Filtering in Discrete Systems
Roger L. Easton · 2010
This chapter considers the practical application of linear imaging systems to modeling impulse responses and transfer functions of some useful 1-D and 2-D discrete linear filters. One goal is to more completely understand the differences between the continuous and discrete cases. The Fourier shift theorem relates the continuous impulse response and transfer function of the 1-D translation operator. The real and imaginary parts of the transfer function are respectively one cycle of a cosine and of a sine with identical amplitudes determined by the DFT normalization. The transfer function is a discrete approximation of the parabola with values at the edges of the array that ensure smooth periodicity. Higher-order discrete derivatives may be derived by repeated discrete convolution of ∂x and discarding any common linear-phase factors. Controlled Vocabulary Terms finite impulse response filters; Fourier transforms