Thursday, June 27, 2019

1D FFT algorithm elements explanation?



Can someone explain to me what each element of the 1D FFT algorithm are (i.e. N,x,n,j,k) ? From here, http://zone.ni.com/reference/en-XX/help/371361H-01/lvanls/fft/#details


(Can't post pictures on stackoverflow just yet.... but its the 1D FFT description)



Answer



Consider a MATLAB/OCTAVE implementation of 1D-DFT/FFT sum: $$X[k] = \sum_{n=0}^{N-1} {x[n]e^{-j \frac {2\pi}{N} k n} } $$ where $n=0,...,N-1$ and $k=0,...,N-1$


Those identifiers are used to denote the following:


$N$: is the FFT length as in $N$-point FFT.


$x[n]$: is the discrete-time signal whose DFT $X[k]$ is to be computed.


$X[k]$: is the complex-valued DFT of the signal $x[n]$


$n$: is the discrete-time index for the signal $x[n]$ in the range $n=0,...,N-1$.


$k$: is the frequency sample index of $X[k]$, in the range $k=0,...,N-1$.



$j$: is the imaginary unit $\sqrt {-1}$ in signal processing terminology.


Note if the signal $x[n]$'s length is shorter than the FFT length, then zero padding is applied on $x[n]$ to make its length $N$. Otherwise if it's longer than $N$, then the samples after $N-1$ are ignored according to MATLAB/OCTAVE implementation.


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