With choosing a sine wave as the orthogonal function in the above expression, all that is left is to solve for the coefficients to construct a square wave and plot the results.
#BIPOLAR SQUARE WAVE MATHEMATICAL REPRESENTATION SERIES#
We will first write a step function of length (L) that, when repeated periodically, will be our representation of a square wave.Īs stated earlier, this function can be rewritten as an infinite series of an orthogonal function φ: Now, let’s take a look at a square wave and how it appears when constructed using Fourier series the same way an oscilloscope would. Theoretically, if an infinite number or terms are used, the Fourier series will cease to be an approximation and take the exact shape of the function. You can see that the more terms that are summed, the closer the approximation becomes to the actual function. This is tougher to picture because a line is not oscillatory, but the addition of multiple sine or cosine terms will begin to take the shape of a line. Let’s look at constructing a linear line using sine and cosine functions. This would be a Fourier series with only one term, and would return the desired function with the magnitude changed. It is easy to picture changing the amplitude of a sine wave by multiplying it by a coefficient. The values of these coefficients determine the function that will be reconstructed. Every function has a unique set of coefficients, An, that are substituted into the sum. You can see that any function can be constructed using an infinite series of terms, and approximated by a finite number of sine waves. Using the above definition, it is possible to solve for the coefficients (An) for any function, and build the function, using orthogonal functions. The definition of orthogonality for a function φ is that it must satisfy the following condition: While these functions are useful for problems in cylindrical or polar coordinates, they do not apply to this discussion. There are a few other functions that satisfy the orthogonality requirement including Bessel functions and Legendre polynomials. Sine and cosine functions can both be used for Fourier series because they both have the property of orthogonality. More specifically, any function of x, f(x), can be built using an infinite series of sine waves with coefficients An and increasing frequencies (n*pi). It’s also worth noting that this method is used by all oscilloscopes, it’s not just a Keysight method.įirst off, you need to understand that any arbitrary function f(x) can be constructed by the sum of simple sine or cosine functions that vary in amplitude and frequency. Understanding this can help you to select the right oscilloscope for your measurement needs. The inherent discontinuity of a square wave presents some problems with this reconstruction method that can be understood by exploring the mathematical theory. It is the most useful way for an oscilloscope to process and measure a signal because it deconstructs the signal into its frequency components for analysis. This method of building a signal is known as Fourier series. In short, this ringing is a phenomenon that presents itself because of the method an oscilloscope uses to construct a signal – by summing the frequency components of the signal. The occurrence of this ringing will be explained from the perspective of the underlying theory (Fourier series as a method for solving partial differential equations), and then relate it back to using an oscilloscope. Have you ever wondered, “What is this ringing I am seeing in my signal? Why is there preshoot and overshoot on a simple square wave? How is a preshoot possible when it appears to be precausal to downstream information?” It will be explained the occurrence of ringings in a signal from the perspective of the underlying theory (Fourier series as a method for solving partial differential equations), and then relate it back to using an oscilloscope.
Wolfram Language & System Documentation Center.Technical Papers SummaryOscilloscopes - Fourier Series of a Square Wave (and Why Bandwidth Matters) "SquareWave." Wolfram Language & System Documentation Center. author="Wolfram Research", title=", note=[Accessed: 05-November-2021 Wolfram Research (2008), SquareWave, Wolfram Language function. Cite this as: Wolfram Research (2008), SquareWave, Wolfram Language function.
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