File Name: fourier series even and odd function .zip
The Fourier series of functions is used to find the steady-state response of a circuit. There are four different types of symmetry that can be used to simplify the process of evaluating the Fourier coefficients. If a function satisfies Eq.
The Fourier series of functions is used to find the steady-state response of a circuit.
This document derives the Fourier Series coefficients for several functions. The functions shown here are fairly simple, but the concepts extend to more complex functions.
Consider the periodic pulse function shown below. It is an even function with period T. The function is a pulse function with amplitude A , and pulse width T p. The function can be defined over one period centered around the origin as:. During one period centered around the origin. This can be a bit hard to understand at first, but consider the sine function. The values for a n are given in the table below. Note: this example was used on the page introducing the Fourier Series.
You can change n by clicking the buttons. As before , note:. The values for a n are given in the table below note: this example was used on the previous page. Note that because this example is similar to the previous one, the coefficients are similar, but they are no longer equal to zero for n even.
In problems with even and odd functions, we can exploit the inherent symmetry to simplify the integral. We will exploit other symmetries later. Consider the problem above. We can then use the fact that for an even function, e t ,. This will often be simpler to evaluate than the original integral because one of the limits of integration is zero. Consider, again, the pulse function.
We can also represent x T t by the Exponential Fourier Series. The average value i. Note: this is similar, but not identical, to the triangle wave seen earlier. Note: this is similar, but not identical, to the sawtooth wave seen earlier. So far, all of the functions considered have been either even or odd, but most functions are neither.
This presents no conceptual difficult, but may require more integrations. For example if the function x T t looks like the one below. Since this has no obvious symmetries, a simple Sine or Cosine Series does not suffice. For the Trigonometric Fourier Series, this requires three integrals. For this reason, among others, the Exponential Fourier Series is often easier to work with, though it lacks the straightforward visualization afforded by the Trigonometric Fourier Series.
From the relationship between the Trigonometric and Exponential Fourier Series. If the function x T t has certain symmetries, we can simplify the calculation of the coefficients. In other words, if you shift the function by half of a period, then the resulting function is the opposite the original function. The triangle wave has half-wave symmetry. See below for clarification. The first two symmetries are were discussed previously in the discussions of the pulse function x T t is even and the sawtooth wave x T t is odd.
The reason the coefficients of the even harmonics are zero can be understood in the context of the diagram below. The top graph shows a function, x T t with half-wave symmetry along with the first four harmonics of the Fourier Series only sines are needed because x T t is odd. The bottom graph shows the harmonics multiplied by x T t. The odd terms from the 1st red and 3rd magenta harmonics will have a positive result because they are above zero more than they are below zero. The even terms green and cyan will integrate to zero because they are equally above and below zero.
Though this is a simple example, the concept applies for more complicated functions, and for higher harmonics. The only funct ion discussed with half-wave symmetry was the triangle wave and indeed the coefficients with even indices are equal to zero as are all of the b n terms because of the even symmetry. In that case the a 0 term would be zero and we have already shown that all the terms with even indices are zero, as expected. Simplifications can also be made based on quarter-wave symmetry , but these are not discussed here.
A periodic function has quarter wave symmetry if it has half wave symmetry and it is either even or odd around its two half-cycles. Since the coefficients c n of the Exponential Fourier Series are related to the Trigonometric Series by.
However, in addition, the coefficients of c n contain some symmetries of their own. In particular,. Since the function is even, we expect the coefficients of the Exponential Fourier Series to be real and even from symmetry properties. Furthermore, we have already calculated the coefficients of the Trigonometric Series , and could easily calculate those of the Exponential Series.
However, let us do it from first principles. The Exponential Fourier Series coefficients are given by. The last step in the derivation is performed so we can use the sinc function pronounced like " sink ". This function comes up often in Fourier Analysis.
The graph on the left shows the time domain function. If you hit the middle button, you will see a square wave with a duty cycle of 0. The graph on the right shown the values of c n vs n as red circles vs n the lower of the two horizontal axes; ignore the top axis for now.
There are several important features to note as T p is varied. As before , note: As you add sine waves of increasingly higher frequency, the approximation improves.
The addition of higher frequencies better approximates the rapid changes, or details, i. Gibb's overshoot exists on either side of the discontinuity. Because of the symmetry of the waveform, only odd harmonics 1, 3, 5, The reasons for this are discussed below The rightmost button shows the sum of all harmonics up to the 21st harmonic, but not all of the individual sinusoids are explicitly shown on the plot. In particular harmonics between 7 and 21 are not shown.
Note: As you add sine waves of increasingly higher frequency, the approximation gets better and better, and these higher frequencies better approximate the details, i.
Even with only the 1st few harmonics we have a very good approximation to the original function. There is no discontinuity, so no Gibb's overshoot. As before, only odd harmonics 1, 3, 5, There is Gibb's overshoot caused by the discontinuities. A function can have half-wave symmetry without being either even or odd.
In mathematics , even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis , especially the theory of power series and Fourier series. Evenness and oddness are generally considered for real functions , that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups , all rings , all fields , and all vector spaces. Thus, for example, a real function could be odd or even, as could a complex -valued function of a vector variable, and so on. The given examples are real functions, to illustrate the symmetry of their graphs.
Notice that in the Fourier series of the square wave 4. This is a very general phenomenon for so-called even and odd functions. Now if we look at a Fourier series, the Fourier cosine series. There are three possible ways to define a Fourier series in this way, see Fig. Of course these all lead to different Fourier series, that represent the same function on [0,L].
The previous page showed that a time domain signal can be represented as a sum of sinusoidal signals i. This page will describe how to determine the frequency domain representation of the signal. For now we will consider only periodic signals, though the concept of the frequency domain can be extended to signals that are not periodic using what is called the Fourier Transform. The next page will give several examples. Consider a periodic signal x T t with period T we will write periodic signals with a subscript corresponding to the period.
This document derives the Fourier Series coefficients for several functions. The functions shown here are fairly simple, but the concepts extend to more complex functions. Consider the periodic pulse function shown below. It is an even function with period T.
Go back to Even and Odd Functions for more information. In some of the problems that we encounter, the Fourier coefficients a o , a n or b n become zero after integration. Finding zero coefficients in such problems is time consuming and can be avoided. With knowledge of even and odd functions , a zero coefficient may be predicted without performing the integration.
PPT V. Fourier transform PowerPoint Presentation, free download.
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A Fourier series contains a sum of terms while the integral formulae for the Fourier coefficients an and bn contain products of the type f(t) cosnt and f(t) sinnt. We.
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Fourier series take on simpler forms for Even and Odd functions. Even function. A function is Even if for all x. The graph of an even function is.Hildegarda M. 17.05.2021 at 14:21
When finding Fourier Series of even or odd functions, we don't need to find all the coefficients.Waldemar V. 18.05.2021 at 09:27
To browse Academia.Bosduverli 19.05.2021 at 00:36
Fourier series for even and odd functions Notice that in the Fourier series of the square wave () all coefficients an vanish, the series.Adelfo P. 20.05.2021 at 18:31
Go back to Even and Odd Functions for more information.