amplitude change in time variable iir filter

Hello,

If I change the coefficients of an iir filter with
the intention to change the resonance frequency, this
will also change the amplitude.  How can I modify the
values in the filter taps to compensate for the
amplitude change.

example:

A 2 pole filter with pole on the unit circle:

     C = 2 cos(freq);

     y[k] = C y[k-1] - y[k-2]

I hit this filter with an impulse and I get a sinewave out.
Now if I change C for a new frequency (while the filter is
"running") the amplitude changes.
If I want to modify y[k-1] and y[k-2] (not in the output,
but just for computing the next value) how do I have to
modify them?
or to put it in another way:
what is the proportion of the amplitude change 
to the frequency change?

gr.
Anton
 

banton wrote:
> Hello,
>
> If I change the coefficients of an iir filter with
> the intention to change the resonance frequency, this
> will also change the amplitude.  How can I modify the
> values in the filter taps to compensate for the
> amplitude change.
>

While it is theoretically possible to change the filter response in a
controllable fashion by direct manipulation of the filter coefficients,
it isn't usually a recommended method due to the complexities of the
math that would be involved.  Instead, it is often times much easier to
go back to a desired 'analog' filter equation and transform it into a
'digital' filter.

Note, this is a topic that has come up before and if you search through
the group archives you should be able to find some additional details
related to this topic.

banton wrote:
> Hello,
>
> If I change the coefficients of an iir filter with
> the intention to change the resonance frequency, this
> will also change the amplitude.  How can I modify the
> values in the filter taps to compensate for the
> amplitude change.
>
> example:
>
> A 2 pole filter with pole on the unit circle:
>
>      C = 2 cos(freq);
>
>      y[k] = C y[k-1] - y[k-2]
>
> I hit this filter with an impulse and I get a sinewave out.
> Now if I change C for a new frequency (while the filter is
> "running") the amplitude changes.
> If I want to modify y[k-1] and y[k-2] (not in the output,
> but just for computing the next value) how do I have to
> modify them?

If you start the above recursion with y[0] = 1 (and assume y[-1] = 0),
then

A sin(freq) = 1

(1 is the height of the impulse) and therefore

A = 1 / sin(freq)

is the amplitude of the resulting sine wave, ie.

y[k] = 1 / sin(freq) sin(freq k + freq) (the unit impulse response).

So if you want to have the same amplitude A with oscillating frequency
equal to freq2 you have to initialize the recursion with

y[0] = 0, y[1] = sin(freq) / sin(freq2),

and start applying the recursion for y[k], k = 2, 3, ...

Does this answer anything?

Regards,
Andor

banton wrote:
> Hello,
>
> If I change the coefficients of an iir filter with
> the intention to change the resonance frequency, this
> will also change the amplitude.

that is to be expected.

>  How can I modify the
> values in the filter taps to compensate for the
> amplitude change.

when you change the value of "C" to change the frequency, you also have
to fudge the filter "states" y[k-1] and y[k-2].

> example:
>
> A 2 pole filter with pole on the unit circle:
>
>      C = 2 cos(freq);
>
>      y[k] = C y[k-1] - y[k-2]
>
> I hit this filter with an impulse and I get a sinewave out.
> Now if I change C for a new frequency (while the filter is
> "running") the amplitude changes.
> If I want to modify y[k-1] and y[k-2] (not in the output,
> but just for computing the next value) how do I have to
> modify them?

ah, you have precisely the right idea.  the answer is simple, i think.


let's say that your amplitude is A (you determined that by your initial
settings for the states y[-1] and y[-2]) and that you're changing from
freq1 to freq2.  so

      C1 = 2*cos(freq1)
and
      C2 = 2*cos(freq2)

also

      y[k] = A * cos(freq*k + phi)

where A and phi are functions of C, y[-1], and y[-2].  i leave it to
the reader to determine what those functions are.  remember that

      y[-1] = A * cos(-freq + phi)
      y[-2] = A * cos(-2*freq + phi)

first determine what your new output would be if you "didn't" change
the freq (or C)

    y[k] = C1*y[k-1] - y[k-2]

now, if you don't want this to click, you want your next y[k] to be the
same value even after changing C1 to C2 and fudging with the states

so

   y[k] = C1*y[k-1] - y[k-2] = C2*y'[k-1] - y'[k-2]

that's one equation, two unknowns (y'[k-1] and  y'[k-2], your "fudged"
states).  we need another.

   y[k] = C1*y[k-1] - y[k-2] = A * cos(freq1*k + phi1)

or

   y[k] = C2*y'[k-1] - y'[k-2] = A * cos(freq2*k + phi2)

we want the contents to the cos() function to be the same in both cases

i'm just now running late and (being on Google, i can't conveniently
save this), so i'll post this now and return to it.