In the text we have analyzed the performance of fading channels under the assumption of receiver CSI. The CSI is obtained in practice by transmitting training symbols. In this exercise, we will study how the loss in degrees of freedom from sending training symbols compares with the actual capacity of the non-coherent fading channel. We will conduct this study in the context of a block fading model: the channel remains constant over a block of time equal to the coherence time and jumps to independent realizations over different coherent time intervals. Formally,

y[m + nTc] = h[n]x[m + nTc ] + w[m + nTc], m = 1,..., Tc , n ≥ 1, (5.138)

where Tc is the coherence time of the channel (measured in terms of the number of samples). The channel variations across the blocks h[n] are i.i.d. Rayleigh.

1. For the IS-856 system, what are typical values of Tc at different vehicular speeds?

2. Consider the following pilot (or training symbol) based scheme that converts the non-coherent communication into a coherent one by providing receiver CSI. The first symbol of the block is a known symbol and information is sent in the remaining symbols (Tc - 1 of them). At high SNR, the pilot symbol allows the receiver to estimate the channel (h[n], over the nth block) with a high degree of accuracy.

Argue that the reliable rate of communication using this scheme at high SNR is approximately

Tc - 1 C(SNR) bits/s/Hz, (5.139)
Tc
where C(SNR) is the capacity of the channel in (5.138) with receiver CSI. In what mathematical sense can you make this approximation precise?

3. A reading exercise is to study [83] where the authors show that the capacity of the original non-coherent block fading channel in (5.138) is comparable (in the same sense as the approximation in the previous part) to the rate achieved with the pilot based scheme (cf. (5.139)). Thus there is little loss in performance with pilot based reliable communication over fading channels at high SNR.