A Digital filter with a 3dB bandwidth of 0.5π is to be designed from 1st orderanalog butterworth filter using BLT and obtain H(z

To design a digital filter with a 3dB bandwidth of 0.5π using a 1st order analog Butterworth filter and the bilinear transform (BLT) method, follow these steps:

1. Determine the transfer function of the 1st order analog Butterworth filter. The transfer function of a 1st order Butterworth filter is given by:

   H(s) = 1 / (s + ωc)

   where s is the Laplace variable and ωc is the cutoff frequency.

2. Convert the transfer function from the Laplace domain to the z-domain using the bilinear transform. The bilinear transform is given by:

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

   where T is the sampling period.

   Substitute the bilinear transform into the transfer function of the analog Butterworth filter:

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

3. Simplify the expression for H(z) by substituting ωc = 0.5π and rearranging the terms.

4. Normalize the transfer function by dividing both the numerator and denominator by the coefficient of the highest power of z.

5. Implement the digital filter using the simplified and normalized transfer function H(z).

By following these steps, you can design a digital filter with a 3dB bandwidth of 0.5π using a 1st order analog Butterworth filter and the bilinear transform method.