```>On Dec 13, 7:42 am, Jerry Avins <j...@ieee.org> wrote:
>> Anatol wrote:
>> >> "Anatol" <uana...@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 >=3D 2fmax.
>> >>> The explanation of the formula for narrow band signals
>> >>> fs >=3D 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 =3D "{Signals and Systems}",
>> >>  author =3D "{Alan~V.~Oppenheim, Alan~S.~Willsky, with
Ian~T.~Young}",
>> >>  publisher =3D "Prentice Hall",
>> >>  year =3D "1983"}
>>
>> >> --
>> >> %  Randy Yates                  % "Bird, on the wing,
>> >> %% Fuquay-Varina, NC            %   goes floating by
>> >> %%% 919-577-9882                %   but there's a teardrop in his
>> > eye..."
>> >> %%%% <ya...@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.
>>
>> Try it without any math at all. The sampling process creates images.
>> Baseband signals have images that begin above Fs/2. Any part of that
>> image spectrum that extends below Fs/2 is an alias. When the initial
>> spectrum is such that the images created by sampling don't overlap,
>> there is no alias. Plot the locations and extents of the images in the
>> sampling paradigm that puzzles you and you will understand.
>>
>> Jerry
>> --
>> Engineering is the art of making what you want from things you can
get.
>>
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
>=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
>=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF
>
>You are thinking of this...
>
>Bounds on the sampling frequency fs
>
>2fh/k  <=3Dfs<=3D 2fl/(k-1) for k=3D1,2...N
>
>Here k can range from 1 to N with k=3DN yielding the smalest value
>fs=3D2fh/N and k=3D1 corresponding to teh Nyquist rate fs>=3D2fh.
>
>fl and fh here being the lowest and highest frequencies of the signal
>you are sampling.
>
>
>
>Hardy

Hi Hardy,

Can you tell me please what is N in these relations ?
Is it bounded ?
Anatol
```
```On Wed, 12 Dec 2007 11:03:58 -0600, Anatol wrote:

>>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.
>>
>>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

The paper covers the specific case of sampling a passband signal with a
carrier that's higher than the sampling rate.

--
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
```
```>Anatol wrote:
>>> "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.
>
>Try it without any math at all. The sampling process creates images.
>Baseband signals have images that begin above Fs/2. Any part of that
>image spectrum that extends below Fs/2 is an alias. When the initial
>spectrum is such that the images created by sampling don't overlap,
>there is no alias. Plot the locations and extents of the images in the
>sampling paradigm that puzzles you and you will understand.
>
>Jerry
>--
>Engineering is the art of making what you want from things you can get.
>&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;
Hi Jerry,

I thinks your comments is the first step in understanding the fact
that aliasing can be avoided for narrow band signals even if
Fs > 2Fmax is not satisfied. Undestanding that, the formula
Fs > 2Fmax/K becomes more "friendly". Determining K, however
is not obvious using only such plots.

Anatol

```
```>On Dec 13, 7:42 am, Jerry Avins <j...@ieee.org> wrote:
>> Anatol wrote:
>> >> "Anatol" <uana...@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 >=3D 2fmax.
>> >>> The explanation of the formula for narrow band signals
>> >>> fs >=3D 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 =3D "{Signals and Systems}",
>> >>  author =3D "{Alan~V.~Oppenheim, Alan~S.~Willsky, with
Ian~T.~Young}",
>> >>  publisher =3D "Prentice Hall",
>> >>  year =3D "1983"}
>>
>> >> --
>> >> %  Randy Yates                  % "Bird, on the wing,
>> >> %% Fuquay-Varina, NC            %   goes floating by
>> >> %%% 919-577-9882                %   but there's a teardrop in his
>> > eye..."
>> >> %%%% <ya...@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.
>>
>> Try it without any math at all. The sampling process creates images.
>> Baseband signals have images that begin above Fs/2. Any part of that
>> image spectrum that extends below Fs/2 is an alias. When the initial
>> spectrum is such that the images created by sampling don't overlap,
>> there is no alias. Plot the locations and extents of the images in the
>> sampling paradigm that puzzles you and you will understand.
>>
>> Jerry
>> --
>> Engineering is the art of making what you want from things you can
get.
>>
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
>=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
>=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF
>
>You are thinking of this...
>
>Bounds on the sampling frequency fs
>
>2fh/k  <=3Dfs<=3D 2fl/(k-1) for k=3D1,2...N
>
>Here k can range from 1 to N with k=3DN yielding the smalest value
>fs=3D2fh/N and k=3D1 corresponding to teh Nyquist rate fs>=3D2fh.
>
>fl and fh here being the lowest and highest frequencies of the signal
>you are sampling.
>
>
>
>Hardy

Thank you Hardy,
But I have a new problem now :)
I have to post a new question to ask you to explain me the formula
2fh/k  <=3Dfs<=3D 2fl/(k-1) for k=3D1,2...N

Anatol

```
```On Dec 13, 7:42 am, Jerry Avins <j...@ieee.org> wrote:
> Anatol wrote:
> >> "Anatol" <uana...@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..."
> >> %%%% <ya...@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.
>
> Try it without any math at all. The sampling process creates images.
> Baseband signals have images that begin above Fs/2. Any part of that
> image spectrum that extends below Fs/2 is an alias. When the initial
> spectrum is such that the images created by sampling don't overlap,
> there is no alias. Plot the locations and extents of the images in the
> sampling paradigm that puzzles you and you will understand.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;

You are thinking of this...

Bounds on the sampling frequency fs

2fh/k  <=fs<= 2fl/(k-1) for k=1,2...N

Here k can range from 1 to N with k=N yielding the smalest value
fs=2fh/N and k=1 corresponding to teh Nyquist rate fs>=2fh.

fl and fh here being the lowest and highest frequencies of the signal
you are sampling.

Hardy
```
```Anatol wrote:
>> "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.

Try it without any math at all. The sampling process creates images.
Baseband signals have images that begin above Fs/2. Any part of that
image spectrum that extends below Fs/2 is an alias. When the initial
spectrum is such that the images created by sampling don't overlap,
there is no alias. Plot the locations and extents of the images in the
sampling paradigm that puzzles you and you will understand.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```>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

```
```>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.
>
>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
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

```
```>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,

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.