By C. Stanley Ogilvy
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Extra info for Through the Mathescope
Example text
HMRx MTx 0 1 h11 [d] . . h1MTx [d] B C .. .. H[d] ¼ @ A . . hMRx 1 [d] . . hMRx MTx [d] “Indexed matrix,” a discrete sampling of each of MRx ÂMTx ÂD timedispersive channels represented in matrix form. Unweildy. H-tensor, discrete representation of the time-dispersive MIMO channel. Makes it easier to do linear operations. H-matrix, consisting of MRx ÂMTx constant complex coefficients. Impulse response depends on t, and h(t) changes over time; the delay t is needed to describe this change. Continuous variable.
B B B uM 21k uMRx 22k Rx B B . B .. @ uMRx MTx 1k uMRx MTx 2k ÁÁÁ u11Dk .. ÁÁÁ ÁÁÁ uMRx 1Dk u12Dk .. uMRx 2Dk .. ÁÁÁ ÁÁÁ 1 C C C C C C C: C C C C C C C C C A (2:47) uMRx MTx Dk where U WB,k is the kth wideband eigenbasis. As with the narrowband case, UWB,k is the basis for a set of correlated scatterers, with fading coefficient lWB,k. Each 42 MULTIPLE ANTENNA CHANNELS AND CORRELATION eigenbasis can be further reduced using the SVD, as shown by UWB,k ¼ M min X l0 [k, ‘]uRxTx,k,‘ uTD,k,‘ , (2:48) l¼1 T , are the lth singular value, left where Mmin ¼ min(MRxMTx, D), u RxTx,k,‘, and u D,k,‘ singular vector, and right singular vector of the kth wideband eigenbasis, respectively.
If we recast H[d ] as a third-order tensor, we can use multilinear algebra to analyze the wideband MIMO channel in a rather compact manner. We define the wideband H-tensor H [ CMRx ÂMTx ÂD to be a tensor whose element hmnd is the complex gain of the channel between receiver m, transmitter n at delay d, for m [ f1, . . , MRxg, n [ f1, . . , MTxg, and d [ f1, . . , Dg. The structure of the H-tensor is illustrated in Fig. 9. 9 Visualizing the structure of the H-tensor. 34 MULTIPLE ANTENNA CHANNELS AND CORRELATION In this way, the wideband H-matrix H[d ] consists of the subset of elements of H corresponding with delay tap d.
Through the Mathescope by C. Stanley Ogilvy
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