Theory 101 - Impulse response of "perfect" LPF

Jeff

When did sinc(0) become 0?

Dirk

jeff227 wrote:
> >Jerry
> >You are imagining the wrong model...
> >It's no surprise that an analogiy based on a flawed model yields a
> >flawed conclusion.
>
>
>
> OK, got it.  However I believe this "mass model" analogy IS correct - it
> has an Fc of zero.  "Sinc" of Fc zero is zero.
>
> Useless information - just trying to gain an understanding of principles.
> 
> Thank you for the replies.

>When did sinc(0) become 0?


If Fc is zero then sin(2*PI*Fc*i)/PI*i is zero for any i > 0.


Rune;

The reason I had "infinite" in my mechanical analogy is because the
"perfect" LPF requires an infinite series (e.g., sinc).  I was trying to
compare the two.  I was trying to understand why in the mechanical case
the mass doesn't move but in the DSP case it creates a sine function.

When you said, "because of the P in LPF" it began to make sense.

I think I answered my own question when I suggested that the mechanical
example has an Fc of zero - there is no passband.  In that case the sinc
value (of Fc = 0) would be a constant, not a sine wave, and the analogy
would make sense.  (Or not?)

Yes I do often try to think outside the box to bring about an
understanding.  When I studied higher math in college many (many) years
ago it was purely academic - no one explained how it was related to
anything.  Much of it made absolutely no sense logically but I learned how
to "solve" the equations and got through it.

DSP is not my occupation, it's an interest.  I am learning as I can from
reading books, forums, etc.  Sometimes, however, I just have to ask a
question (or two or three).

Thank you very much, Rune and everyone, for all your comments.

jeff227 skrev:
> >When did sinc(0) become 0?
>
>
> If Fc is zero then sin(2*PI*Fc*i)/PI*i is zero for any i > 0.
>
>
> Rune;
>
> The reason I had "infinite" in my mechanical analogy is because the
> "perfect" LPF requires an infinite series (e.g., sinc).  I was trying to
> compare the two.  I was trying to understand why in the mechanical case
> the mass doesn't move but in the DSP case it creates a sine function.

You have just touched on the main reason why I don't like
analogies: If one gets them wrong, they confuse way more
than they help.

> When you said, "because of the P in LPF" it began to make sense.
>
> I think I answered my own question when I suggested that the mechanical
> example has an Fc of zero - there is no passband.

Exactly.

> In that case the sinc
> value (of Fc = 0) would be a constant, not a sine wave, and the analogy
> would make sense.  (Or not?)

I don't like analogies; I try to avoid them as best I can for the
very reason that they cause confusion, which seems to be
the case here.

> Yes I do often try to think outside the box to bring about an
> understanding.  When I studied higher math in college many (many) years
> ago it was purely academic - no one explained how it was related to
> anything.

Some times there is nothing to explain. Mathematics is an
abstract discipline that exists on its own terms. Accept that,
and you are half-way to undertsand DSP. I have seen too
many good ideas go down the drains because people insisted
on obtaining a "physical" understanding of mathematical concepts.

> Much of it made absolutely no sense logically but I learned how
> to "solve" the equations and got through it.

What do you mean by "logically"? Maths is logic. Maths
might not be *intuitive*, though.

> DSP is not my occupation, it's an interest.  I am learning as I can from
> reading books, forums, etc.  Sometimes, however, I just have to ask a
> question (or two or three).

Sure. Asking questions is one of the greatest ways to learn.

> Thank you very much, Rune and everyone, for all your comments.

Y're welcome.

Rune

jeff227 wrote:
> Wrong!  A spring, damper and mass is EXACTLY equivalent to a LPF.  In this
> example the spring and damper are zero and the mass is infinite.
> 
> Before you insult my intelligence take some time to think outside the
> box.
> 

Wot I don't understand is this: if the mass is infinite, that means it 
must be >= sizeof(TheUniverse), in which case, how is there anything 
left to hit it with?

Richard Dobson

That's not a sinc function.

sinc(x)=sin(x)/x for x!=0
         = 1 for x=0, which is the limit of sin(x)/x as x->0

Dirk
.

jeff227 wrote:
> >When did sinc(0) become 0?
>
>
> If Fc is zero then sin(2*PI*Fc*i)/PI*i is zero for any i > 0.
>
>
> Rune;
>
> The reason I had "infinite" in my mechanical analogy is because the
> "perfect" LPF requires an infinite series (e.g., sinc).  I was trying to
> compare the two.  I was trying to understand why in the mechanical case
> the mass doesn't move but in the DSP case it creates a sine function.
>
> When you said, "because of the P in LPF" it began to make sense.
>
> I think I answered my own question when I suggested that the mechanical
> example has an Fc of zero - there is no passband.  In that case the sinc
> value (of Fc = 0) would be a constant, not a sine wave, and the analogy
> would make sense.  (Or not?)
>
> Yes I do often try to think outside the box to bring about an
> understanding.  When I studied higher math in college many (many) years
> ago it was purely academic - no one explained how it was related to
> anything.  Much of it made absolutely no sense logically but I learned how
> to "solve" the equations and got through it.
>
> DSP is not my occupation, it's an interest.  I am learning as I can from
> reading books, forums, etc.  Sometimes, however, I just have to ask a
> question (or two or three).
> 
> Thank you very much, Rune and everyone, for all your comments.

jeff227 wrote:
>> When did sinc(0) become 0?
> 
> 
> If Fc is zero then sin(2*PI*Fc*i)/PI*i is zero for any i > 0.
> 
> 
> Rune;
> 
> The reason I had "infinite" in my mechanical analogy is because the
> "perfect" LPF requires an infinite series (e.g., sinc).  I was trying to
> compare the two.  I was trying to understand why in the mechanical case
> the mass doesn't move but in the DSP case it creates a sine function.
> 
> When you said, "because of the P in LPF" it began to make sense.
> 
> I think I answered my own question when I suggested that the mechanical
> example has an Fc of zero - there is no passband.  In that case the sinc
> value (of Fc = 0) would be a constant, not a sine wave, and the analogy
> would make sense.  (Or not?)

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.

   ...

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

>Wot I don't understand is this: if the mass is infinite, that means it 
>must be >= sizeof(TheUniverse), in which case, how is there anything 
>left to hit it with?


Now THAT is thinking outside the box - brilliant!!!

Cheers

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

Rune Allnor wrote:
> jeff227 skrev:
>>> When did sinc(0) become 0?
>>
>> If Fc is zero then sin(2*PI*Fc*i)/PI*i is zero for any i > 0.
>>
>>
>> Rune;
>>
>> The reason I had "infinite" in my mechanical analogy is because the
>> "perfect" LPF requires an infinite series (e.g., sinc).  I was trying to
>> compare the two.  I was trying to understand why in the mechanical case
>> the mass doesn't move but in the DSP case it creates a sine function.
> 
> You have just touched on the main reason why I don't like
> analogies: If one gets them wrong, they confuse way more
> than they help.
> 
>> When you said, "because of the P in LPF" it began to make sense.
>>
>> I think I answered my own question when I suggested that the mechanical
>> example has an Fc of zero - there is no passband.
> 
> Exactly.
> 
>> In that case the sinc
>> value (of Fc = 0) would be a constant, not a sine wave, and the analogy
>> would make sense.  (Or not?)
> 
> I don't like analogies; I try to avoid them as best I can for the
> very reason that they cause confusion, which seems to be
> the case here.
> 
>> Yes I do often try to think outside the box to bring about an
>> understanding.  When I studied higher math in college many (many) years
>> ago it was purely academic - no one explained how it was related to
>> anything.


> 
> Some times there is nothing to explain. Mathematics is an
> abstract discipline that exists on its own terms. Accept that,
> and you are half-way to undertsand DSP. I have seen too
> many good ideas go down the drains because people insisted
> on obtaining a "physical" understanding of mathematical concepts.

More strictly speaking, mathematics is a language[1]. A precise one [3], 
but a language nonetheless. It is the language one must speak fluently 
before attempting DSP; or perhaps before attempting to understand DSP :)

As with all languages, there are varying levels of proficiency and 
required levels of proficiency - one might get by with a mere nodding 
acquaintance if one is a plumber [2], but one needs a more thorough 
linguistic acquaintance to effectively navigate filter theory.

[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'.

[2] Although they tend to be very good at multiplication, especially of 
hours * billable rate.

[3] This might well be disputed, something I won't go into in this thread :)

Cheers

PeteS


> 
>> Much of it made absolutely no sense logically but I learned how
>> to "solve" the equations and got through it.
> 
> What do you mean by "logically"? Maths is logic. Maths
> might not be *intuitive*, though.
> 
>> DSP is not my occupation, it's an interest.  I am learning as I can from
>> reading books, forums, etc.  Sometimes, however, I just have to ask a
>> question (or two or three).
> 
> Sure. Asking questions is one of the greatest ways to learn.
> 
>> Thank you very much, Rune and everyone, for all your comments.
> 
> Y're welcome.
> 
> Rune
> 

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.
>

I think Jerry had it right before when he said the mass/force are a
(good)  analogy for an integrator.  The mass of the mass corresponds to
the gain of the integrator.  And an integrator is a low pass filter but
not a very good one.  The response is 6dB per octave forever.

Now the op can add a few more masses to the analogy and couple them
with some springs and some dashpots and we might get to a better low
pass filter.

Mark