The spectral analysis of sequence and distance data is a new approach to phylogenetic analysis. For two-state character sequences, the character values at a given site split the set of taxa into two subsets, a bipartition of the taxa set. The vector which counts the relative numbers of each of these bipartitions over all sites is called a sequence spectrum. Applying a transformation called a Hadamard conjugation, the sequence spectrum is transformed to the conjugate spectrum. This conjugation corrects for unobserved changes in the data, independently from the choice of phylogenetic tree. For any given phylogenetic tree with edge weights (probabilities of state change), we define a corresponding tree spectrum. The selection of a weighted phylogenetic tree from the given sequence data is made by matching the conjugate spectrum with a tree spectrum. We develop an optimality selection procedure using a least squares best fit, to find the phylogenetic tree whose tree spectrum most closely matches the conjugate spectrum. An inferred sequence spectrum can be derived from the selected tree spectrum using the inverse Hadamard conjugation to allow a comparison with the original sequence spectrum. A possible adaptation for the analysis of four-state character sequences with unequal frequencies is considered. A corresponding spectral analysis for distance data is also introduced. These analyses are illustrated with biological examples for both distance and sequence data. Spectral analysis using the Fast Hadamard transform allows optimal trees to be found for at least 20 taxa and perhaps for up to 30 taxa. The development presented here is self contained, although some mathematical proofs available elsewhere have been omitted. The analysis of sequence data is based on methods reported earlier, but the terminology and the application to distance data are new.

Spectral Analysis of Phylogenetic Data

The discrete, orthogonal Walsh functions can be generated by a multiplicative iteration equation. Using this iteration equation, an efficient Walsh transform computation algorithm is derived which is analogous to the Cooley-Tukey algorithm for the complex-exponential Fourier transform.

Computation of the Fast Walsh-Fourier Transform

Communication theory was founded on the system of sine-cosine functions. A more general theory has become known more recently; it replaces the sine-cosine functions by other systems of orthogonal functions, and the concept of frequency by that of sequency. Of these systems, the Walsh functions are of great practical interest since they lead to equipment that is easily implemented by semiconductor technology. Filters, multiplexing equipment, and a voice analyzer/synthesizer have been built successfully for Walsh functions. Some interesting applications of electromagnetic Walsh waves have been found theoretically.

Applications of Walsh functions in communications

This chapter is a self-contained development of abstract harmonic analysis applied to a single-output combinational logic functions; linear algebra and elementary group theory are the only mathematical prerequisites. New analysis and synthesis techniques are developed, and the groundwork is laid for future extensions to multiple output combinational logic, sequential machines, and real- or complex-valued functions of binary arguments. Harmonic analysis adds novel conceptual insights and unifying principles, improved computational techniques, and new measures of complexity to the traditional approach to switching theory. The first section is a summary of the chapter. Section II surveys classical Fourier transform properties and introduces the canonical expansion of a switching function as an n-dimensional abstract Fourier transform over the finite two-element field. The two most important transform properties are the convolution theorem, which leads to tests for prime implicants and disjunctive decompositions, and spectrum invariance which is basic to further theoretical developments and to a new synthesis technique called encoded input logic. Section III develops a new algorithm which concurrently extracts prime implicants and detects disjunctive decompositions of a switching function. Implicants of both the function and its complement are detected simultaneously, and “core” implicants can be identified. The algorithm which is not sensitive to functional complexity has been programmed for a commercial time-sharing system. Section IV introduces the restricted affine group (RAG) whose elements, called prototype transformations, encode the arguments and outputs of combinational logic functions. This group partitions the space F of all two-valued functions on Zn into 3, 8, and 48 equivalence classes respectively for n=3, 4, and 5. Unique representatives are identified for each class when n=3 and 4 and for 46 of the 48 classes of 5-argument functions. Section V applies the tools of abstract harmonic analysis to the synthesis problem for large truth tables (many-input combinational logic). A general multilevel synthesis approach, called encoded input logic, is introduced which is compatible with large-scale-integrated circuit technology. Both the conventional macrocellular and the newer microcellular array approach are included as special cases. Prototype encoding transformations are used to reduce the complexity of an imbedded normal form realization. Practical synthesis algorithms are based on Fourier analysis. A realistic 6-argument example is treated in detail. Section V concludes with a list of fundamental problems whose solutions would extend the research presented herein.

Harmonic analysis of switching functions

A technique is presented to implement a class of orthogonal transformations on the order of pN log p N operations. The technique is due to Good [1] and implements a fast Fourier transform, fast Hadamard transform, and a variety of other orthogonal decompositions. It is shown how the Kronecker product can be mathematically defined and efficiently implemented using a matrix factorization method. A generalized spectral analysis is suggested, and a variety of examples are presented displaying various properties of the decompositions possible. Finally, an eigenvalue presentation is provided as a possible means of characterizing some of the transforms with similar parameters.

A Generalized Technique for Spectral Analysis

: A discrete time transform was studied and applied to the representation and discrimination of digitized signals. The transform consists of an orthogonal (Hadamard) matrix whose elements are all ones and minus ones. To facilitate implementation, a fast Hadamard transform (FHT) has been developed requiring only NlogN rather than N squared algebraic additions. Several properties of the FHT are revealed, including the nature of its presence in the fast Fourier transform, in which it performs the additive operations as shown by further decomposing the product of matrices representing the FFT.

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D F Guinn

Papers

The Fast Fourier-Hadamard Transform and Its Use in Signal Representation and Classification,