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In this lab, we examine the representation of periodic signals based on Fourier series. Periodic signals can be represented by a linear combination of an infinite sum of sine waves, as expressed by the trigonometric Fourier series representation.Periodic signals can also be represented by an infinite sum of harmonically related complex exponentials, as expressed by the exponential Fourier series representation. In this lab, we analyze both of these series representations. In particular, we focus on how to compute Fourier series coefficients numerically.
This example helps one to gain an understanding of Fourier series decomposition and reconstruction for periodic signals. The first step involves estimating $x(\mathrm{m\Delta t})$ which is a numerical approximation of the analog input signal. Though programming environments deploy discrete values internally, we can obtain a close analog approximation of a continuous-time signal by using a very small $\mathrm{\Delta t}$ . That is to say, for all practical purposes, when $\mathrm{\Delta t}$ is taken to be very small, we get the analog representation or simulation of the signal. In this example, create four input signals using the listed LabVIEW MathScript functions in Table 1.
Waveform type | LabVIEW MathScript function |
Square wave |
square(T) , T denotes period |
Triangular wave |
sawtooth
(T,Width) , Width=0.5 |
Sawtooth wave |
sawtooth
(T,Width) , Width=0 |
Half wave rectified sine wave | $\{\begin{array}{cc}\text{sin}(2\ast \text{pi}\ast f\ast t)& \begin{array}{cc}\text{for}& 0\le t\end{array}<T/2\\ 0& \begin{array}{cc}\text{for}& T/2\le t<T\end{array}\end{array}$ , $f=1/T$ denotes frequencyHalf period is sine wave and the other half is made zero |
Use a
switch
structure to select different types of input waveforms. Set the switch parameter w as the input and connect it to an
Enum Control
(
Controls
→
Modern
→
Ring&Enum
→
Enum) . Edit the Enum Control items to include all the waveform types.
Set Amplitude of input (A), Period of input (T) and Number of Fourier coefficients (N) as control parameters. Determine Fourier coefficients
${a}_{0},{a}_{n}$ and
${b}_{n}$ and reconstruct the signal from its Fourier coefficients using equations provided in Chapter 4. Determine the error between the input and the reconstructed signal by simply taking the absolute values of
$x(t)-\stackrel{\u02c6}{x}(t)$ via the LabVIEW MathScript function
abs
. Finally, determine the maximum and average errors by using the functions
max
and
sum
.
[link] shows the completed block diagram of the Fourier series signal decomposition and reconstruction system.
Display the input signal using a waveform graph. Before displaying the graph, configure it using the function Build Waveform (Functions → Programming → Waveforms → Build Waveforms) . Also display the Fourier coefficients, reconstructed signal and error in the waveform graph, and place several numerical indicators to show the values of the Fourier coefficients, maximum error and average error.
[link] and [link] illustrate the front panel of the Fourier series signal decomposition and reconstruction system, respectively. To display all the outputs within a limited screen area, use a Tab Control (Controls → Modern → Containers → Tab Control) in the front panel. Here the outputs are arranged in two different tabs: Fourier Series and Signal Reconstruction.
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