Example 8.11.

1. Design a second-order digital bandpass Butterworth filter with the following specifications:

   • an upper cutoff frequency of 2.6 kHz and

   • a lower cutoff frequency of 2.4 kHz,

   • a sampling frequency of 8,000 H z.

2. Use MATLAB to plot the magnitude and phase responses.

Solution:

1. Let us find the digital frequencies in radians per second:

   • ? _h = 2 ? f _h = 2 ?(2600) = 5200 ? rad/sec

   • ? _l = 2 ? f _l = 2 ?(2400) = 4800 ? rad/sec, and T = 1/ f _s = 1/8000 sec.

   Following the steps of the design procedure, we have the following:

   1. 

   2. We perform the prototype transformation (lowpass to bandpass) to obtain H( s). From Table 8.3, we pick the lowpass prototype with the order of 1 to produce the bandpass filter with the order of 2, as

      
      

      and applying the lowpass-to-bandpass transformation, it follows that

      
      

   3. Hence we apply the BLT to yield

      
      

      Via algebra work, we obtain the digital filter as

      
      

2. MATLAB Program 8.7 is given for this example, and the corresponding frequency response plots are illustrated in Figure 8.20.

   Program 8.7: MATLAB program for Example 8.11.

   %Example 8.11% Design of the digital bandpass Butterworth filterformat longfs = 8000;[B A] = lp2bp([1], [1 1],sqrt (5.7499*10^8), 4088)% Complete step 2[b a] = bilinear(B,A,fs) % Complete step 3% Plot the magnitude and phase responses%b = [0.0730 ?0.0730]; numerator coefficients fromMATLAB%a...