# Translations:Frequency response of a digital system/1/en

We recall that impulse response ${\displaystyle h_{k}}$ is the system’s response to a unit spike. Thus, the impulse response represents a transient that dies out over time for a stable system. On the other hand, the frequency response can be interpreted as a steady-state response of the system to signals, each with a pure frequency. Let the real variable ${\displaystyle \omega }$ denote frequency (in radians per second), and let ${\displaystyle \Delta t}$ be the discrete time spacing (in seconds). The function ${\displaystyle e^{i\omega \Delta tn}}$ is a steady-state sinusoidal wave of pure frequency ${\displaystyle \omega }$ If we let this steady-state signal be the input to a unit-delay system Z, the steady-state output is the delayed signal ${\displaystyle e^{i\omega \Delta t\left(n-1\right)}}$. The frequency response is the ratio of the steady-state output to the steady-state input ${\displaystyle e^{-i\omega \Delta t}}$; that is,