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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 ω := 2 π 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( ω t + θ ) is periodic with the fundamental frequency | ω | and the fundamental period T = 2 π/ | ω | . • The CT complex sinusoidal signal x ( t ) = e jω t is periodic with the fundamental fre- quency | ω | and the fundamental period T = 2 π/ | ω | . 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 φ k ( t ) = e jkω t where ω is the fundamental frequency of x ( t ), i.e., x ( t ) = ∞ summationdisplay k =-∞ a k e jkω 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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