Let f(x) be defined for the real variable x in (−∞, ∞)

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Let f(x) be defined for the real variable x in (−∞, ∞). The Fourier transform (FT) of f(x) is the function of the real variable ω given by F(ω) = Z ∞ −∞ f(x)e iωxdx. (10.1) Having obtained F(ω) we can recapture the original f(x) from the FourierTransform Inversion Formula: f(x) = 1 2π Z ∞ −∞ F(ω)e −iωxdω. (10.2) Precisely how we interpret the infinite integrals that arise in the discussion of the Fourier transform will depend on the properties of the function f(x). 10.1.1 Decomposing f(x) One way to view Equation (10.2) is that it shows us the function f(x) as a superposition of complex exponential functions e −iωx, where ω runs over the entire real line. The use of the minus sign here is simply for notational convenience later. For each fixed value of ω, the complex number F(ω) = |F(ω)|e iθ(ω) tells us that the amount of e iωx in f(x) is |F(ω)|, and that e iωx involves a phase shift by θ(ω). 75 76CHAPTER 10. PROPERTIES OF THE FOURIER TRANSFORM (CHAPTER 8) 10.1.2 The Issue of Units When we write cos π = −1, it is with the understanding that π is a measure of angle, in radians; the function cos will always have an independent variable in units of radians. By extension, the same is true of the complex exponential functions. Therefore, when we write e ixω, we understand the product xω to be in units of radians. If x is measured in seconds, then ω is in units of radians per second; if x is in meters, then ω is in units of radians per meter. When x is in seconds, we sometimes use the variable ω 2π ; since 2π is then in units of radians per cycle, the variable ω 2π is in units of cycles per second, or Hertz. When we sample f(x) at values of x spaced ? apart, the ? is in units of x-units per sample, and the reciprocal, 1 ? , which is called the sampling frequency, is in units of samples per x-units.