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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 8 33 Copyright © Orchard Publications Using MATLAB for Finding the Fourier Transform of Time Functions The Fourier transform of the waveform of Figure 8.15 is shown in Figure 8.16. Figure 8.16. The Fourier transform of a train of equally spaced delta functions Figure 8.16 shows that the Fourier transform of a periodic train of equidistant delta functions in the time domain, is a periodic train of equally spaced delta functions in the frequency domain. This result is the basis for the proof of the sampling theorem which states that a time function can be uniquely determined from its values at a sequence of equidistant points in time. 8.7 Using MATLAB for Finding the Fourier Transform of Time Functions MATLAB has the built in fourier and ifourier functions to compute the Fourier transform and its inverse. Their descriptions and examples, can be displayed with the help fourier and help ifourier commands. In examples 8.4 through 8.7 we present some Fourier transform pairs, and how they are verified with MATLAB. Example 8.4 (8.97) This time function, like the time function of Subsection 8.6.6, is its own Fourier transform multi- plied by the constant . syms t v w x; ft=exp( t^2/2); Fw=fourier(ft) Fw = 2^(1/2)*pi^(1/2)*exp(-1/2*w^2) pretty(Fw) 1/2 1/2 2 2 pi exp(- 1/2 w ) % Check answer by computing the Inverse using "ifourier" ft=ifourier(Fw) ft = exp(-1/2*x^2) ω Α . . . . . . . . . . F ω () 4 ω 0 3 ω 0 2 ω 0 ω 0 0 ω 0 2 ω 0 3 ω 0 4 ω 0 ft e 1 2 --t 2 2 π e 1 2 -- ω 2 2 π

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Chapter 8 The Fourier Transform 8 34 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Example 8.5 (8.98) syms t v w x; ft=t*exp( t^2); Fw=fourier (ft) Fw = -1/2*i*pi^(1/2)*w*exp(-1/4*w^2) pretty(Fw) 1/2 2 - 1/2 i pi w exp(- 1/4 w ) Example 8.6 (8.99) syms t v w x; fourier(sym(' exp( t)*Heaviside(t)+3*Dirac(t)')) ans = -1/(1+i*w)+3 Example 8.7 (8.100) syms t v w x; u0=sym('Heaviside(t)'); Fw=fourier(u0) Fw = pi*Dirac(w)-i/w We have summarized the most common Fourier transform pairs in Table 8.9.
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