# Complex version of an impulse

Started by April 16, 2004
```Hi all,

I've got an analyic signal for which I'm designing an IIR
filter with purely real valued coefficients. I'd like to
look at the impulse response of this filter, but since the
normal impulse is purely real and the coefficients are all
real, the impulse response is also purely real.

In order to get a complex impulse response, I need a
complex impulse. Is this:

....  0 0 0 0 0 (1+i) 0 0 0 0 0 0 ....

```
```In article da4d20d8.0404152109.7ed6b813@posting.google.com, Impulse at
impulse@e.coolworks.com wrote on 04/16/2004 01:09:

> Hi all,
>
> I've got an analyic signal for which I'm designing an IIR
> filter with purely real valued coefficients. I'd like to
> look at the impulse response of this filter, but since the
> normal impulse is purely real and the coefficients are all
> real, the impulse response is also purely real.
>
> In order to get a complex impulse response, I need a
> complex impulse. Is this:
>
> ....  0 0 0 0 0 (1+i) 0 0 0 0 0 0 ....
>

there is a paper called "the Analytical Impulse" by Andrew Duncan in the AES
Journal that might suggest:

x[n] = d[n] + j*Hilbert{ d[n] }         ("d[n]" is the discrete impulse)

as the thing to bang a complex linear system with.

i dunno.

r b-j

```
```"Impulse" <impulse@e.coolworks.com> wrote in message
> Hi all,
>
> I've got an analyic signal for which I'm designing an IIR
> filter with purely real valued coefficients. I'd like to
> look at the impulse response of this filter, but since the
> normal impulse is purely real and the coefficients are all
> real, the impulse response is also purely real.
>
> In order to get a complex impulse response, I need a
> complex impulse. Is this:
>
>    ....  0 0 0 0 0 (1+i) 0 0 0 0 0 0 ....
>

Yes and no.  Think of superposition.

What if the unit sample were purely imaginary:

...... 0 0 0 0 1*i 0 0 0 0 .....

What would be the difference in the impulse response?  Nothing except there
would be an "i" attached to all of the output samples.

So using 1+i only superimposes two unit samples, one real and one imaginary,
resulting in a superimposed output.  And, the two will appear to be the same
with one real and one imaginary.

"complex" is just a 2-dimensional version of multidimensional.  Look at a
3-dimensional space with unit vectors in x,y and z being scaled by unit
vectors in each dimension: i,j and k.
Numbers (vectors) are represented like this: 3i+2j+4k.

In 2-dimensions we forget about the i vector altogether, we replace j with i
and call the result "complex".  Otherwise, it's the same thing.  And that's
like an analytic signal.

So, why would you need to do anything?  i.e. given the case you've posed.
The impulse response is the impulse response and changing the input doesn't
affect that....

Fred

```
```robert bristow-johnson wrote:

> In article da4d20d8.0404152109.7ed6b813@posting.google.com, Impulse at
> impulse@e.coolworks.com wrote on 04/16/2004 01:09:
>
>
>>Hi all,
>>
>>I've got an analyic signal for which I'm designing an IIR
>>filter with purely real valued coefficients. I'd like to
>>look at the impulse response of this filter, but since the
>>normal impulse is purely real and the coefficients are all
>>real, the impulse response is also purely real.
>>
>>In order to get a complex impulse response, I need a
>>complex impulse. Is this:
>>
>>....  0 0 0 0 0 (1+i) 0 0 0 0 0 0 ....
>>
>
>
>
> there is a paper called "the Analytical Impulse" by Andrew Duncan in the AES
> Journal that might suggest:
>
>     x[n] = d[n] + j*Hilbert{ d[n] }         ("d[n]" is the discrete impulse)
>
> as the thing to bang a complex linear system with.
>
> i dunno.
>
> r b-j

I don't get it. In the real world, an analytic signal has two parts, I
and Q, that exist on two wires. To filter an analytic signal, the parts
must be filtered separately. Inside a computer -- or our heads -- there
is more flexibility, up to a point. The samples of an analytic signal
may be thought of as pairs of samples (labeled I and Q, e.g.) or as
complex numbers with real or imaginary parts. Either way, the parts can
be filtered as separate streams. It seems to me that without great
cleverness, they have to be.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;

```
```"robert bristow-johnson" <rbj@surfglobal.net> wrote in message
news:BCA4EC98.A719%rbj@surfglobal.net...

> there is a paper called "the Analytical Impulse" by Andrew Duncan in the
AES
> Journal that might suggest:
>
>     x[n] = d[n] + j*Hilbert{ d[n] }         ("d[n]" is the discrete
impulse)
>
> as the thing to bang a complex linear system with.

Robert,

Along the lines of my earlier post and Jerry's post, what was the point of
their paper?  Or was it a "special" paper in the April issue?

Fred

```
```Jerry Avins wrote:

...

> I don't get it. In the real world, an analytic signal has two parts, I
> and Q, that exist on two wires. To filter an analytic signal, the parts
> must be filtered separately. Inside a computer -- or our heads -- there
> is more flexibility, up to a point. The samples of an analytic signal
> may be thought of as pairs of samples (labeled I and Q, e.g.) or as
> complex numbers with real or imaginary parts. Either way, the parts can
AND

> be filtered as separate streams. It seems to me that without great
> cleverness, they have to be.
>
> Jerry

--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;

```
```In article ANudnUzyt78Ynx3dRVn-jw@centurytel.net, Fred Marshall at
fmarshallx@remove_the_x.acm.org wrote on 04/16/2004 11:57:

>
> "robert bristow-johnson" <rbj@surfglobal.net> wrote in message
> news:BCA4EC98.A719%rbj@surfglobal.net...
>
>> there is a paper called "the Analytical Impulse" by Andrew Duncan in the
> AES
>> Journal that might suggest:
>>
>> x[n] = d[n] + j*Hilbert{ d[n] }         ("d[n]" is the discrete
> impulse)
>>
>> as the thing to bang a complex linear system with.
>
> Robert,
>
> Along the lines of my earlier post and Jerry's post, what was the point of
> their paper?  Or was it a "special" paper in the April issue?

this is a paper that is around a decade old.  i dunno (or don't remember)
what you would be using it for, in the context of audio or not, but Andrew
got an AES Publications Award (and sits on the review board) mostly because
of that paper.

obviously, the spectrum evaluated for the negative frequencies is zippo and
since multiplying zero times anything is still zero, that would be the case
for the analytical impulse response of any linear system, real coefs or
complex.  what good that's for?  beats the hell outa me.

r b-j

```
```It's as right as :

..... 0 0 0 0 2 0 0 0 .....

or

---- 0 0 0 0 0 0 -e 0 0 0 0 0 -----

or  .... -j 0 0 0 0 0 .....

if you don;t care where it starts and you do something which normalises
afterwards.

Best of Luck - Mike.

"Impulse" <impulse@e.coolworks.com> wrote in message
> Hi all,
>
> I've got an analyic signal for which I'm designing an IIR
> filter with purely real valued coefficients. I'd like to
> look at the impulse response of this filter, but since the
> normal impulse is purely real and the coefficients are all
> real, the impulse response is also purely real.
>
> In order to get a complex impulse response, I need a
> complex impulse. Is this:
>
>    ....  0 0 0 0 0 (1+i) 0 0 0 0 0 0 ....
>