EE422

Homework #25 (Last HW!!)

Don't Turn In!

Please check the FIR example worked in class!

Also, check your exam 3 scores

 

1a)

 Make a Bode plot of the transfer function from the last two homeworks:and find the bandwidth (you may use Matlab)

b)

 For your bilinear transformation design H(z) from Homework #36 problem 1b), make a frequency response plot using z=ejw T using T=10msec. (Matlab is very helpful here!) Find the bandwidth and compare to your plot for part a). For what frequencies is your frequency response valid?

c)

 The problem with using z=ejw T is that the resulting H(jw ) is not a rational function in jw even if H(z) is rational. Thus, all of our well-founded rules about Bode plots using poles and zeroes does not apply! One solution is to use the W-plane transformation where we approximate: . Notice that w has been replaced by the complex variable w indicating an approximation! By making this substitution, H(jw) is now a rational function in jw and all of our Bode Plot rules apply!

 

Repeat part b) but this time make a Bode plot using the w-plane transformation,z=(1+jwT/2)/(1-jwT/2) that you learned in class today using T=10msec. Find the bandwidth and compare to your plot for part a).

 

2a)

 Recall that the the first-order, lowpass filter in 1a) is an approximation to the ideal low pass filter:

 

Design a tenth order, FIR digital filter with a rectangular (boxcar) window and no scaling to realize this filter by using the Matlab command:

 

» b=5/4*fir1(10,1/(25*pi),boxcar(11),'noscale');

 

Note: by default Matlab will use a Hamming window and scale the center of the pass-band to 1.

 

b)

 Make a Bode plot of this FIR filter (the easy way!) using the Matlab commands:

 

» w=logspace(-1,2,200);

» a=1;

» dbode(b,a,.01,w)

 

What is the bandwidth of your filter? What is the D.C. (low frequency) gain of your filter? What should it be?

 

 

c)

 By default Matlab will use a Hamming window and scale the center of the pass-band to 1 when you use the FIR1 command. So redo part b) with the scaled FIR filter obtained from the Matlab command:

 

» b=5/4*fir1(10,1/(25*pi));

 

Is this FIR filter realization a little more faithful to the ideal low-pass filter?