# questions raised by reading and thinking with possibly missing background

Started by December 10, 2005
```Richard Owlett wrote:
> Richard Owlett wrote:
>
>> ...
>> That got me thinking ;<
>
>
> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
> shape in the frequency domain to be "linear phase".
>
> One of the references I was reading stated that "a FIR filter would be
> 'linear phase' if its coefficients were symmetric about the middle
> coefficient."

Not very well put. With an even number of coefficients, there is no
middle one.

> Is that a "sufficient" condition or a "necessary" condition?
> What implication does it have for the passband response?

A filter's phase response is linear *if and only if* it is symmetric or
antisymmetric about its middle. Note: Both [1 1] and [1 0 1] are
symmetric.  Both [1 -1] and [1 0 -1] are antisymmetric. Exception:
adding zeros to one end of an otherwise symmetric or antisymmetric
filter doesn't impair its phase linearity.

Jerry

P.S. Most frequency-altering processes in nature affect phase.
Electronic means to restore a flat response without affecting phase
don't usually sound as good as a more complete correction.
--
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;
```
```Randy Yates wrote:

> abariska@student.ethz.ch writes:
>
>
>>Richard Owlett wrote:
>>
>>>Richard Owlett wrote:
>>>
>>>
>>>>...
>>>>That got me thinking ;<
>>>
>>>What are the *NECESSARY* conditions for a FIR filter of an arbitrary
>>>shape in the frequency domain to be "linear phase".
>>>
>>>One of the references I was reading stated that "a FIR filter would be
>>>'linear phase' if its coefficients were symmetric about the middle
>>>coefficient."
>>>
>>>Is that a "sufficient" condition or a "necessary" condition?
>>
>>We discussed this last June:
>>
>
>
> Consider the filter coefficients determined as
>
> function y = test(x)
> %function y = test(x)
> n = [-25 : 25];
> Fs = 1;
> Ts = 1/Fs;
> t = n*Ts;
> plot(sinc(t+1/7));
>
> These are neither symmetric nor antisymmetric in the sense you defined,
> and yet this is a linear phase filter, is it not?

je ne comprend pas ;]

Can you give me code understood by Scilab so I can know what point you
wish to make ;[

```
```Jerry Avins wrote:

> Richard Owlett wrote:
>
>> Richard Owlett wrote:
>>
>>> ...
>>> That got me thinking ;<
>>
>>
>>
>> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
>> shape in the frequency domain to be "linear phase".
>>
>> One of the references I was reading stated that "a FIR filter would be
>> 'linear phase' if its coefficients were symmetric about the middle
>> coefficient."
>
>
> Not very well put. With an even number of coefficients, there is no
> middle one.

The reference I was reading seemed to treat that as a degenerate case.

>
>> Is that a "sufficient" condition or a "necessary" condition?
>> What implication does it have for the passband response?
>
>
> A filter's phase response is linear *if and only if* it is symmetric or
> antisymmetric about its middle. Note: Both [1 1] and [1 0 1] are
> symmetric.  Both [1 -1] and [1 0 -1] are antisymmetric. Exception:
> adding zeros to one end of an otherwise symmetric or antisymmetric
> filter doesn't impair its phase linearity.

Picking numbers *AT RANDOM*
Are you saying that a "low pass" filter with an overall passband of 10
kHz with "peaks" at 2 kHz, 3.14 kHz, and 7.9631 kHz with relative
amplitudes of 1, 2.91234, and 1.167 *could not* be linear phase?

>
> Jerry
>
> P.S. Most frequency-altering processes in nature affect phase.
> Electronic means to restore a flat response without affecting phase
> don't usually sound as good as a more complete correction.

Huh. I'm not natural (so to speak ;)

```
```Jerry Avins wrote:

> Richard Owlett wrote:
>
>> Richard Owlett wrote:
>>
>>> ...
>>> That got me thinking ;<
>>
>>
>>
>> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
>> shape in the frequency domain to be "linear phase".
>>
>> One of the references I was reading stated that "a FIR filter would be
>> 'linear phase' if its coefficients were symmetric about the middle
>> coefficient."
>
>
> Not very well put. With an even number of coefficients, there is no
> middle one.

The reference I was reading seemed to treat that as a degenerate case.

>
>> Is that a "sufficient" condition or a "necessary" condition?
>> What implication does it have for the passband response?
>
>
> A filter's phase response is linear *if and only if* it is symmetric or
> antisymmetric about its middle. Note: Both [1 1] and [1 0 1] are
> symmetric.  Both [1 -1] and [1 0 -1] are antisymmetric. Exception:
> adding zeros to one end of an otherwise symmetric or antisymmetric
> filter doesn't impair its phase linearity.

Picking numbers *AT RANDOM*
Are you saying that a "low pass" filter with an overall passband of 10
kHz with "peaks" at 2 kHz, 3.14 kHz, and 7.9631 kHz with relative
amplitudes of 1, 2.91234, and 1.167 *could not* be linear phase?

>
> Jerry
>
> P.S. Most frequency-altering processes in nature affect phase.
> Electronic means to restore a flat response without affecting phase
> don't usually sound as good as a more complete correction.

Huh. I'm not natural (so to speak ;)

```
```Jerry Avins wrote:

> Richard Owlett wrote:
>
>> Richard Owlett wrote:
>>
>>> ...
>>> That got me thinking ;<
>>
>>
>>
>> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
>> shape in the frequency domain to be "linear phase".
>>
>> One of the references I was reading stated that "a FIR filter would be
>> 'linear phase' if its coefficients were symmetric about the middle
>> coefficient."
>
>
> Not very well put. With an even number of coefficients, there is no
> middle one.

The reference I was reading seemed to treat that as a degenerate case.

>
>> Is that a "sufficient" condition or a "necessary" condition?
>> What implication does it have for the passband response?
>
>
> A filter's phase response is linear *if and only if* it is symmetric or
> antisymmetric about its middle. Note: Both [1 1] and [1 0 1] are
> symmetric.  Both [1 -1] and [1 0 -1] are antisymmetric. Exception:
> adding zeros to one end of an otherwise symmetric or antisymmetric
> filter doesn't impair its phase linearity.

Picking numbers *AT RANDOM*
Are you saying that a "low pass" filter with an overall passband of 10
kHz with "peaks" at 2 kHz, 3.14 kHz, and 7.9631 kHz with relative
amplitudes of 1, 2.91234, and 1.167 *could not* be linear phase?

>
> Jerry
>
> P.S. Most frequency-altering processes in nature affect phase.
> Electronic means to restore a flat response without affecting phase
> don't usually sound as good as a more complete correction.

Huh. I'm not natural (so to speak ;)

```
```Jerry Avins wrote:

> Richard Owlett wrote:
>
>> Richard Owlett wrote:
>>
>>> ...
>>> That got me thinking ;<
>>
>>
>>
>> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
>> shape in the frequency domain to be "linear phase".
>>
>> One of the references I was reading stated that "a FIR filter would be
>> 'linear phase' if its coefficients were symmetric about the middle
>> coefficient."
>
>
> Not very well put. With an even number of coefficients, there is no
> middle one.

The reference I was reading seemed to treat that as a degenerate case.

>
>> Is that a "sufficient" condition or a "necessary" condition?
>> What implication does it have for the passband response?
>
>
> A filter's phase response is linear *if and only if* it is symmetric or
> antisymmetric about its middle. Note: Both [1 1] and [1 0 1] are
> symmetric.  Both [1 -1] and [1 0 -1] are antisymmetric. Exception:
> adding zeros to one end of an otherwise symmetric or antisymmetric
> filter doesn't impair its phase linearity.

Picking numbers *AT RANDOM*
Are you saying that a "low pass" filter with an overall passband of 10
kHz with "peaks" at 2 kHz, 3.14 kHz, and 7.9631 kHz with relative
amplitudes of 1, 2.91234, and 1.167 *could not* be linear phase?

>
> Jerry
>
> P.S. Most frequency-altering processes in nature affect phase.
> Electronic means to restore a flat response without affecting phase
> don't usually sound as good as a more complete correction.

Huh. I'm not natural (so to speak ;)

```
```Jerry Avins wrote:

> Richard Owlett wrote:
>
>> Richard Owlett wrote:
>>
>>> ...
>>> That got me thinking ;<
>>
>>
>>
>> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
>> shape in the frequency domain to be "linear phase".
>>
>> One of the references I was reading stated that "a FIR filter would be
>> 'linear phase' if its coefficients were symmetric about the middle
>> coefficient."
>
>
> Not very well put. With an even number of coefficients, there is no
> middle one.

The reference I was reading seemed to treat that as a degenerate case.

>
>> Is that a "sufficient" condition or a "necessary" condition?
>> What implication does it have for the passband response?
>
>
> A filter's phase response is linear *if and only if* it is symmetric or
> antisymmetric about its middle. Note: Both [1 1] and [1 0 1] are
> symmetric.  Both [1 -1] and [1 0 -1] are antisymmetric. Exception:
> adding zeros to one end of an otherwise symmetric or antisymmetric
> filter doesn't impair its phase linearity.

Picking numbers *AT RANDOM*
Are you saying that a "low pass" filter with an overall passband of 10
kHz with "peaks" at 2 kHz, 3.14 kHz, and 7.9631 kHz with relative
amplitudes of 1, 2.91234, and 1.167 *could not* be linear phase?

>
> Jerry
>
> P.S. Most frequency-altering processes in nature affect phase.
> Electronic means to restore a flat response without affecting phase
> don't usually sound as good as a more complete correction.

Huh. I'm not natural (so to speak ;)

```
```Jerry Avins wrote:

> Richard Owlett wrote:
>
>> Richard Owlett wrote:
>>
>>> ...
>>> That got me thinking ;<
>>
>>
>>
>> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
>> shape in the frequency domain to be "linear phase".
>>
>> One of the references I was reading stated that "a FIR filter would be
>> 'linear phase' if its coefficients were symmetric about the middle
>> coefficient."
>
>
> Not very well put. With an even number of coefficients, there is no
> middle one.

The reference I was reading seemed to treat that as a degenerate case.

>
>> Is that a "sufficient" condition or a "necessary" condition?
>> What implication does it have for the passband response?
>
>
> A filter's phase response is linear *if and only if* it is symmetric or
> antisymmetric about its middle. Note: Both [1 1] and [1 0 1] are
> symmetric.  Both [1 -1] and [1 0 -1] are antisymmetric. Exception:
> adding zeros to one end of an otherwise symmetric or antisymmetric
> filter doesn't impair its phase linearity.

Picking numbers *AT RANDOM*
Are you saying that a "low pass" filter with an overall passband of 10
kHz with "peaks" at 2 kHz, 3.14 kHz, and 7.9631 kHz with relative
amplitudes of 1, 2.91234, and 1.167 *could not* be linear phase?

>
> Jerry
>
> P.S. Most frequency-altering processes in nature affect phase.
> Electronic means to restore a flat response without affecting phase
> don't usually sound as good as a more complete correction.

Huh. I'm not natural (so to speak ;)

```
```Richard Owlett wrote:

...

> Picking numbers *AT RANDOM*
> Are you saying that a "low pass" filter with an overall passband of 10
> kHz with "peaks" at 2 kHz, 3.14 kHz, and 7.9631 kHz with relative
> amplitudes of 1, 2.91234, and 1.167 *could not* be linear phase?

No. You seem to still be confusing a filter's (or a window's) frequency
response with FIR coefficients.

>> P.S. Most frequency-altering processes in nature affect phase.
>> Electronic means to restore a flat response without affecting phase
>> don't usually sound as good as a more complete correction.
>
>
> Huh. I'm not natural (so to speak ;)

A linear-phase phono equalizer completely louses up the transient
response. A "perfect" linear-phase speaker crossover often sounds much
worse that the minimum-phase analog approximation that it replaced.

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 wrote:

> A linear-phase phono equalizer completely louses up the transient
> response. A "perfect" linear-phase speaker crossover often sounds much
> worse that the minimum-phase analog approximation that it replaced.

Don't look at the transient response and linear phase will sound just as
good as the minimal phase :) We are entering the area of the holy wars
of the blunt-pointed vs sharp pointed. From my experience the only
observable difference results from the implementation issues like
overflows, loss of accuracy, group delay or frequency response mismatch
and such.

BTW, what do you think about Bessel filters, which are the minimum phase
approximations of the linear phase?