HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 523

A plot of the values An vs. ωn is the typical representation of a discrete spectrum for a function. The discrete spectrum will show that the function has components at angular frequencies ωn which are integer multiples of the fundamental angular frequency ω0. Suppose that we are faced with the need to expand a non-periodic function into sine and cosine components. A non-periodic function can be thought of as having an infinitely large period. Thus, for a very large value of T, the fundamental angular frequency, ω0 = 2π/T, becomes a very small quantity, say ∆ω. Also, the angular frequencies corresponding to ωn = n⋅ω0 = n⋅∆ω, (n = 1, 2 now take values closer and closer to each other, suggesting the need for a continuous spectrum of values. The non-periodic function can be written, therefore, as ∫∞ f (x) = [C(ω) ⋅ cos(ω ⋅ x) + S(ω) ⋅ sin(ω ⋅ x)]dω, 0 where and ∫ C(ω) = 1 ⋅ ∞ f (x) ⋅ cos(ω ⋅ x) ⋅ dx, 2π − ∞ ∫ S(ω) = 1⋅ ∞ f (x) ⋅ sin(ω ⋅ x) ⋅ dx 2π −∞ The continuous spectrum is given by A(ω ) = [C(ω )]2 + [S(ω)]2 The functions C(ω), S(ω), and A(ω) are continuous functions of a variable ω, which becomes the transform variable for the Fourier transforms defined below. Example 1 - Determine the coefficients C(ω), S(ω), and the continuous spectrum A(ω), for the function f(x) = exp(-x), for x > 0, and f(x) = 0, x < 0. Page 16-45