This lab was developed based upon Reference [1].
Note before you begin: If you are having trouble viewing these lab instructions, I recommend using Firefox on a desktop computer or laptop (not mobile device)
All MATLAB code examples for this lab are collected here.
The objectives of this lab are to learn how to visualize signals in MATLAB. This requires two tasks:
Sinusoids are the "building blocks" of all (reasonably well-behaved) signals. By adding enough sinusoids - sometimes an infinite number - with the correct amplitudes, frequencies, and relative phase angles, we can create any signal. Sinusoids are described in mathematics using the cosine function (or sin function) or by complex exponentials \(e^{j\theta}\).
In engineering, many systems ,such as RLC circuits and small signal amplifiers, can be accurately described as linear Time Invariant (LTI) systems. It is straightforward to show that if a sinusoidal signal is input to an LTI system, the output will always be a sinusoid with the same frequency (though, in general, the output will have a different amplitude and phase). This is illustrated in Figure 1 for a cosine function. This means that a 100 Hz sinusoidal signal input to an LTI system will always result in a 100 Hz sinusoid output (zero output would be regarded as a 100 Hz sinusoid with zero amplitude). If a 100 Hz sinusoid input results in a 300 Hz output then the system is not LTI. If a 100 Hz sinusoid input results in a square wave output then the system in not LTI. LTI systems are generally much easier to analyze and characterize than nonlinear and time-varying systems. Thus if we can determine how an LTI system will effect sinusoids,we can determine how the system will effect any other signal by first describing the signal as a sum of sinusoids and then determining the effect of the system on the sinusoids separately. This is the motivation for working with sinusoids.
The frequency spectrum of a signal is the frequency content of a signal. Spectral analysis is a widely used and powerful method of data analysis, which determines the spectrum of a time-varying signal.
The spectrum of periodic signals is usually represented as vertical lines, called line spectra, whose frequencies and amplitudes are indicated on the x and y axes, respectively. For example, suppose a signal is represented by: \begin{equation} x(t)=\cos(2\pi10t)+0.5\cos(2\pi15t) \end{equation} The frequency spectrum can be plotted either as single-sided or double-sided. In the single-sided spectrum, the vertical lines represent the amplitudes of the sinusoids (cosines) that comprise a function. The single-sided spectrum for this example is represented in Figure 2. This plot was generated using the stem plotting command in MATLAB. Click here to view the MATLAB script used to generate this plot. The double-sided spectrum is more general than the single-sided spectrum in that is can represent the spectra of both real and complex signals. In the double-sided spectrum, the vertical lines represent the amplitudes of the complex exponentials (\(e^{j2\pi f_0t}\)) that comprise a function. It results from Euler's Equation, which says that we can decompose a sinusoid into a sum of two complex exponential functions: \begin{equation} A\cos(2\pi f_0t) = A\frac{{e^{j2\pi f_0t}+e^{-j2\pi f_0t}}}{2} \end{equation} Looking at the equation above, we see that the amplitude of the double-sided spectrum for \(A\cos(2\pi f_0t)\) is \(A/2\) at frequencies \(f_0\) and \(-f_0\). Therefore, the double-sided spectra for a general function \(x(t)\) looks just like the single-sided spectra at positive frequencies (\(f>0\)). The only difference is that the amplitudes of the double-sided spectrum are one-half as large (i.e. the vertical lines are one-half as long) compared to the single-sided spectrum. If a DC component exist (i.e. a component at (\(f = 0\)) then this spectral component is identical for bot single-sided and double-sided spectra. Figure 3 shows the double-sided spectrum for the signal given in Eq. (1). Click here to view the MATLAB script used to generate this plot. Note the similarities between Figures 2 and 3.
The fundamental frequency is the greatest common divisor of all the frequencies present in the signal. The fundamental frequency of the signal given in Eq. (1) above is 5 Hz, so \(x(t)\) will repeat every 1/5 = 0.2 seconds. Non-periodic signals can have discrete line spectrum as well. These types of signals are called "quasi-periodic" (or sometimes "non-periodic" or "almost periodic"). If in our previous example we change the 15Hz component to \( \sqrt{8} \) Hz it will never repeat, since \( \sqrt{8} \) and 10 do not share a common divisor. However, this signal is still composed of two sinusoids, which are represented as line spectra. On the other hand, if in our previous example we change the 15Hz component to 15.21 Hz it will repeat, since 15.21 and 10 share 0.01 as the common divisor. In this case \(x(t)\) would repeat every 1/0.01 = 100 seconds
In this lab we will be creating (simulating) analog signals on a computer using MATLAB. This takes a bit of careful thinking. True analog signals are continuous signals (continuous in both amplitude and time). However, computers deal with digital signals that are both discrete in time and amplitude. Discrete amplitudes aren't usually a problem, since MATLAB can handle large floating point numbers. However the time instances for a signal are not selected automatically in MATLAB. Meaning that if we chose too few points in time to define our signal, it will not be plotted smoothly. Soon you will learn that the minimum sampling frequency that you can have for a signal is \(2f_{max}\),where \(f_{max}\)= the maximum frequency of the signal being sampled. This prevents Aliasing. You need not worry about aliasing yet, just remember to select enough time samples for your signal so that the plots appear smooth. For a sinusoid a smooth plot will have a sampling frequency 25 - 100 times the frequency of the signal. In the examples you will see that I have selected appropriate sampling frequencies for plotting each signal.
Consider the following sinusoidal signal: \begin{equation} x(t)=A\cos(2\pi f_0t+\phi) \end{equation} We will simulate the signal with the following parameters:
A true sinusoid is everlasting - it started at negative infinity (before the Age of the Dinosaurs) and will continue into the indefinite future forever (long after the Sun consumes the Earth). It is impossible to plot an everlasting signal, so let us choose to plot the sinusoid starting at t = 0 and end at 3 periods (\(3T\)). This is a reasonable time span for the purposes of visual communication (it is not too "zoomed in" nor too "zoomed out"). Since \(f_0 = 100 Hz\), the period \(T = 1/f_0 = 0.01\) seconds. Hence we will plot the signal between 0 seconds and 0.03 seconds.
As mentioned above, \(x(t)\) is a continuous time-wave form. MATLAB cannot plot continuous-time signals directly. Instead it plots the waveform at isolated (discrete) points in time. By default, MATLAB's plot function then automatically connects those points with straight lines. Here we will choose a sampling rate that is 100 times the frequency of the signal: \(f_s = (100)(100) = 10000 Hz\). In other words, we choose to plot the sampled continuous-time signal \(x(t)\) at equally spaced time instances \(T_s = 1/f_s = (100)(100) = 0.001\) seconds. With this choice, our plot will appear to the eye continuous (more-or-less).
The plot is shown in Figure 4 below. The MATLAB script used to generate this example is available here.
In this example we will simulate and plot two signals to demonstrate Euler's equation. Euler's identity is given as: \begin{equation} e^{j\theta}=\cos(\theta)+j\sin(\theta) \end{equation} We will simulate the signal with the following parameters:
In this example we demonstrate the linear combination of sinusoids that are present in a square wave. If you search for "Fourier Series Table", you will find a plethora of tables available online, which present the Fourier Series of many waveforms. The Fourier Series of a square wave is shown below in Figure 6 below. Using this Fourier Series, we simulate in MATLAB how adding up various numbers of sinusoids approximates a square wave. The resulting plots are shown in Figure 7 and Figure 8. The MATLAB script used to generate this example is available here.
Later in this course we will study the Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) in detail. For now, lets take an initial look at how to use MALTAB's FFT function for spectral analysis. Consider the sinusoidal signal given in Equation (1). We shall repeat it here again for convenience: \begin{equation*} x(t)=\cos(2\pi10t)+0.5\cos(2\pi15t) \end{equation*} Figure 9 shows the spectrum of this signal, as computed using using MATLAB's FFT function. Compare the double-sided spectrum shown in Figure 9 with that shown in Figure 3. Qualitatively, they are very similar. The ripples and "signal spread" exhibited in Figure 9 are due to windowing, which we will study more about later in this course. For now, you can think of windowing like this: A true sinusoid is everlasting (it starts at t = -infinity and ends at t = +infinity). However, with MATLAB we can only simulate a sinusoid that exists for a finite time. Hence the FFT of a sinusoid gives, in fact, the spectrum of a truncated sinusoid (not an ideal everlasting sinusoid). This fact gives rise to the differences between Figure 3 and Figure 9. In Figure 9, the time-window was chosen as 1.3333 seconds. If we increase this window, then the spectra will be "sharper" (approaching that shown in Figure 3) The MATLAB script used to generate this example is available here.
For the lab report, you will create a PowerPoint presentation (or use a similar presentation program), save it as a PDF, and submit it on-line according to the instructions given in class. All submitted Matlab code should be well organized and commented with clear comments for easy readability. All submitted plots should be easy to see and well-labeled.
You are to work independently. This is not a team assignment. Feel free to help your fellow classmates understand principles and concepts, but please do not share your work.
Your presentation will have 23 slides. Please include a slide number in the footer of each slide. To earn full credit your presentation must contain the slides in the order asked for in this lab. If you miss a slide, please leave a blank slide in its place so that you have still have exactly 23 slides total. Your first slide should be:
