【文章內(nèi)容簡介】 hysical phenomena, and proceeded to statistical fading models, which are more appropriate for the design and performance analysis of munication schemes. We will in fact see a lot of parallelism in the specific channel modeling technique as well. Our focus throughout is on flat fading MIMO channels. The extensions to frequencyselective MIMO channels are straightforward and are developed in the exercises. 7． 1 Multiplexing capability of deterministic MIMO channels A narrowband timeinvariant wireless channel with nt transmit and nr receive antennas is described by an nr by nt deterministic matrix H. What are the key properties of H that determine how much spatial multiplexing it can support? We answer this question by looking at the capacity of the channel. Capacity via singular value deposition The timeinvariant channel is described by y = Hx+w_ () where x,yand wdenote the transmitted signal, received signal and white Gaussian noise respectively at a symbol time (the time index is dropped for simplicity). The channel matrix H is deterministic and assumed to be constant at all times and known to both the transmitter and the receiver. Here, hij is the channel gain from transmit antenna j to receive antenna i. There is a total power constraint, P, on the signals from the transmit antennas. This is a vector Gaussian channel. The capacity can be puted by deposing the vector channel into a set of parallel, independent scalar Gaussian subchannels. From basic linear algebra, every linear transformation can be represented as a position of three operations: a rotation operation, a scaling operation, and another rotation operation. In the notation of matrices, the matrix H has a singular value deposition (SVD): 11 Where and are (rotation) unitary matrices1 and is a rectangular matrix whose diagonal elements are nonnegative real numbers and whose offdiagonal elements are The diagonal elements are the ordered singular values of the matrix H, where nmin： =min（ nt， nr） . Since the squared singular values _2i are the eigenvalues of the matrix HH* and also of H*H. Note that there are nmin singular values. We can rewrite the SVD as The SVD deposition can be interpreted as two coordinate transformations: it says that if the input is expressed in terms of a coordinate system defined by the columns of V and the output is expressed in terms of a coordinate system defined by the columns of U, then the input/output relationship is very simple. Equation () is a representation of the original channel () with the input and output expressed in terms of these new coordinates. We have already seen examples of Gaussian parallel channels in Chapter 5, when we talked about capacities of timeinvariant frequencyselective channels and about timevarying fading channels with full CSI. The timeinvariant MIMO channel is yet another example. Here, the spatial dimension plays the same role as the time and frequency dimensions in those other problems. The capacity is by now familiar: where P1*,… ,Pnmin*are the waterfilling power allocations: with chosen to satisfy the total power constraint corresponds to an eigenmode of the channel (also called an eigenchannel). Each eigenchannel can support a data stream。 thus, the MIMO channel can support the spatial multiplexing of multiple streams. Figure pictorially depicts the SVDbased architecture for reliable munication. 12 There is a clear analogy between this architecture and the OFDM system introduced in Chapter 3. In both cases, a transformation is applied to convert a matrix channel into a set of parallel independent subchannels. In the OFDM setting, the matrix channel is given by the circulant matrix C in (), defined by the ISI channel together with the cyclic prefix added onto the input symbols. The important difference between the ISI channel and the MIMO channel is that, for the former, the U and V matrices (DFTs) do not depend on the specific realization of the ISI channel, while for the latter, they do depend on the specific realization of the MIMO channel. Physical modeling of MIMO channels In this section, we would like to gain some insight on how the spatial multiplexing capability of MIMO channels depends on the physical environment. We do so by looking at a sequence of idealized examples and analyzing the rank and conditioning of their channel matrices. These deterministic examples will also suggest a natural approach to statistical modeling of MIMO channels, which we discuss in Section . To be concrete, we restrict ourselves to uniform linear antenna arrays, where the antennas are evenly spaced on a straight line. The details of the analysis depend on the specific array structure but the concepts we want to convey do not. SIMO channel The simplest SIMO channel has a single lineofsight (Figure (a)). Here, there is only free space without any reflectors or scatterers, and only a direct signal path between each antenna pair. The antenn