CC BY
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A Low-Cost Optimal Time SIC Pair Generator
I. Voyiatzis1, H. Antonopoulou2, C. Efstathiou1
lDepartment of Informatics, Technological Educational Institute of Athens, Greece 2 Technological Educational Institute of Patras, Computer Technology Institute, Patras, Greece
voyageri@otenet.gr
Abstract - The application of Single Input Change (SIC) pairs of test patterns is very efficient for sequential, i.e. stuck-open and delay fault testing. In this paper a novel implementation for the application of SIC pairs is presented. The presented generator is optimal in time, in the sense that it generates the n-bit SIC pairs in time nx2n, i.e. equal to the theoretical minimum. Comparisons with the schemes that have been proposed in the open literature which generate SIC pairs in optimal time reveal that the proposed scheme requires less hardware overhead
Keywords - Built-In Self Test, Two-Pattern Testing, Delay Fault Testing, Stuck-Open Testing
I. Introduction
WHILE every VLSI design project has its own unique set of goals, there is a fundamental need for reliability in the finished product. Built-In Self Test (BIST) [1] constitutes an attractive and practical solution. Advantages of BIST include the possibility of performing at-speed testing, very high fault coverage, elimination of test generation effort and less reliance on expensive external testing equipment for applying and monitoring test patterns. Therefore BIST reduces the cost of testing. With the increasing complexity of today's VLSI devices (with millions of gates) BIST schemes for embedded modules are increasingly becoming a necessity.
Despite of the fact that exhaustive single-pattern testing provides for 100% fault coverage of detectable single and multiple stuck-at faults without the need for fault simulation or deterministic test pattern generation, it is widely known that a large class of physical defects can not be modeled as stuck-at faults. For example, a transistor stuck-open fault in a CMOS circuit can convert a combinational Circuit Under Test (CUT) into a sequential one [2] while a delay fault (although it does not affect the steady-state operation) may cause circuit malfunction at clock speed [3]. Detection of such faults requires two-pattern tests.
In the literature, two largely known types of two-pattern tests have been investigated, Multiple Input Change (MIC) and Single Input Change (SIC) pairs. SIC pairs are pairs of patterns in which the first pattern differs from the second one in exactly one bit. The utilization of SIC pairs for the detection of stuck-open and delay faults holds some very interesting properties and has been studied by a number of
researchers both theoretically [11] and experimentally [29], [31]-[38]. In the theoretical field, Smith [11] proved that SIC tests are sufficient to detect all robustly detectable path delay faults. In the experimental field, Wang and Gupta [6] proved that SIC pairs provide higher pseudorandom robust path delay fault coverage than MIC pairs. In other words, if a certain number of pairs is applied to the inputs of a Circuit Under Test (CUT), if the pairs are SIC, the achieved fault coverage will be higher than the case in which the pairs are MIC. Gizdarski [40] utilized SIC sequences in order to test delay faults in the address decoders of RAM memories. The above-referenced results, as well as a number of related works [31-38] indicate that the utilization of SIC pairs for testing delay and stuck-open faults compares favorably to the utilization of MIC pairs, since it results in higher fault coverage with fewer test vectors.
In this paper a novel technique is presented for the generation of SIC pairs of patterns. The number of cycles required to generate the SIC pairs is nx2n, i.e. equal to the theoretical minimum. Comparisons with schemes proposed previously for the application of SIC pairs in optimal time indicate that the proposed scheme requires less hardware overhead.
The paper is organized as follows. In Section 2 the proposed scheme is introduced. In Section III the hardware implementation is presented. In Section IV the techniques presented in the literature for the generation of SIC pairs in optimal time are compared. Finally, in Section V we conclude the paper.
II. Implementation of the proposed scheme
Definition 1. We define by Gn = (gn-1,gn-2, ■■■,■■■ ,g1, go)
the 2n-row by n column matrix that is the output of a binary-reflected gray code.
For example, for n=3, G3 = (g2, gb g0) is the 8-row 3 column matrix presented in the sequel.
000
001
011
010
110
111
101
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Definition 2: We define by G1 = D(G), 1<i<n the 2n-row by n column matrix that is generated from G by cyclically shifting the columns one position to the right and inverting the high- and low-order columns.
For example, for G= (g2, gb g0), G1 = T1(G) = (g0’, g2, g1’), G2 = T2(G) = T1(T1(G)) = (g1, g0’, g2’) and G3 = T3(G) = T1(T2(G)) = T1(T1(T1(G))) = (g2 g1 gc) = G . In the following table, we present the matrices G, G1 and G2 for
n=3.
G = (g2 g1 gO) G1 = T1 (G) = (go' g2 gl') G2 = T2(G) = (gl g0' g2')
000 101 011
001 001 001
011 000 101
010 100 111
110 110 110
111 010 100
101 011 000
100 111 010
It is trivial to show that for any value of n, Gnn = Tn(Gn) =
Gn
Hayes [41] proposed a procedure to construct all SIC pairs within nx2n cycles by applying the n sequences Gn, Gn1, Gn2, ... Gnn-1, and presented an intuitive proof for the correctness of the construction. The proposed generator for the generation of the SIC pairs in optimal time is presented in Figure 1.
The module depicted in Figure 1 comprises an n-stage gray counter (an n-stage counter and n-1 2-input XOR gates), an n-stage barrel shifter, a k-stage counter, a k-input OR gate, a k-to-n decoder with enable and a series of 2-input XOR gates. It operates as follows. Initially, both counters are reset to 0. The n-stage counter starts increasing, hence the sequence Gn is generates at the A[n-1:0] outputs of the generator. When the n-stage counter
reaches 2n-1, the k-stage counter increases to 1. Hence, the outputs of the n-stage counter are shifted one position to the right by the barrel shifter, the output of the OR gate is 1 and the decoder output is 00.01. Therefore, the sequence G1 = (g0 gn-1 gn-2 ... g2 g1’) is generated. When this completes, the k-stage counter is increased again, the shifter shifts the outputs of the gray counter two positions to the right, the output of the OR gate becomes 1 again and the decoder output becomes 00.010; therefore, the sequence G2 = (g1 go’ gn-1 gn-2 . g2’) is generated and so on.
Gray counter
A[n-1] A[n-2] A[n-3] A[1] A[0]
Fig. 1. The proposed generator
In order to exemplify the operation of the proposed generator, in Figure 2 we present the operation for the case n=3. During the first phase, Figure 2 (a), the G3 = (g2 g1 g0) sequence is applied to the A[2:0] outputs. During the second phase, Figure 2 (b), the sequence G31 = (g0’ g2 g1’) is applied to the A[2:0] outputs; finally, during the third phase, the sequence G32 = (g1, g0’ g2’) is applied.
a
b
Fig. 2. Operation of the proposed generator for n=3
c
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III. Calculation of the hardware overhead of the
PROPOSED SCHEME
In the application of the proposed scheme, implementing the Gray generator requires an n-bit counter accompanying (n-1) XOR gates; also, the n-bit barrel shifter is required (nxlog2n flip flops) the k-stage counter (k=log2n), the k-to-n decoder with enable, a k-input OR gate, and (n+1) additional XOR gates are required. To calculate the hardware overhead of the k-to-n decoder, we follow a reasoning similar to that used in [27].
A k-to-K decoder can be implemented as follows. Let
k1=L "2 J and k2=~ f-1 Then kj+k2=k. A k-to-K decoder (K=2k) with enable input can be implemented using two subdecoders with enable (k1-to-K1) and (k2-to-K2) and K 2-input NOR gates. The first kj-to-Kj subdecoder is denoted by Da; its inputs are denoted by da0 to dak1-1 and its outputs are denoted by Da0 to DaK1-1; the second subdecoder is denoted by Db; its inputs are denoted by db0 to dbK2-1; its outputs are denoted by Db0 to DbK2-1. All outputs of the two subdecoders are inverted using K1+K2 inverters. Each one of the K gates takes two inputs: the first is an output of the first decoder; the second is an output of the second decoder as follows. Do=Da0 + Db0 ; Dj= Da0 + Db1, ..., DK-1= DK11 + DbK2 l. For example, in Figure 3 we present a 3-to-8 decoder using the above-mentioned procedure.
For the proposed scheme, only n out of 2k outputs are implemented (n<2k). In Table I the Hardware Overhead (in transistors) for various values of n, the inputs of the CUT is presented.
TABLE I
Patterns generated by the module in Figure 1
c’[1:01 C[2:01 G[2:01 S[2:01 N D[0:21 A[2:01
00 000 000 000 0 000 000
001 001 001 001
010 011 011 011
011 010 010 010
100 110 110 110
101 111 111 111
110 101 101 101
111 100 100 100
01 000 000 000 1 100 101
001 001 100 001
010 011 101 000
011 010 001 100
100 110 011 110
101 111 111 010
110 101 110 011
111 100 010 111
10 000 000 000 1 010 011
001 001 010 001
010 011 110 101
011 010 100 111
100 110 101 110
101 111 111 100
110 101 011 000
111 100 001 010
00 000 000 000 0 000 000
In Table I, in the first column we present the value of n and in the second column the value of k=log2n; in the
third and fourth columns we present the k1 and k2 values such that k1+k2=k; in the seventh and eighth columns the hardware overhead of the two sub-decoders is presented; in the ninth column the overhead of the n 2-input gates is presented. In the tenth column, the hardware overhead of the decoder for each value of n is presented. This value will be considered for the calculation of the hardware overhead of the proposed scheme. For the calculations, an да-input NAND/NOR gate is considered to have 2m transistors and an m-input AND 2m+2 transistors [23]. The hardware overheads of the 2x4, 3x8 and 4x16 decoders are 26, 66, and 116 transistors respectively.
Fig. 3. Implementation of 3-to-8 decoder utilizing smaller decoders and 2-input gates
Table II
Calculation of k-to-n decoder (n<2k) with enable hardware overhead (in _____________________________transistors)____________________________
n k Deci Dec2 Dec1 H/W Dec2 H/W n 2- input gates Dec H/W
12 4 2x4 2x4 48 100
16 26 64 116
20 5 3x8 80 172
24 96 188
28 112 204
32 26 128 220
36 6 3x8 144 276
40 160 292
44 176 308
48 192 324
52 208 340
56 224 356
60 240 372
64 66 256 388
68 7 4x16 272 404
72 288 470
76 304 486
80 320 502
84 336 518
88 352 534
92 368 550
96 384 566
100 400 582
104 416 598
108 432 614
112 448 630
116 464 646
120 480 662
124 496 678
128 66 116 512 694
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IV. Comparisons
In this section, the proposed scheme will be compared with the techniques proposed hitherto for the generation of SIC pairs in optimal time, i.e. in exactly nx2n clock cycles [5], [37], [40] in terms of the required hardware overhead.
In PEAT [5], an и-stage NFSR, an и-stage shift register and an и-stage shift register with flip capability are utilized to generate the SIC pairs within (и+1)х2и clock cycles. To implement the technique, the NFSR and и scaи flip-flops with flip capability are implemented. Furthermore, the и flip-flops of the existing register are substituted by scaи flip-flops.
In [37], Das et al presented an optimal solution to the problem of generating SIC pairs, in the sense that the pairs are generated within time equal to the theoretical minimum, i.e. nx2n+1. However, the hardware overhead of [37] is rather high, thus the value of the scheme lies mainly on its high theoretical significance. The hardware overhead of the scheme is, according to [37], 3n+2 flip flops, n XOR gates (2-input), (2и-1) OR gates (2-input), n+1 AND gates (2-input) and 1 NOT gate.
Gizdarksi [40] utilized the algorithm proposed by Hayes in order to generate the SIC pairs to the inputs of the address decoder of a RAM. Gizdarski utilized two sequences, called TS1 and TS2 in [40]; TS2 is utilized in order to detect additional faults in the address decoder (and its generation is more complicated and hardware intensive). Hence the generator for the TS1 sequence is considered for our comparisons. The required hardware includes control logic, an n-bit binary counter, an n-bit register, n 2-input gates, and nxlog2n 2-to-1 multiplexers in a barrel shifter. Since no information is provided in [40]
for the control logic, we shall not take it into account in our comparisons.
For the comparisons, the following are taken into account [23]. A 2-input NAND/NOR gate requires 4 transistors; a 2-input AND requires 6 transistors and a 2-input XOR gate can be implemented by 4 CMOS transistors [42]. The memory elements used are considered to have set/reset capability. Thereby, the flip-flop requires 26 transistors, the scaи flip-flop requires 34 transistors and the scaи flip-flop with flip capability [5] requires 46 transistors.
In T able III we present, for each one of the optimal SIC pair generation techniques (first column) the formulas used for the calculation of the hardware overhead (second column) and the hardware overhead (in transistors, third column). In Figure 4 we present, for various values of n, the ratio of the hardware overhead (in transistors) over n, the number of CUT inputs. From Figure 4 it can be concluded that the proposed scheme presents the least hardware overhead of the schemes that have been proposed in the open literature.
V. Conclusion
In this paper a novel generator for Single Input Change two-pattern tests has been presented. The number of cycles required to generate the SIC pairs is их2и, i.e. equal to the optimal (minimum) time required. Comparisons with the techniques that have been proposed in the literature for the generation of SIC pairs in optimal time revealed that the proposed scheme requires less hardware overhead.
TABLE III
Optimal-time SIC pair generation techniques: Comparison
Hardware Overhead
T echnique Modules T ransistors
Peat [5] Gizdarski [40] DAS [37] Proposed rax(DFF+NOR)+raxDFFscanwithflip+ raxDFFscan Control + nxDFF + nxDFF + nxAND2 + nxlog2nxMUX21 (2n+2)xDFF + nxXOR + (2n-1)xOR2 + (n+1)xAND2 + NOT nxDFF + (n-1)xXOR + log2nxDFF + nxlog2n xMUX21 + log2n-to-n Dec + (n+1)xXOR + log2n-input OR 110xra nx(58+log2n) 84xn+40 nx(38+6xlog2n)+20xlog2n
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Fig. 4. Optimal time SIC pair generation schemes: Comparison
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