There is? It seems that is a result only for normal distributions.
Yup. From the derivative theorem, the variance is proportional to the
second derivative of the transform, evaluated at zero frequency. To
within a constant factor that depends on your Fourier transform definition,
int(-inf to inf) [t**2 * h(t)]
var(h) = ------------------------------- = H''(0)/H(0)
int(-inf to inf) [h(t)]
Because the function is real, its transform is Hermitian, i.e. the real
part is even and the imaginary part is odd. Even-order derivatives of
an odd function vanish at the origin, as do odd-order derivatives of an
even function.
When you convolve g(t) and h(t), the transform is G(f)H(f). The
variance of this is proportional to the normalized second derivative at
the origin, as before. Ignoring a possible constant of proportionality
that we don't care about,
d(GH)/df = GH' + HG' so
d**2/df**2[GH]| H(0)G''(0) + G(0)H''(0) + 2G'H'
var(g*h) = --------------| = -------------------------------
GH |f=0 H(0)G(0)
G''(0) H''(0) 2G'(0)H'(0)
= ------- + -------- + ------------
G(0) H(0) H(0)G(0)
The first term is the variance of G, the second is the variance of H,
and the third is zero if either G or H is an even function.
There is a slightly more subtle condition that will make the third term
zero for any choice of g and h: that the first moment of either h(t) or
g(t) is zero, i.e. that at least one of them has its centroid at t=0,
which makes its first derivative zero at f=0.
It is always possible to satisfy this condition by an appropriate choice
of the time origin, so if you'll allow me to slide over that rather
trivial issue(*), variances add under convolution.
Cheers
Phil Hobbs
(*) The reason for this is the effect of a shift of origin on the
variance. If you convolve a function centred at t=5 with one centredat
t=3, the convolution's centroid is at t=8. If you're computing the
variance as the second moment about t=0, this will make a big
difference, but it doesn't change the shape of the resulting convolution
function. Forcing one of them to have its centroid at 0 gets rid of
this shift of time origin.
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics
160 North State Road #203
Briarcliff Manor NY 10510 USA
+1 845 480 2058
hobbs at electrooptical dot nethttp://electrooptical.net