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where h represents a realization of the channel. Note that in (3.17), the effect of ||v||2 cancels each other between the numerator and the denominator. Hence, without loss of generality, we x ||v||2 = 1. Similarly, since the transmitter transmits with total energy Et||w||2, we assume that ||w||2 = 1 and the total 2 transmit energy is given by Et. Using these assumptions, we obtain gr = Er Et v H hw Et Gr = = 2 2 2 sz sz sz
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(3.18)
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where G = |vHhw|2 is the effective channel gain. In a MIMO system both a transmit beamforming vector and a receive combining vector need to be chosen. Areceiver with v maximizes |vHhw| given w is called a maximumratio combining (MRC) receiver. The MRC vector v follows from the vector norm inequality v H hw v
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Since ||v||2 = 1, thus the MRC vector must satisfy 2 v H hw = hw
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Hence, the MRC vector v given w is easily seen to be the unit vector v = hw/||hw||2. The remaining optimization parameter is w. The optimal w without any design constraints is given by w = arg max hx
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where WnT refer to the set of unit vectors in C nT. However, recall that the channel state information is not available to the transmitter. Instead, there is a low-rate, error-free, zerodelay feedback link for the purpose of conveying the optimal w to the transmitter. Since w can be any unit vector in the space WnT, it is essential to introduce some method of quantization due to the limited feedback channel. A reasonable solution is to let both the receiver and transmitter use a codebook of Q beamforming vectors. The receiver then quantizes the beamforming vector by selecting the best [according to Equation (3.21)] beamforming vector from the codebook and feeding the index of this vector back to the transmitter via the limited feedback link of capacity Cfb = log2Q. Unfortunately, it is not obvious which Q vectors should be included in the codebook. So, the remaining design problem is to select an optimal codebook for the beamforming vector w given by the following problem. Problem 3.1 (Maximum SNR) Find the codebook W in which {w1, w2, . . . , wQ} denote beamforming vectors and Q represents the size of codebook to maximize the average receiving SNR: eH[arg maxx W||hx||2].
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It has been shown [31,130] that an optimal beamforming vector for the maximum ratio transmission (MRT) systems is the dominant right singular vector of H with H de ned as in the beginning of this section. The optimal vector is given by w = arg max x H h H hx
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(3.22)
Note that the optimal solution in this equation is not unique. For instance, if w is an optimal solution, then ejfw is also an optimal solution because |wHhHhw|2 = |e-jfwHhHhejfw|2. Hence, we can deduce that if w w (using the w are both optimal equivalence relation de ned in Section 3.3.2), then w and solutions and will provide the same SNR performance. In other words, all the optimal beamforming vectors w form a line over C nT. This result is summarized in the following lemma [93]. Lemma 3.1 (Isotropic Beamforming Vectors) The line generated by the optimal beamforming vectors for a MIMO Rayleigh fading channel is an isotropically oriented line in C nT passing through the origin. Therefore, the problem of the quantized transmit beamforming in a MIMO communication system reduces to quantizing an isotropically oriented line in C nT and the codebook designing problem is equivalent to the Grassmannian line packing problem with the following criterion. Grassmannian Beamforming Criterion. Design the set of codebook vectors Q {wi}i=1 such that the corresponding codebook matrix W maximizes d (W) = max 1 - w H w1 k