# 4.1 Lab 3: prelab (part 1)

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You will derive the transfer function of a second-order, Direct Form II, infinite impulse response (IIR) filter. Then you will create a fourth-order IIR filter, plot its frequency response, and decompose the fourth-order filter into two second-order sections, choosing an appropriate gain for each stage to prevent overflow.

The transfer function for the second-order section shown in IIR Filtering: Introduction is

$H(z)=G\frac{1+{b}_{1}z^{-1}+{b}_{2}z^{-2}}{1+{a}_{1}z^{-1}+{a}_{2}z^{-2}}$

## Exercise

First, derive the above transfer function. 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 .

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

[B,A] = ellip(4,.25,10,.25) freqz(B,A)

MATLAB's freqz command displays the frequency responses of IIR filters and FIR filters. For more information about this, type help freqz . Be sure to look at MATLAB's definition of the transfer function.
If you use the freqz command as shown above, without passing its returned data to anotherfunction, both the magnitude (in decibels) and the phase of the response will be shown.

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 cannot 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 $\left|\frac{W(z)}{X(z)}\right|< 1$ .

## Preparing for processor implementation

As the processor exercises become more complex, it will become increasingly important to observe good programmingpractices. Of these, perhaps the most important is careful planning of your program flow, memory and register use, andtesting procedure. Write out pseudo-code for the processor implementation of a bi-quad. Make sure you consider the wayyou will store coefficients and states in memory. Then, to prepare for testing, compute the values of $w(n)$ and $y(n)$ for both second-order sections at $n=\{0, 1, 2\}$ using the filter coefficients you calculated in MATLAB. Assume $x(n)=(n)$ and all states are initialized to zero. You may also want to create a frequency sweep test-vector like theone in DSP Development Environment: Introductory Exercise for TI TMS320C54x and use the filter command to find the outputs for that input. Later,you can recreate these input signals on the DSP and compare the output values it calculates with those you find now. Ifyour program is working, the values will be almost identical, differing only slightly because of quantizationeffects, which are considered in IIR Filtering: Filter-Coefficient Quantization Exercise inMATLAB .

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