Beginner question on the DFT and sampling theory

I am having a little trouble with getting a clear picture of
extracting what frequencies are represented in an input stream from
the DFT.

Here's my current picture --

1) Given N=16384 samples at a sampling rate of 5 secs or 0.25 Hz

2) 16384 samples = 81920 secs = 1365.3 min = 22.76 hrs

3) Each frequency sample of the DFT, Xk = c_k / N, the spectral
coefficient in the discrete time fourier series

4) Xk,k=1, is the spectral coefficient for frequency (2*pi/16384) * 1,
f_1 = 1/16384 = 61u
    Xk,k=2, is the spectral coefficient for frequency (2*pi/16384) *
2, f_2 = 2/16384 = 122u
*** I divided the frequency by 2*pi to get f_k, this factor includes
the Nyquist rate, I saw this in a short tutorial

5) X_1 at frequency f_1 of the input stream maps to an actual period
of  5 secs / 61u = 81967.2 secs

6) X_8192 at frequency f_8192 of the input stream maps to an actual
period of 5 secs / 0.5 = 10 secs

7) X_16384 maps to an actuall period of 5 secs

8) Because of the Nyquist rate I can only accurately represented the
time period from 81967.2 secs to 10 secs.

Is this correct?   My main confusion is with extracting frequencies
from the samples and then using the sampling period map back into the
real frequencies.  Any help is greatly appreciated.

Thanks,

Aaron

On 8 Des, 21:36, Aaron <abote...@yahoo.com> wrote:
> I am having a little trouble with getting a clear picture of
> extracting what frequencies are represented in an input stream from
> the DFT.

> Is this correct? &#4294967295; My main confusion is with extracting frequencies
> from the samples and then using the sampling period map back into the
> real frequencies. &#4294967295;Any help is greatly appreciated.

Work with simpler numbers. The shear scale of the numbers
you quote makes the explanation hard to comprehend.

The idea is simple:

With 10 samples and sampling period 0.1 s, you have 1 second
of data. If you compute the 10 pt DFT you will get the
spectrum coefficients at n*fs/N, N = 10, n = 0,1,2,...,9.
Since you have real-valued data, there will be symmetries
involved.

The spectrum coefficients represent the whole time interval.
Change one sample and all the spectrum coefficients will
change. Or change one symmetric pair of spectrum coefficients,
and all the samples will change.

Rune

On Dec 8, 3:36&#4294967295;pm, Aaron <abote...@yahoo.com> wrote:
> I am having a little trouble with getting a clear picture of
> extracting what frequencies are represented in an input stream from
> the DFT.
>
> Here's my current picture --
>
> 1) Given N=16384 samples at a sampling rate of 5 secs or 0.25 Hz
>
> 2) 16384 samples = 81920 secs = 1365.3 min = 22.76 hrs
>
> 3) Each frequency sample of the DFT, Xk = c_k / N, the spectral
> coefficient in the discrete time fourier series
>
> 4) Xk,k=1, is the spectral coefficient for frequency (2*pi/16384) * 1,
> f_1 = 1/16384 = 61u
> &#4294967295; &#4294967295; Xk,k=2, is the spectral coefficient for frequency (2*pi/16384) *
> 2, f_2 = 2/16384 = 122u
> *** I divided the frequency by 2*pi to get f_k, this factor includes
> the Nyquist rate, I saw this in a short tutorial
>
> 5) X_1 at frequency f_1 of the input stream maps to an actual period
> of &#4294967295;5 secs / 61u = 81967.2 secs
>
> 6) X_8192 at frequency f_8192 of the input stream maps to an actual
> period of 5 secs / 0.5 = 10 secs
>
> 7) X_16384 maps to an actuall period of 5 secs
>
> 8) Because of the Nyquist rate I can only accurately represented the
> time period from 81967.2 secs to 10 secs.
>
> Is this correct? &#4294967295; My main confusion is with extracting frequencies
> from the samples and then using the sampling period map back into the
> real frequencies. &#4294967295;Any help is greatly appreciated.
>
> Thanks,
>
> Aaron
Aaron,

There are some flash programs here that help with the DFT. You need to
have a very good grasp of the Fouriers Series before you can
appreciate the DFT.  There are tutorials on the fourier series also
located at this site.

http://www.fourier-series.com/fourierseries2/DFT_tutorial.html


Good luck.

Brent


http://www.fourier-series.com/fourierseries2/DFT_tutorial.html