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:<math>\mathbf{y} = \sum_{i=1}^{K} s_i \mathbf{h}_i + \mathbf{n}</math>
where <math>s_i</math> is the transmitted signal of user <math>i</math>, <math>\mathbf{n}</math> is the <math>N_r \times 1</math> noise vector, <math>\mathbf{h}_k</math> is the <math>N_r \times 1</math> channel vector.
==== Quantify the feedback amount ====
Quantify the amount of the feedback resource required to maintain at least a given throughput performance gap between zero-forcing with perfect feedback and with limited feedback, i.e.,,
:<math>\Delta R = R_{ZF} - R_{FB} \leq log_2 g</math>.
Jindal showed that the required feedback bits of an [[Spatial Correlation|spatially uncorrelated]] channel should be scaled according to SNR of the downlink channel, which is given by<ref name=jindal />:
:<math> B = (M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (g-1) </math>
where ''M'', is the number of transmit antennas and <math>\rho_{b,m}</math> is SNR of the downlink channel.
To feedback ''B'' bits though uplink channel, the throughput performance of the uplink channel should be larger than or equal to 'B'
:<math> b_{FB} \log_2(1+\rho_{FB}) \geq B </math>
where <math>b = \Omega_{FB} T_{FB}</math> is the feedback resource consisted by 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 function of both downlink and uplink chanel conditions. It is reasonable to include the uplink channel status in the calcuration 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. Considedr 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 proportion to the number of transmit antennas
:<math> b_{FB,min}^* = \lim_{\rho_{FB} \to \infty } \frac{(M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (g-1)}{\log_2(1+\rho_{FB})} = M - 1</math>.
It follows the above equation that the feeback resource (<math>b_{FB}</math>) is not necessary to scale according to SNR of the downlink channel, which is almost contradict to the case of the feedback bits. We, hence, see that the whole systematic analysis can reverse the facts resulted from each reductioned situation.
==See also==
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