Lesson_31 - EEL 3135: Signals and Systems Dr. Fred J....

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Unformatted text preview: EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: Fourier Transform Lesson Number: 31 [Section 11-6 to 11-11] Background: In Chapter 11 the basic form of the Fourier transform are established. The Fourier transform pair is defined to be: ( 29 ( 29 ( 29 ( 29 d e j X t x dt e t x j X t j t j -- - = = 2 1 ; 1.a The Fourier transform was said to exist if the signal is absolutely integrable (although this is not a necessary condition). At times, the production of a Fourier transform or series can be a straightforward calculation (e.g., pulse), at other times it requires great finesse. Fortunately most of the important transforms and signal are summarized in Table 11.2 of the text along with properties given in Table 11.3. The Fourier transform of x(t) exists if x(t) is absolutely integrable, square integrable (finite energy), plus some exceptions. The so-called Dirichlet conditions state that: The inverse Fourier transform of X(j ) equals x(t) at points where x(t) is continuous. The inverse Fourier transform of X(j ) converges to the mid-point of x(t) at points where x(t) is discontinuous. Example: What is the inverse Fourier transform of the single sideband signal having a line spectrum ( - )? Figure 1: Single Sideband Signal. The inverse Fourier transform of the line spectrum ( - ) is given by: 1 EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor ( 29 ( 29 ( 29 t j t j t j e d e d e 2 1 2 1 2 1 1 = =- =- - -- It then follows that ( 29 ( 29 2 / 1 - t j e , or ( 29 ( 29 2 - t j e . End of example ------------------- Example Suppose x(t)=sgn(t)={1 if t>0, -1 if t<0, 0 if t=0} (see Figure 2), what is X (j )? Figure 2: sgn Function. Notice that x(t) is technically not absolutely or square integrable. Nevertheless, its Fourier transform exists and can be computed using the subterfuge shown below. Consider approximating the sgn function using the decaying exponentials shown in Figure 2. That is: ( 29 [ ] ) ( ) ( sgn --- = a at at t u e t u e t Then ( 29 ( 29 [ ] ( 29 j a j j a j a t u e t u e t a a a at at 2 2 1 1 ) ( ) ( sgn 2 2 = +- = -- + =-- = - which has a magnitude of 2/ , and peaks at =0. The phase shift equals -90 for 0 and 90 for <0. End of example ------------------------------------------- 2 e-at u(t)-e at u(-t) EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Convolution To this point, convolution has been studied in the time-domain (convolution sum and integrals), and transform domain (transforms). Continuous time convolution is given by: ( 29 ( 29 ( 29 ( 29 ( 29 d t h x t h t x t y- = = - 2....
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Lesson_31 - EEL 3135: Signals and Systems Dr. Fred J....

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