Enery conservation during Sampling

Hi all...
I am pretty new to signal processing...
            Let say we sample a band limited signal... So the spectrum
of the sampled signal will have spectral replications of the band
limited signal and multiples of the Sampling frequency...
            So in this case the resulting signal seems to have more
energy... since its spectrum is large compared to the bandpass signal
and the sampling impulse train... How does this happen...? From where
does this resulting signal get its energy from...? I think mostly
sampling circuits don't provide energy to the signals...!
           Does this have anything to do with Power Spectral density
Vs FFT...? will FFT of the sampled signal be different from that of
its PSD...?

Thanks

Very good. It seems you just found the solution to the energy problem
of America by digitizing a bandlimited signal. It is even better than
a perpetual motion machine... My further question is, why do you want
to conserve it? Let's just use it up.

James
www.go-ci.com

On Thu, 16 Oct 2008 22:24:55 -0700, lordsathish wrote:

> Hi all...
> I am pretty new to signal processing...
>             Let say we sample a band limited signal... So the spectrum
> of the sampled signal will have spectral replications of the band
> limited signal and multiples of the Sampling frequency...
>             So in this case the resulting signal seems to have more
> energy... since its spectrum is large compared to the bandpass signal
> and the sampling impulse train... How does this happen...? From where
> does this resulting signal get its energy from...? I think mostly
> sampling circuits don't provide energy to the signals...!
>            Does this have anything to do with Power Spectral density
> Vs FFT...? will FFT of the sampled signal be different from that of its
> PSD...?
> 
> Thanks

You are confusing mathematics with reality.

Sampling is mathematically modeled as multiplying the continuous-time 
signal with a chain of impulses.  Since an impulse has infinite energy, a 
chain of impulses has infinite power.

This is where your confusion comes in: you can't _really_ multiply a 
continuous-time signal with a chain of impulses, because no one can 
generate a chain of impulses, nor can anyone build a multiplier that 
would do anything useful with an impulse as an input.

The _model_ has you multiplying the signal with a chain of impulses, but 
the _reality_ is that you measure the value of the signal around your 
chosen sampling instant, then do something with it (usually do an analog 
to digital conversion).  If you try to do your mathematical analysis 
while leaving the sampling process all bound up with the reality of what 
you're doing you'll find the going very rough -- so you model the "ideal" 
sampled signal, then you model the rest of the signal processing chain.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html

On 17 Okt, 07:24, lordsathish <lordsath...@gmail.com> wrote:
> Hi all...
> I am pretty new to signal processing...
> &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; Let say we sample a band limited signal... So the spectrum
> of the sampled signal will have spectral replications of the band
> limited signal and multiples of the Sampling frequency...

Nope. The mathematical mappings back and forth between continuous
time (CT) and discrete time (DT) domains are ambiguous. Unless
one imposes some additional constraint (baseband or bandpass
sampling) one can't tell which 'spectral mirror' corresponds to
the physical signal and which are 'artificial'. Unless Nyquist's
condition is satisfied one does not know whether the observed
DT spectrum is at all relevant with respect to the CT spectrum.

> &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; So in this case the resulting signal seems to have more
> energy...

Don't confuse physical energy with mathemathical norm.
One important mathemathical constraint behind the Fourier
transform is that an integral or sum is finite (view
with fixed-width font):

   b                                b
integral |x(t)|^2 dt < ifinite,   sum |x[n]| ^2 < infinite
   a                              n = a

where the exact details vary from case to case. This is a bit
like the constraint that energy in a physical system must
be finite, and is called 'norm' in the context of math.

So in the same way that the energy (or power) of a physical
signal most be finite, the Fourier transform only works
of the norm of the data is finite.

In the case of sampled signals the integral or sum limits
in the formulas above are chosen so that the sum or
integral is take over a bandwidth of Fs, for instance
[-Fs/2, Fs/2>, although other choises are possible.

> since its spectrum is large compared to the bandpass signal
> and the sampling impulse train... How does this happen...?

It's a purely mathemathical artifact.

> From where
> does this resulting signal get its energy from...? I think mostly
> sampling circuits don't provide energy to the signals...!

They don't. ADCs take the signal from physical reality, where
energy must be preserved, and maps it into a mathematical
world where other rules apply. You might think of the ADC
as the mirror in Lewis Carrol's 'Through the looking glass.'

> &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295;Does this have anything to do with Power Spectral density
> Vs FFT...?

Nope.

> will FFT of the sampled signal

'FFT' is a computational algorithm, like addition. The
resulting data is a Discrete Fourier Transform (DFT) of
the data the same way that the 'sum' is the result of
performing an 'addition'. The terms FFT and DFT are as
(little) interchangable as the tems 'addition' and 'sum'.

> be different from that of
> its PSD...?

Yes, but only partially because of sampling.

Formally, the DFT can only be used to compute an *estimate*
for the PSD. For real-valued signals the FFT computes a
complex-valued spectrum, whereas the PSD is real-valued.
So the PSD is fundamentally different from the DFT in that
it contains only some of the the information you get in
the DFT.

Second, if you want to compute a 'true' estimate for the PSD,
where the numbers match those from the physical world, you need
to make the effort to calibrate the ADC to find whatever scale
and conversion factors are involved in the sampling. That's
a big deal in practice, so people don't calibrate the systems
unless there are very good reasons for it.

Rune

Lordsathish: The simple answer is that it is an illusion to believe
that your multiple bandlimited spectra replicated for each multiple of
sampling frequency has "infinite" energy. These bandlimited spectra
are simply the observation if the A/D samples at these frequencies to
the same time domain signals.

James
www.go-ci.com

Rune Allnor wrote:
(snip regarding sampling and energy)

> They don't. ADCs take the signal from physical reality, where
> energy must be preserved, and maps it into a mathematical
> world where other rules apply. You might think of the ADC
> as the mirror in Lewis Carrol's 'Through the looking glass.'
(snip)

I was thinking just last week about one discussed here
not so long ago.  If you consider the sampled signal
of period three

{ -1, 1, 1 }

and reconstruct the band limited signal you find the
amplitude greater than one.  (Or peak-to-peak greater than two.)

It might seem that you can't get an output greater than one
or less than -1 out of that, but you can.  You can also
consider the period four sampled signal { 1, -1 ,-1, 1 }

Compute the output power of the continuous signal and compare
it to the power in the discrete signal.

> Formally, the DFT can only be used to compute an *estimate*
> for the PSD. For real-valued signals the FFT computes a
> complex-valued spectrum, whereas the PSD is real-valued.
> So the PSD is fundamentally different from the DFT in that
> it contains only some of the the information you get in
> the DFT.

> Second, if you want to compute a 'true' estimate for the PSD,
> where the numbers match those from the physical world, you need
> to make the effort to calibrate the ADC to find whatever scale
> and conversion factors are involved in the sampling. That's
> a big deal in practice, so people don't calibrate the systems
> unless there are very good reasons for it.

-- glen

Tim Wescott wrote:
(snip)

> You are confusing mathematics with reality.

> Sampling is mathematically modeled as multiplying the continuous-time 
> signal with a chain of impulses.  Since an impulse has infinite energy, a 
> chain of impulses has infinite power.

Well, delta has finite area, so if you integrate over it you
get a finite power.

> This is where your confusion comes in: you can't _really_ multiply a 
> continuous-time signal with a chain of impulses, because no one can 
> generate a chain of impulses, nor can anyone build a multiplier that 
> would do anything useful with an impulse as an input.

This is true, but you can still figure out the power from the
sampled signal.  If you take an analog signal voltage as a
function of time, the sampled signal still has units of volts.

Similar to integrating the square of the voltage and dividing
by the time, you can sum the square of the sample values and
divide by the number of samples.  To get actual power you need
a load resistance in both cases.

> The _model_ has you multiplying the signal with a chain of impulses, but 
> the _reality_ is that you measure the value of the signal around your 
> chosen sampling instant, then do something with it (usually do an analog 
> to digital conversion).  If you try to do your mathematical analysis 
> while leaving the sampling process all bound up with the reality of what 
> you're doing you'll find the going very rough -- so you model the "ideal" 
> sampled signal, then you model the rest of the signal processing chain.

Hopefully the signal is changing slowly relative to the sampling
window width.  Then the measured samples are a good approximation
of the instantaneous sample value.

-- glen

On Fri, 17 Oct 2008 15:17:15 -0800, glen herrmannsfeldt wrote:

> Tim Wescott wrote:
> (snip)
> 
>> You are confusing mathematics with reality.
> 
>> Sampling is mathematically modeled as multiplying the continuous-time
>> signal with a chain of impulses.  Since an impulse has infinite energy,
>> a chain of impulses has infinite power.
> 
> Well, delta has finite area, so if you integrate over it you get a
> finite power.

But not after you square it to get power.

If you take the delta functional to be a pulse 1/a wide and a tall, then 
in the limit as a -> infinity the area stays at 1 and the height goes to 
infinity.  But when you square it to find the energy, the area = a, and 
goes to infinity as well.

So, infinite energy (or at least an infinite mathematical norm) per 
pulse.  Ergo infinite power...

-- snip --

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html

Tim Wescott wrote:
(snip)

>>Well, delta has finite area, so if you integrate over it you get a
>>finite power.

> But not after you square it to get power.

> If you take the delta functional to be a pulse 1/a wide and a tall, then 
> in the limit as a -> infinity the area stays at 1 and the height goes to 
> infinity.  But when you square it to find the energy, the area = a, and 
> goes to infinity as well.

There has to be an answer to this, but I agree that I don't
know what it is.  Quantum mechanics uses delta functions where
the probability (square of the amplitude) better not be greater
than one.

-- glen