LCS_5_notes - 1 LINEAR CIRCUITS AND SIGNALS Fourier Series...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 15

LCS_5_notes - 1 LINEAR CIRCUITS AND SIGNALS Fourier Series...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online