### 10. IIR Filters

Overview: In this chapter the class of infinite-impulse-response (IIR) filters is introduced. These digital filters involve feedback terms because past values of the output signal are used to compute the present output. The terminology "recursive filter" is also used to name these filters. Their $$z$$-transform system functions are rational functions which have poles and zeros. Insight into the important properties of these filters comes directly from the pole-zero representation.

Databases:

#### Demos - MATLAB5

The Filter Design Demo is a program that designs simple IIR (and FIR) digital filters, along with tutorial visualizations of the filter design process.

Features:

• IIR Design of Butterworth filters
• Numerous plot options: magnitude/phase response, pole-zero diagram, and impulse response
• FIR Design with many different window types
• Parks-McClellan FIR Design showing the iterations of the Remez Exchange method
A short tutorial on first- and second-order IIR (infinite-length impulse response) filters. This demo shows plots in the three domains for a variety of IIR filters with different filter coefficients.
PeZ (pezdemo) is a MATLAB tool for pole/zero manipulation. Poles and zeros can be placed anywhere on a map of the $$z$$-plane. The corresponding time domain ($$n$$) and frequency domain ($$\hat\omega$$) plots will be displayed. When a zero pair (or pole pair) is dragged, the impulse response and frequency response plots will be updated in real time.
The connection between the $$z$$-transform domain of poles and zeros and the time domain, and also the frequency domain is illustrated with several movies where individual poles, or zeros or pole pairs of IIR filters are moved continuously.
A demo that illustrates the connection between the complex $$z$$-plane and the frequency response of a system. The frequency response is obtained by evaluating $$H(z)$$ on the unit circle in the complex $$z$$-plane.

#### Labs - MATLAB6

The objective for this lab is to build an intuitive understanding of the relationship between the location of poles and zeros in the z-domain, the impulse response $$h[n]$$ in the $$n$$-domain, and the frequency response $$H(e^{j\hat\omega})$$ (the $$\hat\omega$$-domain). A graphical user interface (GUI) called PeZ was written in MATLAB for doing interactive explorations of the three domains.
The goal of this lab is to explore the connection between the time domain (n), the frequency domain ($$\hat\omega$$), and the z-transform domain, using the GUI PeZ in MATLAB.
1. Placing Zeros: When placed on the unit circle, zeros of $$A(z)$$ will force the frequency response to be zero which can then be used to null out sinusoids at one frequency.
2. Placing Poles: When placed near the unit circle (but inside), roots of $$B(z)$$ will create peaks in the frequency response which can be used to form BPFs.
3. Designing IIR Notch Filters: requires conjugate zeros on the unit circle (UC) with a poles at the same angle, just inside the UC. The frequency response of the notch is much sharper than a nulling filter which is an FIR filter formed only from the conjugate zeros on the UC.
In this mini-project you will experiment with PeZ to learn the connection between pole-zero placement and frequency response. Given this information you will redo the Tone Removal Mini-Project using an IIR.
For this mini project you will write a simple function that listens to a tone and identifies what note it is. [Files]
For this mini project you will write a simple function that listens to a wav file of a simple song and identifies the notes being played. The wav files have some simple songs on which you can practice. [Files]
You have gotten to the point in your studies that you can understand DSP papers that appear in IEEE publications. The purpose of this project is to read one such paper and reproduce some of its results.

#### Labs - LabVIEW7

The objective for this lab is to build an intuitive understanding of the relationship between the location of poles and zeros in the z-domain, the impulse response $$h[n]$$ in the $$n$$-domain, and the frequency response $$H(e^{j\hat\omega})$$ (the $$\hat\omega$$-domain). A graphical user interface (GUI) called PeZ was written in MATLAB for doing interactive explorations of the three domains.
In this mini-project you will experiment with PeZ to learn the connection between pole-zero placement and frequency response. Given this information you will redo the Tone Removal Mini-Project using an IIR. [Files]
For this mini project you will write a simple function that listens to a tone and identifies what note it is. [Files]
For this mini project you will write a simple function that listens to a wav file of a simple song and identifies the notes being played. The wav files have some simple songs on which you can practice. [Files]
You have gotten to the point in your studies that you can understand DSP papers that appear in IEEE publications. The purpose of this project is to read one such paper and reproduce some of its results.
The goal of this mini-project is to help you understand a simple modem, the FSK modem, referred to by the International Telecommunications Union (ITU) as V.21.
The goal of this mini-project is to build a simple modem receiver. This a follow on for the previous mini-project.