Complex BIFORE transform†

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Abstract
Complex BIFORE (Binary FOurior REpresentation) transform belongs to the family of discrete orthogonal transformations and is analogous to discrete Fourier transform (DFT) when dealing with complex inputs. For real inputs, complex BIFORE transform (CBT) reduces to BIFORE or Hadamard transform (BT or HT) whose bases are Walsh functions. BT has been applied in several phases of information processing and sequency filters and sequency multiplexing equipment have also been built. When dealing with complex signals, CBT has some inherent computational advantages, and can be used to analyse and synthesize complex input functions. In the present paper, CBT is defined and its relationship to BT is shown. Several properties of CBT are developed. Invariance of power spectrum to sequential shift of the sampled data is shown. Using matrix factoring, fast algorithms suitable for digital computation of CBT and its inverse are developed. CBT is extended to multiple dimensions. Fast algorithms and corresponding flow graphs for direct evaluation of the CBT power spectrum without the need to compute the transform coefficients, are developed. The frequency structure of CBT power spectrum represents groups of frequencies based on the half-wave symmetry structure of the signal, unlike the individual frequency representation of the DFT power spectrum. finally, the relationship between the CBT and the DFT power spectra is shown.

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