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/(r) = 2 Ri2- 2 R" * yv- (43°)
i= 1 i = N +1 Ho
If the hypotheses are equally likely and the criterion is minimum Ðã(º), the threshold 17 in the LRT is unity (see 69). From (388) and (390) we see
that this will result in yv = 0. This case occurs frequently and leads to a
simple error calculation. The test then becomes
N Hi 2 N
/i(R) é 2 ë,21 2 R* - WR)- (431)
³ = 1 "î i = H + 1
The probability of error given that Hx is true is the probability that
/0(R) is greater than /i(R). Because the test is symmetric with respect to the two hypotheses,
Pr (c) = ³ Pr (c|ß0 + ³ Pr (ˆ|tf0) = Pr (ˆ|Hx). (432a)
Pr («) = Ã dLlPh ³íÄ^²ß,) Ã Pio\h1(L0\H1) dL0. (432b)
JO J Li
Substituting (406) and (407) in (4326), recalling that N is even, and evaluating the inner integral, we have
Pr(,) “ ³ 2"VTe~L,‘'u‘‘
®i2 + an2 as2 + 2an2 and integrating, (432c) reduces to
N12-1 ¹ , j _ ]\
Pr (º) = aNI2 2 2 1(1 - «Ó- (434)
/=0 \ j I
This result is due to Pierce . It is a closed-form expression but it is tedious to use. We delay plotting (434) until Section 2.7, in which we derive an approximate expression for comparison.
Case 2B. Uncorrelated Signal Components: Unequal Variances. Now,
116 2.7 Performance Bounds and Approximations
It follows easily that
1 Ã N rr 2 2N rr2
**) = Ë 2-ÃÒ—a**’" 2 2Ã* Ê
°n L»=l aSi ³ = N + 1 an aSi- N
As in Case IB, the performance is difficult to evaluate. The approximations developed in Section 2.7 are also useful for this case.
We have discussed in detail the general Gaussian problem and have found that the sufficient statistic was the difference between two quadratic forms:
/(R) = i(RT ~ moT)Qo(R - m0) - i(Rr - m1r)Q1(R - m,). (437)
A particularly simple special case was the one in which the covariance matrices on the two hypotheses were equal. Then
/(R) = ³ AmTQR, (438)
and the performance was completely characterized by the quantity d2:
d2 = AmTQ Am. (439)
When the covariance matrices are unequal, the implementation of the likelihood ratio test is still straightforward but the performance calculations are difficult (remember that d2 is no longer applicable because /(R) is not Gaussian). In the simplest case of diagonal covariance matrices with equal elements, exact error expressions were developed. In the general case, exact expressions are possible but are too unwieldy to be useful. This inability to obtain tractable performance expressions is the motivation for discussion of performance bounds and approximations in the next section.
Before leaving the general Gaussian problem, we should point out that similar results can be obtained for the Af-hypothesis case and for the estimation problem. Some of these results are developed in the problems.
2.7 PERFORMANCE BOUNDS AND APPROXIMATIONS
Up to this point we have dealt primarily with problems in which we could derive the structure of the optimum receiver and obtain relatively simple expressions for the receiver operating characteristic or the error probability.
In many cases of interest the optimum test can be derived but an exact performance calculation is impossible. For these cases we must resort to
bounds on the error probabilities or approximate expressions for these probabilities. In this section we derive some simple bounds and approximations which are useful in many problems of practical importance. The basic results, due to Chernoff , were extended initially by Shannon . They have been further extended by Fano , Shannon, Gallager, and Berlekamp , and Gallager  and applied to a problem of interest to us by Jacobs . Our approach is based on the last two references. Because the latter part of the development is heuristic in nature, the interested reader should consult the references given for more careful derivations. From the standpoint of use in later sections, we shall not use the results until Chapter II-3 (the results are also needed for some of the problems in Chapter 4).
The problem of interest is the general binary hypothesis test outlined in Section 2.2. From our results in that section we know that it will reduce to a likelihood ratio test. We begin our discussion at this point.