Frequency Response Functions and Filtering

A linear time-invariant (LTI) system with input $x(t)$ and output $y(t)$ can be described in frequency domain by their Fourier coefficients (Fourier transforms) $X(\omega)$ and $Y(\omega)$, which are complex variables representing the magnitude and phase of each frequency component $\omega$ contained in the signal. In particular, when the signals contain only a single frequency component $\omega$, as assumed in the discussion for AC circuit analysis, the Fourier coefficients $X(\omega)$ and $Y(\omega)$ of the input and output are essentially the same as their phasor representations $\dot{X}$ and $\dot{Y}$ (with the variable of frequency $e^{j\omega}$ dropped).

The ratio of the output $Y(\omega)$ and the input $X(\omega)$ both represented in frequency domain (phasor form) is called the frequency response function (FRF).

$$H(\omega)=\frac{Y(\omega)}{X(\omega)}=\frac{\dot{Y}}{\dot{X}}$$ (308)