Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL TRANSFORMS◮Prove the Wiener–Kinchin theorem,˜C(k) = √ 2π [˜f(k)] ∗˜g(k). (13.42)Following a method similar to that for the convolution of f and g, let us consider theFourier transform of (13.40):˜C(k) = √ 1 ∫ ∞{∫ ∞}dz e −ikz f ∗ (x)g(x + z) dx2π −∞−∞= √ 1 ∫ ∞{∫ ∞}dx f ∗ (x) g(x + z) e −ikz dz .2π −∞−∞Making the substitution u = x + z inthesecondintegralweobtain˜C(k) = √ 1 ∫ ∞{∫ ∞}dx f ∗ (x) g(u) e −ik(u−x) du2π −∞−∞= √ 1 ∫ ∞∫ ∞f ∗ (x) e ikx dx g(u) e −iku du2π −∞−∞= √ 1 × √ 2π [˜f(k)] ∗ × √ 2π ˜g(k) = √ 2π [˜f(k)] ∗˜g(k). ◭2πThus the Fourier transform of the cross-correlation of f and g is equal tothe product of [˜f(k)] ∗ and ˜g(k) multiplied by √ 2π. This a statement of theWiener–Kinchin theorem. Similarly we can derive the converse theoremF [ f ∗ (x)g(x) ] = √ 1 ˜f ⊗ ˜g.2πIf we now consider the special case where g is taken to be equal to f in (13.40)then, writing the LHS as a(z), we havea(z) =∫ ∞−∞f ∗ (x)f(x + z) dx; (13.43)this is called the auto-correlation function of f(x). Using the Wiener–Kinchintheorem (13.42) we see thata(z) = √ 1 ∫ ∞ã(k) e ikz dk2π −∞= √ 1 ∫ ∞ √2π [˜f(k)]∗˜f(k) e ikz dk,2π−∞so that a(z) is the inverse Fourier transform of √ 2π |˜f(k)| 2 , which is in turn calledthe energy spectrum of f.13.1.9 Parseval’s theoremUsing the results of the previous section we can immediately obtain Parseval’stheorem. The most general form of this (also called the multiplication theorem) is450