characteristic function of a random variable

[claudiocamera]

Hello There!
Studying characteristic function of a random variable in several text I came across with the definition that it is the Fourier transform of the density function.

While the signal of the complex term multiplying in the integral of the fourier transform comes with negative signal {EXP(-Jwx)}, the definition of characteristic function in statistic books presents the complex term with positive sinal {EXP(Jwx)}. The same exchange of signals happens in the inverse functions. Why does it happen ?

 

[LouisSheffield]

It's by definition - from Papoulis:

The characteristic function of a r.v. x is defined by
φ(x) = E{exp(jωx)}
This is the expected value of the complex function cos(ωx)+j sin(ωx)
of x, and it is given by the integral ....

φ(x) = ∫exp(jωx) f(x) dx


(yeah - the one with the +jω sign)


It resembles an inverse Fourier transform (having the self same equation form).
If it is truly defined as per the +jω sign, authors who call that a Fourier transform are merely committing a bit of sloppiness in their notation.

 

[minusinfinity]

Although, the sign doesnot go by defination,

BUT when we say that C.F is the F.T of density function. It actually implies that all the properties of fourier transform can be applied to the C.F.

 

[claudiocamera]

I came across this definition not only in Papoulis, but also in all others statistics books and sites that I looked up. All of them used the φ(x) = ∫exp(jωx) f(x) dx with +jω sign and all of them call the function as Fourier transform. So, Why is it for ?
The answer provided by minusinfinity is very plausible, "It actually implies that all the properties of fourier transform can be applied to the C.F." but why it is not just said by authors, ? Why all of them quote that the expression is actualy the Fourier Transform itself ?

 

[puck]

There is anything wrong with that definition. You only should note that exponentials in analysis and synthesis formulae of Fourier Transforms must be complex conjugate of each other. A general expansion formula in an orthonormal basis is
x = sum(<x,v>v), where <x,v> is the dot product between x and the basis vector v. When you take that inner product you use the complex conjugate of v. But as complex conjugates forms and orthonormal basis too, where you use positive or negative phases is set only by convention. In DSP literature
usually v=exp(jw), with positive sign, so that the Fourier Transform <x,v> appears with negative sign. In Physics and probability, often signs are interchanged. I think that is because reconstruction formulae are less used in that fields. But the point is that there is not conceptual mistake in using one or other sign in defining Fourier Transforms.