I am currently studying Fourier analysis following Richard G. Lyons' book "Understanding Digital Signal Processing". He presents the DFT rectangular form as the following:
$$X(m) = \sum_{n=0}^{N-1} x(n)\left[\cos(2\pi{}nm/N) - i\sin(2\pi{}nm/N)\right]$$
Where
- $x(n)$ is the $n$-th input element ($n=0,1,2,...,N-1$)
- $i=\sqrt{-1}$
- $X(m)$ is the $m$-th frequency component ($m=0,1,2,...,N-1$)
My confusion comes when the author gives an example for the frequency values in an $N$-point DFT. So he proposes a signal sampled at 500 Hz and a 16-point DFT, and then says the hihgest frequency would be at $X(15)$ having 468.75 Hz.
He summarizes that with an equation:
$$f_{\text{analysis}}(m)=\frac{mf_s}{N}$$
A bit later, he says
It's also important to realize, from equation [above], that $X(N/2)$, when $m=N/2$, corresponds to half the sample rate, i.e., the folding (Nyquist) frequency $f_s/2$.
OK, that makes sense. But should not the next component, $m=\frac{N}{2}+1$, be an orthogonal component of and with the same frequency as $m=\frac{N}{2}-1$? Isn't that the reason for the symmetry of the DFT? How can you have a DFT frequency component that is higher than $f_s/2$?
To stick to his example, how can a DFT from a signal sampled at 500 Hz reveal a frequency component that is 468.75 Hz?
"To stick to his example, how can a DFT from a signal sampled at 500 Hz reveal a frequency component that is 468.75 Hz?"
I can answer that one.
A dft signal sampled at 500 Hz means it will give you a distribution, up to 500hz, in the frequency domain, the components can be ANY FREQUENCY within this range. So if you got a 500 Hz sample, you'll get all the components in between.
With that in mind, we revise the concept the author described, X(m) will hold the m'th value , he says that at Mth 15 you get 468.75.
There is a post on the right which i just saw that might be more explanatory to this subject > Discrete Fourier Series: What Happens After N/2?
Good luck to any additional 52 people who read this maybe having the same doubt.