# Sampling theorem for narrow band signals

Started by December 12, 2007
```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 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
```
```>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

```
```>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 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
```
```>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
```