Periodogram returning extra values

On Thu, 30 Oct 2008 15:20:53 -0800, Glen Herrmannsfeldt
<gah@ugcs.caltech.edu> wrote:

>dbd wrote:
>(snip)
>
>> The N point DFT of a real signal returns a real value at DC, N/2 - 1
>> complex values, and an imaginary value at fs/2; a total of N values
>> from N real samples.
>
>And by symmetry, the DFT of an imaginary signal should return an
>imaginary value at DC, N/2-1 complex values, and a real value at fs/2.
>
>I don't believe that there is a mathematical symbol for 'less than
>and partially equal to'.
>
>-- glen

Hmmm.....     <~=

:)


Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.ericjacobsen.org

Blog: http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php

On Oct 30, 9:32 pm, Glen Herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
> dbd wrote:
>
> (snip)
>
> > What non-periodic band limited signal can you
> > reconstruct from N+1 sample points?
> > Doesn't Nyquist deal with bandwidths and
> > rates, not numbers of samples?
>
> One that is N divided by the appropriate sampling rate long.
>
> -- glen

This rectangular pulse is not band limited, nor can it be uniquely
reconstructed from N+1 samples

Dale B. Dalrymple

On Thu, 30 Oct 2008 14:05:37 -0700 (PDT), dbd <dbd@ieee.org> wrote:

   (snipped by Lyons)

>The N point DFT of a real signal returns a real value at DC, N/2 - 1
>complex values, and an imaginary value at fs/2; a total of N values
>from N real samples.
>
>Dale B. Dalrymple

Hi,
  Are you sure it's an imaginary value 
at fs/2?

The complex exponential in the N-point DFT 
equation (when N is even) is:

     exp(-j*pi*n) 

where "n" is the time-domain index.  That 
exponential is real-valued for all "n".

[-Rick-]

On Nov 1, 2:33 pm, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> On Thu, 30 Oct 2008 14:05:37 -0700 (PDT), dbd <d...@ieee.org> wrote:
>
>    (snipped by Lyons)
>
> >The N point DFT of a real signal returns a real value at DC, N/2 - 1
> >complex values, and an imaginary value at fs/2; a total of N values
> >from N real samples.
>
> >Dale B. Dalrymple
>
> Hi,
>   Are you sure it's an imaginary value
> at fs/2?
>
> The complex exponential in the N-point DFT
> equation (when N is even) is:
>
>      exp(-j*pi*n)
>
> where "n" is the time-domain index.  That
> exponential is real-valued for all "n".
>
> [-Rick-]

No, should be real.

Dale B. Dalrymple

Rick Lyons wrote:

> On Thu, 30 Oct 2008 14:05:37 -0700 (PDT), dbd <dbd@ieee.org> wrote:

>>The N point DFT of a real signal returns a real value at DC, N/2 - 1
>>complex values, and an imaginary value at fs/2; a total of N values
>>from N real samples.

>   Are you sure it's an imaginary value at fs/2?

> The complex exponential in the N-point DFT 
> equation (when N is even) is:

>      exp(-j*pi*n) 

> where "n" is the time-domain index.  That 
> exponential is real-valued for all "n".

You want exp(-j*2*pi*n*k/N) where n and k are
integers from 0 and N-1, inclusive.

http://en.wikipedia.org/wiki/Discrete_Fourier_transform#Definition

-- glen

On Nov 1, 6:06 pm, Glen Herrmannsfeldt <g...@ugcs.caltech.edu> wrote:

> ...

>
> You want exp(-j*2*pi*n*k/N) where n and k are
> integers from 0 and N-1, inclusive.
>
> http://en.wikipedia.org/wiki/Discrete_Fourier_transform#Definition
>
> -- glen

Gee, does this mean that glen is suggesting that the Nth term is in
some sense imaginary?

Unfortunately, this is a reference to that great repository of urban
legend, wikipedia. As is often the case in technical areas, the
material in the wiki differs from the version in the references cited.
Take for example Steven Smith's chapter 8, cited in the wiki (see the
third reference in the wiki for the full citation), which refers to
this as "a common bug in DFT programs" on page 148. Now it may be that
the bug is sometimes intentional, but that is a matter of engineering
practice departing from theoretical accuracy.

Dale B. Dalrymple

On Nov 1, 2:33 pm, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> On Thu, 30 Oct 2008 14:05:37 -0700 (PDT), dbd <d...@ieee.org> wrote:
>
>    (snipped by Lyons)
>
> >The N point DFT of a real signal returns a real value at DC, N/2 - 1
> >complex values, and an imaginary value at fs/2; a total of N values
> >from N real samples.
>
> >Dale B. Dalrymple
>
> Hi,
>   Are you sure it's an imaginary value
> at fs/2?

Another way to look at it is you get two real and two imaginary
values at fs/2 (for both the positive and negative frequency),
the two real values are of the same sign, but the two imaginary
values are of the opposite sign, so always cancel, giving you
an imaginary sum of zero.  That's also why the amplitudes in the
bins for DC and fs/2 seem to have an factor of 2X compared to
all of all the other bins.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
  http://www.nicholson.com/rhn/dsp.html