A Simulation Experiment on a Built-In Self Test Equipped with Pseudorandom Test Pattern Generator and Multi-Input Shift Register (MISR)

International journal of VLSI design & Communication Systems (VLSICS) Vol.1, No.4, December 2010
DOI : 10.5121/vlsic.2010.1401 1
A Simulation Experiment on a Built-In Self Test
Equipped with Pseudorandom Test Pattern
Generator and Multi-Input Shift Register (MISR)
Afaq Ahmad
Department of Electrical and Computer Engineering
College of Engineering, Sultan Qaboos University
P. O. Box 33, Postal Code 123; Muscat, Sultanate of Oman
Telephone: (+ 968) 2414 1327
Fax: (+968) 2441 3416
afaq@squ.edu.om
ABSTRACT
This paper investigates the impact of the changes of the characteristic polynomials and initial loadings,
on behaviour of aliasing errors of parallel signature analyzer (Multi-Input Shift Register), used in an
LFSR based digital circuit testing technique. The investigation is carried-out through an extensive
simulation study of the effectiveness of the LFSR based digital circuit testing technique. The results of the
study show that when the identical characteristic polynomials of order n are used in both pseudo-random
test-pattern generator, as well as in Multi-Input Shift Register (MISR) signature analyzer (parallel type)
then the probability of aliasing errors remains unchanged due to the changes in the initial loadings of the
pseudo-random test-pattern generator.
KEYWORDS
LFSR, MISR, BIST, Characteristic Polynomial, Primitive
1.0 INTRODUCTION
Reliability is one of the main considerations in any circuit design. It involves a correct and
predictable behaviour of the circuit according to design specifications over a sufficiently long
period of time. To achieve this goal, the logic-circuit design is aimed at an error-free circuit
operation. Unfortunately, in spite of all possible care being bestowed on design and simulation,
hardware faults resulting from physical defects (e.g. mask defects, manufacturing process flaws)
will occur in the hardware implementation of the logic circuit. Hence, when a fault occurs
anyway, one must be able to detect the presence of the fault and, if desired to pinpoint its
location. This task is accomplished by testing the circuit. System maintenance draws heavily
upon the testing capability of the logic system [1], [2].
Digital circuit manufacturers are well aware of the need to incorporate testability features early
in the design stage, or otherwise they have to incur higher testing costs, subsequently. An
empirical relationship, that have been used for estimating the cost of finding a faulty chip,
indicates that the cost increases by a factor of 10, as fault finding advances from one level to the
next [3] – [7]. However, recent studies have shown that the cost of testing and fault finding, at
system and field level, is higher than this factor of 10 and increases exponentially [7] – [9].
Thus, if a fault can be detected at chip or board level, then significantly larger costs per fault can
be avoided. This is the prime reason that attention has now focused on providing testability at
chip, module or even at board level.
Any test methodology usually consists of
(i) A test strategy for generating the test-stimuli,
(ii) A strategy for evaluating output responses, and
(iii) Implementation mechanisms to realize the appropriate strategies in test-generation and
response evaluation.

2
Present day philosophy to achieve economical and cost effective testing of Very Large Scale
Integration (VLSI) components and systems is to provide on-chip testing. Though these
techniques involve additional chip area for the added test circuitry, they have provided
reasonable testability levels. In fact it has been reported that, for about 20% additional silicon
area required, more than 98% of the chip design can be checked using structured Design For
Testability (DFT) approaches [7] – [11].
As a natural outcome of the structured design approach for DFT, built-in testing has drawn
considerable attention. Usually a built-in test methodology is defined as the one that
incorporates both test-pattern generation and response data compression mechanisms internally
in the chip itself. If this methodology is self-sufficient in detecting the faults of its internal test
circuitry also, then such a methodology is referred to as Built-In Self-Test (BIST) in test
literature. The main emphasis in BIST designs is that to provide close to one hundred percent
testing of combinational circuits [10], [12]. In particular, pseudo-random test-pattern generation
followed by the compression of response data by signature analysis has become a standard form
of testing in BIST environment. Linear Feedback Shift Registers (LFSRs) have been proposed
as an integral part of a sequential logic design, such that they can be used to both generate and
compact the results of a test. Undoubtedly, an LFSR based pseudo-random test-pattern
generation is an extremely simple tool for generating desired sequence as well as the length of
the test-stimuli. Many estimation schemes are readily available for computing the length of test
patterns where the desired sequence of the test-patterns can be obtained by the predetermined
seed of the LFSRs. Further, the effective testing of large circuits uses the concept of ‘pseudo-
exhaustive testing’ where the principle of divide and conquer is applied [13] – [19].
Difficulty arises when the resulting response data obtained from the Circuit Under Test (CUT)
is compressed into small signatures using Signature Analyzer (SAZ). Although, this scheme of
SAZ is easily implemented by an LFSR either in the form of Single Input Shift Register (SISR)
– serial SAZ or MISR- parallel SAZ . But this leads to loss of information, due to the erroneous
response patterns that get compressed into the same signature as the fault-free signature of the
CUT. Thus, some of the faults might go undetected due to this masking phenomenon. This
compression can further reduce the fault-coverage in the BIST scheme. This problem of error
masking is called aliasing [20] – [25].
Methods to determine the extent of fault escape caused by a response compressor are not readily
available. However, various attempts have been made to analyze and improve the basic SAZ's
realization methods [26] – [33]. The end goal of the above schemes, individually, or with a
combination of these, is to reduce the deception volume [26]. Methods that consider both, the
test pattern generator and response data compressor factors in totality and simultaneously, in
analyzing the aliasing behaviour of the circuit, are not available. There is a growing need for
such an approach, which comprehensively looks at aliasing problems with respect to both test
pattern generator and response data compressor and reflects the true aliasing characteristics.
This is the prime justification of the research work in this area. Therefore, towards this direction
a research work have been done [34], [35]. Through these papers, different studies were carried
out to investigate the roles of characteristic polynomials used in Pseudo-Random Test-Pattern
Generator (PRTPG) as well as in SAZ, and initial loading of PRTPG with the behaviour of
aliasing errors of SAZ. The work contained in the papers [34], [35] used separate different
structures (internal and external exclusive-OR types) of LFSRs. However, both the papers
considered SISR type of SAZ. This paper is an extension of the previous research where MISR
type of SAZ is used. In this work external exclusive-OR type and internal exclusive-OR type
LFSR is used in PRTPG and MISR respectively. The results of the study show that the
probability of aliasing errors remains unchanged due to the change in the initial loading when
the identical characteristic polynomials of order n are used in both PRTPG, as well as in SAZ
(MISR type).

3
2.0 MATHEMATICAL CHARACTERIZATION OF LFSR
In this section we consider briefly the mathematical background, definitions and theorems
related to an LFSR. Details can be found in literatures [36] – [39]. Basic definitions and
theorems are given below for the sake of completeness.
Let [A] represent the state transition matrix of order n × n, for an n stage LFSR shown in Figure
1. Assume the state at any time ‘t’ be represented by vector [Q(t)] = [qn(t), ... ,qj(t), ... ,q2(t),
q1(t)] (which is effectively the contents of the LFSR) where each qj represents the state of the jth
stage of the LFSR. Further, let the LFSR feedback stages be numbered from C0 to Cn,
proceeding in the same direction as the shifting occurs i.e. left to right. Let the present state of
the LFSR be represented by [Q(t)] and, one clock later, the next state by [Q(t+1)]; then the
relationship between the two states is given by Equation (1).
Figure 1. An n-bit LFSR model
(1)
where, cj = 0 or 1, for 1≤ j ≤ n-1 and cj = 1, for j = n. (2)
In Equation (2), the values of Cj show the existence or absence of a feedback connection from
the jth
stage of the LFSR. Equation (1) can be written as
[Q(t+1)] = [A][Q(t)] (3)
If [Q] = [Q(0)] represents a particular initial loading of the LFSR, then the sequence of states
through which the LFSR will pass during successive times is given by
[Q(t)], [A][Q(t)], [A]2
[Q(t)], [A]3
[Q(t)], …
Let the matrix ‘period’ be the smallest integer p for which [A]p
=I, where I is an identity matrix.
Then [A]p
[Q(t)] = [Q(t)] for any non zero initial vector [Q(0)], indicating the ‘cycle length (or
period)’ of the LFSR is p.
Thus, on the basis of this property of periodicity of LFSR and Equation (3), it follows that

4
[Q(t)] = [Q(t+p)] = [A]p
[Q(t)] (4)
Definition 1:
The cycle length p, for vector [Q(0)] = 0 is always 1, which is independent of matrix [A].
Definition 2: The period p of an n bit LFSR will only be maximal when p = m = 2n
-1.
For the matrix [A] of the LFSR, the characteristic equation is given by
Determinant [A-λI] = 0. Thus,
F(λ)=
















−
−
−
−
−
−
−
−
−
− −
−
−
λ
λ
λ
λ
λ
λ
λ
λ
λ
λ
1
0
0
0
0
0
1
0
0
0
...
0
0
0
...
0
1
... 1
2
2
1
L
L
M
M
M
M
M
C
C
C
C
C
C i
n
n
n
n
(5)
Definition 3: For the matrix [A] of an LFSR, the polynomial of {determinant [A-λI]}is called
the characteristic polynomial F(λ) of the LFSR and can be written as
F(λ) = j
n
j
j
C λ
∑
=
+
1
1 ; Cn = 1. (6)
Let, T(λ) denote the characteristic polynomial of an n stage LFSR used in PRTPG. Let, {a-1, a-
2,...,a-n+1 ,a-n } be the initial state of the shift register. The sequence of numbers a0, a1, a2, ... aq ...
can be associated with a polynomial, called generating function M(λ), by the rule
M(λ) = a0 + a1 λ + ... + aq λq
+ …
Let {aq} = a0, a1, a2,... represent the output sequence generated by an LFSR used as PRTPG,
where ai = 0 or 1. Then this sequence can be represented as
q
q
q
a
M λ
λ ∑
∞
=
=
0
)
( (7)
From the structure of the type of the LFSR shown in Figure 2, it can be seen that if the current
state of the jth
flip-flop is aq-j, for j = 1, 2 , ... , n , then by the recurrence relation
j
q
n
j
j
q a
C
a −
=
∑
=
1
(8)
Substituting (8) in (7)
q
j
q
q
n
j
j a
C
M λ
λ −
∞
= =
∑∑
=
0 1
)
( (9)
Or, by solving for generating function, it can be shown that M(λ) is given by

5
∑
∑
=
=
+
+
+
+
+
+
= n
j
j
j
n
j
C
M
1
1
1
-
1
-
1
j
-
1
j
-
j
-
j
-
j
j
1
)
a
........
a
(a
C
)
(
λ
λ
λ
λ
λ
λ (10)
∑
∑
=
=
+
+
+
+
+
= n
j
j
j
n
j
C
M
1
1
1
-
j
1
-
1
j
-
j
-
j
1
)
a
........
a
(a
C
)
(
λ
λ
λ
λ (11)
Or,
)
(
)
(
)
(
λ
λ
λ
T
B
M = (12)
Thus, the PRTP (represented by polynomial M(λ), generated by the LFSR can be obtained
through the long division of the function B(λ) by T(λ). Therefore, it can be implied from the
Equation (11) that the generated sequence M(λ) is function of initial loading as well
characteristic polynomial of the LFSR used in the realization of the PRTPG.
3.0 MULTI-INPUT SHIFT REGISTER (MISR) MODEL
Figure 2 shows a typical MISR configuration. This configuration of MISR is based on the
internal-EXOR. In the figure, n denotes the length of the MISR, i.e., the number of FFs in the
register. Also, the LFSR feedback stages be numbered from Cn to C1, are the binary coefficients
of the characteristic polynomial P (of the MISR.
Figure 2. An n-bit MISR model
P(λ) = ∑
=
+
n
j
j
j
C
1
1 λ (13)
Let fi be the content of the i-th FF, then the state of the MISR
(i.e., the sequence r = rn + rn-1 + ………… + r2 + r1) can be represented by the state polynomial
s(λ) = rn λn-1
+ rn-1 λn-2
+ ………… + r2 λ+ r1 (14)
Similarly, an n-bit input sequence (d = Dn + Dn-1 + ………… + D2 + D1) can be represented by
the input polynomial.

6
d(λ) = Dn λn-1
+ Dn-1 λn-2
+ ………… + D2 λ+ D1 (15)
The signature obtained by SAZ (any SISR or MISR) is defined as the final state of the register
after the input sequence d has been entered into the register. Then the MISR signature can be
given as
s(x) = D(x) mod P(x) (16)
Thus, the theory behind the use of the LFSR for SAZ is based on the concept of polynomial
division process, where the remainder left in the register after the last bit of input data is
sampled, corresponds to the final signature. Whereas, the output sequence from the nth
bit of the
LFSR defines the quotient, QO of the division. In general, the shift register is initialized by the
reset or by parallel load function of the register, at a time of fault-free evaluation as well as
every time when a new fault is injected in the CUT. Assume that the CUT is of combinational
nature with n primary inputs and k primary outputs. If the initial state of the LFSR is all 0’s, let
the final state of the LFSR be represented by the polynomial s(λ). Then it can be shown that
these polynomials are related by the equation
)
(
)
(
)
(
)
(
)
(
λ
λ
λ
λ
λ
P
s
QO
P
d
+
= (17)
where, P(λ) is the characteristic polynomial of the LFSR structure used in MISR-SAZ. Hence
an LFSR carries out polynomial division on the input data stream by the characteristic
polynomial, producing an output stream corresponding to the quotient QO(λ) and remainder
s(λ).
3.0 SIMULATION STUDY
The testing model employed in the simulation study for PRTPG and MISR is as described in
section 2 and shown in Figures 1 and 2. Various combinational circuits have been simulated
using the manufacturer's logical diagrams with gate level description. A single stuck-at fault
model is assumed. Where, s-a-0 and s-a-1 faults are postulated on each individual, NL, branches
of the each CUT. In the case of each n-input CUT, identical all possible characteristic
polynomials of order n are individually applied to PRTPG and MISR-SAZ as well. All possible
initial loading of PRTPG, 2n
-1, are exhausted to monitor the aliasing error behaviour of MISR-
SAZ. These characteristic polynomials are generated using the algorithms developed and
reported in papers [2], [40] and [41]. To make it more readable the simulation procedure used to
study the aliasing behaviour is described below.
Simulation procedure
Begin
(For an n-input CUT)
NL = total number of branches in the CUT;
NP = total number of possible characteristic polynomial of order n, over GF(2);
Li = is the periodicity of ith
characteristic polynomial of order n;
NS = total number of possible initial loading {NS = 2n
-1, excluding S =[000..0]};
RC = is the aliasing count;
For i = 1 to NP, do
For r = 1 to NS, do
Begin

7
Choose ith
characteristic polynomial for PRTPG as well as for SAZ {i.e.
Ti(λ) and Pi(λ)}respectively;
Choose rth
initial loading Sr;
Generate test stimuli of length Li;
Choose circuit response dg {fault-free circuit response};
Compute sg {fault-free signature};
For t = 1 to 2NL, do
Begin
Initialize aliasing count RC;
Choose circuit response dft {df1 has fault number 1 inserted, df2 has fault
number 2 inserted, etc.};
Compute signature sft {signature when the tth
fault is inserted};
Compare sft, with sg if sft ≠ sg then, increment aliasing count RC;
End do;
Write aliasing count {one each for sr, Ti (λ), Pi(λ)};
End do;
End.
The above procedure is used to observe the effect of the characteristic polynomials used in
PRTPG as well as in MISR-SAZ along with the initial loading of PRTPG on aliasing counts of
MISR-SAZ. The aliasing counts for the circuits of Table 1 were monitored for the sets of all NP
and NS of order n.
Table 1. Summary of simulated circuits
Circuit
Number
Module of IC Number Circuit
Specifications
(n-inputs, k-outputs)
Number of
Faults
Injected
NP / NS
of order n
C-1 SN-74LS139
DUAL 2-TO-4 LINE
DECODER/
DEMULTIPLEXER
3-inputs; 4-outputs
9-gates
58 2 / 7
C-2 SN-74LS82
2-BIT BINARY FULL
ADDER
5-inputs; 3-outputs
21-gates
148 6 / 31
C-3 SN-74H87
4-BIT TRUE/
COMPLEMENT,
ZEO/ ONE ELEMENT
6-inputs; 4-outputs
14-gates
64 6 / 63
The observed results demonstrates that when the identical characteristic polynomials are used in
both the PRTPG and MISR-SAZ, then any change in initial loading of PRTPG does not change
the value of aliasing count. Due the complexity of the result sets and space only a candidate of
result for circuits summarized in Table 1 are shown in Tables 2 to 4. In Tables 2 – 4 the aliasing
count is shown. These values of aliasing counts remain unchanged for all the possible changes
of initial loading of PRTPG.

8
Table 2. Aliasing errors for all possible NS for circuit C-1
PRTPG T(λ) MISR-SAZ P(λ)
1+λ+λ
4
1+λ
3
+λ
4
1+λ+λ3
9 13
1+λ2
+λ3
13 9
1+λ+λ
4
1+λ
3
+λ
4
1+λ5
+λ6
24 19
1+λ+λ6
6 18
1+λ2
+λ3
+λ5
+λ6
17 22
1+λ+λ4
+λ5
+λ6
21 16
1+λ+λ3
+λ4
+λ6
23 26
1+λ+λ2
+λ5
+λ6
19 28
1+λ+λ
3
1+λ
2
+λ
3
1+λ3
+λ5
4 12
1+λ2
+λ5
17 15
1+λ2
+λ3
+λ4
+λ5
13 9
1+λ+λ3
+λ4
+λ5
14 16
1+λ+λ2
+λ4
+λ5
10 18
1+λ+λ2
+λ3
+λ5
7 11

9
4.0 CONCLUSIONS
It has been demonstrated through this simulation study that the change of the initial loading of
PRTPG does not have any impact on the effectiveness of an LFSR based digital circuit testing
technique that uses identical characteristic polynomials in both the PRTPG and MISR-SAZ as
well. Thus, this result restricts the outright use of the results of findings; that the effectiveness of
LFSR based digital circuit testing techniques can be improved by changing the initial loading of
PRTPG. Thus, for effective use of initial loading of PRTPG of LFSR based digital circuit
testing technique, it is essential to analyze the role of characteristic polynomials used in PRTPG
as well as in MISR-SAZ. Although, our investigation is limited with small sizes of circuits but
the trend of results suggests for further through analytical investigation.
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[ 40] Ahmad A. and Elabdalla A. M., An efficient method to determine linear feedback connections in
shift registers that generate maximal length pseudo-random up and down binary sequences,”
Computer & Electrical Engineering - An Int’l Journal (USA), vol. 23, no. 1, pp. 33-39, 1997
[ 41] Ahmad, A., Al-Musharafi, M.J., and Al-Busaidi S., A new algorithmic procedure to test m-
sequences generating feedback connections of stream cipher’s LFSRs, Proceedings IEEE
conference on electrical and electronic technology (TENCON’01), Singapore, August 19 – 22,
2001, vo. 1, pp. 366 – 369, 2001.
ACKNOWLEDGEMENTS
The acknowledgements are due to authorities of Sultan Qaboos University (Sultanate of Oman)
for providing generous research support grants and environments for carrying out the research
works.

12
Author (Short Biography)
Afaq Ahmad belongs to department of Electrical and Computer Engineering department at
Sultan Qaboos University, Sultanate of Oman. He holds B.Sc. Eng., M.Sc. Eng., DLLR and
Ph.D. degrees. Ahmad did his PhD from IIT Roorkee, India in 1990. Before joining Sultan
Qaboos University, Dr. Ahmad was Associate Professor at Aligarh Muslim University, India.
Prior to starting carrier at Aligarh, he also worked as consultant engineer with Light & Co.,
lecturer with REC Srinagar and senior research fellow with CSIR, India.
Dr. Ahmad is Fellow member of IETE (India), senior member of IEEE Computer Society
(USA) and life member of SSI (India), senior member IACSIT, member IAENG and WSEAS;
He has published over 100 technical papers. At present he is associated as editors and reviewers
of many reputed journals. He has delivered many keynote, invited addresses, extension lectures,
organized conferences, short courses, and conducted tutorials at various universities of globally
repute. He chaired many technical sessions of international conferences, workshops,
symposiums, seminars, and short courses. He has undertaken and satisfactorily completed
many highly reputed and challenging consultancy and project works. His research interests are:
fault diagnosis and digital system testing, data security, graph theoretic approach,
microprocessor based systems and computer programming.
Dr. Ahmad’s field of specialization is VLSI testing and fault-tolerant computing.

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