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									International Summer Academy 2008, Cottbus

Test Generation for Digital Circuits
Ondřej Novák TU Liberec and CTU in Prague, Czech Republic

contact: novako3@fel.cvut.cz

Outline
 Functional testing  Structure oriented testing  Deterministic test algorithms Test compaction  Sequential circuits testing  Fault simulation

International Summer Academy 2008

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Domains and levels of a design
Domain Behavioral L E V Logic E L Circuit
Differential Equations

Structural Blocks Registers Logic Gates Transistors

Physical Chip Macro Cells Standard Cells Masks

System RTL

System Specifications

RTL Specifications Boolean Functions

International Summer Academy 2008

Ondřej Novák

source:Samiha Mourad 3 Santa Clara University

Basic terminology
Digital signals that are useful to test potential faults in a circuit are „test patterns“. A set of bits applied at a certain time is a „test vector“. A test vector has to: - provoke a potential fault condition at a potential fault site - propagate the fault to a primary circuit output to make it detectable. The sum of all test vectors / test patterns to be applied to a specific device is called the test set. Several (mostly two) test vectors need to be applied in a specific order to provoke and propagate certain fault effects. Then this pair (or triple) of vectors is called a test sequence.
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Fault coverage
 FC: Number of faults covered by the test set divided by the total number of faults in a test set.  Problem: The test set does not cover all possible faults (defects)

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Required test properties
A test set should be as short as possible, but it should cover relevant faults.

• For production test many different faults should be found as soon as possible, no fault diagnosis is necessary. Therefore test patterns that cover multiple faults at a time are preferred.
• For diagnostic test (e. g. prototype-test) patterns that may excite specific fault conditions (single faults) are preferred. • Speed: Length of the test set and the speed of test pattern application & response recording cost money • Cost: For complex ICs, test-related costs may be from 30% to 50% of the overall production cost.
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Testing methods

Structural approach

Functional approach

(Pseudo) exhaustive testing

(Weighted) random patterns

Path sensitization

SSBDD, Bool. satisfiability

Boolean difference

Checking sequence

BIST
International Summer Academy 2008

ATPG
Ondřej Novák

not used designs without 7 DFT

Functional approach
 Exercising all possible input vector sequences, checking sequence:
• • • • synchronizing sequence homing sequence distinguishing sequence transition sequence

Pros: zero hw overhead, patterns easy available, realistic test patterns (close to functional states, not overexcite the circuit) Cons: unacceptable high number of tests for complex circuits, final fault coverage is low
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Concatenation of test tables
 Circuit decomposition to basic testable units or composition of a system from testable modules  Creating test sets (tables) for the modules  Concatenation of basic test sets in such a way that every module inputs can be exited with supposed vectors and the output values are observable on system outputs

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Concatenation of test sets
V X Y
M2 M3

Z

M1

V1 ........X1
V2 .......X2 Vn ......Xn Stimul 1 Stimul 2 Stimul m Stimul 1 Stimul 2 Stimul k

sensitive path1
sensitive path 2 sensitive path n X1 ........Y1 X2 .......Y2 Xm ......Ym Stimul 1 Stimul 2 Stimul k

sensitive path 1
sensitive path 2 sensitive path n sensitive path 1 sensitive path 2 sensitive path m Y1 ........Z1 Y2 .......Z2 Yk ......Zk
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Testing with random patterns
 high initial fault coverage efficiency
hard detectable faults fault coverage

deterministic TPG

random TPG

number of patterns
International Summer Academy 2008 Ondřej Novák
figure: E.11 Gramatova, Bratislava

Limits of random test
InputOriented Weighted Rand. Patt. Pattern Oriented Weigthed Rand. Patt.

Circuit

# Fault Count

# Random Patterns

Equi-probable Rand. Patt.

# untested faults
2,000 C2670 2,408 10,000 50,000 290 290 279

# untested # untested faults faults
50 19 6 51 6 2

2,000
c7552 6,876 10,000 50,000
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365
292 218
Ondřej Novák

165
47 25

212
93 29
data: H.-J.Wunderlich, 12 Stuttgart

Random resistant faults
 Random testability can be evaluated from the circuit scheme. The algorithm that estimates the testability cuts reconvergent fan outs and calculates probabilities of signal values (cutting algorithm).

SAF0

International Summer Academy 2008

Ondřej Novák

figure: E.13 Gramatova, Bratislava

Testing a circuit with a random test

Random TPG
Fault list Fault list Fault simulation

Deterministic TPG

Fault simulation Random TPG

Fault simulation is less complex computational problem than test generation, it can help with pattern selection.
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Exhaustive and pseudoexhaustive test
 exhaustively tests a circuit (exhaustive), all subcircuits (pseudoexhaustive)  (n,r) exhaustive test tests exhaustively every possible cone of combinational logic that has r or less inputs. TEST PATTERN GENERATOR  used in BIST
1 2 r1 3 . . . r2 n ru

International Summer Academy 2008

Ondřej Novák

CUT15

Structure-oriented test algorithms
Exercises the minimal set of faults on each line of the circut testing stuck-at-faults, delay faults, shorts ...  condition: single faults  ATPG:
• generates test patterns • finds redundant or unnecessary logic • verifies circuit implementation

 NP complete problem, the complexity is assumed to be exponential in the worst case.
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Structural testing : algorithm types Symbolic – Boolean Difference (reduced information about structure – more functional)  Path sensitization  Boolean Satisfiability (not entirely) structural approach)  Structurally Synthesized Binary Decision Diagrams (SSBDD)
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Boolean Difference
df(x)/dxi =

f ( x 1 ,  , x i  0 ,  , x n )  f ( x 1 ,  , x i  1,  , x n )  1

df(x)/dx for xi =0 and df(x)/dxi pro xi=1 are called residua One of them is equal to the output value for the fault less circuit, the second one to output with a fault on xi A fault is detected when the residua are complement
The test patterns can be derived from the following equations: xi df(x)/dxi = 1 nebo xi ' df(x)/dxi = 1
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Path sensitization
path sensitisation

force to 1

0 1 SAF0

0  1
good output

faulty output

Sensitive path:The circuit output changes its value in case of constant input values and a fault on sensitive path appearance
International Summer Academy 2008 Ondřej Novák
figure: E. Gramatova, 19 Bratislava

Path sensitization
Sensitization or Propagation U=0 G4 Justification W/1 A=0 G1 E=1 B=0 Implication C G2 0 V G5
figure:Samiha Mourad Santa Clara University

G H=1

G 6

Z

G3

F

Justification

Observability and controllability measures (SCOAP, ...) estimating the effort to set or observe log. 1 or 0 are used in path sensitization algorithms in order to find an easiest sensitization path to PO and justification to PIs.
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D – algorithm (Roth, 1964)
 Formally describes conditions of the sensitive path existence  In D algorithm, the ATPG algebra has 5 symbols: D, D´, 0, 1, X  The sensitive path is created as a concatenation of so called D cubes  D cubes that describe the sensitive path through logic gates:

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D algorithm – one pattern creation
 A fault selection  Primitive D cube development  concatenation the D cube of the fault with a D cube of the log. element on the output of the faulty element  Reaching primary output by concatenated cubes(Sensitization)  Finding primary input values (Justification) If not successfull - backtrack, new sensitization step
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Podem (1981)
 reduces the search space by expanding the decision tree only on primary inputs  introduction of so called D frontier, measuring the simplicity of reaching a primary output – heuristics for finding the easiest sensitive path

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FAN (1983)
 Decreases the number of backtracks:
• • • • unique sensitization procedure multiple backtracking procedure immediate implications and justification uses headlines (points, where a circuit can be partitioned)

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Learning ATPG programs
 Socrates (1988) static and dynamic learning. The learning procedure systematically sets circuit signals to different values and discovers what values are other signals implied.  Recursive learning (1995) learning procedure is called recursively

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Search Space Abstractions in ATPGs  Binary decision tree  Binary decision diagram

International Summer Academy 2008

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figure: Dirk Beyer , 26
Vikipedia

Test generation with structurally synthesized BDDs
 Testing all stuck-at faults corresponds to testing all the nodes in each SSBDD  Testing of a node consists of:
• Activating a path from the graph root to the node • Activating two paths from the tested node to terminal nodes corresponding to different logical values

 Effective for structures that can be easily modelled by a graph. For multipliers less effective.
figure: R. Ubar, Talinn

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Boolean Satisfiability
 Boolean satisfiability test generation approach is not structure oriented.  The fault free and faulty circuits are described by equations in conjunctive normal form together with the condition of the fault detection  A test vector can be found as a solution of concatenated conjunctive normal form composed by three original conjunctive forms.  A universal SAT solver can be used instead of an ATPG  The complexity of SAT solution depends on the number of literals in the OR clause of the form (2 SAT relation solvable in polynomial time)  SAT solver based test generators are competitive with the fastest structure oriented algorithms.  Commercial ATPG tool Tafertshofer, speed evaluation: 25057 times faster than D alg.
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Test compaction
 Many combinational ATPGs use random patterns to achieve 60 % fault coverage, then generate deterministic tests to obtain 100 % coverage.  Some patterns are not sufficiently effective.  Fault simulation is performed in reverse order. Fault detecting patters are preserved. Significant test length reduction.  Additional static or dynamic heuristics based method is used for pattern compaction replacing don't care values by 0 or 1. Further reduction by 50 %.
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Sequential circuit test generation
 Time frame expansion – circuit can be tested by a combinational ATPG. Efficient for circuits described at the gate level, parallel computation often used for shortening the test generation time. Less efficient for cyclic structures, multiple-clocks and asynchronous circuits. 9 value logic system required.  Simulation based methods – test vectors randomly chosen and the fault coverage is simulated. Efficient for every circuit that can be simulated. Genetic algorithm can be used for vector selection.
m I SI Copy 1

m
n Copy 2 Z n SO

m
k n SI Copy j Z SO

n

Fault simulation
 Complexity of Fault Simulation: In general, proportional to the following factors F: number of faults P: number of test patterns G: number of gates Complexity approximately O(G3)  More difficult than logic simulation by the factor F.  easier than test generation  The complexity can be greatly reduced by – fault dropping – advanced techniques.
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Effectiveness of simulation techniques

table:: Chien-Mo James Li Graduate Institute of Electronics Engineering National Taiwan University

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Conclusion
 Quality of test vectors and the CPU time of their generation depends on the circuit testability (SCOAP measures).  Path sensitization algorithms can be used for combinational circuits, for sequential circuits with multi clock domains, internal cycles and asynchronous behavior are less effective.  Algorithms based on solution of the SAT problem are the most effective for the biggest circuits.  Fault simulation together with a genetically driven searching of test vectors could be a solution for testing sequential circuits.
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