The capacity bounds derived in Chapter 3 were chosen for their utility and tractability. However, these are not the only possible bounds. In this appendix, we derive alternative bounds that, in some cases, may be tighter than their counterparts in Chapter 3.
In the uplink, the variances of the effective noises in (3.23) and (3.32) depend on the channel estimate, which itself is known by the base station. As a result, the corresponding capacity bounds entail inconvenient expectations of logarithms of stochastic quantities. In Chapter 3, we circumvented this difficulty via the “use and forgetCSI” trick,which converted the problem into one that was amenable to the bounding technique of Section 2.3.2. We point out that the same issue does not occur in the case of the downlink, where the terminal is ignorant of the channel estimate.
Capacity Lower Bound via Jensen's Inequality
Jensen's inequality – see Section C.1 – circumvents the difficulty of taking the expectation of a logarithm by bringing the expectation inside the logarithm. We apply Jensen in two ways: first to produce a lower bound on the expectation, second to produce an upper bound.
The application of Jensen's inequality to (3.26), in the form (C.3), yields
where we have used the identity (B.1). This argument merely reproduces the earlier “use and forget CSI” bound in (3.28), but it is interesting that radically different approaches yield the same bound.
Tightness of the Lower Bound (3.28)
We now show that the expression (3.28) is typically an excellent approximation to (3.26), by finding an upper bound on the expectation of the logarithm (3.26). To this end, we apply Jensen's inequality in the form (C.2) to (3.26),
where we have also used the identity (B.2).