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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)}$$
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