Theory 101 - Impulse response of "perfect" LPF

Jerry Avins wrote:
> jeff227 wrote:
>>> This has no relevance to LPFs. 
>>
>>
>> Wrong!  A spring, damper and mass is EXACTLY equivalent to a LPF.  ...
> 
> But not *every* LPF. Your analogy depends on a "perfect" lowpass, which 
> an inductor is not. An mass hit by an impulse moves at a rate 
> proportional to the strength of the impulse and inversely proportional 
> to the amount of mass. That velocity goes to zero when mass goes to 
> infinity is no more significant here than with any other denominator.
> 
> I'll embellish now what you seem to have overlooked earlier: a lowpass 
> filter passes a range of low frequencies with no attenuation and high 
> frequencies not at all. In practical filters, there is a transition band 
> between those behaviors. The width of the transition in an ideal filter 
> is zero. No isolated mass approaches LP, let alone ideal LP, behavior.
> 
> Jerry

Indeed.

A passive filter can never approach 'brick wall' response for a number 
of reasons, so we turn to active filters and DSP techniques. I'll admit 
we can get pretty good response, but no more. (I say that as someone who 
has designed passive filters).

An active filter (and by analogy DSP filters) work by using some sort of 
feedback, but an infinite brick wall response [i.e. the response of the 
filter goes truly to zero from unity with no transition frequency band] 
requires infinite gain / bandwidth in the loop [as defined in servo 
theory] - not something we've yet learned to do ;)


Cheers

PeteS

PeteS wrote:

   ...

> A passive filter can never approach 'brick wall' response for a number 
> of reasons, so we turn to active filters and DSP techniques. I'll admit 
> we can get pretty good response, but no more. (I say that as someone who 
> has designed passive filters).

   ...

The same techniques that allow engineers to make 100 ms audio L-C delay 
lines also allows them to make some rather sharp audio filters. Think 
how sharp a 5 KHz lowpass you could make with a 100 ms impulse response 
shaped to your design.

Bell Labs demonstrated the ling delay at the 1939 World's Fair. I got a 
trophy because I was able to put on the headphones and speak into the 
microphone without faltering. A reporter asked what special skill 
allowed a seven-year-old to succeed when adults failed. I answered, 
"Easy. I don't listen."

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

OK, I think I understand where my original thoughts were wrong.

If you hit a mechanical mass, no matter how big it is, with an impulse it
will move - maybe very small - but it WILL move.  It will oscillate in a
decaying amplitude forever.  This is an IIR filter, an RC filter, etc.

My confusion was "transition band" vs. "transition frequency" in a FIR
filter.  A "perfect" LP FIR has a perfect (zero width) transition band,
infinite length and a sinc function that "oscillates" forever.

What my analogy of the infinite mass incorrectly represented is NOT the
sinc function - it is Fc, the transistion frequency.  As the mass goes to
infinity Fc goes to zero.  At infinite mass (and zero Fc) the integration
becomes a flat line - there is no change.

Fc vs. transition band.  THAT was my error.

Do I understand it now?

jeff227 wrote:
> OK, I think I understand where my original thoughts were wrong.
> 
> If you hit a mechanical mass, no matter how big it is, with an impulse it
> will move - maybe very small - but it WILL move.  It will oscillate in a
> decaying amplitude forever.  This is an IIR filter, an RC filter, etc.

No, You would need some kind of spring to make it turn around. If the 
mass is isolated and there is no friction to slow it down, it will move 
at constant speed. The dimension of mechanical impulse is force x time. 
If you work it out, you will see that that is the same as mass x . 
velocity, and that is momentum. (f = ma; ft = mat = mv.)

> My confusion was "transition band" vs. "transition frequency" in a FIR
> filter.  A "perfect" LP FIR has a perfect (zero width) transition band,
> infinite length and a sinc function that "oscillates" forever.

Why FIR? What would a "perfect" IIR be, or a "perfect" T-section cascade 
with m-derived end sections?

> What my analogy of the infinite mass incorrectly represented is NOT the
> sinc function - it is Fc, the transistion frequency.  As the mass goes to
> infinity Fc goes to zero.  At infinite mass (and zero Fc) the integration
> becomes a flat line - there is no change.
> 
> Fc vs. transition band.  THAT was my error.
> 
> Do I understand it now?

I don't think so. An infinite mass will move at infinitesimal speed. 
Conservation of momentum would be violated otherwise.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

PeteS wrote:
> Rune Allnor wrote:
>
> [1] I have a BA in mathematics, and it was stated to me that 'one gets a
> BA because mathematics is not science, it is one of the arts'.
>

Hello Pete, whether it is a BA or a BS in math is up to the university
or the underwriting authority. Many schools offer both with the BA
trading a few hours of math for a few hours of a language such as
Spanish, French, German, etc.

IIRC, Newton called Mathematics the Queen of the sciences.

Clay

>jeff227 wrote:

>> If you hit a mechanical mass, no matter how big it is, with an impulse
it
>> will move - maybe very small - but it WILL move.  It will oscillate in
a
>> decaying amplitude forever.  This is an IIR filter, an RC filter, etc.
>
>No, You would need some kind of spring to make it turn around. 

Oy, I am struggling to get this out.  Yes, YES!  It would need a lossless
spring or swing like a pendulum in a vacuum, etc.


>> My confusion was "transition band" vs. "transition frequency" in a FIR
>> filter.  A "perfect" LP FIR has a perfect (zero width) transition
band,
>> infinite length and a sinc function that "oscillates" forever.
>
>Why FIR? What would a "perfect" IIR be, or a "perfect" T-section cascade

>with m-derived end sections?

I was comparing to FIR because the impulse response is easy to see by
simply graphing the coefficients.  I have a much harder time visualizing
the response of an infinite order IIR.


>> Do I understand it now?
>
>I don't think so. An infinite mass will move at infinitesimal speed. 
>Conservation of momentum would be violated otherwise.

OK, agree.  Then what would the impulse response of a fictional, Fc = 0
FIR look like?  Would the first tap be "1" and all the rest zero?

Clay wrote:
> PeteS wrote:
>> Rune Allnor wrote:
>>
>> [1] I have a BA in mathematics, and it was stated to me that 'one gets a
>> BA because mathematics is not science, it is one of the arts'.
>>
> 
> Hello Pete, whether it is a BA or a BS in math is up to the university
> or the underwriting authority. Many schools offer both with the BA
> trading a few hours of math for a few hours of a language such as
> Spanish, French, German, etc.
> 
> IIRC, Newton called Mathematics the Queen of the sciences.

Calculus should be part of every liberal arts curriculum. Innumeracy is 
a branch of illiteracy.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

jeff227 wrote:
>> jeff227 wrote:
> 
>>> If you hit a mechanical mass, no matter how big it is, with an impulse
> it
>>> will move - maybe very small - but it WILL move.  It will oscillate in
> a
>>> decaying amplitude forever.  This is an IIR filter, an RC filter, etc.
>> No, You would need some kind of spring to make it turn around. 
> 
> Oy, I am struggling to get this out.  Yes, YES!  It would need a lossless
> spring or swing like a pendulum in a vacuum, etc.
> 
> 
>>> My confusion was "transition band" vs. "transition frequency" in a FIR
>>> filter.  A "perfect" LP FIR has a perfect (zero width) transition
> band,
>>> infinite length and a sinc function that "oscillates" forever.
>> Why FIR? What would a "perfect" IIR be, or a "perfect" T-section cascade
> 
>> with m-derived end sections?
> 
> I was comparing to FIR because the impulse response is easy to see by
> simply graphing the coefficients.  I have a much harder time visualizing
> the response of an infinite order IIR.
> 
> 
>>> Do I understand it now?
>> I don't think so. An infinite mass will move at infinitesimal speed. 
>> Conservation of momentum would be violated otherwise.
> 
> OK, agree.  Then what would the impulse response of a fictional, Fc = 0
> FIR look like?  Would the first tap be "1" and all the rest zero?

An Fc = 0 filter of any type has no response at all. The analogy to mass 
has led you astray. It's about as relevant here as phlogiston is to 
chemistry. A plausible but misleading analog.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:

(snip)

> Only very vaguely. The analogy isn't close enough to draw conclusions 
> from because a mass is not even close to a brick-wall filter. Two ways 
> to model mechanical objects (springs. masses, dashpots) as electrical 
> components make a mass either an inductor or a capacitor. One inductor 
> does not a filter make, at least not a filter with sharp cutoff.

Inductor or capacitor depending on whether current or voltage is
the analog of displacement.

It reminds me of an undergrad physics demonstration (when I was
in college).   The goal was to make an analogy between shorted or
open transmissions lines and closed or open air filled tubes.

With an oscilloscope in the transmission line and a microphone
and oscilloscope on the air tube, the result came out backwards.

The next week the same demonstration came out, but with a current
probe on the oscilloscope so that the analogy would be correct.

-- glen

Jerry Avins wrote:

(snip)

> But not *every* LPF. Your analogy depends on a "perfect" lowpass, which 
> an inductor is not. 

Superconducting inductors are pretty close, though.

-- glen