24 2 properties fourier trnsform

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Properties of the Fourier Transform 24.2 Introduction In this Section we shall learn about some useful properties of the Fourier transform which enable us to calculate easily further transforms of functions and also in applications such as electronic communication theory. Prerequisites Before starting this Section you should . . . • be aware of the basic definitions of the Fourier transform and inverse Fourier transform ' Learning Outcomes On completion you should be able to . . . & 14 $ • state and use the linearity property and the time and frequency shift properties of Fourier transforms • state various other properties of the Fourier transform HELM (2008): Workbook 24: Fourier Transforms % ® 1. Linearity properties of the Fourier transform (i) If f (t), g(t) are functions with transforms F (ω), G(ω) respectively, then • F{f (t) + g(t)} = F (ω) + G(ω) i.e. if we add 2 functions then the Fourier transform of the resulting function is simply the sum of the individual Fourier transforms. (ii) If k is any constant, • F{kf (t)} = kF (ω) i.e. if we multiply a function by any constant then we must multiply the Fourier transform by the same constant. These properties follow from the definition of the Fourier transform and from the properties of integrals. Examples 1. F{2e−t u(t) + 3e−2t u(t)} = F{2e−t u(t)} + F{3e−2t u(t)} = 2F{e−t u(t)} + 3F{e−2t u(t)} = 2 3 + 1 + iω 2 + iω 2. 4 −3 ≤ t ≤ 3 0 otherwise f ( ...
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project management

Please download the Word doc and use it to complete the Project.:Page 1. Cover page with your name, the class name, and Ly ...

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