By Avella P., Boccia M., Sforza A.

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N − 1. 6), namely, N −1 r Xr ω N , where ωN = ej2π/N , x = = 0, 1, . . , N − 1. r=0 In this setting, the N samples {x0 , x1 , . . , xN −1 } cover the period [0, T ], and the actual sampling time begins at t = 0 and ends at t = T − t = T − T/N , because the rst sample of the next period [T, 2T ] will have to be taken at t = T . Corresponding to {x0 , x1 , . . , xN −1 }, the DFT and IDFT have (discrete) period of N (samples), which is re ected by x +N = x and Xr+N = Xr for 0 ≤ r, ≤ N − 1. Now, since a function x(t) with period T is de n ed over (−∞, ∞) by periodic extension, the sampling period may begin and end anywhere.

1, we recall that Fα = fα t; we thus have the equality Fα = fα when t = 1, which gives both the same numerical values, but their de ni ng relationship dictates that Fα is measured by c ycles per sample, and fα is measured by c ycles per unit time. 3. Although the results derived using analog frequency with unit time spacing ( t = 1) will not be different from those derived using the digital frequency, the explicit incorporation of t in the latter s de n ition provides direct means to interpret and apply the results for values of t other than unity.

That is, T is the common period of f (t) and f1 = 1/T is the fundamental frequency of f (t). Note that f (t) completes one cycle over any interval of length T , including the commonly used [−T/2, T/2]. Depending on the application context, the Fourier series of function f (t) may appear in variants of the following forms: 1. 44) f (t) = A0 + 2 ∞ Ak cos k=1 2πkt 2πkt + Bk sin . T T Note that f (t) has a nonzero DC term, namely, A0 /2, for which we have the following remarks: CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS 24 Remark 1.

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A Branch-and-Cut Algorithm for the Median-Path Problem by Avella P., Boccia M., Sforza A.

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