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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 8 17 Copyright © Orchard Publications Properties and Theorems of the Fourier Transform Since and , Parseval’s theorem is proved. For convenience, the Fourier transform properties and theorems are summarized in Table 8.8. TABLE 8.8 Fourier Transform Properties and Theorems Property Linearity Symmetry Time Scaling Time Shifting Frequency Shifting Time Differentiation Frequency Differentiation Time Integration Conjugate Functions Time Convolution Frequency Convolution Area under Area under Parseval’s Theorem ft () f t [] t d 1 2 π ------ F ω F ω ω d 1 2 π F ω F ω ()ω d == f t 2 = F ω F ω F ω 2 = F ω a 1 f 1 t a 2 f 2 t () … ++ a 1 F 1 ω a 2 F 2 ω Ft 2 π f ω fa t 1 a -----F ω a ---   ft t 0 F ω e j ω t 0 e j ω 0 t F ωω 0 d n dt n --------ft j ω n F ω jt n d n d ω n ---------- F ω f τ ()τ d t F ω j ω ------------ π F0 δω + f t F ω f 1 t f 2 t F 1 ω F 2 ω f 1 t f 2 t 1 2 π ------F 1 ω F 2 ω t d = F ω f0 1 2 π F ω d = 2 t d 1 2 π F ω 2 ω d =

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Chapter 8 The Fourier Transform 8 18 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications 8.4 Fourier Transform Pairs of Common Functions The Fourier transform pair of the most common functions described in Subsections 8.4.1 through 8.4.9 below. 8.4.1 The Delta Function Pair (8.60) Proof: The sifting theorem of the delta function states that and if is defined at , then, By the definition of the Fourier transform and (8.60) follows. Likewise, the Fourier transform for the shifted delta function is (8.61) We will use the notation to show the time domain to frequency domain correspon- dence. Thus, (8.60) may also be denoted as in Figure 8.1. Figure 8.1. The Fourier transform of the delta function 8.4.2 The Constant Function Pair (8.62) δ t () 1 ft ()δ tt 0 t d 0 = t0 = t t d f0 = F ω δ t e j ω t t d e j ω t = 1 == = δ 0 δ 0 e j ω t 0 F ω 0 t 1 ω 0 δ t F ω A2 A πδ ω
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