Fourier Series 2.19 2000/07/21 2004/08/17 22:07:27.213 GMT-5 Don Johnson dhj@rice.edu Roy Ha rha@rice.edu Don Johnson dhj@rice.edu Benjamin Fite bfite@rice.edu Euler Fourier coefficients Fourier series frequency Gauss orthogonality sinusoid square wave Signals can be composed by a superposition of an infinite number of sine and cosine functions. The coefficients of the superposition depend on the signal being represented and are equivalent to knowing the function itself. In Signal Decomposition, we have shown that we could express the square wave as a superposition of pulses. This superposition does not generalize well to other periodic signals: How can a superposition of pulses equal a smooth signal like a sinusoid? Because of the importance of sinusoids to linear systems, you might wonder whether they could be added together to represent a large number of periodic signals. You would be right and in good company as well. Euler and Gauss in particular worried about this problem, and Fourier got the credit even though tough mathematical issues were not settled until later. They worked on what is now known as the Fourier series. Let s t have period T . We want to show that periodic signals, even those that have constant-valued segments like a square wave, can be expressed as sum of harmonically related sine waves. s t a 0 k 1 a k 2 k t T k 1 b k 2 k t T The family of functions called basis functions 2 k t T 2 k t T form the foundation of the Fourier series. No matter what the periodic signal might be, these functions are always present and form the representation's building blocks. They do depend on the signal's period T , and are indexed by k . The frequency of each term is k T . For k 0 , the frequency is zero and the corresponding term a 0 is a constant. The basic frequency 1 T is called the fundamental frequency because all other terms have frequencies that are integer multiples of it. These higher frequency terms are called harmonics: The term at frequency 1 T is the fundamental and the first harmonic, 2 T the second harmonic, etc. Thus, larger values of the series index correspond to higher harmonic-frequency sinusoids. The Fourier coefficients, a k and b k , depend on the signal's waveform. Because frequency is linked to index, the coefficients implicitly depend on frequency. Assuming we know the period, knowing the Fourier coefficients is equivalent to knowing the signal. Assume for the moment that the Fourier series works. To find the Fourier coefficients, we note the following orthogonality properties of sinusoids. k l k l t 0 T 2 k t T 2 l t T 0 t 0 T 2 k t T 2 l t T T 2 k l k 0 l 0 0 k l k 0 l t 0 T 2 k t T 2 l t T T 2 k l k 0 l 0 T k 0 l 0 k l These orthogonality relations follow from the following important trigonometric identities. α β 1 2 α β α β α β 1 2 α β α β α β 1 2 α β α β These identities allow you to substitute a sum of sines and/or cosines for a product of them. Each term in the sum can be integrating by noticing one of two important properties of sinusoids. The integral of a sinusoid over an integer number of periods equals zero. The integral of the square of a unit-amplitude sinusoid over a period T equals T 2 . To use these, let's multiply the Fourier series for a signal ( s t a 0 k 1 a k 2 k t T k 1 b k 2 k t T ) by 2 l t T and integrate. The idea is that, because integration is linear, the integration will sift out all but the term involving a l . t 0 T s t 2 l t T t 0 T a 0 2 l t T k 1 a k t 0 T 2 k t T 2 l t T k 1 b k t 0 T 2 k t T 2 l t T The first and third terms are zero; in the second, the only non-zero term in the sum results when the indices k and l are equal (but not zero), in which case we obtain a l T 2 . If k 0 l , we obtain a 0 T . Consequently, l l 0 a l 2 T t 0 T s t 2 l t T All of the Fourier coefficients can be found similarly. a 0 1 T t 0 T s t k k 0 a k 2 T t 0 T s t 2 k t T b k 2 T t 0 T s t 2 k t T The expression for a 0 is referred to as the average value of s t . Why? The average of a set of numbers is the sum divided by the number of terms. Viewing signal integration as the limit of a Riemann sum, the integral corresponds to the average. Finding the Fourier series coefficients for the square wave is very simple. sqT t . Mathematically, this signal can be expressed as sqT t 1 0 t T2 1 T2 t T The expressions for the Fourier coefficients have the common form a k b k 2 T t 0 T 2 2 k t T 2 k t T 2 T t T 2 T 2 k t T 2 k t T The cosine coefficients a k are all zero, and the sine coefficients are b k 4 k k is odd 0 k is even Thus, the Fourier series for the square wave is sq t k k 1 3 … 4 k 2 k t T Consequently, the square wave equals a sum of sinusoids, but only those having frequencies equal to odd multiples of the fundamental frequency 1 T . The coefficients decay slowly as the frequency index k increases. This index corresponds to the k-th harmonic of the signal's period. Let's find the Fourier series representation for the half-wave rectified sinusoid. s t 2 t T 0 t T 2 0 T 2 t T Begin with the sine terms in the series; to find b k we must calculate the integral b k 2 T t 0 T 2 2 t T 2 k t T Using our trigonometric identities turns our integral of a product of sinusoids into a sum of integrals of individual sinusoids, which are much easier to evaluate. t 0 T 2 2 t T 2 k t T 1 2 t 0 T 2 2 k 1 t T 2 k 1 t T 1 2 k 1 0 Thus, b 1 1 2 b 2 b 3 … 0 On to the cosine terms. The average value, which corresponds to a 0 , equals 1 . The remainder of the cosine coefficients are easy to find, but yield the complicated result a k 2 1 k 2 1 k 2 4 … 0 k is odd Thus, the Fourier series for the half-wave rectified sinusoid has non-zero terms for the average, the fundamental, and the even harmonics.