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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 41 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(jΩ) = x(t)e−jΩtdt (1) 1−∞� ∞ x(t) = X(jΩ)ejΩtdΩ (2)2π −∞ Equation (??) is known as the forward Fourier transform, and is analogous to the analysis equation of the Fourier series representation. It expresses the time-domain 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(jΩ) has units of amplitude density, that is the total “amplitude” contained within a small increment of frequency is X(jΩ)δΩ/2π. Equation (??) defines the inverse Fourier transform. It allows the computation of the time-domain function from the frequency domain representation X(jΩ), and is therefore analogous to the Fourier series synthesis equation. Each of the two functions x(t) or X(jΩ) is a complete description of the function and the pair allows the transformation between the domains. We adopt the convention of using lower-case letters to designate time-domain functions, and the same upper-case letter to designate the frequency-domain function. We also adopt the nomenclature x(t) F X(jΩ)⇐⇒ as denoting the bidirectional Fourier transform relationship between the time and frequency-domain representations, and we also frequently write X(jΩ) = F{x(t)} x(t) = F−1 {X(jΩ)} as denoting the operation of taking the forward F{} , and inverse F−1{} Fourier transforms respectively. 1copyright cD.Rowell 2008 4–11.1 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 1/2π that appears in the inverse transform. Some authors, usually in physics texts, define the so-called unitary Fourier transform pair as 1 � ∞X(jΩ) = x(t)e−jΩtdt√2π −∞1 � ∞ x(t) = X(jΩ)ejΩtdΩ√2π −∞ so as to distribute the constant symmetrically over the forward and inverse transforms. Many engineering texts address the issue of the asymmetry by defining the transform with respect to frequency F = 2πΩ in Hz, instead of angular frequency Ω in radians/s. The effect, through the change in the variable in the inverse transform, is to redefine the transform pair as � ∞ x(t)e−j2πF tdtX(F ) = −∞� ∞ j2πF tdFx(t) = X(F )e−∞ Some authors also adopt the notation of dropping the j from the frequency domain repre-sentation and write X(Ω) or X(F ) as above. Even more confusing is the fact that some authors (particularly in physics) adopt a definition that reverses the sign convention on the exponential terms in the Fourier integral, that is they define � ∞X(jΩ) = x(t)ejΩtdt 1−∞� ∞ x(t) = X(jΩ)e−jΩtdΩ 2π −∞ These various definitions of the transform pair mean that care must be taken to understand the particular definition adopted by texts and software packages. Throughout this course we will retain the definitions in Eqs. (??) and (??). 1.2 Fourier Transform Examples Example 1 Find the Fourier transform of the pulse function x(t) = � a |t| < T/2 0 otherwise. 4–2�t0T / 2- T / 2ax ( t )- 3 003 0X ( jW)Wa TTT T- 2 0- 1 0T1 0T2 0TF o u r i e r T r a n s f o r mSolution: From the definition of the forward Fourier transform � ∞X(jΩ) = x(t)e−jΩtdt (i) −∞� T/2 = a e−jΩtdt (ii) −T/2 � j ����T/2 = a e−jΩt(iii)Ω −T/2 = ja �e−jΩT /2 − ejΩT /2� (iv)Ω sin(ΩT/2) = aT . (v)ΩT/2 The Fourier transform of the rectangular pulse is a real function, of the form (sin x)/x centered around the jΩ = 0 axis. Because the function is real, it is sufficient to plot a single graph showing only X(jΩ) . Notice that while| |X(jΩ) is a generally decreasing function of Ω it never becomes identically zero, indicating that the rectangular pulse function contains frequency components at all frequencies. The function (sin x)/x = 0 when the argument x = nπ for any integer n (n = 0). The main peak or “lobe” of the spectrum X(jΩ) is therefore contained within the frequency band defined by the first two zero-crossings ΩT/2 < π or Ω <| | | |2π/T . Thus as the pulse duration T is decreased, the sp ectral bandwidth of the pulse increases, indicating that short duration pulses have a relatively larger high frequency content. 4–3x ( t )t0at0aTT / 4a Ta T / 4WX ( jW)F o u r i e r T r a n s f o r mF o u r i e r T r a n s f o r mT / 2- T / 2T / 8- T / 8x ( t )X ( jW)WExample 2 Find the Fourier transform of the Dirac delta function δ(t). Solution: When substituted into the forward Fourier transform � ∞Δ(jΩ) = δ( t)e−jΩtdt −∞ = 1 (i) by the sifting property. The spectrum of the delta function is therefore constant over all frequencies. It is this property that makes the impulse a very useful test input for linear systems. Example 3 Find the Fourier transform of the causal real exponential function x(t) = us(t)e−at ( for a > 0). 4–4- 2 0 a- 1 0 a01 0 a 2 0 a- 1 . 5- 1 . 0- 0 . 51 . 5- 2 0 a- 1 0 a01 0 a 2 0 aW| X( jW)|0 1 2 3 4x ( t )a t1 . 00 . 5F o u r i e r T r a n s f o r m0 . 51 . 0e- a t1 / aW| X( jW)|Solution: From the definition of the forward Fourier transform X(jΩ) = � ∞ e−at e−jΩtdt 0� −1 e−(a+jΩ)t��∞ = a + jΩ ��0 1 = a + jΩ which is complex, and in terms of a magnitude and phase function is 1 |X(jΩ)| = √a2 + Ω2 (i) � X(jΩ) = tan−1 �−a Ω� (ii) Other examples are given in the


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