Sampling theorem revisited...

Hi All,

If an analog signal is sampled such that sampling satisfies the
Nyquist criteria, then is it possible to reconstruct the original
signal back from the sampled signals perfectly? Is it really possible
to get back the original signal 100% exactly as it was before?
How do we reconstruct the analog signal given the sampled signal at
the input side.
Which interpolation function is the best function to get the 100%
exact signal at the output side?
100% as in the corelation between the reconstructed signal and the
original signal is equal to the maximum value, where maximum value is
the auto corelation of the original signal.

-a

On 29 Mar 2004 04:33:12 -0800, akd_ecc@yahoo.com (EC-AKD) wrote:

>Hi All,
>
>If an analog signal is sampled such that sampling satisfies the
>Nyquist criteria, then is it possible to reconstruct the original
>signal back from the sampled signals perfectly? Is it really possible
>to get back the original signal 100% exactly as it was before?
>How do we reconstruct the analog signal given the sampled signal at
>the input side.
>Which interpolation function is the best function to get the 100%
>exact signal at the output side?
>100% as in the corelation between the reconstructed signal and the
>original signal is equal to the maximum value, where maximum value is
>the auto corelation of the original signal.

In theory, you could reconstruct the continuous analog signal by
synthesizing all the proper amplitudes and phases of the sine wave
components specified in the FFT (instead of doing an inverse FFT and
trying to interpolate between time-series samples).  In practice,
roundoff error will limit the degree to which the reconstructed signal
represents the original.

-Robert Scott
 Ypsilanti, Michigan
(Reply through this forum, not by direct e-mail to me, as automatic reply address is fake.)

"EC-AKD" <akd_ecc@yahoo.com> wrote in message
news:2f592dd3.0403290433.8283ae1@posting.google.com...
> Hi All,
>
> If an analog signal is sampled such that sampling satisfies the
> Nyquist criteria, then is it possible to reconstruct the original
> signal back from the sampled signals perfectly? Is it really possible
> to get back the original signal 100% exactly as it was before?
> How do we reconstruct the analog signal given the sampled signal at
> the input side.
> Which interpolation function is the best function to get the 100%
> exact signal at the output side?
> 100% as in the corelation between the reconstructed signal and the
> original signal is equal to the maximum value, where maximum value is
> the auto corelation of the original signal.
>
> -a

Hello A,

In theory the answer is yes - in practice maybe. If we want to allow
frequencies up to but not including one half the sampling rate, the
interpolant is just the sinc function. This is well known from WKS sampling
theory. At one half of the sampling rate we have critical sampling and our
measured amplitude will depend on the phase between the sampling clock and
the sinusoid to be sampled. At that magic frequency the samples occur every
pi radians, so we could end up sampling the zero crossings and gain no
information. On the other hand we could be sampling the extrema, so we would
just get plus or minus the amplitude as the sample values.

Usually with practical sampling systems instead of trying to preserve
frequency info of up to 50% of the sampling rate, we relax the requirement
to something like 40 or 45% of the sampling rate. For example CDs
technically can have up to 22.05kHz signals, but in practice, they are
restricted to about 20kHz. Now when the bandwidth requirement is relaxed, we
can find an interpolant that has better convergence properties than the sinc
function. The problem with the sinc is its 1/x type of convergence. This is
very slow and you will need a lot of terms just to get a small error. As you
can guess, the ideal interpolants exist for all time, and in a practical
application, we will truncate the interpolant to exist for only a finite
amount of time. But if the bandwidth is reduced and some ripple is allowed
in the passband, then a reasonably short (in time) interpolant can be found
that also has a very small approximation error.

IHTH,

-- 
Clay S. Turner, V.P.
Wireless Systems Engineering, Inc.
Satellite Beach, Florida 32937
(321) 777-7889
www.wse.biz
csturner@wse.biz

-a:

[snip]
> If an analog signal is sampled such that sampling satisfies the
> Nyquist criteria, then is it possible to reconstruct the original
> signal back from the sampled signals perfectly? Is it really possible
> to get back the original signal 100% exactly as it was before?
> How do we reconstruct the analog signal given the sampled signal at
> the input side.
> Which interpolation function is the best function to get the 100%
> exact signal at the output side?
> 100% as in the corelation between the reconstructed signal and the
> original signal is equal to the maximum value, where maximum value is
> the auto corelation of the original signal.
>
> -a
[snip]

The answer is that it *may* be possible to get the original signal back from
the samples, it all depends...

But it is a very high probablility that you cannot get back the original
signal *exactly*, even for strictly
bandlimited functions.

This is simply because if there is any amount of excess bandwidth [alpha
factor] there are an infinity of
bandlimited interpolation functions [kernels] that can be used to
re-construct the signal from the samples.

Despite common myths and the fact that it is used as the basis for most
sampling theorem proofs, the
function [kernel] "Sinc(x) = sin(x)/x" is not the only possible bandlimited
interpolation function!

Despite its' anachronistic sounding title, and for an interesting and
rigorous viewpoint, with quite useful
and practical explantations of all of this I recommend that you locate and
read the following:

R. A. Gibby and J. W. Smith, "Some extensions of Nyquist's telegraph
transmission theory.", BSTJ,
Vol. 44, No. 7, pp. 1487-1510, Sept. 1965.

Gibby and Smith show that for any system with "excess bandwidth", no matter
how slight, a continuous
waveform may be interpolated through the sample points by an interpolation
function [not Sinc] which
has both a magnitude and phase characteristic each of which can have
somewhat arbitrary nature.

In particular they show for example that given a somewhat arbitrary
magnitude characteristic the associated
phase is only partially prescribed and can have a somewhat arbitrary nature,
e.g. if the phase over the first
frequency interval [0 - pi/T] is arbitrarily specified then a corrective
interpolating phase may often be calculated
over the second frequency interval, which pulls the signal back onto the
sample points!

The Gibby and Smith criteria are not just theoretical.  I have used the
Gibby and Smith "criteria" and formulae
in practice myself to design corrective phase equalizers, [i.e.group delay
equalizers] for somewhat arbitrary
prescribed magnitude analog filters [e.g. Butterworth, Chebyshev, Cauer,
etc...] that when cascaded with
the prescribed filters, provides a perfectly interpolated continuous
waveform through the set of sample points.
The application was for *precision* reconstruction filters at the output of
sampled data systems.

The interesting thing about all of this is that one then finds available a
selelction of bandlimited continuous
waveforms that all pass through the interpolation points but that have
differing characteristics between
the sample points that can be exploited in various ways, e.g. for timing
recovery robustness, etc...

Check out Gibby and Smith, groundbreaking but often neglected work...

--
Peter
Professional Consultant - Signal Processing and Analog Electronics
Indialantic By-the-Sea, FL

In article <2f592dd3.0403290433.8283ae1@posting.google.com>,
EC-AKD <akd_ecc@yahoo.com> wrote:
>If an analog signal is sampled such that sampling satisfies the
>Nyquist criteria, then is it possible to reconstruct the original
>signal back from the sampled signals perfectly?

Almost by definition, if the signal has no frequency components higher
than half the sample rate, then convolving the samples with the Sinc
function will reconstruct the original signal within the limits imposed
by the sample data itself (quantization, length, error rate, etc.).
But this in only in the mathematical sense; physical reconstruction
may require components with impossible characteristics (transducers with
zero mass, etc.)

>Is it really possible
>to get back the original signal 100% exactly as it was before?

No, unless you sample with infinite resolution and zero sample jitter
after a perfect low-pass filter using perfect transducers with zero noise.
(And there is also the shift in wall clock time produced by the two
conversion processes).  However, if you allow for some system noise,
some quantization (say to 16 bits) and a realistic filter roll-off below
the Nyquist frequency, then you can get close to the noise and frequency
distortion limits imposed by these criteria.  In the physical world,
you also need to account for the limits imposed by the transducers.

>Which interpolation function is the best function to get the 100%
>exact signal at the output side?

100% is impossible.  In audio practice, a windowed Sinc reconstruction
filter produces the original signal close to the frequency response
and noise limits allowed by the original audio low-pass filters and
the quantization to 16-bit data (maybe 99.995% exact for mid-range
frequencies).  Essentially, use a resampling filter with as much width
and resolution as you need.


IMHO. YMMV.
-- 
Ron Nicholson   rhn AT nicholson DOT com   http://www.nicholson.com/rhn/ 
#include <canonical.disclaimer>        // only my own opinions, etc.

In article 2f592dd3.0403290433.8283ae1@posting.google.com, EC-AKD at
akd_ecc@yahoo.com wrote on 03/29/2004 07:33:

> If an analog signal is sampled such that sampling satisfies the
> Nyquist criteria, then is it possible to reconstruct the original
> signal back from the sampled signals perfectly?

it is, to whatever finite degree of accuracy you specify.

> Is it really possible
> to get back the original signal 100% exactly as it was before?

it's as possible as performing an infinite summation.  theoretically we know
what to do, but it costs a lot.

> How do we reconstruct the analog signal given the sampled signal at
> the input side.

we get this discussion a lot.  look at:

http://groups.google.com/groups?selm=BB15F9AE.11F3%25rbj%40surfglobal.net

> Which interpolation function is the best function to get the 100%
> exact signal at the output side?

it's called the "sinc function"   sinc(x) = sin(pi*x)/(pi*x)

but if you window it to a finite length, you won't get 100% exact.

> 100% as in the corelation between the reconstructed signal and the
> original signal is equal to the maximum value, where maximum value is
> the auto corelation of the original signal.

dunno.

r b-j

In article <c49isl$k4g$1@blue.rahul.net>,
Ronald H. Nicholson Jr. <rhn@mauve.rahul.net> wrote:
>In article <2f592dd3.0403290433.8283ae1@posting.google.com>,
>EC-AKD <akd_ecc@yahoo.com> wrote:
>>Is it really possible
>>to get back the original signal 100% exactly as it was before?
>
>No, unless you sample with infinite resolution and zero sample jitter
>after a perfect low-pass filter using perfect transducers with zero noise.

...and for an infinite length of time.


IMHO. YMMV.
-- 
Ron Nicholson   rhn AT nicholson DOT com   http://www.nicholson.com/rhn/ 
#include <canonical.disclaimer>        // only my own opinions, etc.

EC-AKD wrote:

> Hi All,
> 
> If an analog signal is sampled such that sampling satisfies the
> Nyquist criteria, then is it possible to reconstruct the original
> signal back from the sampled signals perfectly? Is it really possible
> to get back the original signal 100% exactly as it was before?
> How do we reconstruct the analog signal given the sampled signal at
> the input side.
> Which interpolation function is the best function to get the 100%
> exact signal at the output side?
> 100% as in the corelation between the reconstructed signal and the
> original signal is equal to the maximum value, where maximum value is
> the auto corelation of the original signal.
> 
> -a

The math imposes no limit if you're willing to wait impractically long,
like forever. Practical considerations will probably force you to sample
at least 10% faster than Nyquist would imply, and because no filter is
really perfect, you'll have to put up with a little aliasing.

Add to that the limitations imposed by a finite number of bits and the
inevitable noise (both in the signal path and in the phase of the sample
clock), and absolute perfection simply can't happen. That should be no
surprise.

What you _can_ do is very very good. If you accept only perfection,
prepare to go through life without friends or possessions. :-(

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

"robert bristow-johnson" in  news:BC8DB90A.9E40%rbj@surfglobal.net...
> In article 2f592dd3.0403290433.8283ae1@posting.google.com, EC-AKD at
> akd_ecc@yahoo.com wrote on 03/29/2004 07:33:
>
> we get this discussion a lot.


Don't we, though!  I could refer also to

http://groups.google.com/groups?selm=51304%40prls.UUCP&oe=UTF-8

-- August 1991 -- quantization (especially) and sampling, including
oversampling for quantization, with references;


http://groups.google.com/groups?selm=51443%40prls.UUCP&oe=UTF-8

-- November 1991 -- sampling and physical analogies to quantization noise,
with particular "Nyquist" language tirade.

There are many others on related newsgroups back to 1985 and earlier (some
of them less conveniently archived, but still archived).  But archiving is
not the problem really; nobody ever seems to look back at the existing
corpus of newsgroup commentary, even on perennial subjects like sampling.
(This parallels some contemporary writing in refereed journals, I could
demonstrate.)

-- Max

EC-AKD wrote:

> If an analog signal is sampled such that sampling satisfies the
> Nyquist criteria, then is it possible to reconstruct the original
> signal back from the sampled signals perfectly? Is it really possible
> to get back the original signal 100% exactly as it was before?
> How do we reconstruct the analog signal given the sampled signal at
> the input side.

For a signal of length T, sampled at sampling rate f, there
are N=T f samples.

If you add an additional sample on each end with a value of zero,
at T=0 and T=(N+1)/f only solutions of the form sin(w T)=0
will be allowed, so w(n)=n pi f/(N+1).

With N samples we can solve a system of N equations and N
unknowns, such as values of n between 1 and N,
resulting in a Nyquist frequency of w(N)/(2 pi) or f/2.

You can then see that the reconstructed signal will be slightly
different depending on the exact position of the boundary points.

In the limit N --> infinity, the signal will be reconstructed
exactly.  Also, if you fix the end points you remove the
ambiguity and can reconstruct exactly.

For digitized signals, the sample values are not exact, but
are quantized by the A/D converter.   That adds a small
amount of noise to the reconstruction.

-- glen