Shannon and negative frequencies

In Shannon's paper that sets out the Sampling Theorem:

     http://www.stanford.edu/class/ee104/shannonpaper.pdf

he formally states (using B to represent the Nyquist frequency):

"If a function x(t) contains no frequencies higher than B hertz, it is
completely determined by giving its ordinates at a series of points
spaced 1/(2B) seconds apart."

"A similar result is true if the band does not start at zero frequency
but at some higher value, and can be proved by a linear translation
(corresponding physically to single-sideband modulation) of the zero-
frequency case."

That is, he refers only to nonnegative frequencies.

In the Wikipedia article on Shannon's theorem, the same restriction to
nonnegative frequenceis applies (using Fs as the sampling frequency):

"For a signal X(f) to be band limited, its values X(f) must be zero
for all non-negative f outside the open band of frequencies:

{ (N/2) * Fs, ((N+1)/2) * Fs) }

for some nonnegative integer N."

Can anyone refer me to a place where this result for nonnegative
frequencies is formally extended to the case of negative frequencies
(and so using complex numbers..)?

Thanks,

Chris
============================
Chris Bore
BORES Signal Processing
www.bores.com

On 13 Jan, 18:40, Chris Bore <chris.b...@gmail.com> wrote:
> In Shannon's paper that sets out the Sampling Theorem:
>
> =A0 =A0 =A0http://www.stanford.edu/class/ee104/shannonpaper.pdf
>
> he formally states (using B to represent the Nyquist frequency):
>
> "If a function x(t) contains no frequencies higher than B hertz, it is
> completely determined by giving its ordinates at a series of points
> spaced 1/(2B) seconds apart."
>
> "A similar result is true if the band does not start at zero frequency
> but at some higher value, and can be proved by a linear translation
> (corresponding physically to single-sideband modulation) of the zero-
> frequency case."
>
> That is, he refers only to nonnegative frequencies.
>
> In the Wikipedia article on Shannon's theorem, the same restriction to
> nonnegative frequenceis applies (using Fs as the sampling frequency):
>
> "For a signal X(f) to be band limited, its values X(f) must be zero
> for all non-negative f outside the open band of frequencies:
>
> { (N/2) * Fs, ((N+1)/2) * Fs) }
>
> for some nonnegative integer N."
>
> Can anyone refer me to a place where this result for nonnegative
> frequencies is formally extended to the case of negative frequencies
> (and so using complex numbers..)?

The Nyquist sampling theorem is only valid for real-valued
signals, that is, where H(-w) =3D conj(H(w)).

In the case of complex-valued x(t), there is no requirement
on the spectrum, so consider some signal with non-zero spectrum
constrained to w =3D [0,B]. Using the spectrum wrap-around
property when the signal is sampled, it is a simple graphical
argument to see that a sampling rate F > B suffices (view
with fixed-width font):

      ^
      |  __
      | /  |
------+------------->

      ^
   __ |  __   __
  /  || /  | /  |
------+-----+----+- .. ->
      0     B    2B     w

Rune

On Jan 13, 5:51=A0pm, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 13 Jan, 18:40, Chris Bore <chris.b...@gmail.com> wrote:
>
>
>
>
>
> > In Shannon's paper that sets out the Sampling Theorem:
>
> > =A0 =A0 =A0http://www.stanford.edu/class/ee104/shannonpaper.pdf
>
> > he formally states (using B to represent the Nyquist frequency):
>
> > "If a function x(t) contains no frequencies higher than B hertz, it is
> > completely determined by giving its ordinates at a series of points
> > spaced 1/(2B) seconds apart."
>
> > "A similar result is true if the band does not start at zero frequency
> > but at some higher value, and can be proved by a linear translation
> > (corresponding physically to single-sideband modulation) of the zero-
> > frequency case."
>
> > That is, he refers only to nonnegative frequencies.
>
> > In the Wikipedia article on Shannon's theorem, the same restriction to
> > nonnegative frequenceis applies (using Fs as the sampling frequency):
>
> > "For a signal X(f) to be band limited, its values X(f) must be zero
> > for all non-negative f outside the open band of frequencies:
>
> > { (N/2) * Fs, ((N+1)/2) * Fs) }
>
> > for some nonnegative integer N."
>
> > Can anyone refer me to a place where this result for nonnegative
> > frequencies is formally extended to the case of negative frequencies
> > (and so using complex numbers..)?
>
> The Nyquist sampling theorem is only valid for real-valued
> signals, that is, where H(-w) =3D conj(H(w)).
>
> In the case of complex-valued x(t), there is no requirement
> on the spectrum, so consider some signal with non-zero spectrum
> constrained to w =3D [0,B]. Using the spectrum wrap-around
> property when the signal is sampled, it is a simple graphical
> argument to see that a sampling rate F > B suffices (view
> with fixed-width font):
>
> =A0 =A0 =A0 ^
> =A0 =A0 =A0 | =A0__
> =A0 =A0 =A0 | / =A0|
> ------+------------->
>
> =A0 =A0 =A0 ^
> =A0 =A0__ | =A0__ =A0 __
> =A0 / =A0|| / =A0| / =A0|
> ------+-----+----+- .. ->
> =A0 =A0 =A0 0 =A0 =A0 B =A0 =A02B =A0 =A0 w
>
> Rune- Hide quoted text -
>
> - Show quoted text -

Thanks.

I can see that this is so, but was looking for a formal proof, and
preferably one that was commonly cited when referring to sampling of
complex-valued signals.

I found a useful paper that derives such a proof:

http://www.national.com/an/AN/AN-236.pdf

but with such an important extension of the theory, I thought there
must be a well-known formal reference?

Chris
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
Chris Bore
BORES Signal Processing
www.bores.com

Chris Bore wrote:
> In Shannon's paper that sets out the Sampling Theorem:
> 
>      http://www.stanford.edu/class/ee104/shannonpaper.pdf
> 
> he formally states (using B to represent the Nyquist frequency):
> 
> "If a function x(t) contains no frequencies higher than B hertz, it is
> completely determined by giving its ordinates at a series of points
> spaced 1/(2B) seconds apart."
> 
> "A similar result is true if the band does not start at zero frequency
> but at some higher value, and can be proved by a linear translation
> (corresponding physically to single-sideband modulation) of the zero-
> frequency case."
> 
> That is, he refers only to nonnegative frequencies.
> 
> In the Wikipedia article on Shannon's theorem, the same restriction to
> nonnegative frequenceis applies (using Fs as the sampling frequency):
> 
> "For a signal X(f) to be band limited, its values X(f) must be zero
> for all non-negative f outside the open band of frequencies:
> 
> { (N/2) * Fs, ((N+1)/2) * Fs) }
> 
> for some nonnegative integer N."
> 
> Can anyone refer me to a place where this result for nonnegative
> frequencies is formally extended to the case of negative frequencies
> (and so using complex numbers..)?
> 
> Thanks,
> 
> Chris
> ============================
> Chris Bore
> BORES Signal Processing
> www.bores.com
> 
> 
Chris,

I would venture to guess that he didn't mean it as literally as you're 
taking it.  I think that the negative frequency content is implied.

It was not uncommon to ignore the negative frequencies I think - that 
is, not refer to them explicitly.

But, that's pure conjecture on my part.
Then, the translation of bandwidth B would be from fc by fc-B/2.
Then all the words work and the "pictures" in my mind work out perfectly.

Fred

On Jan 13, 12:40&#4294967295;pm, Chris Bore <chris.b...@gmail.com> wrote:
> In Shannon's paper that sets out the Sampling Theorem:
>
> &#4294967295; &#4294967295; &#4294967295;http://www.stanford.edu/class/ee104/shannonpaper.pdf
>
> he formally states (using B to represent the Nyquist frequency):
>
> "If a function x(t) contains no frequencies higher than B hertz, it is
> completely determined by giving its ordinates at a series of points
> spaced 1/(2B) seconds apart."

Although there is no explicit mention of negative frequencies,
the proof involves integrating the fourier integral over the
frequency interval [-B,+B].

> "A similar result is true if the band does not start at zero frequency
> but at some higher value, and can be proved by a linear translation
> (corresponding physically to single-sideband modulation) of the zero-
> frequency case."
>
> That is, he refers only to nonnegative frequencies.
>
> In the Wikipedia article on Shannon's theorem, the same restriction to
> nonnegative frequenceis applies (using Fs as the sampling frequency):
>
> "For a signal X(f) to be band limited, its values X(f) must be zero
> for all non-negative f outside the open band of frequencies:
>
> { (N/2) * Fs, ((N+1)/2) * Fs) }
>
> for some nonnegative integer N."
>
> Can anyone refer me to a place where this result for nonnegative
> frequencies is formally extended to the case of negative frequencies
> (and so using complex numbers..)?

It can be inferred from the following sentence:

Other generalizations, for example to signals occupying multiple
non-contiguous bands, are possible as well.

Hope this helps.

Greg