ecen314-lec10

ecen314-lec10 - Lecture 10: CT Fourier Series Tie Liu...

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Unformatted text preview: Lecture 10: CT Fourier Series Tie Liu October 4, 2011 1 Periodic CT signals A CT signal x ( t ) is said to be periodic if there exists T > 0 such that x ( t + T ) = x ( t ) , t R where T is called a period . The smallest such T , if it exists, is called the fundamental period and is usually denoted as T . The fundamental frequency is defined as := 2 T Apparently, if T is a fundamental period of x ( t ), then any positive integer multiple T = kT is a period of x ( t ). Examples: The CT sinusoidal signal x ( t ) = cos( t + ) is periodic with the fundamental frequency | | and the fundamental period T = 2 / | | . The CT complex sinusoidal signal x ( t ) = e j t is periodic with the fundamental fre- quency | | and the fundamental period T = 2 / | | . 2 CT Fourier series In the 17th century, Jean Baptiste Joseph Fourier discovered that a wide class of periodic signals x ( t ) can be written as linear combinations of the basic signals k ( t ) = e jk t where is the fundamental frequency of x ( t ), i.e., x ( t ) = summationdisplay k =- a k e jk t Terminologies: { a k } : Fourier series coefficients 1 a : DC a 1 : first harmonic a 2 : second harmonic Two questions: 1. (Existence of Fourier series representation) Which periodic signals can indeed be rep- resented as linear combinations of a complex exponential function and its harmonics? 2. If the Fourier series representation indeed exists, how can we determine the Fourier series coefficients?...
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ecen314-lec10 - Lecture 10: CT Fourier Series Tie Liu...

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