This lab was developed based upon Reference [1].

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All MATLAB code examples for this lab are collected here.

Objectives

The objective of this lab is to explore the concepts of sampling and reconstruction of analog signals. Specifically, we will simulate the sampling process of an analog signal using MATLAB, investigate the effect of sampling in the time and frequency domains, and introduce the concept of aliasing.

Introduction - Sampling and aliasing

An analog-to-digital converter (ADC) converts an analog signal to a digital form. An ADC produces a stream of binary numbers from analog signals by taking the samples of the analog signal and digitizing the amplitude at these discrete times. Prior to the ADC conversion, an analog filter called the prefilter or antialiasing filter is applied to the analog signal in order to deal with an effect known as aliasing. Aliasing causes multiple continuous time signals to yield the exact same sampled discrete time signal.

In this lab we focus our attention in the process of sampling and how to avoid the problem of aliasing. During sampling, an analog signal \(x_a(t)\) is periodically measured every \(T_s\) seconds: \begin{equation} t=nT_s, \quad n = 0,1,2,... \end{equation} where \(T_s\) is called the sampling period, and is the fixed time interval between samples (here we assume a uniform sampling rate that does not change with time.) The inverse of \(T_s\) is called the sampling frequency, that is, the samples per second: \begin{equation} f_s=1/T_s. \end{equation} When sampling an analog signal we must sample fast enough (i.e. be sure \(f_s\) is sufficiently high), so that the samples are a good representation of the original analog signal. If the sampling frequency \(f_s\) is not fast enough then too much information is lost, and it becomes impossible to reconstruct our original analog signal using a digital-to-analog converter (DAC). However, if you do sample fast enough, then, theoretically, it is possible to exactly reconstruct the original signal.

Sampling theorem: For accurate representation of signal \(x_a(t)\) by its time samples \(x[nT]\), two conditions must be met: 1) The signal \(x_a(t)\) must be bandlimited, that is, its frequency content (spectrum) must be limited to contain frequencies up to some maximum frequency \(f_{max}\) and no frequencies beyond that, and 2) the sampling rate \(f_s\) must be chosen to be at least twice the maximum frequency \(f_{max}\), that is

\begin{equation} f_s \geq 2f_{max}. \end{equation}

According to the sampling theorem, before sampling we must make sure the signal is bandlimited (this is the function of the analog prefilter) and that the sampling frequency is at least twice the maximum frequency.

The traditional Nyquist sampling theorem presented above is true for real-valued (i.e. not complex), lowpass (i.e. baseband) signals. For complex lowpass signals, the sampling theorem states that the ultimate minimum sampling rate to avoid aliasing is actually \(f_s = B\), where \(B\) is the double-sided bandwidth. For bandpass signals, things get even more interesting. You can subsample (i.e. sample below the Nyquist rate) to achieve frequency translation to lower frequencies and recover the original signal. See Reference [2] for more details.

Example 1 - Sampling higher than Nyquist rate

Consider the following continuous time signal: \begin{equation} x_a(t)=3\sin(2\pi 0.5t)+2\sin(2\pi 2.5t) \end{equation} The signal contains two frequency components at \(f_1\ = 0.5 Hz\) and \(f_2\ = 2.5 Hz\). We explore the effect of aliasing by sampling \(x_a(t)\) at \(f_s\ = 2.5f_{max} = 6.25 Hz\). This sampling frequency meets the sampling theorem requirement. In MATLAB we explore both time-domain and frequency domains by making the following plots:

Note that the reconstructed time-domain signal is equal to the original time-domain signal, because the sampling theorem is met. The MATLAB script used to generate this example is available here.

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Figure 1. First plot shows the original continuous time signal \(x_a(t)\). Second plot shows the sampled discrete time signal \(x[n]\) with \(f_s=2.5f_{max}=6.25 Hz\). Third plot shows the sampled discrete time signal superimposed on top of the reconstructed time-domain signal.

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Figure 2. First plot shows the two-sided amplitude spectrum of the original time signal. Second plot shows the two-sided amplitude spectrum of the sampled discrete time signal with \(f_s=2.5f_{max}=6.25 Hz\). Third plot shows the two-sided amplitude spectrum of the reconstructed time-domain signal.

Example 2 - Sampling lower than Nyquist rate

Consider the same continuous time signal as in Example 1: \begin{equation} x_a(t)=3\sin(2\pi 0.5t)+2\sin(2\pi 2.5t) \end{equation} The signal contains two frequency components at \(f_1\ = 0.5 Hz\) and \(f_2\ = 2.5 Hz\). We explore the effect of aliasing by sampling \(x_a(t)\) at \(f_s\ = 1 Hz\). This sampling frequency does not meet the sampling theorem requirement. In MATLAB we explore both time-domain and frequency domains by making the following plots:

Note that the reconstructed time-domain signal is not equal to the original time-domain signal, because the sampling theorem is not met. The reconstructed signal is: \begin{equation} x_r(t)=3\sin(2\pi 0.5t)+2\sin(2\pi 0.5t) =5\sin(2\pi 0.5t) \end{equation} The above equation results because a 0.5 Hz sinusoid is an alias of a 2.5 Hz sinusoid. The MATLAB script used to generate this example is available here.

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Figure 3. First plot shows the original continuous time signal \(x_a(t)\). Second plot shows the sampled discrete time signal \(x[n]\) with \(f_s=f_{max}/2.5=1 Hz\). Third plot shows the sampled discrete time signal superimposed on top of the orignal time-domain signal. Forth plot shows the sampled discrete time signal superimposed on top of the reconstructed time-domain signal.

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Figure 4. First plot shows the two-sided amplitude spectrum of the original time signal. Second plot shows the two-sided amplitude spectrum of the sampled discrete time signal with \(f_s=f_{max}/2.5=1 Hz\). Third plot shows the two-sided amplitude spectrum of the reconstructed time-domain signal.

Tasks

Report guidelines

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.

This lab involves generating many plots using MATLAB's subplot function. This will likely result in figures which are "squashed" together. To unsquash the images, please manually stretch (resize) the figure window before placing the figures in your presentation.

Your presentation will have 11 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 11 slides total. Your first slide should be:

  1. Sampling higher than Nyquist rate

    Consider a continuous time domain signal \begin{equation} x_a(t)=2\cos(2\pi 0.5t)+\sin(2\pi 1.5t)+\cos(2\pi 2.5t)+1.5\cos(2\pi 3.5t) \end{equation}
    1. Suppose we sample \(x_a(t)\) with a sampling frequency \(f_s = 10f_{max}\). Create a 3-by-1 plot like Example 1 where you plot the original continuous time signal \(x_a(t)\), the sampled discrete time signal \(x[n]\), and the sampled discrete time signal superimposed over the reconstructed signal. Present your plot and your MATLAB code in Slides 2 - 5. Please program MATLAB to display your name as the title of the plot. (e.g. The MATLAB command would be something like: title('Jane Doe').)

  2. Sampling lower than Nyquist rate

    1. Consider the continuous time signal \(x_a(t)\) expressed in Equation (7). Suppose we sample \(x_a(t)\) with a sampling frequency \(f_s = f_{max}\). Create a 4-by-1 plot like Example 2 where you plot the original continuous time signal \(x_a(t)\), the sampled discrete time signal \(x[n]\), the sampled discrete time signal superimposed over the original continuous time signal, and the sampled discrete time signal superimposed over the reconstructed signal. Present your plot in Slide 6. Please program MATLAB to display your name as the title of the plot. (e.g. The MATLAB command would be something like: title('Jane Doe').)

  3. Anti-aliasing filter

    1. Consider the continuous time signal \(x_a(t)\) expressed in Equation (7). Consider again a sampling frequency \(f_s = f_{max}\). Suppose you first process \(x_a(t)\) with an ideal anti-aliasing low-pass prefilter and then sample the signal. What is the signal after the anti-aliasing low-pass prefilter? Present your expression in Slide 7. Create a 4-by-1 plot where you plot the original continuous time signal \(x_a(t)\), the continuous time signal after it has been processed by the prefilter, the sampled discrete time signal \(x[n]\), and the sampled discrete time signal superimposed over the reconstructed signal. Present your plot Slide 8. Please program MATLAB to display your name as the title of the plot. (e.g. The MATLAB command would be something like: title('Jane Doe').)

  4. The Atari (wrap-around) Effect

    1. Consider a continuous time domain complex signal \begin{equation} x_a(t)=e^{j2\pi10t} \end{equation} Create a 4-by-1 plot where you plot the double-sided spectrum of the sampled signal for the following sampling frequencies: \(f_s = 25 Hz\), \(f_s = 21 Hz\), \(f_s = 19 Hz\), \(f_s = 15 Hz\). Present your plot in Slide 9. Please program MATLAB to display your name as the title of the plot. (e.g. The MATLAB command would be something like: title('Jane Doe').) When creating your signal, sample your signal for a total of 10 seconds or more (this large sampling window will result in "sharp" spectral peaks). For your plot make sure to set axis tight.
    1. Consider a continuous time domain signal \begin{equation} x_a(t)=\cos(2\pi10t) \end{equation} Create a 4-by-1 plot where you plot the double-sided spectrum of the sampled signal for the following sampling frequencies: \(f_s = 25 Hz\), \(f_s = 21 Hz\), \(f_s = 19 Hz\), \(f_s = 15 Hz\). Present your plot in Slide 10. Please program MATLAB to display your name as the title of the plot. (e.g. The MATLAB command would be something like: title('Jane Doe').) When creating your signal, sample your signal for a total of 10 seconds or more (this large sampling window will result in "sharp" spectral peaks). For your plot make sure to set axis tight.
    2. In Slide 11 explain what is meant by the Atari (wrap-around) effect in terms of sampling and aliasing.

References

  1. J. Sprunger and M. Aboy, Computer Explorations in DSP Laboratory 2 - Signals and Aliasing , prepared for EE 430, Oregon Institute of Technology, 2013.
  2. M. Valkama, Complex valued signals and systems - Basic principles and applications to radio communications and radio signal processing, Tampere University of Technology, Presentation.