"At the heart of the bilinear transform is a simple yet profound idea: that every analog story can be retold, with integrity, in the digital world.”, Unknown
The bilinear transform is a powerful tool in digital signal processing that allows us to map analog systems into the digital domain. It provides a one-to-one mapping between the continuous-time frequency domain ($s$-plane) and the discrete-time frequency domain ($z$-plane). In this blog post, we'll explore the characteristics and properties of the bilinear transform.
One of the key features of the bilinear transform is that it maps the $j\Omega$-axis in the $s$-plane into the unit circle in the $z$-plane only once. This means that the entire left half-plane (LHP) of the $s$-plane is mapped inside the unit circle, while the right half-plane (RHP) is mapped outside the unit circle. This property ensures stability preservation when transforming analog systems to digital systems.
Let's consider a simple analog system given by the transfer function $H(s)=\frac{b}{s+a}$. This system corresponds to the differential equation:
$$ \frac{d y(t)}{d t}+a y(t)=b x(t) $$
To obtain the equivalent discrete-time system, we can use the trapezoidal formula to approximate the integral form of the differential equation. This approximation leads to the following difference equation:
$$ y(n T)=\frac{T}{2}[y^{\prime}(n T)+y^{\prime}(n T-T)]+y(n T-T) $$
After further calculations, we arrive at the equivalent discrete-time system:
$$ (1+\frac{a T}{2}) y(n)-(1-\frac{a T}{2}) y(n-1)=\frac{b T}{2}[x(n)+x(n-1)] $$
Taking the $z$-transform of this difference equation yields the system function of the equivalent digital filter:
$$ H(z)=\frac{(b T / 2)(1+z^{-1})}{1+a T / 2-(1-a T / 2) z^{-1}} $$
or equivalently,
$$ H(z)=\frac{b}{\underset{s}{\underbrace{\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}}}+a} $$
This transformation from $H(s)$ to $H(z)$ is known as the bilinear transformation.