# Frequency convolution

Started by October 10, 2005
```Hello everyone. I have a question I cant find an answer to:

Say you have, X(z) the z transform of a sampled time response x[n],
i.e.

Z{x[n]} = X(z)

Does a Y(z) exist s.t.

Z{x[n]^2} = Y(z)

and if so, how does it relate to X(z)? If you dont feel challenged
enough, try this other one: Does a Y2(z) exist such that:

Z{abs(x[n])} = Y2(z)?

For the first point I am aware that multiplication in the time domain
is the same as convolution in the frequency domain, and that
convolution in the z-domain is expressed as a contour integral, but I
have no idea how to interpret these.

My guess is that Y(z) will only have positive coefficients.

```
```Seppo wrote:

> Hello everyone. I have a question I cant find an answer to:
>
> Say you have, X(z) the z transform of a sampled time response x[n],
> i.e.
>
> Z{x[n]} = X(z)
>
> Does a Y(z) exist s.t.
>
> Z{x[n]^2} = Y(z)

Yes; if you can express y[n] = f(x[n]) you can find the z transform of y
given f and x.
>
> and if so, how does it relate to X(z)?

Not necessarily very directly, if you have X(z) in the form of a ratio
of polynomials.

> If you dont feel challenged
> enough, try this other one: Does a Y2(z) exist such that:
>
> Z{abs(x[n])} = Y2(z)?

Yes, see above.
>
> For the first point I am aware that multiplication in the time domain
> is the same as convolution in the frequency domain, and that
> convolution in the z-domain is expressed as a contour integral, but I
> have no idea how to interpret these.
>
> My guess is that Y(z) will only have positive coefficients.
>
If you are doing Fourier analysis in discrete-time then the y[n] =
x[n]^2 case is fairly easy to analyze; the y[n] = abs(x[n]) isn't really
worth it.

If you could find a Y(z) that was a ratio of polynomials then it
wouldn't necessarily have all positive coefficients.  Answering these
questions yourself will give you a deeper understanding of the z
transform, but it's the trip, not the destination, that does the teaching.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```