interpolation and filtering

Hello,

    I know that one way of interpolation is low pass filtering. But I think 
any other type of interpolation (such as spile or linre ,...) can be modeled 
as a low pass filter with different frequency response. Am I wrong? if no, 
where can I read more? for example what is the frequency response of a 
spilne interpolator?



Regards

ma wrote:

> Hello,
> 
>     I know that one way of interpolation is low pass filtering. But I think 
> any other type of interpolation (such as spile or linre ,...) can be modeled 
> as a low pass filter with different frequency response. Am I wrong? if no, 
> where can I read more? for example what is the frequency response of a 
> spilne interpolator?
> 
Any kind of interpolation that you can think of can be modeled as a low 
pass filter.  The term 'frequency response' carries an implicit 
assumption of a system that's linear, and we can probably think of 
interpolation schemes that aren't linear.

I'm 99.44% sure that you could model a spline as a linear filter.  You 
can make sure by writing out the results of using a spline on your input 
data, and checking it to see if it creates a linear system for you.

-- 

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

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

On Tue, 11 Jul 2006 14:48:53 -0700, Tim Wescott <tim@seemywebsite.com>
wrote:

>ma wrote:
>
>> Hello,
>> 
>>     I know that one way of interpolation is low pass filtering. But I think 
>> any other type of interpolation (such as spile or linre ,...) can be modeled 
>> as a low pass filter with different frequency response. Am I wrong? if no, 
>> where can I read more? for example what is the frequency response of a 
>> spilne interpolator?
>> 
>Any kind of interpolation that you can think of can be modeled as a low 
>pass filter.  The term 'frequency response' carries an implicit 
>assumption of a system that's linear, and we can probably think of 
>interpolation schemes that aren't linear.
>
>I'm 99.44% sure that you could model a spline as a linear filter.  You 
>can make sure by writing out the results of using a spline on your input 
>data, and checking it to see if it creates a linear system for you.

Hi Tim,

  that "99.44%" phrase makes me think that you're 
older than I thought.  Darned few people 
remember the phrase "ninety nine and forty four 
one hundredths percent pure".

As far as I remember the product that was claimed to 
be "ninety nine and forty four one hundredths percent 
pure" would float if placed in water.

[-Rick-]

 

Rick Lyons wrote:

   ...

> As far as I remember the product that was claimed to 
> be "ninety nine and forty four one hundredths percent 
> pure" would float if placed in water.

But then, so did "Even-your-best-friends-won't-tell-you" Lifebuoy.

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

Rick Lyons wrote:

> On Tue, 11 Jul 2006 14:48:53 -0700, Tim Wescott <tim@seemywebsite.com>
> wrote:

   -- snip --

>>I'm 99.44% sure that you could model a spline as a linear filter.  You 
>>can make sure by writing out the results of using a spline on your input 
>>data, and checking it to see if it creates a linear system for you.
> 
> 
> Hi Tim,
> 
>   that "99.44%" phrase makes me think that you're 
> older than I thought.  Darned few people 
> remember the phrase "ninety nine and forty four 
> one hundredths percent pure".
> 
> As far as I remember the product that was claimed to 
> be "ninety nine and forty four one hundredths percent 
> pure" would float if placed in water.
> 
> [-Rick-]
> 
>  
> 
AFAIK they still print it on the package.

If it helps I used to read soap packages on my way back from the store 
in the 1970's, when I was too young to be driving.

-- 

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

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

Tim Wescott wrote:

   ...

> AFAIK they still print it on the package.

   ...

I just looked. The package reads "99 44/100% pure. It floats."

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

Tim Wescott wrote:
> ma wrote:
>
> > Hello,
> >
> >     I know that one way of interpolation is low pass filtering. But I think
> > any other type of interpolation (such as spile or linre ,...) can be modeled
> > as a low pass filter with different frequency response. Am I wrong? if no,
> > where can I read more? for example what is the frequency response of a
> > spilne interpolator?
> >
> Any kind of interpolation that you can think of can be modeled as a low
> pass filter.

Not true. The most basic interpolation, linear interpolation, can't be.
It's
not a LTI process (specifically, it's not time-invariant).

--Randy

> > >     I know that one way of interpolation is low pass filtering. But I think
> > > any other type of interpolation (such as spile or linre ,...) can be modeled
> > > as a low pass filter with different frequency response. Am I wrong? if no,
> > > where can I read more? for example what is the frequency response of a
> > > spilne interpolator?
> > >
> > Any kind of interpolation that you can think of can be modeled as a low
> > pass filter.
>
> Not true. The most basic interpolation, linear interpolation, can't be.
> It's
> not a LTI process (specifically, it's not time-invariant).

Linear interpolation is the equivalent of taking your data samples
(spaced by 'T' seconds) and sending them through a continuous-time
filter which has a triangular impulse response and then re-sampling the
output at the new (interpolated) sample times.

The impulse response of this continuous-time filter is:
h(t) = 1 - abs(t)/T   for abs(t) <= T; and h(t) =  0 abs(t) >= T.

At each new sampling instant, the output is a weighted combination of
only two samples, in accordance to the the linear interpolation
formula.  For the case where the new sample falls exactly on the time
index of an old sample, then the new sample equals the old sample (as
one would hope!).

The frequency response of this procedure is determined by the Fourier
Transform of h(t).
The response has a sinc() squared response.

Wilson

Randy Yates wrote:

> Tim Wescott wrote:
> > ma wrote:
> >
> > > Hello,
> > >
> > >     I know that one way of interpolation is low pass filtering. But I think
> > > any other type of interpolation (such as spile or linre ,...) can be modeled
> > > as a low pass filter with different frequency response. Am I wrong? if no,
> > > where can I read more? for example what is the frequency response of a
> > > spilne interpolator?
> > >
> > Any kind of interpolation that you can think of can be modeled as a low
> > pass filter.
>
> Not true. The most basic interpolation, linear interpolation, can't be.
> It's not a LTI process (specifically, it's not time-invariant).

Randy,

if you have a sequence x[n], and you want to generate a new sequence
y[n] where the values y[n] lie on a connecting line between x[n] and
x[n-1], this can be achieved with a simple two tap FIR filter. I guess
this is not what you meant with linear interpolation?

Regards,
Andor

ma wrote:

> Hello,
>
>     I know that one way of interpolation is low pass filtering. But I think
> any other type of interpolation (such as spile or linre ,...) can be modeled
> as a low pass filter with different frequency response. Am I wrong? if no,
> where can I read more? for example what is the frequency response of a
> spilne interpolator?

They are commonly known as "spline" interpolators. There is a paper
somewhere which discusses general polynomial interpolation filters and
their frequency responses ... <click, click> ... it moved ....  ah,
here:

http://www.biochem.oulu.fi/~oniemita/dsp/deip.pdf

Regards,
Andor