> [...]
> 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
Hi Wilson,
Yes, you are correct. I made a mistake.
I guess I had a strange view of linear interpolation in mind. Assuming
we're interpolating by M, then I was thinking of a process that
*began* with a sequence of values at the upsampled rate, then computed
values m*M + 1, m*M + 2, ..., m*M + M - 1 from the points m*M and
(m+1)*M, where m is an arbitrary integer.
Normally the values m*M are the samples from the original sequence,
but in my gedanken they don't have to be. In case they are, you
get the linear interpolation we all know and love.
If, however, you shifted this sequence by 1 (and assuming the
in-between values were zero), you'd get zeros out with this
scheme. This is why I said it wasn't LTI, and thus cannot be
represented as a linear filter.
But I tend to think in strange ways...
--Randy
Reply by Andor●July 13, 20062006-07-13
ma wrote:
> "Andor" wrote in message
> news:1152784565.872351.298120@b28g2000cwb.googlegroups.com...
> >
> > 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
> >
>
> Thanks Andor,
> This is a very nice paper on 1D interpolation. Is there any paper
> similar to this one but about 2D interpolations?
"Andor" <andor.bariska@gmail.com> wrote in message
news:1152784565.872351.298120@b28g2000cwb.googlegroups.com...
>
> 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
>
Thanks Andor,
This is a very nice paper on 1D interpolation. Is there any paper
similar to this one but about 2D interpolations?
Best regards
Reply by Andor●July 13, 20062006-07-13
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
Reply by Andor●July 13, 20062006-07-13
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
Reply by Wilson●July 12, 20062006-07-12
> > > 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
Reply by Randy Yates●July 12, 20062006-07-12
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
Reply by Jerry Avins●July 12, 20062006-07-12
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.
�����������������������������������������������������������������������
Reply by Tim Wescott●July 12, 20062006-07-12
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-]
>
>
>
> 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.
�����������������������������������������������������������������������