Lab 8: IIR Filtering

Lab 8: IIR Filtering

Nasir Ahmed

Introduction:

In this lab, we learn the concept of the IIR filter. In the first part of the lab we are to use MATLAB to design a fourth order IIR filter through the cascading of two second order IIR filters. We learn how to derive difference equations and from these equations, how the transfer function is derived. We next learn how to implement this on a fixed point processor (TI C6211).

Exercise 8.1)

From the block diagram of figure 1, we verify that the transfer function is indeed correct.

‘w[n]=x[n]G-a1*w[n-1]-a2*w[n-2]’

‘y[n]=w[n]+b1*w[n-1]+b2*w[n-2]’

if we simply replace w(n-k) with w*z^(-k), and solve for w, we get:

w=xG/(1+a1z^-1+a2z2^-2)

and y=w(1+b1z^-1+b2z^-2), hence

Y(z)/X(z)=G(1+b1z^-1+b2z^-2)/(1+a1z^-1+a2z2^-2)

Next, we designed a fourth order elliptical IIR lowpass filter, with a cutoff frequency of 1kHz. The MATLAB command is as follows:

[B,A]=ellip(4,0.25,10,0.25);

freqz(B,A);

Next, by find the roots of B, A, we discovered that the roots were in pairs of complex conjugates. So, we found the polynomial of the complex conjugates, used one set of numerator and denominator as one system, and the other set as the second portion of the cascaded system. The result is 2 second order systems in cascade to represent a first order system. However, next we must determine the gain factor G1*G2=G to make sure that all intermediate signal do not over flow and are in the range (-1,1). This was accomplished with the following MATLAB script:

[B,A]=ellip(4,0.25,10,0.25);

%freqz(B,A);

zer=roots(B);

pol=roots(A);

num1=poly(zer(1:2));

num2=poly(zer(3:4));

den1=poly(pol(1:2));

den2=poly(pol(3:4));

x=zeros(1,500);

x(1)=1;

temp1=filter(num1,den1,x);

G1=1/sum(temp1);

temp2=filter(num2,den2,x);

G2=1/sum(temp2);

G=G1*G2;

G1=0.0373

G2=0.4049

So, now we have found the gain factors G1,G2 of the first and second system respectively.

8.2 (IIR filter implementation):

Next, we need to implement this IIR filter in the code composer studio in C. This was simply a modification of the FIR filter. In fact, the code is much simpler than the FIR filter. This is because all we need to do in the code is store some previous elements (the memory), and the coefficients, and perform the difference equation calculations. This is straight forward as can be seen in the attached code.

Conclusion/Recommendations:

This lab was very useful in teaching us the basics of IIR filtering. We learned some implementation issues of the IIR filter in the DSP. We also learned some more filter design in MATLAB. In terms of length, this lab was a perfect length, it took us about 4 hours. However, I wish there was some more background on IIR filtering in the lab manual. I didn’t really understand the need for G1, G2 from the handout itself. It took more contemplation. Additionally, section 8.1 took us longer than 8.2! This was because I felt the instructions in 8.1 weren’t sufficient.