File Name: fourier series and fourier transform formula .zip
We begin with a brief review of Fourier series.
- 1.5: The Power of the Fourier Transform for Spectroscopists
- Basic Fourier Series Formula List in PDF
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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.
1.5: The Power of the Fourier Transform for Spectroscopists
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.
We can represent any such function with some very minor restrictions using Fourier Series. In the early 's Joseph Fourier determined that such a function can be represented as a series of sines and cosines. In other words he showed that a function such as the one above can be represented as a sum of sines and cosines of different frequencies, called a Fourier Series.
There are two common forms of the Fourier Series, " Trigonometric " and " Exponential. For easy reference the two forms are stated here, there derivation follows. The Fourier Series is more easily understood if we first restrict ourselves to functions that are either even or odd. We will then generalize to any function. The following derivations require some knowledge of even and odd functions, so a brief review is presented. Examples are shown below. An even function, x e t , can be represented as a sum of cosines of various frequencies via the equation:.
This is called the "synthesis" equation because it shows how we create, or synthesize, the function x e t by adding up cosines. An example will demonstrate exactly how the summation describing the synthesis process works.
Consider the following function, x T and its corresponding values for a n. Note: we have not determined how the a n are calculated; that derivation follows, that calculation comes later. The example above shows how the harmonics add to approximate the original question, but begs the question of how to find the magnitudes of the a n. Start with the synthesis equation of the Fourier Series. When we integrate this function, the result is zero because we are integrating over an an integer greater than or equal to one number of oscillations.
This simplifies our result to. So the entire summation reduces to. We switched m to n in the last line since m is just a dummy variable. We now have an expression for a n , which was our goal. All of the integrals but the third one will go to zero because the integration is over an integer number of oscillations as will all of the omitted terms.
The third integral becomes a 2 T , as was expected. This is called the orthogonality function of the cosine. It is similar to orthogonality of vectors. Consider two vectors and their dot product. We say the vectors x and y are orthogonal if their dot product the sum the elementwise products of the vectors' elements is zero.
If we switch integral for sum since the function is a continuous function of time we say functions are orthogonal if the integral of the product of the two functions is zero. This leads to the result stated above, that a 0 is the average value of the function. An odd function can be represented by a Fourier Sine series to represent even functions we used cosines an even function , so it is not surprising that we use sinusoids. Note that there is no b 0 term since the average value of an odd function over one period is always zero.
The derivation closely follows that for the a n coefficients. Any function can be composed of an even and an odd part. Given a function x t , we can create even and odd functions. We can use a Fourier cosine series to find the a n associated with x e t and a Fourier sine series to find the b n associated with x o t. Given a periodic function x T , we can represent it by the Fourier series synthesis equations. We determine the coefficients a n and b n are determined by the Fourier series analysis equations.
A more compact representation of the Fourier Series uses complex exponentials. In this case we end up with the following synthesis and analysis equations:. The derivation is similar to that for the Fourier cosine series given above. Note that this form is quite a bit more compact than that of the trigonometric series; that is one of its primary appeals. Other advantages include: a single analysis equation versus three equations for the trigonometric form , notation is similar to that of the Fourier Transform to be discussed later , it is often easier to mathematically manipulate exponentials rather sines and cosines.
A principle advantage of the trigonometric form is that it is easier to visualize sines and cosines in part because the c n are complex number,, and the series can be easily used if the original x T is either purely even or odd. This begs the question of how the c n terms are related to the a n and b n terms. In the following discussion it is assumed that x T is real so a n and b n are real. To start consider only the constant terms.
So c 0 is also the average of the function x T. Likewise if we consider only those parts of the signal that oscillate once in a period of T seconds we get. Obviously the left side of this equation is real, so the right side must also be real. So using Euler's identities ,. As stated earlier, there are certain limitations inherent in the use of the Fourier Series. These are almost never of interest in engineering applications. In particular, the Fourier series converges. However this discontinuity becomes vanishingly narrow and it's area, and energy, are zero , and therefore irrelevant as we sum up more terms of the series.
After you have studied Fourier Transforms , you will learn that there is an easier way to find Fourier Series coefficients for a wide variety of functions that does not require any integration. Aside: Even and Odd functions The following derivations require some knowledge of even and odd functions, so a brief review is presented.
Other important facts about even and odd functions: the product of two even functions is an even function. Aside: Orthogonality of functions you may skip this if you would like to - it is not necessary to proceed. Examine the figure to the right.
Important things to note include:.
Basic Fourier Series Formula List in PDF
With appropriate weights, one cycle or period of the summation can be made to approximate an arbitrary function in that interval or the entire function if it too is periodic. As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.
getting from Fourier series to the Fourier transform is to consider nonperiodic phenomena (and thus just about any general function) as a limiting case of periodic.
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Jean Baptiste Joseph Fourier was a French mathematician, physicist and engineer, and the founder of Fourier analysis. Fourier series are used in the analysis of periodic functions. The Fourier transform and Fourier's law are also named in his honour. Graphically, even functions have symmetry about the y-axis,whereas odd functions have symmetry around the origin. Intuition: The area beneath the curve on [-p, 0] is the same as the area under the curve on [0, p], but opposite in sign.
Fourier transform is a mathematical technique that can be used to transform a function from one real variable to another.
Statement of the Problem
They are defined by the formulas. The complex form of Fourier series is algebraically simpler and more symmetric. Therefore, it is often used in physics and other sciences. We can transform the series and write it in the real form. Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website.
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