Fading Broadcast Channel Capacities: III Scalar Fading Channels

In wireless broadcast multicast services (BMS), a standard assumption is that each receiver knows something about the channel h, usually referred as channel side information (CSI) or channel quality information (CQI). This is a pretty reasonable assumption when the channel is fading slowly inside the design boundary since there are pilot symbols available for the receiver to estimate CQI. Since the channel and transmitted signals are independent to each other, the ergodic capacity of the fading channel with receiver side information is given by

C_fading( SNR, h ) = E log( 1 + |h|² SNR ) ≤ C_AWGN( E(|h|²) SNR ).

This means fading hurts or reduces the capacity in general if the transmitter knows nothing of the fading. This is different to the case that assumes the transmitter can estimate channel through CQI feedback and therefore can do some precoding on broadcast signals.

Since a statistic analysis on a log(*) probability function is non-trivial, one approach is to apply the well-know Maclaurin expansion on C_fading( SNR, h ) and obtain a polynomial series of it

C_fading( SNR, h ) = -ln^-12 Σ_n=1[ (-SNR)ⁿE(|h|ⁿ)/n ].

From here, it is much easy for us to find some interesting results in the following.

Low SNR Region

C_fading( SNR, h ) ≈ ln^-12 SNR E(|h|) - ln^-12 SNR² E(|h|)² K( h^1/2)/2

where K() denotes Kurtosis function.
High SNR Region

C_fading( SNR, h ) ≈ E log( |h|² SNR )

It is hard to evaluate the above in general since the random valuable h is inside the nonlinear function log( ). However, a simple close-form solution may be possible for some special cases, including Rayleigh channel model, Weibull channel model, and Nakagami-m channel model

Quantify Channel Fading

There are many literatures discussing how to quantify the amount of fading a channel may have. The parameter, Kurtosis, is one of them. Kurtosis essentially is the normalized fourth moment of a realvalued random valuable and indicates the ”peakedness” of a probability distribution. A high kurtosis means a large variance due to infrequent extreme deviations, which results in a sharp ”peak” and fat ”tails”. Another similar measurement is the amount of fading of a real-valued random valuable, which indicates the severity of channel fading in communications. The Kurtosis and amount of fading of major fading distributions are compiled in Table 1. A similar compilation can be found in [Shamai 01].

Table 1. The Kurtosis and amount of fading of various fading channel models