Important Properties of convolution | Signals and Systems
Concept of convolution has wide ranging applications such as its usage in digital image processing for the purpose of filtering, improving certain features of images and many other signal processing applications.
Convolution is a mathematical operator which relates input and output of an LTI system. It is only with the help of convolution operator that we can generate exact output from input and impulse response. But the only problem is convolution is a little bit complex operator. As an alternative to convolution, we study transformation theory where we reduce convolution relation between input and impulse response in to multiplication by transferring them in to different domain. After finding the output in that domain we again bring the output back to original domain.
Having said that there is an alternative to convolution given the complex nature of convolution operator, there are some important properties of convolution which are in fact very important shortcuts to conventional method of convolution by which we can easily determine the output.
Following are some important properties of convolution
When discrete time signals x[n] and impulse response h[n] are convolved, the resultant signal y[n] has following properties
Lower limit of y[n] is sum of lower limits of x[n] and that of h[n] and upper limit is sum of upper limits of x[n] and that of h([n] respectively.
Length (l) of y[n] is l = l1+l2-1. Where l1 is length of x[n] and l2 is length of h[n] Sum of samples of y[n] is equal to product of sum of x[n] and that of h[n].
When two continuous time signals x(t) and h(t) are convolved, the resultant signal y(t)
Lower limit of y(t) is sum of lower limits of x(t) and that of h(t) and upper limit of y(t) is sum of upper limits of x(t) and that of h(t) respectively.
Width of y(t) is sum of widths of x(t) and h(t).
Area of y(t) is product of areas of x(t) and h(t).
Keywords: Convolution, Properties of convolution, Transform theory, Impulse response, LTI Systems, Signal processing, Image processing, Analog Filter and Digital Filter