# DSP Riddle: Infinitely Narrow Notch Filter

Started by May 19, 2006
```Folks,

is it possible to construct a causal and realizable n-th order IIR
filter, where n > 1 is an integer, which acts as an infinitely narrow
notch filter? This filter must have frequency response

H(w) = 1, for w =/= w0
= 0, for w == w0,

for some arbitrary 0 <  w0 < pi.

Assume for the moment that the filter can be computed with infinite
precision.

Have a nice weekend!

Regards,
Andor

```
```Andor wrote:

> Folks,
>
> is it possible to construct a causal and realizable n-th order IIR
> filter, where n > 1 is an integer, which acts as an infinitely narrow
> notch filter? This filter must have frequency response
>
> H(w) = 1, for w =/= w0
>      = 0, for w == w0,
>
> for some arbitrary 0 <  w0 < pi.
>
> Assume for the moment that the filter can be computed with infinite
> precision.
>
> Have a nice weekend!
>
> Regards,
> Andor

I would say no. If such discrete time filter would be realized the impulse
response would always have zero energy (i.e., h[n]=0, for all n) and thus
it would not be realizable as an IIR filter with the given frequency
response.

--
Jani Huhtanen
Tampere University of Technology, Pori
```
```Jani Huhtanen wrote:

> Andor wrote:
>
>> Folks,
>>
>> is it possible to construct a causal and realizable n-th order IIR
>> filter, where n > 1 is an integer, which acts as an infinitely narrow
>> notch filter? This filter must have frequency response
>>
>> H(w) = 1, for w =/= w0
>>      = 0, for w == w0,
>>
>> for some arbitrary 0 <  w0 < pi.
>>
>> Assume for the moment that the filter can be computed with infinite
>> precision.
>>
>> Have a nice weekend!
>>
>> Regards,
>> Andor
>
> I would say no. If such discrete time filter would be realized the impulse
> response would always have zero energy (i.e., h[n]=0, for all n) and thus
> it would not be realizable as an IIR filter with the given frequency
> response.
>

--
----
Jani Huhtanen
Tampere University of Technology, Pori
```
```Andor wrote:

> Folks,
>
> is it possible to construct a causal and realizable n-th order IIR
> filter, where n > 1 is an integer, which acts as an infinitely narrow
> notch filter? This filter must have frequency response
>
> H(w) = 1, for w =/= w0
>      = 0, for w == w0,
>
> for some arbitrary 0 <  w0 < pi.
>
> Assume for the moment that the filter can be computed with infinite
> precision.
>
> Have a nice weekend!
>
> Regards,
> Andor
>
Yes, but it would take an infinite amount of time to verify that it
worked correctly.

Practically, you could make a Kalman filter to detect your tone and
subtract it out -- that would be an IIR filter (just time varying), and
it would notch out the tone to the best of it's ability for any given time.

But then practically you wouldn't have infinite precision.

--

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

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html
```
```Jani Huhtanen said the following on 19/05/2006 15:44:
> Jani Huhtanen wrote:
>
>> Andor wrote:
>>
>>> is it possible to construct a causal and realizable n-th order IIR
>>> filter, where n > 1 is an integer, which acts as an infinitely narrow
>>> notch filter? This filter must have frequency response
>>>
>>> H(w) = 1, for w =/= w0
>>>      = 0, for w == w0,
>>>
>>> for some arbitrary 0 <  w0 < pi.
>>>
>>> Assume for the moment that the filter can be computed with infinite
>>> precision.
>>>
>> I would say no. If such discrete time filter would be realized the impulse
>> response would always have zero energy (i.e., h[n]=0, for all n) and thus
>> it would not be realizable as an IIR filter with the given frequency
>> response.
>

Even so, I think you may be along the right lines (alternatively, there
may be a massive flaw in the following argument ;) ).

If H(w) is realisable, then we can also realise a filter G(w) = 1 - H(w)
with the following structure:

+----+  +1
--+------>|H(w)|----->O------->
|       +----+      ^
|                   | -1
+-------------------+

G(w) is a BPF with an infinitely narrow passband, which would then have
zero energy (as you originally suggested).

Therefore G(w) is unrealisable, implying that H(w) is also unrealisable.

QED?

--
Oli
```
```Andor wrote:
>
> is it possible to construct a causal and realizable n-th order IIR
> filter, where n > 1 is an integer, which acts as an infinitely narrow
> notch filter? This filter must have frequency response
>
> H(w) =3D 1, for w =3D/=3D w0
>      =3D 0, for w =3D=3D w0,
>
> for some arbitrary 0 <  w0 < pi.
>
> Assume for the moment that the filter can be computed with infinite
> precision.

you can get arbitrarily close for n=3D2.

zeros =3D exp(=B1j*w0)
poles =3D (1-e)*exp(=B1j*w0)

where 0 < e << 1.

but, from a Lebesgue measure POV, there is no difference betwee the
specification you made and one where H(w) =3D 1 for all w.

r b-j

```
```Oli Filth wrote:

> Jani Huhtanen said the following on 19/05/2006 15:44:
>
>> Jani Huhtanen wrote:
>>
>>> Andor wrote:
>>>
>>>> is it possible to construct a causal and realizable n-th order IIR
>>>> filter, where n > 1 is an integer, which acts as an infinitely narrow
>>>> notch filter? This filter must have frequency response
>>>>
>>>> H(w) = 1, for w =/= w0
>>>>      = 0, for w == w0,
>>>>
>>>> for some arbitrary 0 <  w0 < pi.
>>>>
>>>> Assume for the moment that the filter can be computed with infinite
>>>> precision.
>>>>
>>> I would say no. If such discrete time filter would be realized the
>>> impulse
>>> response would always have zero energy (i.e., h[n]=0, for all n) and
>>> thus
>>> it would not be realizable as an IIR filter with the given frequency
>>> response.
>>
>>
>
>
> Even so, I think you may be along the right lines (alternatively, there
> may be a massive flaw in the following argument ;) ).
>
> If H(w) is realisable, then we can also realise a filter G(w) = 1 - H(w)
> with the following structure:
>
>           +----+  +1
> --+------>|H(w)|----->O------->
>   |       +----+      ^
>   |                   | -1
>   +-------------------+
>
> G(w) is a BPF with an infinitely narrow passband, which would then have
> zero energy (as you originally suggested).
>
> Therefore G(w) is unrealisable, implying that H(w) is also unrealisable.
>
> QED?

Doesn't your argument imply that H(w) has zero phase shift in the notch?
I see H(w) being unrealizable simply because there are no infinities is
reality. The slopes of the filter edges are unrealizably steep. (I don't
have to prove it because I just /*know*/.) Andor will unveil it all on
Monday. I (a)wait with (a)bated breath.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Jerry Avins said the following on 19/05/2006 18:17:
> Oli Filth wrote:
>
>> If H(w) is realisable, then we can also realise a filter G(w) = 1 - H(w)
>> with the following structure:
>>
>>           +----+  +1
>> --+------>|H(w)|----->O------->
>>   |       +----+      ^
>>   |                   | -1
>>   +-------------------+
>>
>> G(w) is a BPF with an infinitely narrow passband, which would then have
>> zero energy (as you originally suggested).
>>
>> Therefore G(w) is unrealisable, implying that H(w) is also unrealisable.
>
> Doesn't your argument imply that H(w) has zero phase shift in the notch?

Yes, I believe it does.  I guess this leads back to the phase-linear IIR
discussions that keep cropping up here!

> I see H(w) being unrealizable simply because there are no infinities is
> reality. The slopes of the filter edges are unrealizably steep. (I don't
> have to prove it because I just /*know*/.)

Indeed, but isn't the "fun" in such riddles the proof? :)

--
Oli
```
```> H(w) = 1, for w =/= w0
>    = 0, for w == w0,

Do you mean within the band 0 < w < w_s / 2?  A
digital filter always has infinite images, so otherwise
you couldn't implement such a thing.

--Randy

```
```Tim Wescott <tim@seemywebsite.com> wrote in
news:8KSdnUsGjuGof_DZnZ2dnUVZ_t6dnZ2d@web-ster.com:

> Andor wrote:
>
>> Folks,
>>
>> is it possible to construct a causal and realizable n-th order IIR
>> filter, where n > 1 is an integer, which acts as an infinitely narrow
>> notch filter? This filter must have frequency response
>>
>> H(w) = 1, for w =/= w0
>>      = 0, for w == w0,
>>
>> for some arbitrary 0 <  w0 < pi.
>>
>> Assume for the moment that the filter can be computed with infinite
>> precision.
>>
>> Have a nice weekend!
>>
>> Regards,
>> Andor
>>
> Yes, but it would take an infinite amount of time to verify that it
> worked correctly.

Since BT >= 1, I agree with Tim that it would take an infinite amount of
time since as B=>0, T=>infinity,

however,

You make a notch filter by placing a zero on the unit circuit. To narrow
the bandwidth you place a pole at the same angle but with a radius less
than 1. As the pole gets closer to the unit circle, the bandwidth becomes
increasingly narrower, but not zero. If the pole was placed exactly on
the unit circle (which is sort of causal) , you would get perfect pole
zero cancellation, which doesn't yield the desired notch.

I don't think a using higher order filter will change the result.

So from a purely heuristic point of view, I don't think you can create
Andor's filter.

--
Al Clark
Danville Signal Processing, Inc.
--------------------------------------------------------------------
Purveyors of Fine DSP Hardware and other Cool Stuff
Available at http://www.danvillesignal.com
```