In signal processing, the frequency response is a measure of how a system responds to different frequencies of input signals. It is a mathematical representation of the system's behavior in the frequency domain, and it describes how much each frequency component of an input signal is attenuated or amplified by the system. If the input signal of a [[discrete-time LTI system]] is a [[periodic signal]] represented by a [[discrete Fourier series (DFS)]]: $ \tilde{x}(n)=\frac{1}{N}\sum_{k=0}^{N-1} \tilde{X}(k) e^{j\frac{2\pi}{N}kn} $ We can use the [[transfer function]] to compute the output signal, using $z=e^{j \omega}$ and $\omega=\frac{2\pi}{N}k$: $ \tilde{y}(n) = \frac{1}{N} \sum_{k=0}^{N-1} \underbrace{ \tilde{X}(k) H\left( e^{j\frac{2\pi}{N}k} \right) }_{ \tilde{Y}(k) } e^{j\frac{2\pi}{N}kn} $ were $ H(e^{j\omega}) = \sum_{n=-\infty}^{+\infty} h(n) e^{-j\omega n} $ is the **frequency response** of the discrete-time LTI system with impulse response $h(n)$. The parameter $\omega$ is the angular frequency in rad/s. The frequency response is a particular case of the [[transfer function]] $H(z)$ when $z=e^{j\omega}$, that is, when the input signal is $x(n)=e^{j\omega n}$. The transfer function can be used to relate the Fourier coefficients $\tilde{Y}(k)$ of the output signal $\tilde{y}(n)$ with the coefficients of the input signal: $ \tilde{Y}(k) = H(e^{j\frac{2\pi}{N}k}) \tilde{X}(k) $