DOLPHIN INTEGRATION TAMES-2 workshop 23/05/2004 Corsica1 Behavioural Error Injection, Spectral Analysis and Error Detection for a 4 th order Single-loop

TAMES-2 workshop 23/05/2004 Corsica 1

DOLPHININTEGRATION

Behavioural Error Injection, Spectral Analysis and Error

Detection for a 4th order Single-loop Sigma-delta Converter

Using Walsh transformsKostas Georgopoulos, Martin Burbidge,

Andreas Lechner and Andrew Richardson

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DOLPHININTEGRATION

TAMES-2 workshop 23/05/2004 Corsica © Lancaster University

Presentation Overview

The Sigma-Delta A/D Converter

The Walsh functions and Walsh series

Motivation for work

The FFT error simulation analysis

The Walsh error simulation analysis

Conclusions

Future Work

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The Sigma-Delta A/D Converter (I)

A device comprised by three stages

Anti-aliasing filter

Sigma-Delta modulator

Decimation phase

Analogue Input

Digital Output

Anti-aliasing Filtering

Modulator

Decimation Process – Digital Filtering

Modulated Input Signal – High frequency bit stream

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The Sigma-Delta A/D Converter (II)

Sampling at a frequency much higher then the Nyquist

, where fs is sampling frequency and

fi is the input frequency

-

+ x(t) yi

y(t)

INTEGRATOR A/D

D/A

Clock, fs

ii

S

ff

fOSR

2

1

2

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The Sigma-Delta A/D Converter (III)

Transfer function: , where L is the order of modulator

Noise floor is moved out of bandwidth of interest by noise shaping

s1

I1

I2 s2

I3

s3

I4 s4 X(in)

Y(out)

sum

SMASH 4.4.0 - Fast Fourier Transform - Tue Dec 02 11:11:32 2003

BH7(filter_1000.00)

f = 99.06Hz, df = 5.958KHz, y = -8.564, dy = -32.06, period = 167.9us, slope = -0.00538

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Bandwidth of interest, 0-24 kHz

)(*)1()()( 1 zEzzXzzY qLL

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The Walsh Functions (Theory)

Walsh functions form an ordered set of rectangular orthogonal waveforms

Only two amplitude values, +1 and –1

Fast Walsh transforms exist

Any given signal can be represented through the combination of two or more Walsh functions

1

0 1/2 1

-1

T

SAL(4,T)

CAL(3,T)

SAL(3,T)

CAL(2,T)

SAL(2,T)

CAL(1,T)

SAL(1,T)

CAL(0,T)

WAL(7,T)

WAL(6,T)

WAL(5,T)

WAL(4,T)

WAL(3,T)

WAL(2,T)

WAL(1,T)

WAL(0,T)

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The Walsh Series (I)

The Walsh series is similar to the Fourier Series expansion

where parameter α determines the amplitude or weighting of each Walsh function and

),().0()(1

10 tnWALatWALatf

N

nn

T

dttWALtfT

a0

0 ),0()(1

2

T

n dttnWALtfT

a0

),()(1

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The Walsh Series (II)

Walsh functions can also be expressed in terms of even and odd waveform symmetry

, where

Walsh functions SAL and CAL can be visualised as the respective sine and cosine basis functions in Fourier Series

),(),2( tnCALtnWAL

),(),12( tnSALtnWAL

2....2,1

Nn

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The Walsh Series (III)

Employing SAL and CAL functions a Walsh Series similar to the sine-cosine series is given

where

f(t) is the sum of a series of square-wave shaped functions

)),(),((),0()(2/

1

2/

10 tjCALbtiSALatWALatf j

N

i

N

ji

T

j dttjCALtfT

b0

),()(1

T

i dttiSALtfT

a0

),()(1

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Signal Reconstruction using Walsh

A simple case of a sine-wave signal approximated with 3 Walsh functions

41WAL(1,t)

17WAL(5,t)

8WAL(13,t)

[41WAL(1,t) + 17WAL(5,t) + 8WAL(13,t)]

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Motivation & MethodologyFFT converges rapidly to sine wave hence use for classic dynamic performance testing

Walsh converges rapidly to square wave:Idea to use square wave for input to modulator

Walsh transform of bit-stream should give single spectral peak

All other peaks in spectrum are due to noise and non-idealities

Higher potential for on-chip transform of fewer samples

Methodology:

Determine modulator behavior and model parameters that lead to performance failure in FFT domain

Analyse effect of these failure modes on Walsh results

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Use of initial C-based model provided by Dolphin

Ideal model FFT S/(N + THD) results:

Input 2.5 Vpk sine @ 1 kHz

BW approx 24 kHz (150 Hz to 24 kHz)

S/(N+THD) approx 100dB

Next step: Analyse how Walsh transforms could compare to FFT

FFT Analysis Setup

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Fault Set For FFT

Input Offsets

Integrator Gain Variations

Corrupted feedback paths

Presence of noise on the modulator input

Integrating capacitor mismatch

1

1 2

2

CS VON

1

1 2

2

CS

CI

VOP

VO

CI

VI

Faulty capacitor

Gain mismatch

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Effect of Integrator Offset

40 mV offset on the modulator input, S(N/THD) 99,6 dB

50 mV offset on the modulator input, 50 dB drop in S(N/THD)

SMASH 4.4.0 - Fast Fourier Transform - Wed Mar 03 17:44:45 2004

BH7(filter_1000.00)BH7(filter_1000.00)

f = 3.015KHz, df = 7.394KHz, y = -13.7, dy = -24.61, period = 135.3us, slope = -0.00333

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50 mV offset

40 mV offset

Bandwidth of interest, 46 – 24 kHz

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Analyses of Walsh Testing

Setup:

Square wave test stimulus, @ 1.5kHz, 1.9 to 2.3 V amplitude

Bit-stream frequency of 3.072 MHz, analysis on the bit stream with 16384 and 65536 (1-bit) samples.

Analyses:

Investigation into test stimulus accuracy requirements

Assessment of Walsh-based modulator performance tests

Analysis of Walsh test coverage against modulator failure modes

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Test Stimulus Accuracy (I)

Finite rise/fall time: No significant effect

Overshoots:

4% of maximum amplitude, i.e. 0.08 V for 2 V input

16384 samples

Overshoots Damping (ms) SNR (dB)

Ideal - 97.6

Max. amplitude 2.05 V 0.06 95,9

0.14 94.7

Max. amplitude 2.15 V 0.06 95.5

0.14 94.6

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Test Stimulus Accuracy (II)SNR with respect to different input amplitudes

Analysis for both 16384 and 65536 samplesSquare wave amplitude (V) # samples SNR (dB)

1.9 16384 93.9

2.0 97.6

2.1 94.8

2.2 93.2

2.3 9.7

1.9 65536 104.4

2.0 107.5

2.1 107.6

2.2 102.4

2.3 3.4

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Ideal Walsh Sequency Power Spectrum

BW = 24 kHz (0 kHz to 24 kHz)

0 dB

-130 dB

16384 samples

65536 samples

-120 dB

-12 dB

Walsh transform for 2Vpk square wave @ 1.5 kHz

SNR = 95 dBSNR = 107 dB

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DOLPHININTEGRATION


Gain Deviation in 2nd Integrator

Gain deviation of 7.4%

-72 dB

-120 dB

0 dB

Ideal

Deviated output


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SMASH 4.4.0 - Generic - Tue Mar 23 13:03:01 2004

WALLOGWALLOG

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Walsh Noise Test (I)

0 dB

-130 dB

For both cases N = 65536

Noise modelled at input can cause catastrophic failure (SNR ~30dB)


Ideal

Catastrophic failure

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Walsh Noise Test (II)FFT – Smoother transition to performance failure

Walsh – Sudden transition to catastrophic failure

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1.0E-08 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02Maximum noise amplitude (V)

SN

R (d

B)

FFT 2.5 V

Walsh,N=2^12, 2.0 V

Walsh,N=2^12, 2.1 V

Walsh,N=2^12, 2.2 V

Walsh,N=2^14, 2.1 V

Walsh,N=2^14, 2.2 V

Transition for 216 samples

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Summary and Future WorkSummary

Failure Insertion in C-based models => FFT results

Usage of Walsh Transforms with square wave inputs for spectral analysis

Initial Potential for Walsh SNR test assessed

Test Stimulus Requirements

Challenges and Limitations Identified

Future Work

Expansion of existing fault simulation data for Walsh applicability

Investigation into hardware implementation and test stimulus generation

Investigation into hybrid test solution

Documents

DOLPHIN INTEGRATION TAMES-2 workshop 23/05/2004 Corsica1 Behavioural Error Injection, Spectral Analysis and Error Detection for a 4 th order Single-loop