Question about cross-spectrum and convolution

Hi, I have a question regarding the concept of cross-spectrum, which is
defined as the Fourier transform of convolution between two signals X
and Y:

FT(X*Y)


Isn't it the same as:

FT(X) FT(Y)

My simple clue is:

X*Y = IFT( FT(X) FT(Y) )

So, FT(X*Y) = FT(X) FT(Y).

Am I right?

Thanks

Ze Ji wrote:
> Hi, I have a question regarding the concept of cross-spectrum, which is
> defined as the Fourier transform of convolution between two signals X
> and Y:
>
> FT(X*Y)

Here "*" means convolution, right?

> Isn't it the same as:
>
> FT(X) FT(Y)

No. The cross spectrum is defined as

  FT(X) x conjugate(FT(Y))

where "x" means multiplication.

> My simple clue is:
>
> X*Y = IFT( FT(X) FT(Y) )
>
> So, FT(X*Y) = FT(X) FT(Y).
>
> Am I right?

Except for the conjugate:Yes, you are right. It is not very difficult
to work out the details. Most intermediate-level texts on DSP
contain a chapter named "Properties of the Fourier Transform"
where this is one of the standard derivations.

Rune