EEE30004 Digital Signal Processing - Lab: Spectrograms of AM and FM Signals
EEE30004 Lab: Spectrograms of AM and FM Signals
Pre-Lab: Read the Pre-Lab and do all the exercises in the Pre-Lab section.
Lab Homework Questions: The Lab-
Homework Sheet has a few lab related questions that can be answered at your own pace. The completed Lab-HW
1 Pre-Lab 1.1 Objective
The objective of this lab is to introduce more complicated signals that are related to the basic sinusoid. These signals which implement frequency modulation (FM) and amplitude modulation (AM) are widely used in communication systems such as radio and television. In addition, they can be used to create interesting sounds that mimic musical instruments. There are a number of demonstrations on the companion website that provide examples of these signals for many different conditions, e.g., FM Synthesis. The resulting signal can be analyzed to show its time-frequency behavior by using the spectrogram.
This lab studies signal synthesis for AM and FM signals, and their time-frequency content as shown in a spectrogram. An underlying objective of the lab is to learn more about the spectrogram. There are several specific steps that will be considered in this lab:
- Synthesize a beat-note signal with a MATLAB M-file, and display its spectrogram.
- Study the frequency resolution of the spectrogram for two closely spaced sinusoids.
- Spectrogram: Make empirical observations of the spectrogram as the section length is changed.
- Synthesize a linear-FM chirp with a MATLAB M-file, and display its spectrogram.
- Spectrogram: Create a spectrogram that displays negative frequencies, as well as positive ones.
- Synthesize a frequency-modulated (FM) signal to match a given spectrogram. i.e, match specific time-frequency spectral content.
1.2 Overview
In this lab, we will extend our treatment of sinusoidal waveforms to more complicated signals composed of sums of sinusoidal signals, or sinusoids with changing frequency, i.e., frequency-modulated sinusoids.
1.3 Amplitude Modulation
If we add several sinusoids, each with a different frequency we cannot use the phasor addition theorem, but we can still express the result as a summation of terms with complex amplitudes via:
1.3.1 Beat Control GUI
To assist you in your experiments with beat notes and AM signals, the MATLAB GUI tool calledbeatcon has been created. The M-file beatcon.m is part of the DSP-First (or SP-First) Toolbox. This user interface controller will exhibit the basic signal shapes for beat signals and play the signals. A small control panel will appear on the screen with buttons and sliders that vary the different parameters for the beat signals. It can also call a user-written function named beat.m. Experiment with the beatcon control panel and use it to produce a beat signal with two frequency components: one at 690 Hz and the other at 700 Hz. Use a longer duration than the default to hear the beat frequency sound.
1.4 Frequency Modulated Signals
In this lab, we will examine signals whose frequency content varies as a function of time. Recall that in a constant-frequency sinusoid (1) the argument of the cosine is .2?f0t C'/ which is also the exponent of the complex exponential. We define the argument of the cosine as the angle function.
A generalization is available if we adopt the following notation for the class of signals with time-varying angle functions: where is the angle function. The time derivative of the angle function in (3) gives a frequency with a quadratic angle function Mathematically, we define in (3) as to an ending one.
1.5 Chirp, or Linearly Swept Frequency
A linear-FM chirp signal is a sinusoid whose instantaneous frequency changes linearly from a starting value The derivative of yields an instantaneous cyclic frequency (4) that changes linearly versus time.
The frequency variation in (5) produced by the time-varying angle function is called frequency modulation, so these signals are called FM signals. Finally, since the linear variation of the frequency (5) can produce an audible sound similar to a siren or a bird chirp, linear-FM signals are also called chirps.
1.6 MATLAB Synthesis of Chirp Signals
In MATLAB signals can only be synthesized by evaluating the signal’s defining formula at discrete instants of time. These are called samples of the signal. For the chirp we use the following:
Determine the range of frequencies (in hertz) that will be synthesized by the MATLAB script above, i.e., determine the minimum and maximum frequencies (in Hz) that will be heard. This will require that you relate the parameters ?, f0, and ' to the minimum and maximum frequencies. Examine the MATLAB spectrogram to determine the instantaneous (cyclic) frequency fi.t/ versus time. Zoom in to verify the correct starting and ending frequencies.
The spectrogram usually shows only the frequency components for f ? 0, but with the “tiny imagi- nary part” trick plotspec will show the negative frequency components. We will called this a two- sided spectrogram. Since the chirp signal is real-valued, the spectrum must have conjugate symmetry, so the magnitudes of the negative frequency components are a mirror image of those in the positive frequency region.
4. (d) Use soundsc() to listen to the signal in order to determine whether the signal’s frequency content is increasing or decreasing. Notice that soundsc() needs to know two things: the vector containing the signal samples, and the rate at which the signal samples are to be played out. This rate should be the same as the rate at which the signal values were created (fSamp in the code above).
5. (e) The test case above generates a chirp sound whose frequency starts low and chirps up. Modify the parameters so that the chirp starts at 3500 Hz and chirps down to 500 Hz.
1.8 Review Topic: Spectrograms
The main issue in this lab will be the dependence of the spectrogram on the choice of section length. A spectrogram is formed by taking successive short sections of a signal and performing an FFT analysis of each of those sections to get the spectrum. Since this is done repeatedly, the result is the spectrum versus time, where time is the location of the short sections.
2 Lab Exercise
For the lab exercise, you will synthesize some AM and FM signals, and then verify that these signals have the correct frequency content by using the spectrogram. The objective is to learn enough to be able to discuss the connection between the time-domain definition of the signal and its frequency-domain content.
2.1 Beat Notes and Frequency Resolution
In the section on beat notes in Chapter 3 of the text, we discussed signals formed as the product of two sinusoidal signals of slightly different frequencies; i.e., where fc is the (high) center frequency, and f is the (low) frequency that modulates the envelope of the signal. An equivalent representation for the beat signal is obtained by rewriting the product as a sum:
Beat Note Spectrograms
Beat notes have a simple time-frequency characteristic in a spectrogram. Even though a beat note signal, when defined as a product in (8), may be viewed as a single frequency signal whose amplitude varies with time, the spectrum requires an additive combination as in (9) which turns out to be the sum of two sinusoids with different constant frequencies.
Although some of the mathematical details require further study beyond this course, it is not difficult to appreciate the following issue: there is a fundamental trade-off between knowing which frequencies are present in a signal’s spectrum and knowing how those frequencies vary with time. As discussed previously, a spectrogram estimates the frequency content over short sections of the signal; this is the Section Length changes in the frequency content. The tradeoff, however, is that shorter sections may not provide enough
If we make the section length very short we can track rapid changes in the signal parameter. data to do an accurate frequency measurement. On the other hand, long sections allow the spectrogram to perform excellent frequency measurements, but fail to track sudden frequency changes. This trade-off between the section length (in time) and the frequency resolution is akin to Heisenburg’s Uncertainty Principle in physics. We can summarize this discussion by stating the following hypothesis:
- (a) Use the MATLAB code written in Section 2.1.1 to create and plot a beat signal defined via: with a duration of 5 s, and a sampling rate of fs D 8000 samples/s. The frequency f? should be set to 4 Hz, but will be varied in later parts. The phases can be random.
- (b) When determine the locations of the two spectrum lines that you expect to see in the spectrogram. In other words, derive (mathematically) the spectrum of the signal defined in part (a).
- (c) Make the spectrogram of b.t/ using a (window) section length of LSECT D 256 using the Comment on what you see. Are there two spectral lines, i.e., (horizontal lines across the spectrogram)? If necessary, use the zoom tool (in the MATLAB figure window), or zoom on, to examine the important regions of the spectrogram.
- (d) It should not be possible to see both spectrum lines with LSECT D 256. In order to get both lines a longer section length is needed, so try doubling the section length. Try LSECT D 512, then LSECT D 1024, and so on until you can discern two spectrum lines. Then reduce the value of LSECT little by little to get the smallest LSECT that will work. Getting a value of LSECT to the nearest 500 is sufficient.
2.1.3 Inverse Relationship: Section Length vs. Frequency Resolution
The shortest section length when you are able to discern the two spectrum lines was determined in the previous section.
- (a) The shortest (window) section length of LSECT samples has been converted into a (window) section duration in seconds (via the sampling rate). Compare the inverse of this (window) section duration to the frequency separation of the spectrum lines.
- (b) Now change f? to 16 Hz and repeat the resolution experiment in Sect. 2.1.2(d). That is, find a section length that will resolve the two frequency components which are now farther apart. Use the value of C and (10) to predict the section length that you will need. Verify that this section length will work correctly.
2.2 Spectrogram for a Chirp with Negative Instantaneous Frequency
Use the code provided in the pre-Lab section as a starting point in order to write a MATLAB script or function that will synthesize a “chirp” signal. Then use that M-file in this section.
- (a) What happens when we make a signal that “chirps down” and the instantaneous frequency goes neg- ative? Generate a chirp signal that starts at 2000 Hz when t D 0 s, and chirps down to 1000 Hz, at t D 1:5 s. Use fs D 8000 Hz. Determine the parameters needed in
- (b) Generate the chirp signal in MATLAB and make a spectrogram with LSECT D 200 to verify that you have the correct starting and ending frequencies. For LSECT D 200, determine the section duration TSECT in secs.
- (c) Will you hear negative frequency? Use a spectrogram that contains negative frequencies to explain your answer. If possible listen to the signal and describe what you will hear; then explain in terms of the two-sided spectrogram.
2.2.1 Section Length in Chirp Spectrogram
When we have a signal whose frequency is not constant, we can study how the temporal features of the spectrogram depend on the section length LSECT. As we saw in the previous section, the section length has to be short to capture the temporal changes in the signal. However, let’s see what happens with a long section length.
2.3 Lab-HW: Matching Unknown Spectrograms
Now you are given a spectrogram in Fig. 1, and you must synthesize a signal that will match that time- frequency plot. Explain and discuss your work.
- Define a time signal x.t/ whose spectrogram will match the given spectrogram. This signal definition should be a simple mathematical formula.
Note: you might have to iterate with the following two steps to get a good approximation. - Generate samples of the signal over the appropriate time interval using fs D 4000 Hz.
- Choose the section length in plotspec carefully so that your spectrogram is an excellent match.
