vishwani d. agrawal auburn university, dept. of elec. & comp. engg. auburn, al 36849, u.s.a....

Vishwani D. AgrawalAuburn University,

Dept. of Elec. & Comp. Engg.Auburn, AL 36849, U.S.A.

Nitin YogiNVIDIA Corporation,

Santa Clara, CA 95050

20th April 201028th IEEE VLSI Test Symposium

Santa Cruz, CA

Introduction

Propose an information and noise analysis method for digital signals using Hadamard transform.

Analysis identifies information (spectrally structured) and noise (random) contents of the signal in relative measures.

The analysis is useful in a variety of applications like test generation, BIST, test compression, etc.

We illustrate its application to test generation.

4/20/2010 2VTS'10, Santa Cruz, CA

June 12, 2009 Nitin Yogi - Doctoral Defense 3

Test Vectors and Bit-streams

Circuit Under Test (CUT)In

pu

t 1

Inp

ut

2

Inp

ut

3

Inp

ut

4

Inp

ut

5

Inp

ut

J

Vector 1 →Vector 2 →Vector 3 →Vector 4 →Vector 5 → Vector 6 →

Vector 7 →

Vector 8 →

OutputsT

ime in clocks

A digital binary bit-

stream signal vector

1 1 0 1 0 . . 10 0 1 0 1 . . 11 0 0 1 1 . . 0

1 1 0 1 0 . . 1

0 0 1 1 0 . . 00 1 1 0 1 . . 01 0 1 1 1 . . 11 0 0 0 1 . . 0

Hadamard Transform

1 1 1 1 1 1 1 11 -1 1 -1 1 -1 1 -11 1 -1 -1 1 1 -1 -11 -1 -1 1 1 -1 -1 11 1 1 1 -1 -1 -1 -11 -1 1 -1 -1 1 -1 11 1 -1 -1 -1 -1 1 11 -1 -1 1 -1 1 1 -1

H(8) =

• Hadamard transform transforms a digital signal from time domain to frequency-related domain.

• Uses Walsh functions, which are a complete orthogonal set of basis functions that can represent any arbitrary bit-stream.

• Can be used for binary signals by using the representation {0,1} -> {-1,1}

Example of Hadamard matrix of order 8

w0

w1

w2

w3

w4

w5

w6

w7

Wal

sh fu

nctio

ns (o

rder

3)

time


Hadamard transformation

where:X Time domain digital signal vector of stream of N bitsH(N) Hadamard transform matrix of order NS Hadamard transform (Walsh spectrum) of X

XNHN

S )(1


SNHN

X )(1

Forward transformation (time domain to spectral domain):

Reverse transformation (spectral domain to time domain):

Properties of Hadamard Transform• Orthogonality and symmetry

• Energy conservation

where:

H(N) Hadamard transform matrix of order N X Binary bit-stream vector in time domain S Hadamard transform in spectral domain (Walsh

spectrum)

)()()( NINNHNH T

1

0

21

0

2 )()(N

j

N

k

jSkX


Energy Analysis

Total energy in a {-1,+1} binary signal of length N clocks in time domain:

Total energy in spectrum:

NkXN

k

N

k

1

0

1

0

2 )1()(

NkXjSN

k

N

j

1

0

21

0

2 )()(


Analysis of random binary bit-streamsValues of spectral components of random binary bit-

streams can be approximated as Gaussian distribution Mean (µ) = 0 Standard deviation (σ)

1)(1 21

0

2

r

N

jr SjS

N

where: Sr(j) jth spectral component of a random binary bit-stream of length N

Square of the mean of Sr(j)2

rS


Equal to N(by energy

conservation)

Equal to 0(since mean = 0)

Spectral coefficients of random bit-stream• 500 samples of random binary bit-streams of length 64 were generated• Distribution of values of spectral components analyzed

• Mean = 0.0035 ≈ 0• Standard deviation = 1.000425 ≈ 1

1σ (68.27%)

2σ (95.45%)

3σ (99.73%)


Spectral components below a magnitude of

2σ or 3σ can be treated as noise components

Generating spectral bit-streams1. Perform Hadamard transform on binary bit-stream.

2. Filter out noise-like spectral components having magnitudes less than a spectral threshold(Energy conservation of the transfom transfers the energy of filtered components to noise).

3. Perform reverse Hadamard transform to obtain time-domain values in the range (-1,+1) for bits in the bit-stream.

4. Time-domain values are normalized to range (0,1) and used as probabilities of logic 1 in new random bit-streams.


Generating spectral bit-streams

26

-2

2

2-2-22

1-11

1

1-11

-1

1 1 1 1 1 1 1 11 -1 1 -1 1 -1 1 -11 1 -1 -1 1 1 -1 -11 -1 -1 1 1 -1 -1 11 1 1 1 -1 -1 -1 -11 -1 1 -1 -1 1 -1 11 1 -1 -1 -1 -1 1 11 -1 -1 1 -1 1 1 -1

Hadamard Matrix H(3)

{-1,+1} binary bit stream (X)

Hadamard transform (S)

=

Example {0,1} binary bit-stream

1√8

1√8

101

1

1010

{0,1} converted to {-1,+1}


Example of generating spectral bit-streams

06/√8

0

0

0000

0.75-0.750.75

-0.75

0.75-0.750.75

-0.75

1 1 1 1 1 1 1 11 -1 1 -1 1 -1 1 -11 1 -1 -1 1 1 -1 -11 -1 -1 1 1 -1 -1 11 1 1 1 -1 -1 -1 -11 -1 1 -1 -1 1 -1 11 1 -1 -1 -1 -1 1 11 -1 -1 1 -1 1 1 -1

26

-2

2

2-2-22

Hadamard transform (S) of 8-bit binary bit-

stream

Spectral threshold

= 2 σ = 2

06/√8

0

0

0000

0.8750.1250.875

0.125

0.8750.1250.8750.125

Probabilities for generating

bit-streams


0’s are filtered spectral

coefficients

Time-domain Reverse transformation

Unfiltered spectral component

1√8

1√8

Normalization for probabilities

[X(k)+1] 2

Generated spectral bit-streams

Original {0,1} binary bit-

stream

1-11

1

1-11

-1

Unfiltered spectral

component

1-11

-1

1-11

-1

0.875

0.125

0.875

0.125

0.875

0.125

0.875

0.125

Probabilities for generating

bit-stream

-1-11

-11

-11

-1

1-11

-1111

-1

111

-11

-11

-1

1-11

11

-11

-1

111

-11

-1-1-1

Randomly generated bit-streams from probabilities

1 bit difference between original bit-stream & unfiltered spectral component

Generated bit-streams exhibit similar correlation with the unfiltered spectral component as the original bit-stream.

Few bits changed by noise are shown in red.


Application of analysisApplication of spectral information analysis:

Test generation, BIST, Test data compression, etc.

Illustration of effectiveness of analysis using test generationTest vectors generated for RTL faults (PIOs & flip-

flops)Generate spectral vectors & fault grade on circuitCompare with random, weighted random & randomly

perturbed vectors Analysis applied to ISCAS’89 benchmark circuits

s1488, s5378 & s38417


Spectral coefficients & power analysis for s1488

Inputs Spectral Coefficient Amplitude Power Noise PowerInput 1 w0 0.75 0.56 0.44

Input 2 w1 0.5 0.25

0.61w13 -0.38 0.14

Input 3 w1 0.56 0.32

0.49w19 0.44 0.19

Input 4 w0 0.63 0.39

0.47w22 -0.38 0.14

Input 5

w1 -0.5 0.25

0w5 0.5 0.25w19 0.5 0.25w23 0.5 0.25

Input 6 w4 0.5 0.25

0.5w22 0.5 0.25

Input 7 w1 0.5 0.25

0.47w5 -0.38 0.14w30 0.38 0.14

Input 8 w4 0.38 0.14

0.58w12 0.38 0.14w22 0.38 0.14


32 vectors were generated to detect RTL faults (PIOs & FFs) & analyzed using H(32)

Gate level coverage for s1488


Gate level coverage for s5378


Gate level coverage for for s38417


ConclusionProposed an information analysis framework to

distinguish noise from signal contentIllustrated effectiveness of method for the application

of test generationThe method can easily be extended to other

applications like BIST and test compression. See,N. Yogi and V. D. Agrawal, “BIST/Test-Compressor Design using

Combinational Test Spectrum,” Proc. 13th IEEE VLSI Design & Test Symp. (VDAT), July 2009, pp. 443-454.

N. Yogi and V. D. Agrawal, “Sequential Circuit BIST Synthesis using Spectrum and Noise from ATPG Patterns” Proc. 17th IEEE Asian Test Symp. (ATS), Nov 2008, pp. 69-74.

There is potential for further applications.


4/20/2010 VTS'10, Santa Cruz, CA 20

4/20/2010 VTS'10, Santa Cruz, CA 21

Circuit name

RTL-ATPG spectral tests FlexTest gate-level ATPG

Coverage (%)

No. of vectors

CPU* (secs)

Coverage (%)

No. of vectors

CPU* (secs)

s1488 95.65 512 103 98.42 470 131

s5378 76.49 2432 2088 76.79 835 4439

N. Yogi and V. D. Agrawal, “Spectral RTL Test Generation for Gate-Level Stuck-at Faults,” Proc. 15th IEEE Asian Test Symp. (ATS), Nov 2006, pp. 83-88.* Sun Ultra 5, 256MB RAM

Circuit name

FlexTest gate-level ATPG BIST gate-level fault coverage (%)

Coverage (%)

No. of vectors

64k random vectors

64k weighted random vectors

Spectral BIST (64k vectors)

s1488 97.31 736 92.13 97.11 97.11

s5378 77.06 739 74.39 76.84 78.28

s38417 49.62 55110 13.42 15.87 54.59

N. Yogi and V. D. Agrawal, “Sequential Circuit BIST Synthesis using Spectrum and Noise from ATPG Patterns,” Proc. 17th IEEE Asian Test Symp. (ATS), Nov 2008, pp. 69-74.

Test generation:

BIST:

vishwani d. agrawal auburn university, dept. of elec. & comp. engg. auburn, al 36849, u.s.a....

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