interpolating every 4th sample

Assume I have a 48Khz samplerate audio stream, and every fourth sample is
lost.  The original material
was filtered to about 22KHz.

I was surprised to see that every 4th sample lost causes an alias around
6KHz as well as 12KHz.

What I want to do is to "guess" a new value for the missing sample that will
minimize the error components,
especially those aliased around 6KHz.  I know there is no perfect solution.

What is an optimal solution (besides better error correction) to conceal
this error?

Bruce

On 1 Mrz., 08:55, "doh ..." <s...@nospam.com> wrote:
> Assume I have a 48Khz samplerate audio stream, and every fourth sample is
> lost.  The original material
> was filtered to about 22KHz.
>
> I was surprised to see that every 4th sample lost causes an alias around
> 6KHz as well as 12KHz.
>
> What I want to do is to "guess" a new value for the missing sample that will
> minimize the error components,
> especially those aliased around 6KHz.  I know there is no perfect solution.
>
> What is an optimal solution (besides better error correction) to conceal
> this error?

It certainly isn't optimal, but why not use polynomial interpolation,
using some neighbours from each side? For example, using two
neighbours on each side gives you a third order polynomial
interpolation function. Try out different orders to see which gives
you best result for your signal.

Another approach would be to use a periodic function as interpolation
kernel. Again using the same number of neighbours from each side,
compute the periodic function that passes through the neighbour
values.

Regards,
Andor

Andor wrote:


>>Assume I have a 48Khz samplerate audio stream, and every fourth sample is
>>lost.  The original material was filtered to about 22KHz.
>>
>>What I want to do is to "guess" a new value for the missing sample that will
>>minimize the error components,
>>especially those aliased around 6KHz.  I know there is no perfect solution.
>>
> It certainly isn't optimal, but why not use polynomial interpolation,
> using some neighbours from each side? For example, using two
> neighbours on each side gives you a third order polynomial
> interpolation function. Try out different orders to see which gives
> you best result for your signal.

The polynomial interpolation is simple, however the results will be 
wildly wrong at the high frequencies. The interpolation using LPC or 
Kalman filter is much better for this application.


Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

doh ... wrote:
> Assume I have a 48Khz samplerate audio stream, and every fourth sample is
> lost.  The original material
> was filtered to about 22KHz.
> 
> I was surprised to see that every 4th sample lost causes an alias around
> 6KHz as well as 12KHz.
> 
> What I want to do is to "guess" a new value for the missing sample that will
> minimize the error components,
> especially those aliased around 6KHz.  I know there is no perfect solution.
> 
> What is an optimal solution (besides better error correction) to conceal
> this error?

Who stole your samples? Replacing every fourth sample with zero amounts 
to multiplying -- modulating, that is -- by the the signal 11101110.... 
Modulation theory predicts the results you see, so it oughtn't surprise 
you after thinking on it.

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

Jerry Avins wrote:


>> Assume I have a 48Khz samplerate audio stream, and every fourth sample is
>> lost.

> Who stole your samples?

The professor, probably...


VLV

On 1 Mrz., 15:15, Vladimir Vassilevsky <antispam_bo...@hotmail.com>
wrote:
> Andor wrote:
> >>Assume I have a 48Khz samplerate audio stream, and every fourth sample is
> >>lost.  The original material was filtered to about 22KHz.
>
> >>What I want to do is to "guess" a new value for the missing sample that will
> >>minimize the error components,
> >>especially those aliased around 6KHz.  I know there is no perfect solution.
>
> > It certainly isn't optimal, but why not use polynomial interpolation,
> > using some neighbours from each side? For example, using two
> > neighbours on each side gives you a third order polynomial
> > interpolation function. Try out different orders to see which gives
> > you best result for your signal.
>
> The polynomial interpolation is simple, however the results will be
> wildly wrong at the high frequencies. The interpolation using LPC or
> Kalman filter is much better for this application.

The reason I didn't mention LPC was that the prediction equations are
seriously disturbed by the zeros. It's well possible that the zeros
get modeled right into the data, and the LPC extrapolation for the
deleted sample will be close to zero again (I've seen this happen in a
similar case).

For very short segments such as one single sample, polynomial
interoplation usually works quite well for audio. If the OP is not
happy with the results, I would go for the periodic interpolation,
which can incorporate much more samples than polynomial interpolation,
and gives a pretty good bandlimited approximation for the missing data
points.

Regards,
Andor

Andor wrote:


>>>>Assume I have a 48Khz samplerate audio stream, and every fourth sample is
>>>>lost.  The original material was filtered to about 22KHz.

>>>It certainly isn't optimal, but why not use polynomial interpolation,
>>>using some neighbours from each side?

>>The polynomial interpolation is simple, however the results will be
>>wildly wrong at the high frequencies. The interpolation using LPC or
>>Kalman filter is much better for this application.
> 

> The reason I didn't mention LPC was that the prediction equations are
> seriously disturbed by the zeros.

It is probably possible to modify the equations so zeroes will be 
ignored. This would be a good exercise in math.

> It's well possible that the zeros
> get modeled right into the data, and the LPC extrapolation for the
> deleted sample will be close to zero again (I've seen this happen in a
> similar case).

Instead of zeroes, the interpolated data from the past can be used for 
the prediction. There is a certain potential for ill behavior in the 
singular cases, however it is unlikely with the realistic audio signals.


> For very short segments such as one single sample, polynomial
> interoplation usually works quite well for audio.

That's what they do in the CD players.

> If the OP is not
> happy with the results, I would go for the periodic interpolation,
> which can incorporate much more samples than polynomial interpolation,
> and gives a pretty good bandlimited approximation for the missing data
> points.

Agreed. The interpolation in the frq. domain is pretty straightforward 
and the approximation would be quite good.


VLV

On 1 Mrz., 16:06, Vladimir Vassilevsky <antispam_bo...@hotmail.com>
wrote:
> Andor wrote:
> >>>>Assume I have a 48Khz samplerate audio stream, and every fourth sample is
> >>>>lost.  The original material was filtered to about 22KHz.
> >>>It certainly isn't optimal, but why not use polynomial interpolation,
> >>>using some neighbours from each side?
> >>The polynomial interpolation is simple, however the results will be
> >>wildly wrong at the high frequencies. The interpolation using LPC or
> >>Kalman filter is much better for this application.
>
> > The reason I didn't mention LPC was that the prediction equations are
> > seriously disturbed by the zeros.
>
> It is probably possible to modify the equations so zeroes will be
> ignored. This would be a good exercise in math.

That would limit you to prediction order 2 or smaller. Larger order
predictors necessarily contain zero samples in any prediction
equation. If a second order predictor is used, each batch of 3
consecutive non-stolen samples yiels an equation, and one can use the
standard least-squares to compute the prediction coefficients. It
remains that a second-order predictor probably doesn't do much better
than polynomial interpolation.

>
> > It's well possible that the zeros
> > get modeled right into the data, and the LPC extrapolation for the
> > deleted sample will be close to zero again (I've seen this happen in a
> > similar case).
>
> Instead of zeroes, the interpolated data from the past can be used for
> the prediction. There is a certain potential for ill behavior in the
> singular cases, however it is unlikely with the realistic audio signals.

Yes, an iterative approach could be possible. First interpolate using
low-order predictor or polynomials, then use higher-order predictors.

>
> > For very short segments such as one single sample, polynomial
> > interoplation usually works quite well for audio.
>
> That's what they do in the CD players.

I thought CD players sample-and-hold ...

>
> > If the OP is not
> > happy with the results, I would go for the periodic interpolation,
> > which can incorporate much more samples than polynomial interpolation,
> > and gives a pretty good bandlimited approximation for the missing data
> > points.
>
> Agreed. The interpolation in the frq. domain is pretty straightforward
> and the approximation would be quite good.

Yes, it's probably the best way to proceed for this problem.

Regards,
Andor

Andor wrote:


>>>The reason I didn't mention LPC was that the prediction equations are
>>>seriously disturbed by the zeros.
>>
>>It is probably possible to modify the equations so zeroes will be
>>ignored. This would be a good exercise in math.
> 
> 
> That would limit you to prediction order 2 or smaller. Larger order
> predictors necessarily contain zero samples in any prediction
> equation.

Why it has to be that way? It seems possible to restate the linear 
prediction problem so the every 4-th sample will be skipped.

VLV

Vladimir Vassilevsky wrote:
> 
> 
> Andor wrote:
> 
> 
>>>> The reason I didn't mention LPC was that the prediction equations are
>>>> seriously disturbed by the zeros.
>>>
>>> It is probably possible to modify the equations so zeroes will be
>>> ignored. This would be a good exercise in math.
>>
>>
>> That would limit you to prediction order 2 or smaller. Larger order
>> predictors necessarily contain zero samples in any prediction
>> equation.
> 
> Why it has to be that way? It seems possible to restate the linear 
> prediction problem so the every 4-th sample will be skipped.

Certainly. And a polynomial interpolation that uses the three samples on 
each side of a stolen one is probably adequate and simple enough so it's 
probably what the klepto-professor had in mind.

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