In signal processing, the transfer function is a mathematical representation of how an input signal is transformed into an output signal by a linear and time-invariant (LTI) system. Since the [[discrete-time complex exponential signal]] is an [[eigenfunction of an LTI system]]: $ x(n)=z^{n} \rightarrow{} y(n) = H(z) z^{n} $ The $H(z)$ function is called the transfer function of the [[discrete-time LTI system]]. It is computed using the system's [[impulse response]] $h(n)$: $H(z) = \sum_{n=-\infty}^{+\infty} h(n) z^{-n}$ In the general case, $H(z)$ is a complex value: $ \begin{aligned} H(z) & = \Re (H(z)) + j \Im (H(z)) \\ & = |H(z)| e^{j \angle H(z)} \end{aligned} $ It will be seen later that the transfer function $H(z)$ is the [[z-transform]] of the system's [[impulse response]] $h(n)$. If $x(n)=z^{n}$ the output of the LTI system is: $ y(n) = H(z) x(n) $