Download (direct link):
J-‘- = -ÅÛòãã (449)
and for a Gaussian a priori density
Jp = Ëà-\ (450)
where Aa is the covariance matrix. The term in (449) is analogous to (104) in Section 4.2. Thus it follows easily that (449) reduces to,
We recall that this is a bound with respect to the correlation matrix Re in the sense that
Jr - Re 1 (452)
is nonnegative definite. If the a posteriori density is Gaussian, Re-1 = Jr.
A similar result is obtained for unbiased estimates of nonrandom variables by letting JP = 0. The conditions for the existence of an efficient estimate carry over directly. Equality will hold for the ith parameter if and only if
mt)} -At= 2 è A) P WO - *t, À)] dt. (453)
j — 1 j Tt VSl j
To illustrate the application of this result we consider a simple example.
Additive White Gaussian Noise Channel 373
Example. Suppose we simultaneously amplitude- and frequency-modulate a sinusoid with two independent Gaussian parameters a, N(0, aa), and b, N(0, ab). Then
/2Å\Ó³ T T
r(t) = s(t, Ë, ß) + W(0 = ^ j 5 sin (a>ct + PAt) + Wit); ~<t<j (454)
The likelihood function is
In Ë²ÊÎÌ, Ë] = ^ f™2 [*/) - (“)*Bsin K? + Mo]
x (j Á sin (a>cf + PAt) dt. (455)
Ä) = (y)'/! Â fit cos (a,cf + 0At) (456)
+ Ø (457)
Because the variables are independent, JP is diagonal.
The elements of Jr are
2 ÃÒ²2 7/7 1
Jll = ?a,b B2p2t2 COS2 (à,ñ´ + ^0 Ë +
•/Vo J-T/2 -» Oa
T2 2F 1
= + <458>
7 ËÃ/2 0 17 1 917 1
/22 = ^ ?a,b V Sin2 («cf + PAt) + Ë + -^> (459)
jVo J-TI2 ³ -/>0 <*Ü
2 ã /»ã/2 2F 1
= -^ã ^a.b|J "j7" &Pt sin (î)ñ/ + PAt) cos (a)ct + PAt) dt J ? 0. (460)
Thus the J matrix is diagonal. This means that
äà-«ëã(^ + ..-5 Ö-ã)" («1>
?[(«-«•]> (³ + (4È)
Thus we observe that the bounds on the estimates of a and b are uncorrelated. We can show that for large E/N0 the actual variances approach these bounds.
We can interpret this result in the following way. If, each time the experiment was conducted, the receiver were given the value of b, the performance in estimating a would not be improved over the case in which the receiver was required to estimate b (assuming large EjNo).
We observe that there are two ways in which Ji2 can be zero. If
Ã A B)H, À Â)ë, Q (46J)
J - 772 vA cJtS
before the expectation is taken, it means that for any value of A or Â the partial derivatives are orthogonal. This is required for ML estimates to be uncoupled.
374 4.7 Summary and Omissions
Even if the left side of (463) were not zero, however, the value after taking the expectation might be zero, which gives uncoupled MAP estimates.
Several interesting examples of multiple parameter estimation are included in the problems.
The results can be modified in a straightforward manner to include other cases of interest.
1. Nonrandom variables, ML estimation.
2. Additive colored noise.
3. Random phase channels.
4. Rayleigh and Rician channels.
5. Multiple received signals.
Some of these cases are considered in the problems. One that will be used in the sequel is the additive colored noise case, discussed in Problem 4.6.7. The results are obtained by an obvious modification of (447) which is suggested by (226).
* - <4* ã
1Ô) ~ g(z)] dz, ³ = 1, 2,..., M, (464)
A = a. man
g(z) - g(z) A f ' Qn(z, u)[r(u) - s(u, amap)] du, Ã* < z < Tf. (465)