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Re: [Discuss-gnuradio] Handling of IQ files

From: Marcus Müller
Subject: Re: [Discuss-gnuradio] Handling of IQ files
Date: Sun, 20 Mar 2016 18:56:31 +0100
User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0

Hi Henry,

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.

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
Sent: ‎Sunday‎, ‎March‎ ‎20‎, ‎2016 ‎12‎:‎22‎ ‎PM
To: address@hidden

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.

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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