Sampling theorem for narrow band signals

Hello, 
Could someone explain me please how the sampling theorem 
formula for narrow band signals is obtained. 
In the literature and on the web one can find a good 
explanation of the sampling theorem of band limited signals,
fs >= 2fmax.
The explanation of the formula for narrow band signals 
fs >= 2fmax/k 
is not so intuitive and not very clear.
Could you give me a link to the sampling theorem 
for narrow band signals or some hints about how 
this formula was obtained or how it can be justified.
Thank you,
Anatol

"Anatol" <uanatol@yahoo.com> writes:

> Hello, 
> Could someone explain me please how the sampling theorem 
> formula for narrow band signals is obtained. 
> In the literature and on the web one can find a good 
> explanation of the sampling theorem of band limited signals,
> fs >= 2fmax.
> The explanation of the formula for narrow band signals 
> fs >= 2fmax/k 
> is not so intuitive and not very clear.
> Could you give me a link to the sampling theorem 
> for narrow band signals or some hints about how 
> this formula was obtained or how it can be justified.
> Thank you,
> Anatol

Hi Anatol,

If you can get to a library and find the book Signals and Systems
[signalsandsystems], you will find a good model of the sampling process
there that can be used to understand any type of sampling, whether
bandpass, narrowband, or otherwise. In order to understand it you will
first need to know about the following

  1. convolution
  2. the Dirac delta function and its sifting property
  3. the Fourier transform and some of its properties

Good luck.

--Randy

@BOOK{signalsandsystems,
  title = "{Signals and Systems}",
  author = "{Alan~V.~Oppenheim, Alan~S.~Willsky, with Ian~T.~Young}",
  publisher = "Prentice Hall",
  year = "1983"}

-- 
%  Randy Yates                  % "Bird, on the wing,
%% Fuquay-Varina, NC            %   goes floating by
%%% 919-577-9882                %   but there's a teardrop in his eye..."
%%%% <yates@ieee.org>           % 'One Summer Dream', *Face The Music*, ELO
http://www.digitalsignallabs.com

On Wed, 12 Dec 2007 04:03:52 -0600, "Anatol" <uanatol@yahoo.com>
wrote:

>Hello, 
>Could someone explain me please how the sampling theorem 
>formula for narrow band signals is obtained. 
>In the literature and on the web one can find a good 
>explanation of the sampling theorem of band limited signals,
>fs >= 2fmax.
>The explanation of the formula for narrow band signals 
>fs >= 2fmax/k 
>is not so intuitive and not very clear.
>Could you give me a link to the sampling theorem 
>for narrow band signals or some hints about how 
>this formula was obtained or how it can be justified.
>Thank you,
>Anatol

Hi Anatol,
   your question is a bit confusing to me.
Can you tell us where you encountered that:

  fs >= 2fmax/k

expression?  (Was it on the Internet?)

[-Rick-]

>"Anatol" <uanatol@yahoo.com> writes:
>
>> Hello, 
>> Could someone explain me please how the sampling theorem 
>> formula for narrow band signals is obtained. 
>> In the literature and on the web one can find a good 
>> explanation of the sampling theorem of band limited signals,
>> fs >= 2fmax.
>> The explanation of the formula for narrow band signals 
>> fs >= 2fmax/k 
>> is not so intuitive and not very clear.
>> Could you give me a link to the sampling theorem 
>> for narrow band signals or some hints about how 
>> this formula was obtained or how it can be justified.
>> Thank you,
>> Anatol
>
>Hi Anatol,
>
>If you can get to a library and find the book Signals and Systems
>[signalsandsystems], you will find a good model of the sampling process
>there that can be used to understand any type of sampling, whether
>bandpass, narrowband, or otherwise. In order to understand it you will
>first need to know about the following
>
>  1. convolution
>  2. the Dirac delta function and its sifting property
>  3. the Fourier transform and some of its properties
>
>Good luck.
>
>--Randy
>
>@BOOK{signalsandsystems,
>  title = "{Signals and Systems}",
>  author = "{Alan~V.~Oppenheim, Alan~S.~Willsky, with Ian~T.~Young}",
>  publisher = "Prentice Hall",
>  year = "1983"}
>
>-- 
>%  Randy Yates                  % "Bird, on the wing,
>%% Fuquay-Varina, NC            %   goes floating by
>%%% 919-577-9882                %   but there's a teardrop in his
eye..."
>%%%% <yates@ieee.org>           % 'One Summer Dream', *Face The Music*,
ELO
>http://www.digitalsignallabs.com
>
Thank you Randy,

Yes, the convolution product of the Fourier transform 
of the original signal with the shifted Dirac function 
explaines very well the formula Fs > 2Fmax, but is does 
not explain for me the formula Fs > 2Fmax/K. 
I believe there must be a short and clear idea 
encoded in this formula. The reasoning about the 
greatest integer K that is smaller then Fmax/Fband
is not sufficient for me.

Anatol


Anatol

I hope this sheds some light:

Assume my signal is periodic, it repeats once per second.
Further, let its bandwidth be one kHz. 
Since you asked for narrowband signals, it starts for example at 10 kHz
upwards.

Because of periodicity, there exists only a finite number of sine waves
with a full number of cycles within that bandwidth:
The first one is 10 kHz - 10000 cycles per second.
The second one goes through exactly one more cycle more within the same
time, in other words 10001 Hz.
Then 10002 Hz and so on. The last one within the BW is 10999 Hz.

Each "sine wave" requires an amplitude and a phase (or use sin/cos
amplitude, if that is more convenient). We have two parameters per wave.
Result: 1000 Hz bandwidth, 2000 samples required for one second of
signal.

Two real-valued samples can be replaced by one complex valued sample, in
other words, 1 kHz bandwidth, 1000 complex valued samples per second.

-mn

"Anatol" <uanatol@yahoo.com> writes:
> [...]
> Yes, the convolution product of the Fourier transform 
> of the original signal with the shifted Dirac function 
> explaines very well the formula Fs > 2Fmax, but is does 
> not explain for me the formula Fs > 2Fmax/K. 
>
> I believe there must be a short and clear idea 
> encoded in this formula. The reasoning about the 
> greatest integer K that is smaller then Fmax/Fband
> is not sufficient for me.

I missed the "K" in your original post. Where have you seen this
formula?

The actual relationship is that Fs > 2B, where B is the bandwidth
of the signal. It doesn't matter whether the bandwidth is at baseband
or not, and Fmax/FBand does not have to be integer. Not in general.

I hate to say the formula is wrong - you may be stating it out of
context.
-- 
%  Randy Yates                  % "Maybe one day I'll feel her cold embrace,
%% Fuquay-Varina, NC            %                    and kiss her interface, 
%%% 919-577-9882                %            til then, I'll leave her alone."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO   
http://www.digitalsignallabs.com

On Wed, 12 Dec 2007 04:03:52 -0600, Anatol wrote:

> Hello,
> Could someone explain me please how the sampling theorem formula for
> narrow band signals is obtained. In the literature and on the web one
> can find a good explanation of the sampling theorem of band limited
> signals, fs >= 2fmax.
> The explanation of the formula for narrow band signals fs >= 2fmax/k
> is not so intuitive and not very clear. Could you give me a link to the
> sampling theorem for narrow band signals or some hints about how this
> formula was obtained or how it can be justified. Thank you,
> Anatol

By "Sampling Theorem" I assume you mean the "Nyquist-Shannon Sampling 
Theorem"?  All that says is that you need to sample at over 2x the signal 
bandwidth -- it gives you no clue as to how, nor does it restrict you to 
simple sampling (i.e. you can sample at over 1x the bandwidth as long as 
you get two independent samples of the signal, such as in-phase and 
quadrature parts from a mixer, or the signal and it's derivative, etc.).

As Rick pointed out it's good to define your variables -- I can guess 
what fs and fmax are, but I don't know for sure.

Try this for an answer to your question:  http://www.wescottdesign.com/
articles/Sampling/sampling.html.  Let me know (preferably here) if it 
helps, or if it doesn't.

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

>I hope this sheds some light:
>
>Assume my signal is periodic, it repeats once per second.
>Further, let its bandwidth be one kHz. 
>Since you asked for narrowband signals, it starts for example at 10 kHz
>upwards.
>
>Because of periodicity, there exists only a finite number of sine waves
>with a full number of cycles within that bandwidth:
>The first one is 10 kHz - 10000 cycles per second.
>The second one goes through exactly one more cycle more within the same
>time, in other words 10001 Hz.
>Then 10002 Hz and so on. The last one within the BW is 10999 Hz.
>
>Each "sine wave" requires an amplitude and a phase (or use sin/cos
>amplitude, if that is more convenient). We have two parameters per wave.
>Result: 1000 Hz bandwidth, 2000 samples required for one second of
>signal.
>
>Two real-valued samples can be replaced by one complex valued sample, in
>other words, 1 kHz bandwidth, 1000 complex valued samples per second.
>
>-mn
>

Hi mn,
Thank you for your answer. Now I see that both formulas

Fs > 2Fmax
Fs > 2Fmax/K, where K = ]Fmax/Fband[

can be replaced a single formula 

Fs > 2Fband,

where Fband is the signal bandwidth.

At first sight it was shocking for me to see from the formula
Fs > 2Fmax/K
that we have to sample a hight frequency signal (Fmax) like 
a lower frequency signal (Fmax/K). 
Your hints about the fact that a sine wave must be sampled 
at least 2 times and that we have Fband of such sine waves
in a periodic, band limited signal, make clear that we can 
inteprete 
Fs > 2Fmax/K
like 
Fs > 2Fband.

In conclusion, I see two approaches in explaining the
sampling theorem.
1. The first approach (based on frequency domain). It uses the 
fact that we have to avoid aliasing of the periodic spectrum 
of the sampled signal. This approach explaines very well the 
formula 
Fs > 2Fmax,
but it failes to explain 
Fs > 2Fmax/K
as simply as it does for the first formula.
2. The second approach (based on temporal domain). 
It uses the bandwidth (number of sine waves in case the signal 
is periodic) of a signal and the fact that we need to sample 
a sine wave at least twice.
This approach explains the formula
Fs > 2Fband.
Thus, the formula 
Fs > 2Fmax/K, where K = ]Fmax/Fband[
becomes clear now.

Anatol

>On Wed, 12 Dec 2007 04:03:52 -0600, Anatol wrote:
>
>> Hello,
>> Could someone explain me please how the sampling theorem formula for
>> narrow band signals is obtained. In the literature and on the web one
>> can find a good explanation of the sampling theorem of band limited
>> signals, fs >= 2fmax.
>> The explanation of the formula for narrow band signals fs >= 2fmax/k
>> is not so intuitive and not very clear. Could you give me a link to
the
>> sampling theorem for narrow band signals or some hints about how this
>> formula was obtained or how it can be justified. Thank you,
>> Anatol
>
>By "Sampling Theorem" I assume you mean the "Nyquist-Shannon Sampling 
>Theorem"?  All that says is that you need to sample at over 2x the signal

>bandwidth -- it gives you no clue as to how, nor does it restrict you to

>simple sampling (i.e. you can sample at over 1x the bandwidth as long as

>you get two independent samples of the signal, such as in-phase and 
>quadrature parts from a mixer, or the signal and it's derivative, etc.).
>
>As Rick pointed out it's good to define your variables -- I can guess 
>what fs and fmax are, but I don't know for sure.
>
>Try this for an answer to your question:  http://www.wescottdesign.com/
>articles/Sampling/sampling.html.  Let me know (preferably here) if it 
>helps, or if it doesn't.
>
>-- 
>Tim Wescott
>Control systems and communications consulting
>http://www.wescottdesign.com
>
>Need to learn how to apply control theory in your embedded system?
>"Applied Control Theory for Embedded Systems" by Tim Wescott
>Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

Hello Tim Wescott,

I saw your article yesterday, when looking for the 
answer of my question. 
However, I was looking for the formuls Fs > 2Fmax/K.
I have not seen such a formula in the artice, so I skipped it. 
Now I see it is an interesting practical paper on sampling.

Thanks,
Anatol

>On Wed, 12 Dec 2007 04:03:52 -0600, "Anatol" <uanatol@yahoo.com>
>wrote:
>
>>Hello, 
>>Could someone explain me please how the sampling theorem 
>>formula for narrow band signals is obtained. 
>>In the literature and on the web one can find a good 
>>explanation of the sampling theorem of band limited signals,
>>fs >= 2fmax.
>>The explanation of the formula for narrow band signals 
>>fs >= 2fmax/k 
>>is not so intuitive and not very clear.
>>Could you give me a link to the sampling theorem 
>>for narrow band signals or some hints about how 
>>this formula was obtained or how it can be justified.
>>Thank you,
>>Anatol
>
>Hi Anatol,
>   your question is a bit confusing to me.
>Can you tell us where you encountered that:
>
>  fs >= 2fmax/k
>
>expression?  (Was it on the Internet?)
>
>[-Rick-]
>
>
Hi Rick,

I found this formula in a french textbook.
Now I see that it is equivalent to the formula
Fs > 2*Fband.

Anatol