Block diagonal structure in discrete transforms

Abstract

The author investigates and summarises some of the computational tasks of discrete transforms in which block diagonal structure plays a dominant role. Walsh-Hadamard transform (WHT) based algorithm designs for various well known discrete transforms are presented; it can be proved that, owing to their block diagonal structure, the WHT based discrete transforms are more efficient than those of the conventional radix-r algorithms for transforms of length N<or=64. It is proved that block diagonal structures exist in the running Walsh-Hadamard transform, the running discrete Hartley transform (DHT), and the running discrete cosine transform (OCT). With regard to block diagonal structure in the transform conversions between DHT and DCT, some existing research results are summarised, and an efficient architecture for generating multiple discrete transform simultaneously is proposed. The two-stage DFT algorithm proposed by O.K. Ersoy (1987) is extended to that of the DHT and it is proved that two stage DHTs possess a somewhat more interesting 'balanced-block-diagonal' structure. In the context of VLSI system design, two factors are of particular importance: the regularity of processor cells and local communication between processors. The hardware implementation of the block diagonal algorithm, for moderate N, just meets the above requirements. An example of the WHT/DHT is also included.