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tributed by x1(2) due only to zero padding at the transmitter. Hence, there is a diversity order of nR - nT + 2 for this symbol. Similarly, there is no interference when extracting x1(1) and there is a diversity order of nR - nT + 3. Once all the coded symbols of layer 1 is collected, they are passed to the channel decoder. The decoded bits are reencoded and remodulated. When we want to extract x1(2) for layer 2, the contribution from x2(1) is known and can be subtracted. Hence, in a similar manner, each layer sees the same composition of diversity order from nR - nT + 1 to nR. Since ZF-SIC relies on the successful decoding of the previous layers, there is error propagation problem; therefore, it is critical to make sure that the decoding order is in descending order of ||Pkhk||. In other words, the optimal decoding order is given by p = (p1, . . . , pnT) such that Pp k h p k Pp k -1 h p k -1 "k [2, nT ]
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Note that the ZF-SIC structure is not applicable to the coded V-BLAST architecture. This is because to successfully decode a particular layer, we need to collect the soft information from the entire frame. Yet, the entire coded frame is delivered through various transmit antennas. Hence, it is not possible to cancel the contribution from any particular transmit antenna before the entire coded symbols are collected for soft decision. MMSE Receiver. In addition to zero forcing, another possible linear processing receiver structure is the minimum mean-square error (MMSE) receiver. While ZF is asymptotically optimal at high SNR, there is a performance penalty at small SNR. As we shall illustrate in this section, MMSE can achieve a better performance even in the small SNR region. The MMSE receiver for LST designs is similar to the format in Figure 4.9a except that the ZF blocks are replaced by MMSE blocks. The basic MMSE processing is illustrated below. The soft information corresponding to the transmit symbol xk, namely, vk, is given by premultiplying the received signal y with a nR 1 weight vector w: vk = w *y k (4.19)
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The optimal weight is the one that minimizes the mean-square error ek = e[|w*y - xk|2]. The MMSE weight is given by the following optimization k problem: w k = arg min e w *y - xk k
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"k [1, nT ]
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The MMSE receiver makes an optimal tradeoff between suppressing the spatial interference and minimizing the channel noise. The optimal solution of the weight vector wk is given by
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P0 2 w k = s z I nR + nT
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i h ih* h k i k
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Hence, the equivalent channel between the transmitted symbol xk and the MMSE receiver processing vk is given by * vk = w *y = h k A k1 h k x k + A -1 z k k P0 2 where A k = h * s z I nR + k nT is given by
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i i k h i h * nT -1
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. Hence, the overall ergodic capacity
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nT P0 * -1 C MMSE = e log 2 1 + hk Ak hk nT k =1
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P0 * -1 h k A k h k , the capacity CMMSE is strictly nT larger than the zero-forcing capacity CZF. Similarly, the MMSE receiver can be modi ed into MMSE-SIC structure as illustrated in Figure 4.9b. With interference cancellation, the equivalent channel at the kth-stage processing is given by vk = w *y = h *A k1h k xk + A k1z k k (4.24)
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P0 2 where A k = h * s z I nR + k nT by
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i i =k +1 h i h* nT
. The overall ergodic capacity is given
nT P0 - C MMSE-SIC = e log 2 1 + h* A k1h k k nT k =1
Yet, from the chain rule of mutual information, we have I (X; Y H) = I ( X k ; Y H, X 1 , . . . , X k -1 )