Bilinear Transformation

**$z$ transform**

The $z$ transform is important tool for obtaining transfer function of digital filters . Stability of digital filters can be identified based on the location of poles in $z$ plane. For a stability of digital filters it is important to have poles inside unit circle

The $z$ transform is defined as: $$ H(z) = \sum\limits_{n = - \infty }^\infty {h[n]z^{ - n} } $$ [1]
** Transfer function of discrete-time systems**

The primary goal of $z$ transform is to find the transfer function of linear discrete-time systems. Once, we are able to find the transfer function, then we can analyze the zeros and poles of the transfer function in the $z$ plane.:

The transfer function of discrete-time systems is given as : $$ H(z) = \frac{\sum\limits_{i = 0 }^{M-1} {b[i]z^{-i} }}{\sum\limits_{j = 0 }^{N-1} {a[j]z^{-j} }}=\frac{B(z)}{A(z)} $$

- Numerator coefficients: $b[i]$
- Denominator coefficients: $a[j]$
- The transfer function of non-recursive part of the system: $B(z)$
- The transfer function of recursive part of the system (feedback): $A(z)$ [1]

**Digital filter from analog prototype**

Bilinear transformation is one of the useful ways to convert the analog prototype filter into a digital one. As per the [1] "This method is entirely a frequency-domain method, and, as a result, some of the optimal properties of the analog filter are preserved. The bilinear transformation is a change of variables (a mapping) that is linear in the numerator and denominator" "In Bilinear transformation ,mapping take place from s plane from analog frequency domain to unit cirlcle in z plane. It maps the infinite interval of the analog frequency axis onto the finite interval of the digital frequency axis. There is no folding or aliasing of the prototype frequency response, but there is a compression of the frequency axis that becomes extreme at high frequencies. This nonlinear compression is called frequency warping".[1] In the design of digital filter by bilinear transformation approach frequency warping plays important role, the algoritm of derivation of digital equivalent filter from its analog prototype filter is explained as below

First it replaces protoype frequency scale by ${s}$ with ${Ks}$

Then, it follows following mathematical operation [1]

The usual form is: $$s=\frac{2}{T}\frac{z-1}{z+1}$$

The prewarping scale is given by: $$u_{0}= \frac{2}{T}{\text{tan}}(\frac{\omega_{c}T}{2})$$

Combining prewarping scale and the bilinear transformation gives $${\Omega_{c}= \frac{2K}{T}{\text{tan}}(\frac{\omega_{c}T}{2})}$$

Solving for $K$ $$s= \frac{\Omega_{c}}{{\text{tan}}(\frac{\omega_{c}T}{2})}\frac{z-1}{z+1}$$

Prototype filters are designed with a normalized cutoff frequency, $\omega_{c} =1$. Through bilinear transformation digital IIR filer is obtained, which has same number of zeros as that of poles.[1]