Opracowanie językowe dr inż. J. Jarnicki Internet Engineering Czesław Smutnicki Discrete Mathematics – Discrete Convolution

CONTENTS Fundamentals Properties Discrete convolution Applications DFT and inverse DFT FFT and inverse FFT Differential calculus

FUNDAMENTALS

COMPLEX ROOTS OF ONE i main -1 1 -i

COMPLEX ROOTS OF ONE. PROPERTIES

DFT

INVERSE DFT

CONVOLUTION APPLICATIONS Convolution and related operations are found in many applications of engineering and mathematics. In electrical engineering, the convolution of one function (the input signal) with a second function (the impulse response) gives the output of a linear time-invariant system (LTI). At any given moment, the output is an accumulated effect of all the prior values of the input function, with the most recent values typically having the most influence (expressed as a multiplicative factor). The impulse response function provides that factor as a function of the elapsed time since each input value occurred. In digital signal processing and image processing applications, the entire input function is often available for computing every sample of the output function. In that case, the constraint that each output is the effect of only prior inputs can be relaxed. Convolution amplifies or attenuates each frequency component of the input independently of the other components. In statistics, as noted above, a weighted moving average is a convolution. In probability theory, the probability distribution of the sum of two independent random variables is the convolution of their individual distributions. In optics, many kinds of "blur" are described by convolutions. A shadow (e.g., the shadow on the table when you hold your hand between the table and a light source) is the convolution of the shape of the light source that is casting the shadow and the object whose shadow is being cast. An out-of-focus photograph is the convolution of the sharp image with the shape of the iris diaphragm. The photographic term for this is bokeh.

CONVOLUTION APPLICATIONS Similarly, in digital image processing, convolutional filtering plays an important role in many important algorithms in edge detection and related processes. In linear acoustics, an echo is the convolution of the original sound with a function representing the various objects that are reflecting it. In artificial reverberation (digital signal processing, pro audio), convolution is used to map the impulse response of a real room on a digital audio signal (see previous and next point for additional information). In time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse. In radiotherapy treatment planning systems, most part of all modern codes of calculation applies a convolution-superposition algorithm. In physics, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance. In kernel density estimation, a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian. (Diggle 1995). In computational fluid dynamics, the large eddy simulation (LES) turbulence model uses the convolution operation to lower the range of length scales necessary in computation thereby reducing computational cost.

Thank you for your attention DISCRETE MATHEMATICS Czesław Smutnicki