Sying-Jyan WangDept. Computer Science and Engineering

National Chung-Hsing University

OutlineFundamental

Information theoryData compression

Digital testing: an introductionCode-based test data compression: a case

studyMaximum compression Improving compression rate

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Compressibility of a Data SetData compression (source coding)

The process of encoding information using fewer bits than an unencoded representation would use through use of specific encoding schemes.

How much compression can you get?Encoding schemeData

How to quantify the compressibility of a data set?

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Shannon Entropy (1/4)A measure of the uncertainty associated with a

random variable

Quantify the information contained in a message in the sense of an expected value (usually in bits), or

A measure of the average information content one is missing when one does not know the value of the random variable

Self-information: log p(x)5

C.E. Shannon, "A Mathematical Theory of Communication", Bell System Technical Journal, vol. 27, pp. 379-423, 623-656, July, October, 1948

XxXxxpxp

xpxpXH )(log)(

)(

1log)()(

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Shannon Entropy (2/4)Self-information

How much information carried by a particular value of a random variable?

XXXXXXXXX XXXX XXXX XX XXXXXX XXXXXXXXX.

XXXXXXXXX XXXX XeXX XX XXXXXX XXXXXXXXX.

XXXXXXXXX XXXX wXnt XX XXXXXX XXXXXXXXX.

Professor Wang wXnt to Taipei yesterday.

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Shannon Entropy (3/4)Tossing a coin with

known, probabilities of coming up heads or tailsMaximum entropy

of the next toss -- if the coin is fair 1-bit information

A double-headed coin 0-bit (no) information

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Shannon Entropy (4/4)Shannon's entropy: an absolute limit on the best

possible lossless compression of any communicationA source produces a sequence of letters chosen from

among A, B,C, D with probabilities 1/2, 1/4, 1/8, 1/8, successive symbols being chosen independently

H = 7/4 bis per symbolEncoding:

A 0 B 10 C 110 D 111

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AAAA BB DC

Entropy in ThermodynamicsA measure of the unavailability of a system’s

energy to do work; also a measure of disorder; the higher the entropy the greater the disorder.

A measure of disorder; the higher the entropy the greater the disorder.

In thermodynamics, a parameter representing the state of disorder of a system at the atomic, ionic, or molecular level; the greater the disorder the higher the entropy.

A measure of disorder in the universe or of the availability of the energy in a system to do work.

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Eight Allotropes of Carbon(From Wikipedia)

(a)Diamond(b)Graphite(c) Lonsdaleite(d)Fullerene (C60)(e)Fullerene (C540)(f) Fullerene (C70)(g)Amorphous carbon(h)Carbon nanotube

(CNT)Allotropy is a behavior exhibited by certain chemical elements: these elements can exist in two or more different forms, known as allotropes of that element. In each allotrope, the element's atoms are bonded together in a different manner. Allotropes are different structural modifications of an element.[1]

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Carbon Allotropes Spans a Range of ExtremesSynthetic diamond nanorods are the hardest materials known.

Graphite is one of the softest materials known.

Diamond is the ultimate abrasive.

Graphite is a very good lubricant.

Diamond is an excellent electrical insulator.

Graphite is a conductor of electricity.

Diamond is the best known thermal conductor

Some forms of graphite are used for thermal insulation

Diamond is highly transparent.

Graphite is opaque.

Diamond crystallizes in the cubic system.

Graphite crystallizes in the hexagonal system.

Amorphous carbon is completely isotropic.

Carbon nanotubes are among the most anisotropic materials ever produced. 11

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Amorphous Carbon vs. Carbon Nanotube

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Shannon’s Entropy vs. Entropy: An AnalogyVarious crystalline structures of the same

element exhibit different propertiesHow about different structure of the same set

of data?Can we put the data into a format that is easier

to compress?f f–1

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Digital Testing: A Quick Overview (1/3)All manufactured

components are subject to physical defects

Digital testing is the process to verify whether a chip (system) meets its specification

RequirementsInput vectors only

applied to the chip input pins

Only circuit outputs are observable

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TEST VECTORS

MANUFACTUREDCIRCUIT

COMPARATOR

CIRCUIT RESPONSE

PASS/FAILCORRECTRESPONSES

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Digital Testing: A Quick Overview (2/3)Why Testing is Difficult?

Test application time explodes for exhaustive testing of VLSI For a combinational circuit with 50 inputs, we need 250 =

1.126x1015 test patterns. Assume one test per 10-7sec, it takes 1.125x108sec = 3.57yrs. to test such a circuit.

Test generation of sequential circuits are even more difficult. Lack of Controllability and Observability of

Flip-Flops (Latches)How to generate input test vectors (patterns)

Functional testingFault-oriented test generation

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Digital Testing: A Quick Overview (3/3)Fault models

Modeling the effects of physical defects on the logic function and timing.

Common fault models Stuck-at fault, bridging fault, memory fault, etc.

Single stuck-at fault modelThe most widely used fault modelAssumptions

Only one line is faulty. Gates are fault-free.

Faulty line permanently set to 0 or 1. Fault can be at an input or output of a gate.

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X

X1

0

X0

1

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Scan Based Design-for-Testability (DFT)Sequential circuits are

hard to testScan based design

Modify all flip-flops to provide Parallel load Scan shift

Provide controllability and observability of internal state variables for testing.

Turn the sequential test problem into a combinational one.

State V

ector

CombinationalLogic

X Z

y Y

ScanIn

ScanOut

D QDI

SI

Test Compaction & CompressionWhy?

Modern chips are large and require a lot of test data More than 1 Gb for a chip

Automatic Test Equipments (ATE) are expensive Need to reduce test time

Test data contain a lot of X bits May be up to 97%

CompactionReduce number of test patterns by merging compatible

patternsCompression

Reduce data volume through coding and architectural approaches

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Code-Based Test Data CompressionPartition test data into symbols, and replace

symbols by codewordsFixed-to-fixed

Lengths of both symbols and codewords are fixed Linear decompressor, dictionary code

Fixed-to-variable Huffman code

Variable-to-fixed Run-length code

Variable-to-variable Golomb, FDR, VHIC

AAAA BB DC

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Fixed-Symbol-Length Codes (1/2)Notations:

p0, p1, pX: Probabilities of 0, 1, and X in a test set

Entropy analysisAssume pX = 0Symbol length is nEntropy: n-ii

n

i

n-iini ppppCpH 00

0000 1log1

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p0

En

tro

py

H

n=2

n=4

n=6

n=8

n=10

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Fixed-Symbol-Length Codes (2/2)Maximum compression rate:

No compression when p0 = p1 = 0.5

.gthsymbol_len

entropygthsymbol_len

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p0

Ma

xim

um

Co

mp

ress

ion

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Variable-Symbol-Length Codes (1/3)Most variable-symbol-length codes are based on run-

length encodingRuns of both 0’s and 1’s are encoded

Theorem 1: The expected run length of 0’s is 1/p1, and the expected run length of 1’s is 1/p0.

Only runs of 0’s are encoded Corollary 2: The expected run length 1/p1 in case X-bits are

assigned to 0’s.

Why?Since a run of 0’s is interrupted by a 1, the expected run

length of 0’s is decided by the occurrence frequency of 1’s. Similarly, the expected run length L1 = 1/p0 in case X-bits

are assigned to 1’s.

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Variable-Symbol-Length Codes (2/3)Entropy Analysis

Assume 0-filling, and let y = p0+pX

1

11

00

log log logPr logPr

p

ypylRLlRLH

l

0

1

2

3

4

5

6

7

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

p1

Eb

tro

py H

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Variable-Symbol-Length Codes (3/3)Maximum compression rate:

No compression when p0 = p1 = 0.5

._lengthavg_symbol

entropy_lengthavg_symbol

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p0

Ma

xim

um

Co

mp

ress

ion

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ImplicationsNo compression if the data are truly randomHigher compression rate if probabilities of 0’s and 1’s

are skewedExtremely skewed data unlikely: low information rateHow to improve maximum compression rate?

X-filling X’s are redundant information that can be compressed

ATPG Generate patterns with desired distribution

Partition scan cells Rearrange data to create desired distribution

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Scan Cells Partition (1/4)In a given test set

The number of specified 1’s and 0’s in each scan cell may not be balanced A not-so-random distribution in the cell

By putting “similar” cells together, we can create symbols with smaller entropy Higher maximum compression rate Easier-to-compress data

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Scan Cells Partition (3/4)Bi-partitioned scan chain

is not good enough…Not naturalTwo parts have to be

encoded separatelyTo achieve even better

encoding efficiencyIntroduce an inversion

between the two partsFrom the encoder’s

point of view, there is only one partition

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Scan Cells Partition (4/4)Multiple scan chains

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Routing ConsiderationHow does physical order of scan cells affect

compressibility?1000 random moves in the initial chain

S9234 S15850 S38417

Not PartitionedPartitioned

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Results (1/5)Mintest patternsUse bit-raising to create

X’sPartition scan cells

0-dominant cells: #0 #1 >

1-dominant cells: #1 #0 >

Balanced cells: | #1 #0 > | Balanced cells are

assigned according to geometric adjacency

Cells in a partitioned are connected for minimum routing length

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Results (3/5)Routing Lengths

S38417 Shortest length = 0 = 5

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Results (4/5)Shift power

Partitioned scan chains Lower entropy implies

more homogeneous symbols

Fewer signal transitions Lower shift power for

input patterns0-filling

More 0’s in output response

Lower overall shift power

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Results (5/5)How about the compression rate of another test set?

Assume the same partitioned scan chain orderGenerate a new test set by ATLANTA2/3 of X’s in chain-0 are assigned 02/3 of X’s in chain-1 are assigned 1

About C.E. Shannon (1916-2001)Bachelors, U. Michigan, 1936

EE & MathematicsMaster, MIT, 1937

A Symbolic Analysis of Relay and Switching Circuits On designing electric computer using Boolean algebra

“… possibly the most important, and also the most famous, master's thesis of the century.” -- Howard Gardner, of Harvard University

PhD, Mathematics, MIT, 1940 An Algebra for Theoretical Genetics

Institute of Advanced Study, 1940-1941AT&T Bell Labs, 1941-1956

Information theoryVisiting Professor, MIT, 1956Tenured Professor, MIT, 1958, for only several semestersGambling & Investing using Kelly criterion with Ed Thorpe

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