Minimum useful sinc length

Hello
Do any of you smart, crusty old types have a rule of thumb for figuring
out the smallest length for which a sinc interpolator is more effective
than something like a linear (or in my 2D case, bilinear) interpolator? 
Ie, "n samples works when there are k periods in that length n window", or
something along that line.  I suppose for the moment it is reasonable to
say that the data is bandlimited.

On Jun 4, 5:46&#4294967295;pm, "alexryu" <ryu.a...@gmail.com> wrote:
..
> Do any of you smart, crusty old types have a rule of thumb for figuring
> out the smallest length for which a sinc interpolator is more effective
> than something like a linear (or in my 2D case, bilinear) interpolator?
> Ie, "n samples works when there are k periods in that length n window", or
> something along that line. &#4294967295;I suppose for the moment it is reasonable to
> say that the data is bandlimited.

i think some commercial ASRC chips have had 64-tap FIR filters which
implies a sinc(x) (with the pi in it) ranging from -32 to +32.  now
it's unlikely that just windowing that sinc will be the absolute best
(in stopband attenuation which affects SNR).  designing this "sinc-
like" function (or h(t)) with something like Park-McClellan or least-
square error fit might be better than a windowed sinc(), but if you do
make it a windowed sinc, probably use the Kaiser window with that beta
factor around 5 (i don't remember exactly).

now there is also the number of places that you sample that windowed
sinc.  this means the number of different fractional delays or
phases.  now, if you linearly interpolate between neighboring discrete
delays, you probably only need to sample that at an upsample rate of
512 (32 K words total).  but linearly interpolating means you evaluate
the FIR calculation twice.  making it 128 MACs per sample-channel
instead of 64. otherwize, if you have a lot of memory to waste, you
can upsample by a factor of 512K (32 Meg words) and use "drop-sample"
interpolation (which is none at all).  then it's only 64 MACs per
sample-channel, less computation for more space.

r b-j

alexryu wrote:
> Hello
> Do any of you smart, crusty old types have a rule of thumb for
> figuring out the smallest length for which a sinc interpolator is
> more effective than something like a linear (or in my 2D case,
> bilinear) interpolator? Ie, "n samples works when there are k periods
> in that length n window", or something along that line.  I suppose
> for the moment it is reasonable to say that the data is bandlimited.

I think it's important to focus on the interpolation factor and the 
underlying data / sample rate.  You probably know a lot of this but just to 
be complete:

If the data is bandlimited and the band limit is related to the original 
sample rate then sincs can be used to "reconstruct" which means infinite 
interpolation factor.
- Convolution in time with a sinc.
- Perfect lowpass filter of the samples in frequency.
Now, if you're doing a practical version of this then the sincs will be 
time-limited and your question surely applies.
Let's start with the idea of a localized interpolation about a trio of 
samples:
- each of the samples in the entire sequence contribute a sinc component in 
between the samples to be interpolated.
- if the distant samples are equal for some reason then their respective 
sinc contributions will largely cancel because the sidelobes alternate in 
sign at the frequency of the samples.
So, you might want to decide how much energy is distant and how much 
difference distant samples should really make.  To get a feel for this, take 
a couple of adjacent sincs and add them together to see what the residual 
is.  Of course, if there's an antisymmetric doublet of samples then the sinc 
sidelobe magnitudes will add.

Perhaps that's not what you have in mind.  So, here's another case:
Let's say that the interpolation factor is 2 - meaning you want to double 
the sample rate.
A common way of doing this in frequency is to repeat the frequency samples 
so there are twice as many and then lowpass filter the result to take out 
the "middle" part at the old fs or the new fs/2.
The time domain equivalent is to intersperse the samples with zeros and then 
lowpass the result to get interpolated values in place of the zero values. 
This could be done by convolving with a sinc-like sequence which is at the 
new sample rate.  The filter unit sample response looks like this:

                o
                |
                |
                |
                |
                |             <
                |
                |
              o | o
              | | |
      o       | | |       o
      |       | | |       |
    o---o---o---+---o---o---o-
          | time ->   |
          o           o

This is the unit sample response of a halfband filter.  The length of the 
filter can be anything you want.  Note that a length 3 halfband filter also 
looks like a 1/2  1  1/2 filter which is a linear interpolator - when you're 
"on" the sample it's weighted by 1, when you're "off" one of the original 
samples the output is the sum of 1/2 X each of the adjacent samples.

There's a halfband filter .exe program at:
ftp://ftp.mission-systems-inc.com/outgoing/Halfband/

email me for login.

Fred

alexryu wrote:

> Hello
> Do any of you smart, crusty old types have a rule of thumb for
> figuring out the smallest length for which a sinc interpolator
> is more effective than something like a linear (or in my 2D
> case, bilinear) interpolator?

This article has a formula for the minimum length as a function of 
passband and transition widths, attenuation, and the specific applied 
window's parameter:

http://www.hindawi.com/getarticle.aspx?doi=10.1155/asp.2005.1910

Also of possible interest:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.79.2898
http://dx.doi.org/10.1109/42.932749


Martin

-- 
Quidquid latine scriptum est, altum videtur.

alexryu wrote:
> Hello
> Do any of you smart, crusty old types have a rule of thumb for figuring
> out the smallest length for which a sinc interpolator is more effective
> than something like a linear (or in my 2D case, bilinear) interpolator? 
> Ie, "n samples works when there are k periods in that length n window", or
> something along that line.  I suppose for the moment it is reasonable to
> say that the data is bandlimited.

I do, but I would need to be addressed a little more 
respectfully before I considered sharing it with you.

On Jun 5, 8:58&#4294967295;pm, John Monro <johnmo...@optusnet.com.au> wrote:
> alexryu wrote:
...
> > Do any of you smart, crusty old types have a rule of thumb for figuring
> > out ...
...
> I do, but I would need to be addressed a little more
> respectfully before I considered sharing it with you.

are you serious, John?  this wasn't bad.

i've seen worse.



     ... here at comp.dsp.

r b-j

robert bristow-johnson wrote:
> On Jun 5, 8:58 pm, John Monro <johnmo...@optusnet.com.au> wrote:
>> alexryu wrote:
> ...
>>> Do any of you smart, crusty old types have a rule of thumb for figuring
>>> out ...
> ...
>> I do, but I would need to be addressed a little more
>> respectfully before I considered sharing it with you.
> 
> are you serious, John?  this wasn't bad.
> 
> i've seen worse.
> 
> 
> 
>      ... here at comp.dsp.
> 
> r b-j

You're quite right Robert, it wasn't bad at all.

It did irritate me a little though -- enough for me to 
decide not to contribute.  I thought I would be a good idea 
to tell alexryu why, but maybe I could have phrased it in a 
way that was more friendly to the young whippersnapper :=)

Regards,
John

John Monro wrote:
> robert bristow-johnson wrote:
>> On Jun 5, 8:58 pm, John Monro <johnmo...@optusnet.com.au> wrote:
>>> alexryu wrote:
>> ...
>>>> Do any of you smart, crusty old types have a rule of thumb for
>>>> figuring out ...
>> ...

I just thought it was a nod to the likelihood that some of us might be .....

Otherwise, to me it was metaphorical where "smart, crusty old" just referred 
to "wise" or "experienced" which need not really mean "old".

Fred 

I would like to thank everyone for their helpful advice, and apologize to
those smooth-skinned, clear-eyed practitioners who possess wisdom far
beyond their years that I might have inadvertently offended.

alexryu wrote:
> I would like to thank everyone for their helpful advice, and
> apologize to those smooth-skinned, clear-eyed practitioners who
> possess wisdom far beyond their years that I might have inadvertently
> offended.

Or "thin skinned", eh?  :-)