Windowed sinc

  When making a filter for a time domain interpolation of a signal using 
a windowed sinc kernel, what should be the right way of aligning the 
window towards sinc?

I can think of three possibilities:

  1) The center of the window could be set at 0.
  2) The center of the window could be set to the value of the time 
domain shift of the interpolator.
  3) The center if the window could be aligned to the "center of mass" 
of the set of sinc coefficients.

What is the best option ?

Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

On Sep 2, 7:55&#4294967295;pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> &#4294967295; When making a filter for a time domain interpolation of a signal using
> a windowed sinc kernel, what should be the right way of aligning the
> window towards sinc?
>
> I can think of three possibilities:
>
> &#4294967295; 1) The center of the window could be set at 0.
> &#4294967295; 2) The center of the window could be set to the value of the time
> domain shift of the interpolator.
> &#4294967295; 3) The center if the window could be aligned to the "center of mass"
> of the set of sinc coefficients.
>
> What is the best option ?
>
> Vladimir Vassilevsky
> DSP and Mixed Signal Design Consultanthttp://www.abvolt.com

I would align the center of the window with the center of the sinc
then sample as necessary.

Dale B. Dalrymple

On 9/2/2010 7:55 PM, Vladimir Vassilevsky wrote:
>
> When making a filter for a time domain interpolation of a signal using a
> windowed sinc kernel, what should be the right way of aligning the
> window towards sinc?
>
> I can think of three possibilities:
>
> 1) The center of the window could be set at 0.
> 2) The center of the window could be set to the value of the time domain
> shift of the interpolator.
> 3) The center if the window could be aligned to the "center of mass" of
> the set of sinc coefficients.
>
> What is the best option ?
>
> Vladimir Vassilevsky
> DSP and Mixed Signal Design Consultant
> http://www.abvolt.com
>
>

Vlad,

Not sure I understand the question - although I probably do well 
understand the answer!

I get that you want to use a windowed sinc to interpolate a signal in time.

What I don't get is "aligning the window towards sinc"... ??

Also, I presume that you have a windowed sinc that is sampled at a rate 
higher than the time signal sequence.  Thus it behaves as an 
interpolator when convolved.

The original "window" is a perfect lowpass filter ("brick wall") and, 
thus, the windowing function serves to smooth the edges of the brick 
wall, maybe a lot.  So the window and the lowpass filter functions are 
centered at f=0.  I don't think it's an option.

Help me understand better and I'll try to do better as well.

Fred

On Thu, 2 Sep 2010 20:19:55 -0700 (PDT), dbd <dbd@ieee.org> wrote:

>On Sep 2, 7:55=A0pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
>> =A0 When making a filter for a time domain interpolation of a signal usin=
>g
>> a windowed sinc kernel, what should be the right way of aligning the
>> window towards sinc?
>>
>> I can think of three possibilities:
>>
>> =A0 1) The center of the window could be set at 0.
>> =A0 2) The center of the window could be set to the value of the time
>> domain shift of the interpolator.
>> =A0 3) The center if the window could be aligned to the "center of mass"
>> of the set of sinc coefficients.
>>
>> What is the best option ?
>>
>> Vladimir Vassilevsky
>> DSP and Mixed Signal Design Consultanthttp://www.abvolt.com
>
>I would align the center of the window with the center of the sinc
>then sample as necessary.
>
>Dale B. Dalrymple

I think I agree with Dale, but there's an easy experiment to sort it
out.

Since the effect of the multiplication of the window in the time
domain is convolution in the frequency domain, in order to assess the
effect of the positions of the window just take the FFT of the
combinations you're interested in.   Intuitively from this perspective
the window can easily be applied centered over the "window" of
non-zero sinc samples.

It's then easy to move it around and see what the effects are in the
frequency domain.



Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.abineau.com

On Sep 2, 10:55&#4294967295;pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> &#4294967295; When making a filter for a time domain interpolation of a signal using
> a windowed sinc kernel, what should be the right way of aligning the
> window towards sinc?
>
> I can think of three possibilities:
>
> &#4294967295; 1) The center of the window could be set at 0.
> &#4294967295; 2) The center of the window could be set to the value of the time
> domain shift of the interpolator.
> &#4294967295; 3) The center if the window could be aligned to the "center of mass"
> of the set of sinc coefficients.
>
> What is the best option ?

i'm sorta with Fred.

Vlad, is it reasonable to expect that interpolating the time-reversed
signal should result in the same as what you get when interpolating
the original signal, except time reversed?

i don't get why any interpolation wouldn't be symmetrical and constant
delay (w.r.t. frequency), unless maybe you wanted less delay for some
frequencies.  but i don't really get the unsymmetrical interpolater
response.

r b-j

dbd wrote:
> On Sep 2, 7:55 pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> 
>>  When making a filter for a time domain interpolation of a signal using
>>a windowed sinc kernel, what should be the right way of aligning the
>>window towards sinc?
>>
>>I can think of three possibilities:
>>
>>  1) The center of the window could be set at 0.
>>  2) The center of the window could be set to the value of the time
>>domain shift of the interpolator.
>>  3) The center if the window could be aligned to the "center of mass"
>>of the set of sinc coefficients.
>>
>>What is the best option ?
>>
> I would align the center of the window with the center of the sinc
> then sample as necessary.
> 
> Dale B. Dalrymple

I.e. the option (2). That was my thought, too; however there is a 
subtlety. Let's say we have signal sampled at the points -3, -2, -1, 0, 
1, 2, 3 and we need to get the interpolated value at the point 0.5. So 
we calculate the coefficients as Window(x) sinc(x) centered at 0.5. We 
have 3 filter coefficients at the right from 0.5, and 4 coefficients at 
the left from 0.5. I can either make the window of the length = 6 and 
drop the extra coefficient from the left, or make the window of the 
length = 7, which is not going to be symmetric. When working with the 
windows of the length ~ 10, the +/- one coefficient could make 
substantial difference.

VLV

Fred Marshall  <fmarshall_xremove_the_xs@xacm.org> wrote:

>On 9/2/2010 7:55 PM, Vladimir Vassilevsky wrote:

>> When making a filter for a time domain interpolation of a signal using a
>> windowed sinc kernel, what should be the right way of aligning the
>> window towards sinc?

>> I can think of three possibilities:

>> 1) The center of the window could be set at 0.
>> 2) The center of the window could be set to the value of the time domain
>> shift of the interpolator.
>> 3) The center if the window could be aligned to the "center of mass" of
>> the set of sinc coefficients.
>>
>> What is the best option ?

>Not sure I understand the question - although I probably do well 
>understand the answer!

I am opposite; I understand the question, but do not know the answer
and would have to simulate it.

To me, (1) only makes sense if the effect of the windowing is to
interpolate from an odd number of points.

(2) is how I would initially choose to do it, and (3) is interesting 
and I would want to see the results.


Steve

On Sep 2, 9:18&#4294967295;pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> ...
> there is a
> subtlety. Let's say we have signal sampled at the points -3, -2, -1, 0,
> 1, 2, 3 and we need to get the interpolated value at the point 0.5. So
> we calculate the coefficients as Window(x) sinc(x) centered at 0.5. We
> have 3 filter coefficients at the right from 0.5, and 4 coefficients at
> the left from 0.5. I can either make the window of the length = 6 and
> drop the extra coefficient from the left, or make the window of the
> length = 7, which is not going to be symmetric. When working with the
> windows of the length ~ 10, the +/- one coefficient could make
> substantial difference.
>
> VLV

The continuous weighted sinc designed waveform is symmetric. Your
interpolator samples can only be symmetric if you are calculating the
output value at the time of an input sample or at a time half way
between input sample times.

Dale B. Dalrymple

On Sep 3, 1:29&#4294967295;am, dbd <d...@ieee.org> wrote:
> On Sep 2, 9:18&#4294967295;pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
>
> > ...
> > there is a
> > subtlety. Let's say we have signal sampled at the points -3, -2, -1, 0,
> > 1, 2, 3 and we need to get the interpolated value at the point 0.5. So
> > we calculate the coefficients as Window(x) sinc(x) centered at 0.5. We
> > have 3 filter coefficients at the right from 0.5, and 4 coefficients at
> > the left from 0.5. I can either make the window of the length = 6 and
> > drop the extra coefficient from the left, or make the window of the
> > length = 7, which is not going to be symmetric. When working with the
> > windows of the length ~ 10, the +/- one coefficient could make
> > substantial difference.
>
> The continuous weighted sinc designed waveform is symmetric.

at least it *should* be.  if you want to have the property of "reverse
time invariance".  Vlad, i asked you a question.  care to address it?

> Your
> interpolator samples can only be symmetric if you are calculating the
> output value at the time of an input sample or at a time half way
> between input sample times.

not if you're exactly half way *and* you are using 4 coefficients on
the left and 3 coefficients on the right.

r b-j

robert bristow-johnson wrote:

> On Sep 3, 1:29 am, dbd <d...@ieee.org> wrote:
> 
>>On Sep 2, 9:18 pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
>>
>>
>>>...
>>>there is a
>>>subtlety. Let's say we have signal sampled at the points -3, -2, -1, 0,
>>>1, 2, 3 and we need to get the interpolated value at the point 0.5. So
>>>we calculate the coefficients as Window(x) sinc(x) centered at 0.5. We
>>>have 3 filter coefficients at the right from 0.5, and 4 coefficients at
>>>the left from 0.5. I can either make the window of the length = 6 and
>>>drop the extra coefficient from the left, or make the window of the
>>>length = 7, which is not going to be symmetric. When working with the
>>>windows of the length ~ 10, the +/- one coefficient could make
>>>substantial difference.
>>
>>The continuous weighted sinc designed waveform is symmetric.

Continuous has to be sampled to make a filter, hence the coefficients 
are generally not going to be symmetric.

> at least it *should* be.  if you want to have the property of "reverse
> time invariance".  Vlad, i asked you a question.  care to address it?

"Vlad, is it reasonable to expect that interpolating the time-reversed
signal should result in the same as what you get when interpolating
the original signal, except time reversed?"

If the filter is symmetric, yes. Otherwise the filter has to be time 
reversed also. What is the point?

>>Your
>>interpolator samples can only be symmetric if you are calculating the
>>output value at the time of an input sample or at a time half way
>>between input sample times.
> 
> not if you're exactly half way *and* you are using 4 coefficients on
> the left and 3 coefficients on the right.

VLV