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The one with the smallest Euclidean norm requires that these latter values be chosen as zero. In this case, the minimum norm solution becomes 8 = v c o l ( f ( l ) / a 1 , . . f(p)/a,,O,.. 7 KRONECKER PRODUCTS Let A = [a,,]&1 and B = [ / I % ~ ] &be = ~m x m and n x n matrices, respectively. , Horn and Johnson (1994)): I I In other words, each entry of A is replaced by a scaled multiple of B. In particular, if A is the identity matrix, I,, then I, 5 B is a block diagonal matrix with B repeated along its diagonal: I, 8 B = diag{B, B , .

Then z2 yz = 1 so that a: and y are dependent. 5E sin28 = 0, so that a: and y are uncorrelated. 3 COMPLEX-VALUED RANDOM VARIABLES Although we have focused so far on real-valued random variables, we shall often encounter applications that deal with random variables that assume complex values. ), of its real and imaginary parts. This means that we can regard a complex random variable as a function of two real random 5 variables. 3 ~ i in terms of the means of its real and imaginary parts. 7) where the symbol * denotes complex conjugation.

This means that R, is nonnegative definite, as claimed. For real-valued data, the symbol * is replaced by the transposition symbol T, and R, is defined as R, = A E(z-2)(~-2)~ I I In this case, the covariance matrix R, therefore becomes symmetric rather than Hermitian. In other words. it now satisfies R, = R j Moreover. R, continues to be nonnegative-definite, which now means that a T R z a 2 0 for any realvalued column vector a. 2 (Transmissions over a noisy channel) Consider the setting of Fig.

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