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In this paper, the trellis representation of nonlinear codes is studied from a new perspective. We introduce the new concept of entropy/length profile (ELP). This profile can be considered as an extension of the dimension/length profile (DLP) to nonlinear codes. This elaboration of the DLP, the entropy/length profiles, appears to be suitable to the analysis of nonlinear codes. Additionally and independently, we use wellknown information-theoretic measures to derive novel bounds on the minimal covering of a bipartite graph by complete subgraphs. We use these bounds in conjunction with the ELP notion to derive both lower and upper bounds on the state complexity and branch complexity profiles of (nonlinear) block codes represented by any trellis diagram. We lay down no restrictions on the trellis structure, and we do not confine the scope of our results to proper or one-to-one trellises only. The basic lower bound on the state complexity profile implies that the state complexity at any given level cannot be smaller than the mutual information between the past and the future portions of the code at this level under a uniform distribution of the codewords. We also devise a different probabilistic model to prove that the minimum achievable state complexity over all possible trellises is not larger than the maximum value of the above mutual information over all possible probability distributions of the codewords. This approach is pursued further to derive similar bounds on the branch complexity profile. To the best of our knowledge, the proposed upper bounds are the only upper bounds that address nonlinear codes. The novel lower bounds are tighter than the existing bounds. The new quantities and bounds reduce to well-known results when applied to linear codes.

In this paper we report results of a research in which we studied the problem of determining the threshold signal to noise ratio ( S N R ) between large and small errors estimation of the direction of arrival ( D O A ) of a radiating, farfield source in the presence of another. Using the Barankin lower bound ( B R ) we examine the conditions under which achievable mean square error ( m s e ) performance of any unbiased D O A estimator deviates substantially from the Carmer-Rao lower bound (CRB). We present expressions for the threshold S N R as a function of the source-array geometry and the sources S N R where one and two sources, of known/unknown spectral parameters and DOAs, are present.

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