Zero-forcing precoding: Difference between revisions

Revision as of 19:22, 14 February 2022

Revision as of 08:26, 24 February 2022

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==Mathematical description==

In a multiple antenna downlink system which comprises of <math>N_t</math> transmit antenna access points and <math>K</math> single receive antenna users, such that <math>K \leq N_t</math>, the received signal of user <math>k</math> is described as

:<math>y_k = \mathbf{h}_k^T \mathbf{x} + n_k, \quad k=1,2, \ldots, K</math>

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:<math>\mathbf{s} = [s_1, \ldots, s_K]^T</math> is the <math>K \times 1</math> transmit signal.

A ''zero-forcing precoder'' is defined as a precoder where <math>\mathbf{w}_i</math> intended for user <math>i</math> is orthogonal to every channel vector <math>\mathbf{h}_j</math> associated with users <math>j</math> where <math>j \neq i</math>. That is,

:<math>\mathbf{w}_i \perp \mathbf{h}_j \quad \mathrm{if} \quad i \neq j.</math>

Thus the interference caused by the signal meant for one user is effectively nullified for rest of the users via zero-forcing precoder.

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:<math>y_k = \mathbf{h}_k^T \sum_{i=1}^K \sqrt{P_i} s_i \mathbf{w}_i + n_k = \mathbf{h}_k^T \mathbf{w}_k \sqrt{P_k} s_k + n_k, \quad k=1,2, \ldots, K</math>

The orthogonality condition can be expressed in matrix form as

:<math>\mathbf{H}^T \mathbf{W} = \mathbf{Q}</math>

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:<math> b_{FB} \log_2(1+\rho_{FB}) \geq B </math>

where <math>b = \Omega_{FB} T_{FB}</math> is the feedback resource consisted byof multiplying the feedback frequency resource and the frequency temporal resource subsequently and <math>\rho_{FB}</math> is SNR of the feedback channel. Then, the required feedback resource to satisfy <math>\Delta R \leq \log_2 g</math> is

:<math> b_{FB} \geq \frac{B}{\log_2(1+\rho_{FB})} = \frac{(M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (g-1)}{\log_2(1+\rho_{FB})} </math>.

Note that differently from the feedback bits case, the required feedback resource is a function of both downlink and uplink channel conditions. It is reasonable to include the uplink channel status in the calculation of the feedback resource since the uplink channel status determines the capacity, i.e., bits/second per unit frequency band (Hz), of the feedback link. Consider a case when SNR of the downlink and uplink are proportion such that <math>\rho_{b,m} / \rho_{FB}) = C_{up,dn}</math> is constant and both SNRs are sufficiently high. Then, the feedback resource will be only proportional to the number of transmit antennas