What is the relation between Analog filter Frequency and digital filter Frequency when BLT Method is used for filter design ?

The relation between analog filter frequency and digital filter frequency when using the bilinear transform (BLT) method for filter design can be explained step by step as follows:

1. Start with the analog filter transfer function, which is typically represented in the s-domain (Laplace domain) and has a cutoff frequency denoted as ωc.

2. Apply the bilinear transform, which is a mathematical mapping that converts the s-domain transfer function to the z-domain (digital domain). The bilinear transform is given by the equation:

   s = 2/T * (1 - z^-1) / (1 + z^-1)

   where s is the Laplace variable, z is the z-transform variable, and T is the sampling period.

3. Substitute the bilinear transform equation into the analog filter transfer function, replacing s with the expression from step 2. This results in the digital filter transfer function.

4. The cutoff frequency of the digital filter, denoted as ωc_digital, is related to the analog cutoff frequency ωc through the bilinear transform. The relationship is given by:

   ωc_digital = 2/T * tan(ωc*T/2)

   This equation shows that the digital cutoff frequency is dependent on the analog cutoff frequency and the sampling period.

In summary, when using the BLT method for filter design, the relationship between analog filter frequency and digital filter frequency is determined by the bilinear transform equation. The digital cutoff frequency is calculated based on the analog cutoff frequency and the sampling period.