Look up the Discrete Fourier Transform in its Matrix form: You take a vector of samples
and
multiply it with a matrix 
to get the
discrete Fourier transform of , which I
shall denote 
:
, with
(i.e. a square matrix, representing the N-point
DFT)
(i.e. column vectors).Remember how you do matrix-vector multiplication:
To get the result's element in its $n$-th result row and the $m$-th result column, you take row $n$ from the left factor (i.e. a row of $W$, which has $N$ rows, so $n\in\{1,\dots,N\}$), and multiply it point-wise with $m$-th column of the right factor ($x$, which only has 1 column, so $m\equiv 1$) and calculate the sum.
On 20.03.2016 17:45, Henry Barton
wrote:
So correlating is digitally mixing something with a predetermined sequence? I’ve been struggling to figure out what correlation means and this seems right. I guess, then, if you have a PSK waveform you might multiply it by something, starting at various offsets in the signal, to try and decode it?
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Yes, pretty much. With the DFT (and the continuous one) you are correlating
the input waveform with harmonically related, complex sinusoids; essentially
for each harmonic you mix it down to DC then sum (integrate). The FFT is
different (I actually don't know how it works, other than it operates on 2^n
samples), but the output is the same.
Lou
Henry Barton wrote
> I’ve read up on the FFT and DSP and I must say I’m impressed that
> multiplying two waveforms is the digital equivalent of heterodyning. Am I
> right in my understanding that finding frequency components (FFT-ing) is
> simply multiplying a series of known sine waves by your input waveform?
--
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