Asymptotic performance limit of optimal linear coding

We consider optimal linear coding of several classes of finite power discrete time continuous amplitude sources on several classes of channels. We show that the asymptotic performance limit of linear coding exists for sources and channels for which the limiting eigenvalue distribution functions of their source and channel correlation matrices exist and are continuous and monotone increasing functions. We identify examples of such sources and channels, such as autoregressive sources with fixed or stationary ergodic coefficients, and finite impulse response (FIR) single input single output (SISO) and multiple input multiple output (MIMO) frequency selective channels with fixed or stationary ergodic coefficients. We also introduce a perfectly matched source - channel pair, as a pair for which the source limiting eigenvalue distribution function is equal to the channel limiting eigenvalue distribution function. For such perfectly matched source - channel pairs, we prove that the linear coding performance limit is equal to the Shannon optimal performance theoretically attainable (OPTA) bound, i.e, we cannot do any better with any other coding scheme.