Assignment 1: Signals and Spectra
This assignment is going to examine the concept of spectrum and consider the limitations of spectral representations and how time-frequency analysis can help improve spectral analysis.
a) In MATLAB generate seven 1 second length cosinusoids, use a sampling rate of 22050 Hz so you can listen to them using soundsc() (if you wish).
- Each of the seven cosines will have an amplitude of 1 with frequencies and phases relating to the numbers in your student number (see table below).
|
Student Number
|
Frequency
|
Phase
|
|
1
|
1100 Hz
|
0.1π rad
|
|
2
|
1000 Hz
|
0.2π rad
|
|
3
|
900 Hz
|
0.3π rad
|
|
4
|
800 Hz
|
0.4π rad
|
|
5
|
700 Hz
|
0.5π rad
|
|
6
|
600 Hz
|
0.6π rad
|
|
7
|
500 Hz
|
0.7π rad
|
|
8
|
400 Hz
|
0.8π rad
|
|
9
|
300 Hz
|
0.9π rad
|
|
0
|
1200Hz
|
0 rad
|
By hand, calculate the expected frequency spectrum of these seven sinusoids (individually), label each spectral component with their corresponding complex amplitudes.
Use the fft() function in MATLAB to confirm your calculations by plotting the actual spectra of these signals. Label all MATLAB figures including x and y axes.
b) In this task you will concatenate the seven cosines from task a) into one 7 sec long vector. To concatenate vectors in MATLAB use:
x=[x1 x2 x3 x4 x5 x6 x7];
Use calculations and/or your conceptual understanding of spectra to predict what you'd expect the spectrum of this signal to look like. Confirm this hypothesis by again using fft() in MATLAB to plot the actual spectrum.
If MATLAB differs from your predictions explain why this is so.
c) In this task create a new variable and add together your seven cosines: x=x1 + x2 + x3 + x4 + x5 + x6 + x7;
Again use calculations and/or your conceptual knowledge of the topic to make a prediction about what you'd expect the spectrum of this signal to look like.
Now use MATLAB to confirm your hypothesis.
Discuss:
1. Is this signal periodic? If so what is the fundamental period/frequency? If it's not periodic explain why it's not.
2. Explain how the spectra in parts b and c are the same or different. Explain why there may be differences between the two (you will need to take into account the Fourier Transform property of modulation and possibly phasor addition in your explanation).
3. Discuss techniques on how time information can be retained while still being able to observe the frequency content of a signal.
d) This task is going to look at implementing a time-frequency analysis on the two signals generated in parts b) and c) and consider how this can give us more information about a signal.
- Use spectrogram() in MATLAB to obtain a time-frequency plot of both the concatenated signal and the added signal. E.g.
spectrogram(x, 1024, [], 1024, fs)
Discuss:
1. Contrast the spectrogram graphs to the graphs obtained from using fft() and explain what extra information you've been able to obtain about the signals in parts b) and c).
2. Explain how spectrograms can be used in non-stationary signals to help obtain time and frequency information. Give examples of some practical signals this technique could be useful in analysing.