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Unformatted text preview: UCLA Fall 2011 Systems and Signals Lecture 14: Review II November 14, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1 Continuous Time Fourier Series Consider a complex exponential with s = jω , φ 1 ( t ) = e jω t . It’s periodic with period T = 2 π ω and has frequency f = ω 2 π (angular frequency ω ). Associated harmonically related exponentials: φ k ( t ) = e jkw t , k = 0 , ± 1 , ± 2 ,... . All periodic in T! For example, the fundamental period of φ 2 ( t ) is T 2 , so φ 2 ( t ) = φ 2 ( t + T ) = φ 2 ( t + T 2 ) . ω is often called fundamental frequency or first harmonic . A linear combination of harmonically related complex exponentials is of the form: x ( t ) = ∞ X k =∞ a k e jkω t EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 2 Linear combination of harmonically related complex exponentials scaled by { a k } is the Fourier series representation of x ( t ) . x ( t ) = ∞ X k =∞ a k e jkω t We represent the signal in terms of spectral coefficients or frequency components a k ’s. In general { a k } are complexvalued. This is also called a synthesis equation. When the frequency representation given by a k is available, we can compute (synthesize) x ( t ) . EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3 Continuous Time Fourier Series We now have a pair of equations x ( t ) = ∞ X k =∞ a k e jkω t (Synthesis) a k = 1 T Z T x ( t ) e jkω t dt (Analysis) Memorize them. Analysis equation allows us to decompose a continuous time periodic signal into complex exponentials and to compute their coefficients  set { a k } . Synthesis equation represents a periodic signal as a superposition of scaled complex exponentials. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4 Convergence of the Fourier Series Given a signal x ( t ) and the coefficients a k defined by a k = Z T x ( t ) e jkω t dt does it follow that x ( t ) = ∞ X k =∞ a k e jkω t ?? To answer this consider the truncated Fourier series: ˆ x N ( t )ˆ= N X k = N a k e jkω t EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5 and the resulting instantaneous approximation error N ( t ) = ˆ x N ( t ) x ( t ) and the integral square error or mean squared error ε N ˆ= Z T  N ( t )  2 dt = Z T  ˆ x N ( t ) x ( t )  2 dt We say that the truncated Fourier series converges to the signal if the integral square error E N → as N → ∞ . Often abbreviate this as x ( t ) = ∞ X k =∞ a k e jkω t , but this does not mean that this relation necessarily holds for all t , only in the integral average sense is it true. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6 Properties of Fourier Series Fourier series have many important properties that simplify evaluation of the transform for many signals....
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 Fourier Series, Jin Hyung Lee

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