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Zero-forcing (or null-steering) precoding is a method of spatial signal processing by which a multiple antenna
==Mathematical description==
In a multiple antenna downlink system which comprises
:<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>\Delta R = R_{ZF} - R_{FB} \leq \log_2 g</math> .
Jindal showed that the required feedback bits of a [[Spatial Correlation|spatially uncorrelated]] channel should be scaled according to SNR of the downlink channel, which is given by:<ref name="Jindal_ZF" />
:<math> B = (M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (g-1) </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
:<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
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==Performance==
If the transmitter knows the downlink [[channel state information]] (CSI) perfectly, ZF-precoding can achieve almost the system capacity when the number of users is large. On the other hand, with limited [[channel state information]] at the transmitter (CSIT) the performance of ZF-precoding decreases depending on the accuracy of CSIT. ZF-precoding requires the significant feedback overhead with respect to signal-to-noise-ratio (SNR) so as to achieve the full multiplexing gain.<ref name="Jindal_ZF">{{cite journal |
==See also==
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