Hi Henry,
> So correlating is digitally mixing something with a predetermined
> sequence?
um, no. Not in the sense that I'd use "correlating".
Mixing in this context is simple point-wise multiplication.
> 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?
A correlation of two digital signals is pretty much that, you take the
first signal, multiply it with the second, _sum that up_, and then shift
either signal, rinse, repeat. You get a vector of /correlation
coefficients/ which can be interpreted as "how similar is the first
signal to the second being shifted by a different sample delays".
Generally, if you have a known pulse shape (i.e. your transmitter takes
the transmit symbol, in case of BPSK that would be either +1 or -1, and
multiplies it with the pulse shape) you can correlate against that pulse
shape, and you'll get the factor with which the pulse was multiplied
(which is your transmit symbol combined with whatever your channel does
to the signal).
Because we were talking about the DFT/FFT:
Look up the Discrete Fourier Transform in its Matrix form: You take a
vector of samples $x$ and multiply it with a matrix $W$ to get the
discrete Fourier transform of $x$, which I shall denote $X$:
$X=Wx$, with
$W \in \mathbb N ^{N\times N}$ (i.e. a square matrix, representing the
N-point DFT)
$x,X \in \mathbb N ^N$ (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.
Now, point-wise multiplication & summation is what is defined as /dot
product/, so what happens here is that you take your sample vector $x$
and calculate the dot products with each row of the DFT Matrix $W$. The
rows of the DFT matrices happen to be nothing else but samples of
complex sinusoids.
Regarding Lou's:
> 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.
Exactly, an FFT is just a specific implementation of the DFT (because
you really don't want to do vector-matrix multiplication if you can
avoid it, because you'd end up doing $N^2$ multiplications and
summations), and is mathematically equivalent. In a first step, the fact
that the DFT matrix is pretty regular and all entries appear twice is
used to save calculations. Modern FFTs do all kind of cool optimizations
that I don't understand the least, but what I learnt is that today, the
Radix-2 algorithm isn't the only FFT algorithm in existence, and so it
turns out that for example the FFTW library, which GNU Radio uses to do
its FFTs, can deal with any DFT that has a prime factorization with
"nearly no" prime factors >7 [1].
So, yeah, DSP often gets a lot easier if you just consider it an
exercise in linear algebra :)
Cheers,
Marcus
[1] http://www.fftw.org/doc/Complex-DFTs.html#Complex-DFTs
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?
>
> Sent from Windows Mail
>
> *From:* madengr <mailto:rfeng...@me.com>
> *Sent:* Sunday, March 20, 2016 12:22 PM
> *To:* discuss-gnuradio@gnu.org <mailto:Discuss-gnuradio@gnu.org>
>
> 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?
>
>
>
>
>
> --
> View this message in context:
> http://gnuradio.4.n7.nabble.com/Handling-of-IQ-files-tp58915p58935.html
> Sent from the GnuRadio mailing list archive at Nabble.com.
>
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