On Tue, 04 Jan 2005 08:20:11 -0800, Vec <vec2u@valoudari.edu.mx>
wrote:
>Hello
>
>I need help, in DFT, the frequency domain�s independent variable can be
>refereed to in many ways, one way is being a fraction of the sampling rate.
>could some one please explain why X[ ] independent variable runs between
>0 and 0.5, if you say because discrete data can only contain frequencies
>between DC and � the sampling rate, then if the sampling rate (x[ ] has)
>128 points then 128 * � is 64 then why X [ ] has 65 samples?
>I must be missing something.
If you look at the DFT it runs around in a circle so the values repeat
after going from 0 to pi. The sample point are points on the circle.
Regards,
Steve Drake
Reply by Fred Marshall●January 3, 20052005-01-03
"Vec" <vec2u@valoudari.edu.mx> wrote in message
news:41d9b8ed$1_1@news.iprimus.com.au...
> Hello
>
> I need help, in DFT, the frequency domain�s independent variable can be
> refereed to in many ways, one way is being a fraction of the sampling
> rate.
> could some one please explain why X[ ] independent variable runs between 0
> and 0.5, if you say because discrete data can only contain frequencies
> between DC and � the sampling rate, then if the sampling rate (x[ ] has)
> 128 points then 128 * � is 64 then why X [ ] has 65 samples?
> I must be missing something.
>
> Thanks
>
And I might have added:
While fs/2 is the mirroring point in frequency, it need not have a sample
located there.
If the number of time samples is even, e.g. 64, then so is the number of
frequency samples out of a DFT.
If the number of frequency samples is 64, the frequency sampling interval is
fs/64.
64 samples covers the spectral range zero to fs-fs/64 and F(0)=F(fs) so
F(fs) need not be included because it would be redundant.
Note that the range 0 to fs is split into 64 intervals that are spanned by
65 samples if fs is included.
If it is in 64 intervals then fs/2 is in exactly 32 intervals which is
spanned by 33 samples if fs/2 is included. The remaining 31 samples run
from fs/2+fs/64 to fs-fs/64.
SO: If the number of samples is even, as is often the case for a DFT, then
there is a sample located at fs/2.
If the number of time samples is odd, e.g. 65, then so is the number of
frequency samples out of a DFT.
If the number of frequency samples is 65, the frequency sampling interfval
is fs/65.
65 samples covers the spectral range zero to fs-fs/65 and F(0)-F(fs) as
above.
Note that the range 0 to fs is split into 65 intervals that are spanned by
66 samples if fs is included.
If it is in 65 intervals, then fs/2 lies in the middle of the 33rd interval
and is not spanned by an integral number of samples.
SO: If the number of samples is odd, then there is not a sample located at
fs/2 and the nearest sample to fs/2 is at fs/2-fs/2N. The spectrum of a
real temporal sequence still mirrors at fs/2.
Fred
Reply by Matt Timmermans●January 3, 20052005-01-03
"Vec" <vec2u@valoudari.edu.mx> wrote in message
news:41d9b8ed$1_1@news.iprimus.com.au...
> [...] (x[ ] has) 128 points then 128 * � is 64 then why X [ ] has 65
> samples?
> I must be missing something.
I assume you're using real (i.e., not complex) input data. In the DFT
output, the frequency samples are complex numbers, i.e., each contains two
real numbers, except for the DC and fs/2 which can have only one non-zero
real component each. The total count of real numbers in X[] is then 1 +
63*2 + 1 = 128, as you would expect.
--
Matt
Reply by Fred Marshall●January 3, 20052005-01-03
"Tim Wescott" <tim@wescottnospamdesign.com> wrote in message
news:10tjikljh67jh9d@corp.supernews.com...
> Vec wrote:
>
>> Hello
>>
>> I need help, in DFT, the frequency domain�s independent variable can be
>> refereed to in many ways, one way is being a fraction of the sampling
>> rate.
>> could some one please explain why X[ ] independent variable runs between
>> 0 and 0.5, if you say because discrete data can only contain frequencies
>> between DC and � the sampling rate, then if the sampling rate (x[ ] has)
>> 128 points then 128 * � is 64 then why X [ ] has 65 samples?
>> I must be missing something.
>>
>> Thanks
>>
> The discrete data can only contain frequencies between DC and 1/2 the
> sampling rate _inclusive_, and points 0 through 64 is a set with 65
> members.
>
> Note: Your rule only applies when you are sampling real data. If you are
> sampling complex data (or I & Q data) then you need all 128 points.
>
> --
>
> Tim Wescott
> Wescott Design Services
> http://www.wescottdesign.com
With no disagreement, said a slightly different way:
A DFT means the temporal data is sampled and the spectrum repeats at periods
of fs.
-fs/2 to fs/2 covers one period of the spectrum.
If the temporal data is real, then the spectral data is even.
Accordingly, the spectral data from zero to fs/2 is a mirror image of the
data from zero to -fs/2, and the data from fs/2 to fs is the same as the
data from -fs/2 to zero.
So, all of the information in the spectral data is present from zero to
fs/2.
If we normalize the sequences so that the time sampling interval is assumed
to be 1 second, then fs=1Hz and fs/2 is 0.5Hz. This is handy because often
we are dealing with regularly spaced "temporal" samples or maybe just
regularly spaced unlabeled samples - in which case using a sample interval
of "1.0" is convenient. Then the spectrum from 0 to 0.5 is all we need.
Fred
Reply by Vec●January 3, 20052005-01-03
Hello
I need help, in DFT, the frequency domain�s independent variable can be
refereed to in many ways, one way is being a fraction of the sampling rate.
could some one please explain why X[ ] independent variable runs between
0 and 0.5, if you say because discrete data can only contain frequencies
between DC and � the sampling rate, then if the sampling rate (x[ ] has)
128 points then 128 * � is 64 then why X [ ] has 65 samples?
I must be missing something.
Thanks
Reply by Vec●January 3, 20052005-01-03
Hello
I need help, in DFT, the frequency domain�s independent variable can be
refereed to in many ways, one way is being a fraction of the sampling rate.
could some one please explain why X[ ] independent variable runs between
0 and 0.5, if you say because discrete data can only contain frequencies
between DC and � the sampling rate, then if the sampling rate (x[ ] has)
128 points then 128 * � is 64 then why X [ ] has 65 samples?
I must be missing something.
Thanks