Fourier Series Tutorial

Transcription

1 Fourier Series Tutorial INTRODUCTION This document is designed to overview the theory behind the Fourier series and its alications. It introduces the Fourier series and then demonstrates its use with a detailed examle. The Fourier Series MATLAB GUI can be used to follow along with the examle shown here. BASIC CONCEPTS A Fourier series is a method of reresenting a comlex eriodic signal using simler signals. These simle signals are sinusoids which are summed to roduce an aroximation of the original signal. The aroximation becomes more accurate as more terms are used. The basic form of a Fourier series is x() t = a 0 + a1 cos( ω0t + θ1 ) + a 2 cos( 2ω0t + θ2 ) a N cos( Nω0t + θn ), (1) where x(t) = the aroximation of the original signal, a 0 = a constant, which roduces a DC offset, a 1, a 2, a N = constant terms which change the amlitude of the sinusoidal signals, ω 0 = the dominant frequency of the signal, and θ 1, θ 2, θ N = hase shifts. This form shown with only cosines, but sines or a combination of sines and cosines can be used. APPROXIMATING A SQUARE WAVE Fourier series will be demonstrated by showing how to aroximate a square wave, such as that shown in Fig. 1, using three sine waves. 1

2 Fig. 1. Square wave. Sine waves can be summed, and the resulting signal examined, using a simle SIMULINK model such as the one shown in Fig. 2. Fig. 2. SIMULINK model to add multile sine waves. This model is controlled by the Fourier Series MATLAB GUI. The oeration of the GUI is fairly straightforward, but for more details see the Fourier Series GUI documentation. This model can sum u to five sine waves, but for the sake of simlicity only three will be used for this demonstration. 2

3 If three sine waves with the aroriate amlitudes and frequencies are summed and lotted, we see that the resulting lot resembles the original square wave, as seen in Fig. 3. Fig. 3. Square wave aroximation. If the Fourier series were summed from 0 to infinity (an infinite number of terms), the result would be an exact square wave. Using the Fourier Series GUI, try adjusting the frequencies and magnitudes of three sine waves to aroximate the square wave (hint: do not adjust the hase). This is difficult to do without knowing the correct values to use. The following information about the Fourier Series will hel you choose the correct values for the sine waves. Determination of constants To aroximate a articular signal using a Fourier series, the correct constants must be determined. The Fourier series can be written as 1 () 2nπt 2nπt f t = a 0 + a n cos + bn sin, (2) 2 n= 1 where a 0, a n, and b n are constants, and n = 1, 2, 3, etc. Note that here the series has been written with both sine and cosine terms, rather than using only hase-shifted cosine terms as in (1). The constant terms a 0, a n, and b n are calculated using 2 / 2 a 0 = f ()dt t, (3) / 2 2 / 2 2nπt a n = f () t cos dt for n = 1,2,3,, and (4) / 2 3

4 2 / 2 2nπt bn = f () t sin dt for n = 1,2,3,, (5) / 2 where f(t) = the eriodic function to be aroximated, and = the eriod of the function. We will now calculate a 0, a n, and b n for the square wave shown in Fig. 1. The analytical descrition of the function is 1 if 0.05 < x < 0 f () t = (6) 1 if 0 < x < 0.05 and the eriod () is 0.1 seconds. Therefore 2 / a f () 0 = t dt 1dt 1dt 0 = / = 0.05, (7) 0 which makes sense because the original signal has no DC bias. Then, the coefficient of the cosine term is = 2 / 2 2nπt 2 0 2nπt π () = n t a n f t cos dt 1cos dt 1cos dt / sin( 20n t) sin( 20n t) 05 = = 0, n π + π (8) π for n = 1, 2, 3,, and the coefficient of the sine term is = 2 / 2 2nπt 2 0 2nπt π () = n t bn f t sin dt 1sin dt sin dt / = 2 2 [ ( )] [ ( ) ] 0 if n is even 1 cos nπ = 1 1 n = (9) nπ nπ 4 ( nπ) if n is odd For a square wave, as for all odd functions, the coefficient of the sine term (a n ) vanishes. For even functions the coefficient of the cosine term (b n ) vanishes. To review, an even function is one where f ( x) = f ( x), (10) so the function is symmetrical about the y-axis. For an odd function, f ( x) = f ( x). (11) In addition, the Fourier series for a square wave only has non-zero terms when n is an odd number. The equation which describes our square wave is then () 4 f t = sin( 2π f nt). (12) n= 1,3,5,... nπ Plugging in the frequency of the square wave, 10 Hz, the first three terms of the series are f () t = sin( 20πt) + sin( 60πt) + sin( 100πt) π 3 5. (13) For the sake of utting these values in the GUI, the amlitudes and frequencies of the first three sine waves are given in decimal form in Table 1. 4

5 Table 1: Amlitude and frequency of first three terms. Term Frequency Amlitude 1 10 Hz Hz Hz Go back to the Fourier Series GUI and insert these values; the result resembles a square wave. The general trend of a summation of sine waves can be redicted by looking at the slowest sine wave. The signal with the lowest frequency, determines the general shae of the sum. In the revious examle, we saw that the lot aroximated a 10 Hz sine wave. The lowest frequency was 10 Hz. Test this idea using the GUI, and also try to relicate the other samle signals in the GUI. 5