omplex-valued matrix of rank K then A = UEVH denotes the singular value decomposition of A. The M X M unitary matrix U is composed of the eigenvectors of AAH, that is, U = (U1,U2,...,Um), where AAHu = o2ui. The NXN unitary matrix V is com- posed of eigenvectors of AHA that is, =(v1,V2,...,Vn), where AHvi = o2vi The elements of the MXN are the square roots of the eigenvalues of matrix AHAo as the singular values of matrix A may be written as A = values of matrix A is the rank of A. The singular values of matrix A are positive real numbers which satisfy . 556 Chapter 4 antennas are required to be closely spaced. The optimal water-filling allocation strategy is obtained when the power allocated to each spatial subchannel is non-negative. In the design of wireless communication systems, the main objective is to exploit the trans- mission schemes whose performance can approach the channel capacity as much as possi- ble. Therefore, it is important to understand the underlying concepts and various information theoretic definitions of channel capacity and what can be pragmatically achieved under realistic channel conditions and transceiver implementations. Let us begin our concise study with the most generic definition of channel capacity. We denote the input and output of a memoryless SISO wireless channel with the random variables X and Y, respectively, the channel capacity is defined as C = max p(x) I(X; Y) where I(X; Y) represents the mutual information between X and Y. Shannon's theorem [19] provides an operational meaning to the definition of the instantaneous capacity as the number of bits that can be transmitted reliably over the channel with vanishing probability of error. The mutual infor- mation is maximized with respect to all possible transmit signal statistical distributions p(x). Mutual information is a measure of the amount of information that one random variable contains about another variable. The mutual information between X and Y can also be writ- ten as I(X;) = H(Y) - H(Y|X) where H(Y|X) represents t