-gain on modulator output isminimal. This condition translates to Adc(π + 2) ≫ OSR.However, the derivation assumes the integrator is linear.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−160

−140

−120

−100

−80

−60

−40

−20

0

Normalized Frequency, f/fs

Magnit

ude

indB

0 0.1

−120

−80

−40

0

Normalized Frequency, f/fs

Magnit

ude

indB

8th Order Elliptic Filter, downsample by 32

NTF

Sinc4, downsample by 4

Fig. 22. Decimation in 2-steps, Sinc4,Elliptic Filter

1000 2000 3000 4000 5000 6000 7000

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

A/D Spectrum after Decimation

Frequency in Hz

Spec

trum

indB

A/D SpectrumCumulate SNR

SNR = 133.8 dBENOB = 21.94 bits

Fig. 23. Decimated A/D Spectrum

103

104

105

106

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

Frequency in Hz

Modula

tor

Spec

trum

indB

Spectre-Model and Cumulate SNR

Band-Edge

-122 dB, 20 bit ENOB

Fig. 24. PSD-Spectre Model

−

+

C1

C2

Φ1

Φ1

Φ2

Φ2

Α

-V/A

Vin

V

C1 C2

Φ1

Φ2

Vin[n-1]C1 V[n-1](1+1/A)

C1

C2

V[n-1/2](1/A) V[n-1/2](1+1/A)C2

Φ1 Φ2 Φ1 Φ2Φ2

103

104

105

106

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0PSD of a 3rd-Order Sigma-Delta Modulator

Frequency [Hz]

PSD

[dB

]

Adc = 25

Adc = 38

Adc = 100

Adc = 500

Adc = 1000

Fig. 25. Finite DC-Gain Effect

25 30 35 40 45 50 55 6090

95

100

105

110

115

120

125

130

DC-Gain in dB

SN

Rin

dB

SNR Vs DC-Gain

Fig. 26. Finite DC-Gain Effect

B. Bandwidth and Slew-Rate

The finite-bandwidth of the integrator causes a gain-error dependant on the bandwidth of the integrator [5].However, If the bandwidth of the integrator is very-low themodulator might become unstable. When compared to theeffect of slew-rate on modulator output , the bandwidthcauses only a linear gain error. The slew-rate effect causesnon-linearities at the modulator output. The integratoroutput histogram is shown in the figure 28. If the inte-grators are allowed to slew for only 20% of the Ts/2, theslew-rate requirement is 51 V/µs. The Slew-rate causesnon-linear distortion in the modulator output spectrum.Fig. 29. The output of the integrator is expressed as thefollows,

vo(kT + t) = vo(kT ) +C1

C2(1 + 1Adc

)u(kT )(1 − e−t/τ )

The maximum rate of the above expression is C1u(kT )C2×τ .

Therefore, the integrator will slew if the above maximumrate is larger than the integrator slew-rate.

104

105

−180

−160

−140

−120

−100

−80

−60

−40

−20

Frequency in Hz

Modula

tor

Spec

trum

indB

PSD Vs GBW, CIFF

GBW = 4 MHz

GBW = 5 MHz

GBW = 10 MHz

GBW = 15 MHz

Fig. 27. Gain-Bandwidth Effect

C. Analog-Noise

Considering the switch-ON resistance and the amplifiernoise in fig 13, the total noise power across C1 is given by,

v̄2c1 =

kT

C1

(

1 +x

1 + x+

4

3(1 + x)

)

, x = 2Rongm1

By having larger gmRon product, the contribution from theamplifier can be reduced [1]. Further, the noise-transferfunction from each integrator input to the output is calcu-lated from the figure(30), as follows,

B(z) =a1z

−1

1 − z−1+

a2c2z−2

(1 − z−1)2+

c2c3a3z−3

(1 − z−1)3(2)

−0.5 0 0.50

2000

4000

x1 - state

No

ofhit

s

Integrator Output Histogram

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30

2000

4000

x2 - state

No

ofhit

s

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20

2000

4000

x3 - state

No

ofhit

s

Fig. 28. integrator output histogram

104

105

−160

−140

−120

−100

−80

−60

−40

−20

Frequency in Hz

Pow

erSpec

tralD

ensi

ty

Modulator Spectrum - Effect of Slew Rate

SR = 1 V/µ sSR = 5 V/µ sSR = 40 V/µ s

Fig. 29. Slew-Rate Effect

C(z) =a2z

−1

1 − z−1+

a3c3z−2

(1 − z−1)2(3)

D(z) =a3z

−1

1 − z−1(4)

(5)

vn1(z)

vo(z)=

B(z)

1 + B(z),vn2(z)

vo(z)=

C(z)

1 + B(z)(6)

vn3(z)

vo(z)=

D(z)

1 + B(z)(7)

(8)

The noise transfer function from the first-integrator dom-inates the total noise power at the output of the modulator.

D. Capacitor Mismatch - D/A

D.1 Butterfly Randomization

Butterfly randomization was implemented for a four-level D/A with 7 unary elements for 10-bit and 12-bit linear

Z-1

1 - Z-1c2

u

c1

Z-1

1 - Z-1c3

1 - Z-1a3

g1

x1 x2x3 y v

b1 b4

Z-1

D/A

a2

a1

vn,1

vn,2 vn,3 vn,4

Fig. 30. CIFF-Noise

0 5 10 15 200

2

4Thermometer-Coding

0 5 10 15 200

2

4First-Order Shaping

Fig. 31. Element Selection Logic

feedback D/A Converter [5]. The modulator output spec-trum is shown in fig 33. The SNR determined by just themismatch error and oversampling ratio is

SNR =3M

OSR × σ2x

D.2 First-Order Mismatch Shaping

First-Order Mismatch Shaping switching sequence isshown in the figure 31 and mismatch shaped 8-bit and 10-bit linear feedback D/A. The first-order mismatch shapingand DWA gives similar switching sequences [1].

E. Digital Truncation Errors

For 120 dBFS truncation noise, the word lengths re-quired for the Sinc-Filter are

∆2

12 × OSR≤ 10−12

× 0.25/2 ⇒ Word − Length = 19 (9)

For Second-Integrator

∆2

12 × OSR3≤ 10−12

× 0.25/2 ⇒ Word − Length = 12 (10)

The Decimated A/D spectrum for different word-lengthsis shown in fig 34

0.5 1 1.5 2 2.5 3 3.5

x 10−3

−140

−120

−100

−80

−60

−40

−20

DAC−Mismatch and Butterfly Randomization (Signal Band)

Normalized Frequency

Sm

ooth

ed M

odul

ator

PS

D in

dB

10−bit D/A12−bit D/A10−bit D/A w/ Butterfly Randomization

Fig. 32. Butterfly Randomization

10−2

−160

−140

−120

−100

−80

−60

−40

−20

Sm

ooth

edP

SD

indB

Normalized Frequency

First-Order Mismatch Shaping - CIFF 3rd Order Modulator Output

8−b D/A

8−b D/A, 1st Shaping

ideal DAC10−b D/A

10−b D/A, 1st Shaping

Fig. 33. First-Order Shaped Modulator Spectrum

103

104

−160

−140

−120

−100

−80

−60

−40

−20

Frequency in Hz

A/D

Outp

ut

Spec

trum

indB

A/D Output Spectrum after Decimation with Sinc4 Filter

No Truncation Error

16-bit Word Length

20-bit Word Length

10-bit Word Length

Fig. 34. A/D Spectrum - Digital Truncation Errors

Architecture CIFF

Order 3rd

OSR 128Bandwidth 0-20 kHz

Clock Frequency 5.12 MHzAccuracy 21 bitsPeak SNR 122 dBMax InputAmplitude 0.7 Vp

TotalCapacitance 183.2 pF

Quantizer Levels 5Adc ≥ 100

Slew-Rate 25V/µsBandwidth ≥ 7 MHz

D/A Mismatch 10-bitDecimation Hogenauer,Elliptic

Register Length 19,12

TABLE VII

Performance Summary

References

[1] R. S. G. C. Temes, Understanding Delta-Sigma Data Converters.IEEE Press, 2005.

[2] R. Gregorian and G. C. Temes, Analog MOS Integrated Circuitsfor Signal Processing. Wiley Series on Filters, 1986.

[3] E. Hogenauer, “An economical class of digital filters for decima-tion and interpolation,” Acoustics, Speech and Signal Processing,IEEE Transactions on, vol. 29, no. 2, pp. 155–162, Apr 1981.

[4] G. Suarez, M. Jimenez, and F. Fernandez, “Behavioral Model-ing of Switched Capacitor Integrators with Application to ∆ −ΣModulators,” vol. 2, Aug. 2006, pp. 709–713.

[5] F. Maloberti, Data Converters. Springer, 2007.

IX. Conclusion

A 3rd order CIFF Audio-band Modulator was designedin cadence and MATLAB. Spectre macro-models were usedfor simulating the high-level modulator. The various non-ideal effects like slew-rate, bandwidth, capacitor mismatchwere analyzed using SIMULINK models. The performancesummary is presented above.

%Simulate SNR

clear all;

set(0,’DefaultAxesFontSize’,16,...

’DefaultAxesFontWeight’,’bold’,...

’DefaultAxesFontName’,’times’,...

’DefaultAxesLineWidth’,2,...

’DefaultAxesGridLineStyle’,’:’,...

’DefaultAxesMinorGridLineStyle’,’:’,...

’DefaultLineLineWidth’,2,...

’DefaulttextInterpreter’,’latex’);

nlev = 4;

amp=[-120:5:-20 -17:1:0];

Nfft = 2^14;

finhigh = 31/Nfft;

finlow = 3/Nfft;

t=[0:4*Nfft-1];

OSR=128;

f0=0;

% for order = 1:8

% for bit = 1:4

% nlev=2^bit;

% H=synthesizeNTF(order,128,1,1.5,0);

% [snr,amp]=simulateSNR(H,128,amp,0,nlev,1/(4*128),14);

% snr_peak(order,bit)=max(snr);

% end

% end

% i=1:8

% for j=1:4

% plot(i,snr_peak(:,j));

% hold on

% end

H=synthesizeNTF(3,128,1,2,0);

H1=synthesizeNTF(3,128,0,2,0);

[snr,amp]=simulateSNR(H,128,amp,0,nlev,finhigh,13);

plot(amp,snr,’r’)

[peak_snr1,peak_amp1]=peakSNR(snr,amp);

hold on

[snr,amp]=simulateSNR(H,128,amp,0,nlev,finlow,13);

plot(amp,snr)

[peak_snr,peak_amp]=peakSNR(snr,amp);

xlabel(’Amplitude in dB’)

ylabel(’SQNR in dB’)

title(’SQNR Vs Amplitude , OSR = 128, $H_{\infty}$=2’)

axis([-120 0 10 140])

grid on

figure

%NTF Diagram

[num,den,fs]=tfdata(H,’v’);

[h,w]=freqz(num,den,linspace(1e-3,1,10000));

[num,den,fs]=tfdata(H1,’v’);

[h1,w1]=freqz(num,den,linspace(1e-3,1,10000));

semilogx(w,20*log10(abs(h)),w,20*log10(abs(h1)));

xlabel(’Normalized Frequency’);

ylabel(’dB’);

title(’NTF w/ and w/o Zero Optimization for 3-rd Order Modulator’)

axis([1e-3 0.5 -140 10]);

% Modulator Output and Spectrum

%

form=’CIFF’;

[a,g,b,c]=realizeNTF(H,form);

%b(2:end)=0; %Maximally flat STF;

ABCD=stuffABCD(a,g,b,c,form);

[Ha Ga] = calculateTF(ABCD);

f=linspace(0,0.5,1000);

z=exp(2i*pi*f);

magHa=dbv(evalTF(Ha,z));

magGa=dbv(evalTF(Ga,z));

figure

plot(f,magHa,’b’,f,magGa,’r’)

xlabel(’Normalized Frequency’)

ylabel(’dB’)

title(’NTF and STF for 3-rd Order Modulator’);

figure

plotPZ(Ha,’b’)

hold on

xlabel(’Real Axis’)

ylabel(’Imaginary Axis’)

title(’Pole-Zero Plot of NTF’)

%Time Domain Simulation

u=0.5*(nlev-1)*sin(2*pi*finhigh*t);

v=simulateDSM(u,H,nlev);

n=1:3000;

figure

stairs(t(n),v(n),’b’);

hold on

stairs(t(n),u(n),’m’);

v1=v(end-2^14+1:end);

spec=fft(v1.*hann(Nfft))/(Nfft*(nlev-1)/4);

snr=calculateSNR(spec(1:ceil(Nfft/(2*OSR))+1),31);

NBW=1.5/Nfft;

f=linspace(0,0.5,Nfft/2+1);

Sqq = 4*(evalTF(H,exp(2i*pi*f))/(nlev-1)).^2/3;

figure

plot(f,dbv(spec(1:Nfft/2+1)),’b’);

hold on

plot(f,dbp(Sqq*NBW),’m’);

%Stability of the modulator with quantizer gain

figure

l1 = 1- 1/Ga;

f=linspace(0,0.5,1000);

z=exp(2i*pi*f);

magl1=evalTF(l1,z);

subplot(2,1,1),plot(f,dbv(magl1));

subplot(2,1,2),plot(f,180/pi*angle(magl1));

%Dynamic Range Scaling

xlim = 0.6

[ABCDs umax]=scaleABCD(ABCD,nlev,f0,[3 3 3])

[a1 g1 b1 c1] = mapABCD(ABCDs,form)

figure

u1=0.01:0.01:0.82;

for i=1:82

u=0.01*i*(nlev-1)*sin(2*pi*finhigh*t);

[v xn xmax y] = simulateDSM(u,ABCD,nlev,zeros(3,1));

x1(i)=xmax(1,1);

x2(i)=xmax(2,1);

x3(i)=xmax(3,1);

end

u1=0.01:0.01:0.82;

for i=1:82

u=0.01*i*(nlev-1)*sin(2*pi*finhigh*t);

[v xn xmax y] = simulateDSM(u,ABCDs,nlev,zeros(3,1));

x4(i)=xmax(1,1);

x5(i)=xmax(2,1);

x6(i)=xmax(3,1);

end

%Fractional Realizations

a2=[1.6 .875 0.7];

b2=[0.8 0 0 1];

c2=[0.8 1 0.25];

g2=0;

ABCDf=stuffABCD(a2,g2,b2,c2,form);

u1=0.01:0.01:0.82;

for i=1:82

u=0.01*i*(nlev-1)*sin(2*pi*finhigh*t);

[v xn xmax y] = simulateDSM(u,ABCDf,nlev,zeros(3,1));

x7(i)=xmax(1,1);

x8(i)=xmax(2,1);

x9(i)=xmax(3,1);

end

% hold on

subplot(3,1,1),plot(u1,x1,’b’,u1,x2,’r’,u1,x3,’g’);

hold on

subplot(3,1,2),plot(u1,x4,u1,x5,u1,x6)

hold on

subplot(3,1,3),plot(u1,x7,u1,x8,u1,x9)

hold on

lim1=ones(length(u1))*3;

hold on

subplot(3,1,1),plot(u1,lim1,’m’)

hold on

subplot(3,1,2),plot(u1,lim1,’m’)

hold on

subplot(3,1,3),plot(u1,lim1,’m’)

%NTF and STF after fractional realization

[Ha Ga] = calculateTF(ABCDf);

f=linspace(0,0.5,1000);

z=exp(2i*pi*f);

magHa=dbv(evalTF(Ha,z));

magGa=dbv(evalTF(Ga,z));

figure

plot(f,magHa,’b’,f,magGa,’r’)

xlabel(’Normalized Frequency’)

ylabel(’dB’)

title(’NTF and STF for 3-rd Order Modulator’);

figure

plotPZ(Ha,’b’)

hold on

xlabel(’Real Axis’)

ylabel(’Imaginary Axis’)

title(’Pole-Zero Plot of NTF’)

% Simulate ESL

sigma_d=1/256

v = (v+nlev-1)/2; % scale v to [0,M]

mtf1 = zpk(1,0,1,1); %First-order shaping

sv1 = simulateESL(v,mtf1,nlev);

echo off

T = 20;

figure

subplot(211);

plotUsage(thermometer(v(1:T),nlev));

set(gcf,’NumberTitle’,’off’);

set(gcf,’Name’,’Element Usage’);

title(’Thermometer-Coding’)

subplot(212);

plotUsage(sv1(:,1:T));

title(’First-Order Shaping’);

ideal = v;

% DAC element values

e_d = randn(nlev,1);

e_d = e_d - mean(e_d);

e_d = sigma_d * e_d/std(e_d);

ue = 1 + e_d;

% Convert v to analog form, assuming no shaping

thermom = zeros(nlev+1,1);

for i=1:nlev

thermom(i+1) = thermom(i) + ue(i);

end

conventional = thermom(v+1)’;

% Convert sv to analog form

dv1 = ue’ * sv1;

window = ds_hann(Nfft);

spec = fft(ideal.*window)/(nlev*Nfft/8);

spec0 = fft(conventional.*window)/(nlev*Nfft/8);

spec1 = fft(dv1.*window)/(nlev*Nfft/8);

figure; clf

plotSpectrum(spec0,31,’r’);

hold on;

plotSpectrum(spec1,31,’b’);

plotSpectrum(spec,31,’g’);

sigma_d=1/1024

%v = (v+nlev-1)/2; % scale v to [0,M]

mtf1 = zpk(1,0,1,1); %First-order shaping

sv1 = simulateESL(v,mtf1,nlev+1);

echo off

ideal = v;

% DAC element values

e_d = randn(nlev,1);

e_d = e_d - mean(e_d);

e_d = sigma_d * e_d/std(e_d);

ue = 1 + e_d;

% Convert v to analog form, assuming no shaping

thermom = zeros(nlev+1,1);

for i=1:nlev

thermom(i+1) = thermom(i) + ue(i);

end

conventional = thermom(v+1)’;

% Convert sv to analog form

dv1 = ue’ * sv1;

window = ds_hann(Nfft);

spec3 = fft(conventional.*window)/(nlev*Nfft/8);

spec4 = fft(dv1.*window)/(nlev*Nfft/8);

plotSpectrum(spec3,31,’m’);

plotSpectrum(spec4,31,’c’);

axis([1e-3 0.5 -200 0]);

x1 = 2e-3; x2=1e-2; y0=-180; dy=dbv(x2/x1); y3=y0+3*dy;

plot([x1 x2 x2 x1],[y0 y0 y3 y0],’k’)

text(x2, (y0+y3)/2,’ 60 dB/decade’)

hold off;

grid;

ylabel(’PSD’);

xlabel(’Normalized Frequency’);

legend(’thermometer’,’rotation’,’ideal DAC’);