Integral transforms, operational calculus Discrete operational calculus Theorem

Convolution property of the Z-transform

Builds on Bilateral Z-transform Unilateral Z-transform Region of convergence Convolution Cauchy product

Statement

Let $x = (x_n)_{n \in \Z}$ and $y = (y_n)_{n \in \Z}$ be sequences of complex numbers with bilateral Z-transforms $X$ and $Y$ . Let $z$ be a point of both regions of convergence. Then for every $n \in \Z$ the series $(x * y)_n = \sum_{k \in \Z} x_k \, y_{n-k}$ converges absolutely, $z$ lies in the region of convergence of $x * y$ , and $\mathcal{Z}\{x * y\}(z) = X(z) \, Y(z).$ If moreover $x_n = y_n = 0$ for all $n < 0$ , then $(x * y)_n = \sum_{k=0}^{n} x_k y_{n-k}$ for $n \ge 0$ , and the same identity holds for the unilateral Z-transforms.

Remarks

The double series $\sum_{n} \sum_{k} \abs{x_k} \abs{y_{n-k}} \abs{z}^{-n}$ equals $\left(\sum_k \abs{x_k} \abs{z}^{-k}\right)\left(\sum_m \abs{y_m} \abs{z}^{-m}\right)$ , which is finite. So each inner sum converges absolutely, and the terms can be regrouped, which gives the product formula. In the one-sided case this is the Cauchy product of two absolutely convergent series in $z^{-1}$ . Example: for $x_n = y_n = 1$ ( $n \ge 0$ ), $(x * y)_n = n + 1$ , and $\left(z/(z-1)\right)^2 = \sum_{n \ge 0} (n+1) z^{-n}$ for $\abs{z} > 1$ . If the two regions do not meet, the convolution may fail to exist: for $x_n = y_n = 1$ for all $n \in \Z$ , each sum $\sum_k x_k y_{n-k}$ diverges.

Going deeper

For a student

The transform of the impulse response of a system is its transfer function, and the theorem says that the output transform is the input transform times the transfer function. It is the discrete counterpart of the convolution theorem for the Laplace transform. Together with the shift property, it reduces linear constant-coefficient difference equations to algebra.

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Bilateral Z-transform concept
Cauchy product concept
Convolution concept
Region of convergence concept
Unilateral Z-transform concept
Convolution property of the Z-transform statement

Tag	`FFR8NPWG64`
URI	`m2://FFR8NPWG64`
Kind	statement, theorem
Writing	W0 stub
MSC	44A55, 44A35
Status	This copy of the site has no build data. Run `m2 status FFR8NPWG64`.
About	Z-transform, Convolution
Uses	Bilateral Z-transform, Unilateral Z-transform, Region of convergence, Convolution, Cauchy product