On the Growth Rate of Input-output Weight Distribution of Convolutional Encoders

In this paper, exact formulæ of the input-output weight distribution function and its exponential growth rate are derived for truncated convolutional encoders. In particular, they are expressed in terms of generating functions of error events associated with a minimal realization of the encoder. Although explicit analytic expressions can be computed for relatively small truncation lengths, the explicit expressions become prohibitively complex to compute as truncation lengths and weights increase. Fortunately, a very accurate asymptotic expansion can be derived using the Multidimensional Saddle Point method (MSP-metohd). This approximation is substantially easier to evaluate and is used to obtain an expression for the asymptotic spectral function and to prove continuity and concavity in its domain (convex and closed). Finally, this approach is able to guarantee that the sequence of exponential growth rate converges uniformly to the asymptotic limit and to estimate the speed of this convergence.