Electrical Engineering Department
For complex-valued functions f, g defined on the set Z of integers, the discrete convolution of f and g is given by:
When multiplying two polynomials, the coefficients of the product are given by the convolution of the original coefficient sequences, extended with zeros where necessary to avoid undefined terms; this is known as the Cauchy product of the coefficients of the two polynomials.
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![(f * g)[n] stackrel{mathrm{def}}{=} sum_{m=-infty}^infty f[m], g[n - m]](https://upload.wikimedia.org/wikipedia/en/math/6/1/3/613174e3c16ef60cfa48baee47da2e4a.png)
![= sum_{m=-infty}^infty f[n-m], g[m].](https://upload.wikimedia.org/wikipedia/en/math/c/f/5/cf5f95ecd62414e97c94350009febd8d.png)
