Matrix differentiation - Signal Processing and Optimization for Transceiver Systems - page 681

which proves Eq. (20.52) indeed. In the preceding derivation the second
equality follows from Eq. (20.38), the third from Table 20.3 (entry 9), the
fourth from Tr[ AB ]=Tr[ BA ] , and the last line from Table 20.2.

Thus the stationarity condition ∂ψ/∂ T ∗ = 0 yields

Λ ( I + TT † Λ ) − 2 T = μ T . (20 . 53)

Since T is nonsingular, this yields Λ ( I + TT † Λ ) − 2 = μ I , which shows that
I + TT † Λ = μ

1 / 2 .

The solution Q = TT † therefore takes the form
Q = μ

− 1 / 2

− Λ − 1 ,

(20 . 54)

which is a diagonal matrix. Thus there is no loss of generality in restricting Q
to be a diagonal matrix in the above problem. The constant μ should be chosen
such that the condition Tr( Q )= c is satisfied.
The careful reader will notice that the solution Q calculated above is not
guaranteed to be positive definite, since the diagonal elements on the right-hand
side of Eq. (20.54) are not guaranteed to be positive. This problem can be solved
by a more careful formulation which uses inequality constraints (positivity) as
in Chap. 22. A more elegant approach to this problem based on Schur-convex
functions was presented by Witsenhausen (see Salz [1985]). This is given in
Chap. 13 (Sec. 13.5), and is applicable even when the optimum point is not an
interior point; it is also applicable when Q is only positive semidefinite rather
than positive definite.

Example 20.19. Optimum noise canceller
Figure 20.1 shows a signal y ( n ) with additive noise e 1 ( n ). There is another pure
noise source e 2 ( n ) also shown in the figure. We wish to add an appropriately
transformed version of e 2 ( n ) to the noisy signal, so that the result y ( n )isas

close to y ( n ) as possible in the mean square sense. This system is called the
noise canceller.
If the pure noise component e 2 ( n ) has some correlation with e 1 ( n ), then
indeed such a cancellation is possible to some extent. As an extreme example,
if e 2 ( n )= e 1 ( n ) , then we only have to choose A = − I ,andthenoise e 1 ( n )is

completely cancelled, that is, y ( n )= y ( n ). Similarly if e 2 ( n )= Te 1 ( n ) , where

T has a left inverse T # , then choosing A = − T # makes y ( n )= y ( n ). These are

examples where e 2 ( n ) is completely correlated to e 1 ( n ), that is, we can estimate
the noise e 1 ( n ) with no error based on measurement of e 2 ( n ) . In practice, there
are situations when a separate measurement of correlated noise is available,
although the correlation will only be partial. For example, y ( n )+ e 1 ( n )might
be the noisy speech from a microphone and e 2 ( n ) might be the speech-free noise
measured with another microphone far away from the speaker.