Here, we discuss the fundamental limit of the clasical Shannon theory in terms of performance evaluation of complex wireless networking systems. Then, we investigate the concept of the evolved Shannon theory and effective spectral efficiency (or net capacity).

Using the (conventional) Shannon theory, the channel capacity of an wireless link is denoted as

- $ C_{SU} = \log_2 (1+ \mathrm{SNR} ) $

where the wireless link consists of a single antenna transmitter and a single reception user.

To see the fundamental limit, discuss the performance evaluation of the perfect channel state information (CSI) and the limited feedback ZF-BF scheme for MU-MIMO networks, subsequently.

The $ C_{SU} $ ignores the performance loss by overhead signaling regardless whether we put realistic SINR instead of SNR or not.

## Feedback Signaling Overhead Edit

The capacity of the feedback system ($ C_{ZF-BF} $) can be improved by increasing the amount of feedback signaling in a given SINR regime. However, the overall performance can be decreased if the effective data transmission time is reduced as the feedback time increases where the effective data transmission time is given by

- $ T_{DL}/T =( T - T_{FB} ) / T $.

Because the transmission time is reduced, the overall performance can be redefined as effective spectral efficiency

- $ R_{eff} = (1-T_{FB}/T) * R_{ZF} $

where $ R_{ZF} $ is the spectral efficiency of limited feedback ZFBF. Hence, the optimization of the feedback time is necessary to achieve the maximum effective spectral efficiency.

## Pilot Signalling Overhead Edit

A receiver demodulates the received signal using the estimated channel, which is measured based on pilot signal. The observed channel from the pilot signal is given by

- $ \mathbf{s}_k = \sqrt{\beta P} \mathbf{h}_k + \mathbf{z}_k $

where $ \beta $ denoted the number of pilot symbols, $ P $ is the pilot power and $ \mathbf{z}_k $ is the noise signal. Using the minimum-mean-square-error (MMSE) estimation, the MMSE estimate of channel is obtained as

- $ \tilde{\mathbf{h}}_k = E[\mathbf{h}_k \mathbf{s}_k^H] E[\mathbf{s}_k \mathbf{s}_k^H]^{-1} s_k = frac{\sqrt{\beta P}}{N_0 + \beta P} \mathbf{s}_k $