Design a three order Discrete Time Butterworth Filter with cutoff frequency of 6 KHz and sampling frequency of 10^6 samples/sec by Bilinear Transformation.

Steps to Solve

1. Identify Parameters Define the order of the filter, cutoff frequency, and sampling frequency.

   • Cutoff frequency ($f_c$): 6 kHz
   • Sampling frequency ($f_s$): 10^6 samples/sec
   • Filter order ($N$): 3

2. Calculate Analog Cutoff Frequency Use the formula for the analog cutoff frequency ($\omega_c$): $$ \omega_c = 2\pi f_c = 2\pi \times 6000 , \text{rad/sec} \approx 37699.11 , \text{rad/sec} $$

3. Calculate Normalized Cutoff Frequency The normalized cutoff frequency ($\Omega_c$) is calculated as: $$ \Omega_c = \frac{\omega_c}{2} \tan\left(\frac{\omega_c}{2f_s}\right) $$ Substituting the values: $$ \Omega_c = \frac{37699.11}{2} \tan\left(\frac{37699.11}{2 \times 10^6}\right) $$

4. Calculate Poles of the Filter The poles are calculated using the formula: $$ P_k = \pm \Omega_c e^{j\frac{\pi(2N + 2k + 1)}{2N}} \quad \text{for } k = 0, 1, 2 $$ For $N = 3$, compute the values for $k = 0, 1, 2$.

5. Define H(S) for Even Order For the even order filter, the transfer function $H(S)$ is given by: $$ H(S) = \frac{N/N_{i=1}^N \Omega_c^{N}}{\prod_{k=1}^{N/2}(s^2 + \Omega_c b_s + \Omega_c^2)} $$

6. Define H(S) for Odd Order For the odd order filter, use: $$ H(S) = \frac{N/N_{i=1}^N \Omega_c^{N}}{\prod_{k=1}^{(N-1)/2}(s^2 + \Omega_c b_s + \Omega_c^2)} $$

7. Calculate b_s The parameter $b_s$ is calculated: $$ b_s = 2 \sin\left(\frac{(2S-1)}{2N}\pi\right) $$

8. Final Discrete Filter Transfer Function The discrete transfer function $H(z)$ is defined as: $$ H(z) = H(S) \cdot \frac{z^{-1}}{1 + z^{-1}} $$