So, let’s begin with what the Fourier series is. In this article, we will discuss the Fourier series, formulas, and uses and applications of the Fourier series. Examples of the Fourier series are trigonometric functions like sin x and cos x with period 2 and tan x with period. The period of any trigonometric term in the infinite series is an integral multiple, or harmonic, of the period T of the periodic function. It can be done by using a process called Fourier analysis. What was discovered was that a periodic function can be represented by an infinite sum of sine or cosine functions that are related harmonically. This representation of a periodic function is the starting point for finding the steady-state response to periodic excitations of electric circuits. To do this, a square wave whose frequency is the same as the center frequency of a bandpass filter is chosen.Īn analysis of heat flow in a metal rod led the French mathematician Jean Baptiste Joseph Fourier to the trigonometric series representation of a periodic function. Using a periodic signal like a square wave to test the quality factor of a bandpass or band reject filter. This article will detail a brief overview of a Fourier series, calculating the trigonometric form of the Fourier coefficients for a given waveform and simplification of the waveform when provided with more than one type of symmetry.Īny periodic signal can be represented as a sum of sinusoids where the sinusoids' frequencies are composed of the frequency of the periodic signal and integer multiples of that frequency. Problems involving fluid flow, mechanical vibration, and heat flow all use different periodic functions. Moreover, non-sinusoidal periodic functions are important in analyzing non-electrical systems. ) are needed to approximate the function this is because of the symmetry of the function.As you might be aware, electronic oscillators are extremely useful in laboratory testing equipment and are specifically designed to create non-sinusoidal periodic waveforms. As before, only odd harmonics (1, 3, 5.Looking at this sketch: The net area of the square wave from L to L is zero. It is basically an average of f(x) in that range. a 0 is the net area between L and L, then divided by 2L. There is no discontinuity, so no Gibb's overshoot. Example: This Square Wave: L (the Period is 2 ) The square wave is from h to +h Now our job is to calculate a 0, a n and b n.Even with only the 1st few harmonics we have a very good approximation to the original function. Conceptually, this occurs because the triangle wave looks much more like the 1st harmonic, so the contributions of the higher harmonics are less. Example 8: Compute the exponential series of the following signal. The amplitudes of the harmonics for this example drop off much more rapidly (in this case they go as 1/n 2 (which is faster than the 1/n decay seen in the pulse function Fourier Series (above)).As you add sine waves of increasingly higher frequency, the approximation gets better and better, and these higher frequencies better approximate the details, (i.e., the change in slope) in the original function. The exponential Fourier series spectra of a periodic signal ( ) are the plots of the magnitude and angle of the complex Fourier series coefficients.Note: this is similar, but not identical, to the triangle wave seen earlier. If x T(t) is a triangle wave with A=1, the values for a n are given in the table below (note: this example was used on the previous page). Concept: The complex Exponential Fourier Series representation of a periodic signal x(t) with fundamental period To is given by, (xleft( t right). During one period (centered around the origin) The periodic pulse function can be represented in functional form as Π T(t/T p).
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