Hello everyone
As everyone is waiting for
John to publish his version with the
correlation functionality, I will try to explain
Here what a correlator is.
There will not be much math, just the
necessary.
It starts well we will start from the equation of
cross correlation.
We see that the cross correlation function is a function
of time it is composed of a multiplication + an
integral, "Teta" can be replaced by t, "to" is a
time shift of one of the functions.
In electronics we can make this function by
putting a multiplier + integrator.
This circuit does not output the correlation function but the
correlation coefficient if at the input the frequencies are
fixed, it will be necessary to slide a signal (phase variation)
to obtain the correlation function.
To get rid of the division by time, we can
put a low-pass filter in place of the integrator,
this will be sufficient in the frequency range of your analysis.
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1st exercise: extracting a signal drowned in noise.
Let's imagine that we have a "Morse" signal, I don't know
how to say it in English, but this drawing ".--..-.---..."
will tell you something.
. and - will be replaced by a 1000Hz frequency train of
respective duration.
"." short 1000Hz train
"-" long 1000Hz train
You don't see this signal in the noise, so you are unable
to decode it.
We know that this signal contains 1000Hz trains.
So we put this signal on one input and a 1000Hz signal of infinite duration on the other.
A bit of math:
if there is no "morse" signal.
Let's see what happens for one of the noise frequencies called Fnoise.
we multiply Fnoise by our 1000Hz reference signal.
so at the output of the multiplier we have ::
A*sin(2*pi*Fnoise*t)*B*sin(2*pi*1000*t). A and B the amplitudes.
Yes but
sin A * sin B = 1/2((cos(A-B) - cos(A+B))
the 2nd part, we see, is a periodic signal.
If A-B and A+B belong to our measurement frequency range.
the output of the integral or the low-pass filter will give "0" so
no signal.
So for all the noise frequencies will be eliminated,
except the noise at 1000Hz which I hope will be very small.
Now we multiply a frequency train by our 1000Hz reference signal
at the output of the multiplier we will have:
A*sin(2*pi*1000)*B*sin(2*pi*1000), I'll spare you the phase
it's the mind that counts.
so
(sin A)^2 = 1/2 -1/2 cos 2*A .
We see that this time a continuous value "1/2" will be detected at the
integrator output, therefore signal detection.
In summary, it is only a small trigonometric function that switches the
visibility and decoding of our "morse" signal.
I think that now if you extrapolate this example to the cross correlation that
we are trying to set up we understand better.
replace the 2 previous signals with the same signal from the 2 channels of our ADC.
The first signal will be our noisy signal, the 2nd our reference signal, for
each frequency of our reference signal we apply what was done above.
I'll let you think about it.
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2nd example: let's build a spectrum analyzer with a correlator.
Yes ... it exists in HF, in the 80s (Yes, Yes, I'm old ...)
A person had made an HF spectrum analyzer in a small box
that was connected to an oscilloscope in this way, one channel contained
the output signal of the correlator the other channel a ramp representing the
frequency.
Let's take our correlator again.
On one of the inputs you have your signal to analyze, on the other a
sinus reference signal whose frequency increases linearly with time.
For each frequency of the reference signal you apply what was
said in the detection of a signal drowned in noise.
The analyzer will output a peak each time a frequency line is detected
in the signal to analyze.
So you get a spectrum analyzer ....
A spectrum analyzer is nothing more than a system that can extract signals
from noise either by FFT or by correlation.
I'll let you imagine all that can be done with a correlator.
Look at this paper from Zurich Industry that only talks about correlators:
Zurick Industry correlator application link