EE313 Linear Systems & Signals - Mini-Project #2 Hints - Fall 2024

Mini-Project #2 assignment

Theme: The assignment is to design, analyze, and simulate averaging and nulling FIR filters.
Takeaways: The project covers three key ideas concerning linear time-invariant systems:
- Time domain: Output signal is the convolution of the input signal and the filter’s impulse response.
- Frequency domain: Output signal is the product of the input signal and the filter’s frequency response. The filter’s frequency response is the Fourier transform of the impulse response
- Filtering: The magnitude of the frequency response can be designed to pass, attenuate, and amplify bands of frequencies as well as eliminate individual frequencies. The phase of the frequency response can be designed to delay all frequencies by the same amount in the time domain (linear phase) or assign a different delay to each frequency component. Practical filters have different frequency responses than ideal filters.
Scope and format are described at the bottom of the homework hints page.
Download the Signal Processing First dltidemo (Discrete-Time Linear Time-Invariant Demo)
- Unzip it in your MATLAB folder.
- In MATLAB, change directories to the dltidemo folder
- Type in the MATLAB command dltidemo and click YES to add the folder to the path
Section 1.6. In addition to lab assignment, compute and plot the poles and zeros for the three-coefficient nulling filter in terms of a null frequency at 0.75 π, plot the magnitude response and pole-zero diagram, and connect the values of the poles and zeros to the magnitude response of the nulling filter.
Section 1.7. In Ideal Filters section, part (b) asks you to use as the input signal
```
    x[n] = 1.5 + 0.9 cos(0.55π n)
```
What's the discrete-time frequency that corresponds to 1.5? It's 0 rad/sample because 1.5 does not oscillate. Mathematically, 1.5 cos(0 n) = 1.5. That is, 1.5 is a cosine with discrete-time frequency of 0 rad/sample, phase of 0 rad, and amplitude of 1.5.
Marker board work for ideal filtering applied to x[n].
The Practical Filters section asks to "Right-click to get values from the frequency response plot". For me, I need to hold down the Control key while Right-Clicking.
Section 2.1(b). The Designing Averaging Filters handout analyzes seven-point averaging filters in the time, frequency, and z domains. It factors the frequency response into polar form, which reveals the phase function to be a line vs. discrete-time frequency.
Section 2.1(d). The input signal is
```
    x[n] = 1.8 cos(0.1π(n-2)) = 1.8 cos(0.1π n - 0.2π))
```
which is a cosine with discrete-time frequency 0.1π, phase of 0.2π, and amplitude of 1.8.
The output of the LTI system will be a cosine of the same frequency; however, the amplitude will be scaled by the magnitude response of the LTI system evaluated at the discrete-time frequency 0.1π and the phase will be shifted by the phase response of the LTI system evaluated at the discrete-time frequency 0.1π.
The output is in the form
```
    y[n] = A cos(w₀ (n - n₇)) 
```
where n₇ is an integer. If we distribute the multiplication in the argument of the cosine, we obtain
```
    y[n] = A cos(w₀ n - w₀ n₇) 
```
where the phase is -w₀ n₇, and
```
    Delay(w) = -d/dw ( phase(w) ) = n₇
```
We can estimate the slope of the phase to estimate n₇. We can also obtain calcualte the exact value of n₇ using the Designing Averaging Filters handout.
Section 2.1(e). The discrete-time Fourier transform of a rectangular pulse can written using a Dirichlet function. Please see Signal Processing First, Section 6.7, pages 145-146.
Section 2.3. This section relies on the Matlab file
speechbad.zip
Please download and extra the contents to create the file “speechbad.mat”. This is a binary file format specific to Matlab that contains a copy of one of more Matlab variables. Load it into Matlab using
```
load speechbad
```
to define a corrupted speech segment in the Matlab variable xxbad. By loading the file, three variables will be defined:
```
xxbad
fs
f_interference
```
The sampling rate, fs, is 8000 Hz. You can use the sound command to play it:
```
sound(xxbad, fs)
```
Please submit a narrative report whose sections follow the same sections as in the assigned lab exercise. The audience for your report is other students in the course. Here are several examples of narrative reports for previous mini-project #2 assignments for this course:
- Fall 2018 Octave Band Filtering for Audio Signals: Assignment and Report
- Fall 2021 Wireless Localization: Assignment and Report
- Fall 2023 Image Filtering: Assignment and Report

Last updated 11/06/24. Send comments to (Mailbox)

bevans@ece.utexas.edu