# Detection, Estimation modulation theory part 1 - Vantress H.

ISBN 0-471-09517-6

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An even simpler case is one in which the total probability of error is the criterion. Then we choose an sm such as fi(sm) = 0. From Fig. 2.40, we see that sm = i. Using (484) and (485) we have

where the approximation is very good for d > 6.

This example is a special case of the binary symmetric hypothesis problem in which ^(s) is symmetric about s = When this is true and the criterion is minimum Pr (c), then /*(?) is the important quantity.

The negative of this quantity is frequently referred to as the Bhatta-charyya distance (e.g., [29]). It is important to note that it is the significant quantity only when sm = In our next example we look at a more interesting case.

Example 2. This example is Case 1A of the general Gaussian problem described on p. 108:

¹ = d\

Substituting into (479) and (482), we have

Pp = erfc* [.y V fi(i')] = erfc» (sd)

(492)

(493)

and

PM = erfc, [(1 - tfv'jiM] = erfc, [(1 - *)</].

(494)

(495)

Ml) = ln Ã (496)

J - oo

(497)

Substituting (497) into (499) gives,

—7=----------- ªÕÐ

(V2tt axsal s)

[

sR2 (1 - s)R2

2 of 2®02 .

•] dR (498) (499)

or

(gQ^W)1-*

sK2

A case that will be of interest in the sequel is

128 2.7 Performance Bounds and Approximations

Substituting (500) into (499) gives

g.{(i-,)i„(i+?)-,.[³ *(.-,)?]}. This function is shown in Fig. 2.41.

M.) - f [-In (' + g) +

..,.4 _ N Ã I2

M; 2 [l + (1 - s)(as2la^)\ ’

and

(501)

(502)

(503)

By substituting (501), (502), and (503) into (479) and (482) we can plot an approximate receiver operating characteristic. This can be compared with the exact ROC in Fig. 2.35a to estimate the accuracy of the approximation. In Fig. 2.42 we show the comparison for N = 4 and 8, and <7s2/<7n2 = 1. The lines connect the equal threshold points. We see that the approximation is good. For larger N the exact and approximate ROC are identical for all practical purposes.

Fig. 2.41 [a(j) for Gaussian variables with unequal variances.

Applications 129

Fig. 2.42 Approximate receiver operating characteristics.

Example 5. In this example we consider first the simplified version of the symmetric hypothesis situation described in Case 2A (p. 115) in which N — 2.

2 I D 2 D 2 I D 2\

and

where

Then

= crs2 + <7n 0O2 = <7n2.

n(s) = S In <7n2 + (1 — s) In (<7n2 + as2) - In (<7n2 + <rs2j)

(504)

/«.ãã÷ 1 - / Ri2 + R22 R32 + R*2\

/>r|Hl(R|tfi) (2jr)2CTi2ffo2 exp ^ 2(^2 2^2 j

1 ² RS + Ra* Ëç2 + RS\ ,ñëñ÷

/’riHo( I î) (2*7)4 2°o2 exp I 2cr02 2<T!2 )’ ( }

(506)

+ (1 — s) In ñò„2 + J In (a„2 + <rs2) - In [<7„2 + ffs2(l - i)]

-('+$ -Ê1+Ø+(ço7,

The function fx(s) is plotted in Fig. 2.43a. The minimum is at s — This is the point of interest at which minimum Pr (e) is the criterion.

130 2.7 Performance Bounds and Approximations

Fig. 2.43a (x(5) for the binary symmetric hypothesis problem.

Thus from (473), a bound on the error is,

(1 + as2/an2)

Pr (0<±;

(1 + as2/2an2)2

The bound in (508) is plotted in Fig. 2.436.

Example ÇÀ. An interesting extension of Example 3 is the problem in which

0

<72

Ks =

<*2

aNI2

aNI 2-

(508)

(509)

Applications 131

ÑÃ„2 ^

Fig. 2.436 Bound on the probability of error (Pr(c)).

The r,’s are independent variables and their variances are pairwise equal. This is a special version of Case 2B on p. 115. We shall find later that it corresponds to a physical problem of appreciable interest.

Because of the independence, /x(s) is just the sum of the n{s) for each pair, but each pair corresponds to the problem in Example 3. Therefore

-« - I In (l + ?) - "¯ '« {(' + • ?)(l + (1 - •) ?)}• (510)

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