# Properties of Convolution in Signals and Systems

## Convolution

Convolution is a mathematical tool for combining two signals to produce a third signal. In other words, the convolution can be defined as a mathematical operation that is used to express the relation between input and output an LTI system.

Consider two signals $\mathit{x_{\mathrm{1}}\left( t\right )}$ and $\mathit{x_{\mathrm{2}}\left( t\right )}$. Then, the convolution of these two signals is defined as

$$\mathrm{ \mathit{\mathit{y\left(t\right)=x_{\mathrm{1}}\left({t}\right)*x_{\mathrm{2}}\left({t}\right)\mathrm{=}\int_{-\infty }^{\infty }x_{\mathrm{1}}\left(\tau\right)x_{\mathrm{2}}\left(t-\tau\right)\:d\tau=\int_{-\infty }^{\infty }x_{\mathrm{2}}\left(\tau \right)x_{\mathrm{1}}\left(t-\tau\right)\:d\tau }}}$$

## Properties of Convolution

Continuous-time convolution has basic and important properties, which are as follows −

• Commutative Property of Convolution − The commutative property of convolution states that the order in which we convolve two signals does not change the result, i.e.,

$$\mathrm{\mathit{x_{\mathrm{1}}\left(t\right)*x_{\mathrm{2}}\left(t\right)=x_{2}(t)*x_{\mathrm{1}}\left(t\right)}}$$

• Distributive Property of Convolution −The distributive property of convolution states that if there are three signals $\mathit{x_{\mathrm{1}}\left( t\right )}$,$\mathit{x_{\mathrm{2}}\left( t\right )}$$\mathrm{and} \mathit{x_{\mathrm{3}}\left( t\right )}, then the convolution of \mathit{x_{\mathrm{1}}\left( t\right )} is distributive over the addition \mathit{\left [x_{\mathrm{2}}\left( t\right ) \mathrm{+}\mathit{x_{\mathrm{3}}\left( t\right )}\right ]}i.e.,$$\mathrm{\mathit{x_{\mathrm{1}}\left(t\right)*\left [x_{\mathrm{2}}\left(t\right)\mathrm{+}x_{\mathrm{3}}\left(t\right) \right ]\mathrm{=}\left [ x_{\mathrm{1}}\left(t\right)*x_{\mathrm{2}}\left(t\right) \right ]\mathrm{+}\left [x_{\mathrm{1}}\left(t\right)*x_{\mathrm{3}}\left(t\right)\right ]}}$$• Associative Property of Convolution − The associative property of convolution states that the way in which the signals are grouped in a convolution does not change the result, i.e.,$$\mathrm{\mathit{x_{\mathrm{1}}\left(t\right)*\left [ x_{\mathrm{2}}\left(t\right)*x_{\mathrm{3}}\left(t\right) \right ]=\left [ x_{\mathrm{1}}\left(t\right)*x_{\mathrm{2}}\left(t\right) \right ]*x_{\mathrm{3}}\left(t\right)}}$$• Shift Property of Convolution − The shift property of convolution states that the convolution of a signal with a time shifted signal results a shifted version of that signal, i.e., if$$\mathrm{ \mathit{x_{\mathrm{1}}\left(t\right)*x_{\mathrm{2}}\left(t\right)=y\left(t\right)}}$$Then, according to the shift property of convolution, we have,$$\mathrm{ \mathit{x_{\mathrm{1}}\left(t\right)*x_{\mathrm{2}}\left(t-T_{\mathrm{0}}\right)=y\left(t-T_{\mathrm{0}}\right)}}$$Similarly,$$\mathrm{\mathit{x_{\mathrm{1}}\left(t-T_{\mathrm{0}}\right)*x_{\mathrm{2}}\left(t\right)=y\left(t-T_{\mathrm{0}}\right)}}$$Therefore,$$\mathrm{\mathit{x_{\mathrm{1}}\left(t-T_{1}\right)*x_{\mathrm{2}}\left(t-T_{\mathrm{2}}\right)=y\left(t-T_{\mathrm{1}}-T_{\mathrm{2}}\right)}}$$• Width Property of Convolution − Let the duration of the signal \mathit{x_{\mathrm{1}}\left( t\right )} and \mathit{x_{\mathrm{2}}\left( t\right )} is T1 and T2 respectively. Then, the width property of the convolution states the duration of the signal obtained by convolving \mathit{x_{\mathrm{1}}\left( t\right )} and \mathit{x_{\mathrm{2}}\left( t\right )}is equal to \mathit{\left (T_{\mathrm{1}}\mathrm{+}T_{\mathrm{2}} \right )} • Convolution of Signal with Impulse − This property of convolution states that the convolution of an arbitrary signal \mathit{x\left (t\right )} with a unit impulse signal is the signal itself, i.e.,$$\mathrm{x\left(t\right)*\delta\left(t\right)=x\left(t\right)}$\$

Updated on: 08-Nov-2023

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