Concept of Transfer Function
A filter is fundamentally characterised by a transfer function, denoted as h(t) (H(w)).
The figure below illustrates a filter that takes an input signal x(t) and produces an output signal y(t).
Time Domain : \(\ y\left(t\right)=h\left(t\right)\ast x\left(t\right)=\int_{\tau=-\infty}^{\infty}{h\left(\tau\right).x\left(t-\tau\right).d\tau}\)
\(\quad\) \(\quad\) \(\quad\) \(\quad\) \(\quad\) \(\quad\) \(=\int_{\tau=-\infty}^{\infty}{x\left(\tau\right).h\left(t-\tau\right).d\tau}\)
Frequency Domain : \(y\left(t\right)=h\left(t\right)\ast x\left(t\right)\) \(\rightarrow\) \(Y\left(w\right)=H\left(w\right)X\left(w\right)\)
Any linear filter is fully characterised by its frequency response in amplitude, denoted as\( \left|H(w)\right|\) , and its phase response, represented by \(\varphi\left(w\right)=\arg(H\left(w\right))\).
In the frequency domain, the relationship between the output Y\(\left(w\right)\) and the input \(X\left(w\right)\) of a linear filter is given by :
\(\left|Y(w)\right|=\left|H(w)\right|\left|X(w)\right|\) and \(\arg(\ Y\left(w\right))=arg\ (H(w))+\ arg\ (X\left(w\right))\)
This allows us to express the frequency response H\left(w\right) as :
\(H\left(w\right)=\left|H(w)\right|e^{j\varphi(w)}\)