# Hilbert transform of ARMA filter

Started by February 26, 2005
```Suppose I have an ARMA filter of order N with known A and B coefficients.
Does there exist another ARMA filter of order N whose impulse response
(and output) is the Hilbert transform of the original filter?  If so, what
is the expression relating the two sets of coefficients?

Thanks!

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```
```You can just multiply the two transfer functions, ie. multiply each AR
coefficient with j.

```
```Sorry. Multiply each AR coefficient with -j, or the MA coefficients
with j.

```
```Andor <an2or@mailcircuit.com> wrote:
> Sorry. Multiply each AR coefficient with -j, or the MA coefficients
> with j.

Which of course isn't true :)

Try your transformation on only MA (FIR) process and check hilbert transform
orthogonality property.

```
```Yeah, that was silly. It's one of my s-z domain confusions. scuse me!

```
```I don't completely understand your reply.  What do you mean by "your
transformation" and what do you mean by "check hilbert transform
orthogonality"?

I understand the operations required in the Fourier domain, and I was
hoping this would transfer to a simple operation on ARMA coefficients
(e.g., a rotation, as the multiplication by would imply).

Thanks,
David

pisz_na.mirek@dionizos.zind.ikem.pwr.wroc.pl wrote:
> Andor <an2or@mailcircuit.com> wrote:
> > Sorry. Multiply each AR coefficient with -j, or the MA coefficients
> > with j.
>
> Which of course isn't true :)
>
> Try your transformation on only MA (FIR) process and check hilbert
transform
> orthogonality property.

```
```djklein <kleinsound@yahoo.com> wrote:
>
> I don't completely understand your reply.  What do you mean by "your
> transformation"

Andor's proposition

> and what do you mean by "check hilbert transform
> orthogonality"?

Original signal and Hilbert transformed are orthogonal.

> I understand the operations required in the Fourier domain, and I was

Frequency domain isn't necessary but often used. Why? Because it's simple
method but has drawbacks (errors in time domain).

Hilbert transform is defined as time domain integral.

> hoping this would transfer to a simple operation on ARMA coefficients
> (e.g., a rotation, as the multiplication by would imply).

No, doing in frequency domain you will transform a "transfer function" of
arma filter. After IFFT you get an impulse response - where are your new
coefficients??? You must fit new arma to new impulse response.