Lab 3: prelab (part 1)

The transfer function for the second-order section shown in Lab 3: IIR Filters Overview is

Exercise

First, derive the above transfer function from the block diagram. Begin by writing the difference equations for $w(n)$ in terms of the input and past values ( $w(n-1)$ and $w(n-2)$ ). Then write the difference equation for $y(n)$ also in terms of the past samples of $w(n)$ . After finding the two difference equations, compute the corresponding Z-transforms and use the relation $H(z)=\frac{Y(z)}{X(z)}=\frac{Y(z)W(z)}{W(z)X(z)}$ to verify the IIR transfer function in [link] .

Next, design the coefficients for a fourth-order filter implemented as the cascade of two bi-quad sections. Write aMATLAB script to compute the coefficients. Begin by designing the fourth-order filter and checking the responseusing the MATLAB commands

Next you must find the roots of the numerator, zeros , and roots of the denominator, poles , so that you can group them to create two second-order sections. The MATLAB commands roots and poly will be useful for this task. Save the scripts you use to decompose yourfilter into second-order sections; they will probably be useful later.

Once you have obtained the coefficients for each of your two second-order sections, you are ready to choose a gain factor, $G$ , for each section. As part of your MATLAB script, use freqz to compute the response $\frac{W(z)}{X(z)}$ with $G=1$ for each of the sets of second-order coefficients. Recall that on the DSP we do not represent numbers greaterthan or equal to 1.0. If the maximum value of $\left|\frac{W(z)}{X(z)}\right|$ is or exceeds 1.0, an input with magnitude less than one could produce $w(n)$ terms with magnitude greater than or equal to one; this is overflow . You must therefore select a gain values for each second-order section such that theresponse from the input to the states, $\frac{W(z)}{X(z)}$ , is always less than one in magnitude. In other words, set the value of $G$ to ensure that $\left|\frac{W(z)}{X(z)}\right|< 1$ .

After finishing Part 1, move on to Lab 3: Prelab (Part 2) , where you explore and learn how to mitigate the effects of quantization.