How determine the path of signal?

I am digitizing the sampled signal be using a uniform quantizer, so the
step (q) between two consective quantization levels is fixed and is "q =
Vin/2^N" where 'q' is the quantization step, 'Vin' is the total range of
quantizer and 'N' are the effective number of bits of quantizer.

Now I want to determine the path which the signal is following between two
consective quantization levels. i.e weather it is a straight line, a curve
etc.

Thanks in advance for your ideas. 

		
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qaisar wrote:
> I am digitizing the sampled signal be using a uniform quantizer, so the
> step (q) between two consective quantization levels is fixed and is "q =
> Vin/2^N" where 'q' is the quantization step, 'Vin' is the total range of
> quantizer and 'N' are the effective number of bits of quantizer.
>
> Now I want to determine the path which the signal is following between two
> consective quantization levels. i.e weather it is a straight line, a curve
> etc.
>
> Thanks in advance for your ideas.
>
>
> This message was sent using the Comp.DSP web interface on
> www.DSPRelated.com

You should arrange to take extra samples of the signal at infinitely
close time intervals and using an infinite number of quantisation
levels so that you can plot the now continuous path that the signal
takes between your original samples.

Ian

"qaisar" <alsaeed86@yahoo.com> wrote in message 
news:YqydnaulpK0gt9PeRVn-jw@giganews.com...
>I am digitizing the sampled signal be using a uniform quantizer, so the
> step (q) between two consective quantization levels is fixed and is "q =
> Vin/2^N" where 'q' is the quantization step, 'Vin' is the total range of
> quantizer and 'N' are the effective number of bits of quantizer.
>
> Now I want to determine the path which the signal is following between two
> consective quantization levels. i.e weather it is a straight line, a curve
> etc.
>
> Thanks in advance for your ideas.

I'm not sure about between quantization levels (that doesn't make sense to me), 
but between samples, it may be possible to know what path the signal took.  If 
(and this is very important) the signal is frequency-limited to < 1/2 of the 
sample rate (the Nyquist criterion), the original signal can be reconstructed 
and you can know what path it took between the samples (Google for 
reconstruction filter, sample rate conversion, or bandlimited interpolation.  I 
don't think this answers the question you asked, but maybe is a step in the 
right direction. 

qaisar wrote:
> I am digitizing the sampled signal be using a uniform quantizer, so the
> step (q) between two consective quantization levels is fixed and is "q =
> Vin/2^N" where 'q' is the quantization step, 'Vin' is the total range of
> quantizer and 'N' are the effective number of bits of quantizer.
> 
> Now I want to determine the path which the signal is following between two
> consective quantization levels. i.e whether it is a straight line, a curve
> etc.
> 
> Thanks in advance for your ideas. 

In order for samples to contain information about all of the signal, the 
interval between them must be less than half a period of the highest 
frequency component of the signal. (See "Nyquist-Shannon Sampling 
Theorem.) A bandlimited signal is limited in how fast it can change 
value or slope, and that limitation allows one to infer its value 
between samples.

The signal you deal with is quantized; in other words, you do not know 
the exact value of the signal at the sampling instants. With a perfect 
sampler, the difference between the true and quantized values can be +/- 
half a quantization step. In most cases, the difference varies randomly 
from sample to sample; in those cases, the effect of quantizing can be 
modeled as noise.

Your first paragraph makes it clear that this is a new field for you. We 
usually try not to solve homework problems, but we are glad to help.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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"qaisar" <alsaeed86@yahoo.com> wrote in message 
news:YqydnaulpK0gt9PeRVn-jw@giganews.com...
>I am digitizing the sampled signal be using a uniform quantizer, so the
> step (q) between two consective quantization levels is fixed and is "q =
> Vin/2^N" where 'q' is the quantization step, 'Vin' is the total range of
> quantizer and 'N' are the effective number of bits of quantizer.
>
> Now I want to determine the path which the signal is following between two
> consective quantization levels. i.e weather it is a straight line, a curve
> etc.
>
> Thanks in advance for your ideas.

The problem you're trying to solve is one of "interpolation".
The statement of the problem "determine the path" is fuzzy because it 
doesn't state the degree of accuracy, how many time points on the path, etc.

So, given that you have two points, all you can generate is a straight line.
Given that you have three points, all you can generate is a 2nd order curve.
and so forth.
All of these will be approximations which are good or bad to one degree or 
another.

A few of the answers deal with bandlimited situations for good reason.
If the underlying signal that was sampled was perfect (no quantization 
errors, etc.) and bandlimited to less than half the sampling frequency, then 
it can be (theoretically) perfectly reconstructed (i.e. interpolated) by 
convolving with infinite sinc functions related to the sample rate (which is 
an "assumed" bandwidth that's greater than the actual bandwidth).
Well, that is, except for the quantization noise.  You can't go backwards 
perfectly ... can't interpolate perfectly ... due to the quantization.

It's a nice thing to think about.  Beyond that, you might ask: "why do you 
care?"  "what is the application?"  From this, you can better state the 
problem and get better solutions.

Fred 

>
>"qaisar" <alsaeed86@yahoo.com> wrote in message 
>news:YqydnaulpK0gt9PeRVn-jw@giganews.com...
>>I am digitizing the sampled signal be using a uniform quantizer, so the
>> step (q) between two consective quantization levels is fixed and is "q
=
>> Vin/2^N" where 'q' is the quantization step, 'Vin' is the total range
of
>> quantizer and 'N' are the effective number of bits of quantizer.
>>
>> Now I want to determine the path which the signal is following between
two
>> consective quantization levels. i.e weather it is a straight line, a
curve
>> etc.
>>
>> Thanks in advance for your ideas.
>
>The problem you're trying to solve is one of "interpolation".
>The statement of the problem "determine the path" is fuzzy because it 
>doesn't state the degree of accuracy, how many time points on the path,
etc.
>
>So, given that you have two points, all you can generate is a straight
line.
>Given that you have three points, all you can generate is a 2nd order
curve.
>and so forth.
>All of these will be approximations which are good or bad to one degree
or 
>another.
>
>A few of the answers deal with bandlimited situations for good reason.
>If the underlying signal that was sampled was perfect (no quantization 
>errors, etc.) and bandlimited to less than half the sampling frequency,
then 
>it can be (theoretically) perfectly reconstructed (i.e. interpolated) by

>convolving with infinite sinc functions related to the sample rate (which
is 
>an "assumed" bandwidth that's greater than the actual bandwidth).
>Well, that is, except for the quantization noise.  You can't go backwards

>perfectly ... can't interpolate perfectly ... due to the quantization.
>
>It's a nice thing to think about.  Beyond that, you might ask: "why do
you 
>care?"  "what is the application?"  From this, you can better state the 
>problem and get better solutions.
>
>Fred 
>
>
**************************************************************************

Thanks for your discussion and ideas and specially thanks to jerry for the
comments.  

*Dears my signal is band limited and I am satisfying the Shannon's
theorem.

I want to interpolate the data, at the mid of two consecutive samples and
at the same time I want to estimete the interpolation error. As you know
that if I know the path of signal I can reduce the error of estimation.

Any how as jerry said "A bandlimited signal is limited in how fast it can
change value or slope, and that limitation allows one to infer its value 
between samples".

I am agreed with jerry, as slope of signal can be viewed as its frequency.
But can you please give me some detail that how this information allows one
to infer its value between samples.

Qaisar.



		
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qaisar wrote:
>>"qaisar" <alsaeed86@yahoo.com> wrote in message 
>>news:YqydnaulpK0gt9PeRVn-jw@giganews.com...

   ...

> But can you please give me some detail that how this information allows one
> to infer its value between samples.

Fred already told you: sinc interpolation. His question about what you 
want the result for is pertinent. To draw a graph, just about any spline 
will do. To use the waveform rather than represent it as numbers, the 
usual way of creating an analog signal does exactly what you want. I 
don't think it's reasonable to expect a closed-form equation for the 
time response, so you will always be left with a table of numbers. If 
the original samples are too far apart for your liking, use one of the 
traditional upsampling methods.

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