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Unformatted text preview: 1 LINEAR CIRCUITS AND SIGNALS Fourier Series Department of Electrical and Computer Engineering University of Miami I. PRELIMINARIES A. Periodic Functions A function f ( t ) that satisfies the condition f ( t ) = f ( t ˜ T ) , ∀ t , is said to be periodic. The smallest smallest nonzero value of ˜ T for which this property is true is said to be the period T ; ω o = 2 π/T rad/s or f o = 1 /T Hz is said to be the fundamental frequency. B. Dirichlet Conditions Given a periodic function f ( t ) , it in each period (a) f ( t ) has a finite number of discontinuities; (b) f ( t ) has a finite number of maxima and minima; and (c) f ( t ) is absolutely convergent, i.e., Z t + T t = t  f ( t )  dt < ∞ , then f ( t ) is said to satisfy the Dirichlet conditions. Example Let us consider some functions: • sin[ t ] is periodic with period 2 π and it satisfies the Dirichlet conditions. • e jnω o t is periodic with period 2 π/nω o and it also satisfies the Dirichlet conditions. • ∞ X n =∞ F n e jnω o t , where F n are constants, is also periodic and its period is 2 π/ω o . The Dirichlet conditions are satisfied here too. • Not all sums of sinusoids are periodic. For this sum to be periodic, there must exist a value for ω o , the fundamental frequency, such that the frequencies of all the components can be 2 expressed as an integer multiple of ω o . In other words, all possible ratios of the frequencies must be rational. – 2 cos[ πt ] + 4 cos[2 t ] is not periodic. – 1 + cos[8 πt ] + 7 . 6 sin[10 πt ] is periodic with ω o = 2 π rad/s . • g ( t ) =  tan[ t ]  is periodic with period π , but it does not satisfy the Dirichlet conditions. II. FOURIER SERIES (FS) Consider a function f ( t ) which is periodic with period T and satisfies the Dirichlet conditions. We will demonstrate an important feature of such functions. A. Exponential FS We claim that the function f ( t ) can be expressed as a weighted sum of e jnω o t terms: f ( t ) = ∞ X n =∞ F n e jnω o t , with ω o = 2 π T . (1) Indeed, let us derive the coefficients F n . Using R T to denote that the integral is carried out over one period, we have Z T f ( t ) e jnω o t dt = Z T ∞ X m =∞ F m e jmω o t e jnω o t dt = ∞ X m =∞ F m Z T e j ( m n ) ω o t dt, (2) where we assume that the infinite series is uniformly convergent so the integral and summation operations can be interchanged. Observe that Z T e j ( m n ) ω o t dt = T, for m = n ; , otherwise . (3) Hence 1 T Z T f ( t ) e jnω o t dt = F n . (4) This yields us the exponential FS: f ( t ) = ∞ X n =∞ F n e jnω o t , where F n = 1 T Z T f ( t ) e jnω o t dt. (5) Note the following: 3 • The exponential FS may be expressed as f ( t ) = ∞ X n =∞  F n  e j ( nω o t + ∠ F n ) . (6) The plots of  F n  versus n and ∠ F n versus n are referred to as the magnitude spectrum and phase spectrum of the exponential FS’s coefficients, respectively....
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This note was uploaded on 10/28/2010 for the course EEN 307 taught by Professor Kamalpremeratne during the Spring '10 term at University of Miami.
 Spring '10
 KAMALPREMERATNE

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