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  "body": "\n## Statement\n\nLet $x = (x_n)_{n \\in \\Z}$ and $y = (y_n)_{n \\in \\Z}$ be sequences of complex numbers with [[bilateral Z-transforms|3DBAMXA80B]] $X$ and $Y$.\nLet $z$ be a point of both [[regions of convergence|1P8WX3EGHF]].\nThen 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\n$$\\mathcal{Z}\\{x * y\\}(z) = X(z) \\, Y(z).$$\nIf moreover $x_n = y_n = 0$ for all $n \u003c 0$, 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|EPG6JPVJE7]].\n\n## Remarks\n\nThe 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.\nSo each inner sum converges absolutely, and the terms can be regrouped, which gives the product formula.\nIn the one-sided case this is the [[Cauchy product|0RVD9R2V99]] of two absolutely convergent series in $z^{-1}$.\nExample: 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 $\\abs{z} \u003e 1$.\nIf the two regions do not meet, the [[convolution|2Z29JN3VES]] 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.\n\n## Explanations\n\n### L1 human\n\nConvolution of sequences is the operation that a linear time-invariant filter performs on its input.\nThe Z-transform turns this operation into ordinary multiplication of functions.\nSo passing a signal through a filter amounts to multiplying its transform by the transform of the filter.\n\n### L2 human\n\nThe 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.\nIt is the discrete counterpart of the convolution theorem for the Laplace transform.\nTogether with the shift property, it reduces linear constant-coefficient difference equations to algebra.\n",
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