In this section we present the basic properties of the Fourier transform

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In this section we present the basic properties of the Fourier transform. Proofs of these assertions are left as exercises. Exercise 10.1 Let F(ω) be the FT of the function f(x). Use the definitions of the FT and IFT given in Equations (10.1) and (10.2) to establish the following basic properties of the Fourier transform operation: • Symmetry: The FT of the function F(x) is 2πf(−ω). For example, the FT of the function f(x) = sin(?x) πx is χ?(ω), so the FT of g(x) = χ?(x) is G(ω) = 2π sin(?ω) πω . • Conjugation: The FT of f(x) is F(−ω). • Scaling: The FT of f(ax) is 1 |a| F( ω a ) for any nonzero constant a. • Shifting: The FT of f(x − a) is e iaωF(ω). • Modulation: The FT of f(x) cos(ω0x) is 1 2 [F(ω + ω0) + F(ω − ω0)]. • Differentiation: The FT of the nth derivative, f (n) (x) is (−iω) nF(ω). The IFT of F (n) (ω) is (ix) nf(x). • Convolution in x: Let f, F, g, G and h, H be FT pairs, with h(x) = Z f(y)g(x − y)dy, 10.3. SOME FOURIER-TRANSFORM PAIRS 77 so that h(x) = (f ∗ g)(x) is the convolution of f(x) and g(x). Then H(ω) = F(ω)G(ω). For example, if we take g(x) = f(−x), then h(x) = Z f(x + y)f(y)dy = Z f(y)f(y − x)dy = rf (x) is the autocorrelation function associated with f(x) and H(ω) = |F(ω)| 2 = Rf (ω) ≥ 0 is the power spectrum of f(x). • Convolution in ω: Let f, F, g, G and h, H be FT pairs, with h(x) = f(x)g(x). Then H(ω) = 1 2π (F ∗ G)(ω). Definition 10.1 A function f : R → C is said to be even if f(−x) = f(x) for all x, and odd if f(−x) = −f(x), for all x. Note that a typical function is neither even nor odd. Exercise 10.2 Show that f is an even function if and only if its Fourier transform, F, is an even function. Exercise 10.3 Show that f is real-valued if and only if its Fourier transform F is conjugate-symmetric, that is, F(−ω) = F(ω). Therefore, f is real-valued and even if and only if its Fourier transform F is real-valued