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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing Continuous and Discrete Fall Term 2008 Lecture 4 1 Reading: • 1 Review of Development of Fourier Transform: We saw in Lecture 3 that the Fourier transform representation of aperiodic waveforms can be expressed as the limiting behavior of the Fourier series as the period of a periodic extension is allowed to become very large, giving the Fourier transform pair ∞ X ( j Ω) = x ( t ) e − j Ω t dt (1) 1 −∞ ∞ x ( t ) = X ( j Ω) e j Ω t d Ω (2) 2 π −∞ Equation ( ?? ) is known as the forward Fourier transform, and is analogous to the analysis equation of the Fourier series representation. It expresses the timedomain function x ( t ) as a function of frequency, but unlike the Fourier series representation it is a continuous function of frequency. Whereas the Fourier series coeﬃcients have units of amplitude, for example volts or Newtons, the function X ( j Ω) has units of amplitude density, that is the total “amplitude” contained within a small increment of frequency is X ( j Ω) δ Ω / 2 π . Equation ( ?? ) defines the inverse Fourier transform. It allows the computation of the timedomain function from the frequency domain representation X ( j Ω), and is therefore analogous to the Fourier series synthesis equation. Each of the two functions x ( t ) or X ( j Ω) is a complete description of the function and the pair allows the transformation between the domains. We adopt the convention of using lowercase letters to designate timedomain functions, and the same uppercase letter to designate the frequencydomain function. We also adopt the nomenclature x ( t ) F X ( j Ω) ⇐⇒ as denoting the bidirectional Fourier transform relationship between the time and frequency domain representations, and we also frequently write X ( j Ω) = F{ x ( t ) } x ( t ) = F − 1 { X ( j Ω) } as denoting the operation of taking the forward F{} , and inverse F − 1 {} Fourier transforms respectively. 1 copyright c D.Rowell 2008 4–1 1.1 Alternate Definitions Although the definitions of Eqs. ( ?? ) and ( ?? ) ﬂow directly from the Fourier series, definitions for the Fourier transform vary from text to text and in different disciplines. The main objection to the convention adopted here is the asymmetry introduced by the factor 1 / 2 π that appears in the inverse transform. Some authors, usually in physics texts, define the socalled unitary Fourier transform pair as 1 ∞ X ( j Ω) = x ( t ) e − j Ω t dt √ 2 π −∞ 1 ∞ x ( t ) = X ( j Ω) e j Ω t d Ω √ 2 π −∞ so as to distribute the constant symmetrically over the forward and inverse transforms....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.161 taught by Professor Derekrowell during the Fall '08 term at MIT.
 Fall '08
 DerekRowell

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