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Figure 12.24 Two-rail code checker for duplication check.
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pair, referred to as z0 ; z1 . The output pair must be complementary if and only if each and every input pair is all complementary [TAMI84]. A two-rail code checker for m 2 is shown in Figure 12.25 [CART70]. This is a twolevel AND-OR realization. The table in Figure 12.25 shows this circuit to be self-testing for single faults and also code disjoint. Lemma 12.3 [FUJI87a] Let the input to the checker be Ai a0;i ; a1;i and Bi b0;i ; b1;i , and the output be Ci 1 c0;i 1 ; c1;i 1 . Here Ai , Bi , Ci 1 2 f0; 1; Wg,
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Figure 12.25 A two-rail code checker for m 2. Source: [CART70]. 1970 IEEE.
CODING FOR LOGIC AND SYSTEM DESIGN
where 0 0; 1 , 1 1; 0 , W 2 f 0; 0 ; 1; 1 g. Then Ci 1 can be expressed as the following truth table:
Bi Ai
0 1 W 0 0 1 W 1 1 0 W W W W W
Ci+1 = Ai
0=(0,1) 1=(1,0) W={(0,0),(1,1)}
, meaning Ci 1 Ai ~ Bi ,
The operation performed in the checker is expressed as de ned in the truth table.
Lemma 12.4 Every single fault in the checker can be detected if the inputs Ai and Bi have the following four distinct patterns: Ai ; Bi f 0; 0 ; 0; 1 ; 1; 0 ; 1; 1 g; 0 0; 1 ; 1 1; 0 :
Multi-input two-rail code checker can be implemented by interconnecting such twolevel blocks to form multilevel trees of arbitrary size [ITOH82]. For example, a tree with eight input pairs formed by interconnecting seven blocks is shown in Figure 12.26. Corollary 12.2 [FUJI87a]
1. For Ai ; Bi 2 f0; 1g, the two-input two-rail code checker is equal to modulo-2 adder on f0; 1g. 2. As for the multi-input two-rail code checker implemented with the two-input tworail code checkers, the output of this checker takes the values W if at least one input to this checker is equal to W, where W f 0; 0 ; 1; 1 g.
a0 b0 a1 b1 a2 b2 a3 b3 a4 b4 a5 b5 a6 b6 a7 b7
: 2-Input two-rail code checker z0 z1
Figure 12.26 Eight-input two-rail code checker.
SELF-TESTING CHECKERS
(0 (0 (1 (1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 0 0) 0) 1) 1)
4 Test patterns
Q0 = Q1 Q1 = Q2 Q2 = Q0
Q2 Q0 Q1
0 = (0, 1) 1 = (1, 0)
0 0 1 1
Q0 =
0 1 1 0
0 1 0 1
() () ()
Q1 = Q2 =
0 1 1 0
0 0 1 1
0 1 0 1
Figure 12.27 Example of four test patterns generated systematically for 8-input two-rail code checker.
In general, 2m test patterns are suf cient to diagnose such multiple-input trees if each two-rail block has no more than m input pairs [ANDE71]. Therefore the two-rail tree shown in Figure 12.26 is completely diagnosed by the four test patterns. These test patterns are systematically obtained in [ANDE71]. Figure 12.27 shows an example of the four test patterns systematically obtained for the eight-input two-rail code checker. In this test pattern generation every two-input two-rail code checker has four distinct input patterns de ned in Lemma 12.4. In normal operation, however, all these patterns may not be applied to the multiple-input tree circuit. This is because these checkers are placed at the output of the circuit under check, as shown in Figure 12.24. That is, they are embedded, and hence a restricted number of patterns may be given to the checkers. Even for this situation some techniques have been proposed to satisfy the self-testing condition [HUGH84, FUJI87a]. The following theorem deals with the self-testing property of the multi-input two-rail code checkers. Theorem 12.6 [FUJI87a] If the M-input two-rail code checker having one input V shown in Figure 12.28 satis es the following conditions, 1. Primary input Ai 6 constant, Ai 6 W, i 0; 1; . . . ; M 1. 2. Independent of Ai s, input V v; v can take any value in f0; 1g, then every single fault in this checker can be detected. Proof The output of the i-th level two-input two-rail code checker can be expressed as follows: Ci 1 Ai ~ Ci ; C0 V; i 0; 1; . . . ; M 1;