Impulse-Equivalent Sequences and Arrays

Abstract

The delta function is important in discrete signal and image processing as it is the exemplary perfect point spread function in the spatial or time domain. It provides 100% contrast resolution for all frequencies up the Nyquist frequency in the Fourier domain (perfect modulation transfer function). The construction of spatially finite discrete functions that mimic these properties of the delta function becomes of great value when we want to synchronise signals in time or localise patterns in space. Equivalently, sparse binary arrays are templates for spectrally-neutral functions that provide unbiased sub-sampling patterns for compressed sensing applications. Here a method is described that constructs exact, impulse-equivalent functions by combining complementary sequences based on difference sets. A large variety of these sequences can be prepared that are comprised of simple real integer alphabets, whilst imposing few length restrictions. These ‘perfect’ periodic sequences mimic delta functions through their strong peak, low off-peak, aperiodic auto-correlation. Families of distinct sequences can be produced that exhibit low cross-correlations. These sequences can be used to build discrete impulse-equivalent arrays in higher dimensions. We provide some 2D examples.