INTEGRAL TRANSFORMS
Prove the Wiener–Kinchin theorem,
̃C(k)=√
2 π[ ̃f(k)]∗ ̃g(k). (13.42)Following a method similar to that for the convolution offandg, let us consider the
Fourier transform of (13.40):
C ̃(k)=√^1
2 π∫∞
−∞dz e−ikz{∫∞
−∞f∗(x)g(x+z)dx}
=
1
√
2 π∫∞
−∞dx f∗(x){∫∞
−∞g(x+z)e−ikzdz}
.
Making the substitutionu=x+zinthesecondintegralweobtain
C ̃(k)=√^1
2 π∫∞
−∞dx f∗(x){∫∞
−∞g(u)e−ik(u−x)du}
=
1
√
2 π∫∞
−∞f∗(x)eikxdx∫∞
−∞g(u)e−ikudu=
1
√
2 π×
√
2 π[ ̃f(k)]∗×√
2 π ̃g(k)=√
2 π[ ̃f(k)]∗ ̃g(k).Thus the Fourier transform of the cross-correlation offandgis equal tothe product of [ ̃f(k)]∗and ̃g(k) multiplied by
√
2 π. This a statement of theWiener–Kinchin theorem. Similarly we can derive the converse theorem
F[
f∗(x)g(x)]
=1
√
2 π̃f⊗ ̃g.If we now consider the special case wheregis taken to be equal tofin (13.40)
then, writing the LHS asa(z), we have
a(z)=∫∞−∞f∗(x)f(x+z)dx; (13.43)this is called theauto-correlation functionoff(x). Using the Wiener–Kinchin
theorem (13.42) we see that
a(z)=1
√
2 π∫∞−∞̃a(k)eikzdk=1
√
2 π∫∞−∞√
2 π[ ̃f(k)]∗ ̃f(k)eikzdk,so thata(z) is the inverse Fourier transform of
√
2 π| ̃f(k)|^2 , which is in turn calledtheenergy spectrumoff.
13.1.9 Parseval’s theoremUsing the results of the previous section we can immediately obtainParseval’s
theorem. The most general form of this (also called themultiplication theorem)is