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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 8 9 Copyright © Orchard Publications Properties and Theorems of the Fourier Transform Accordingly, we can state that a necessary and sufficient condition for to be real , is that . Also, if it is known that is real, the Inverse Fourier transform of (8.3) can be simplified as fol- lows: From (8.13), Page 8 2, (8.36) and since the integrand is an even function with respect to , we rewrite (8.36) as (8.37) 8.3 Properties and Theorems of the Fourier Transform The most common properties and theorems of the Fourier transform are described in Subsections 8.3.1 through 8.3.14 below. 8.3.1 Linearity If is the Fourier transform of , is the transform of , and so on, the linear- ity property of the Fourier transform states that (8.38) where is some arbitrary real constant. Proof: The proof is easily obtained from (8.1), Page 8 1, that is, the definition of the Fourier transform. The procedure is the same as for the linearity property of the Laplace transform in Chapter 2. 8.3.2 Symmetry If is the Fourier transform of , the symmetry property of the Fourier transform states that (8.39) ft () F ω F ω = f Re t 1 2 π ------ F ω () ω t cos F Im ω ω t sin [] ω d = ω f t 2 1 2 π F ω t cos F ω ω t sin ω d 0 = 1 π -- A ω ω t ϕω + cos ω d 0 1 π --Re F ω e j ω t + ω d 0 = = F 1 ω f 1 t F 2 ω f 2 t a 1 f 1 t a 2 f 2 t () … a n f n t ++ + a 1 F 1 ω a 2 F 2 ω a n F n ω + a i F ω Ft 2 π f ω

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Chapter 8 The Fourier Transform 8 10 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications that is, if in , we replace with , we obtain the Fourier transform pair of (8.39). Proof: Since then, Interchanging and , we obtain and (8.39) follows. 8.3.3 Time Scaling If is a real constant, and is the Fourier transform of , then, (8.40) that is, the time scaling property of the Fourier transform states that if we replace the variable in the time domain by , we must replace the variable in the frequency domain by , and divide by the absolute value of . Proof: We must consider both cases and . For , (8.41) We let ; then, , and (8.41) becomes For , F ω () ω t ft 1 2 π ------ F ω e j ω t ω d = 2 π F ω e j ω t ω d = t ω 2 π f ω Ft e j ω t t d = aF ω fa t 1 a -----F ω a ---   t at ωω a F ω a a a0 > < > F t {} t e j ω t t d = τ = t τ a = F f τ f τ e j ω τ a - τ a -- d 1 a f τ e j ω a τ τ d 1 a --F ω a == = < F t t e j ω t t d =
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 8 11 Copyright © Orchard Publications

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