```Hi Alex and All,

> By the way as pulse shape is assumed to be known perfectly though the best
> procedure to determine pulse timing and its amplitude to be as fallows.
> One should maximize a correlator betwin pulse function stored in memory
> and real signal. This maximization yelds a incoming time of pulse.
I think this has to be modified. We have a pulse, detected by a power level
above threshold. The known pulse shape is symmetric, it is a large
value at t=0 and weak oscillations surrounding the pulse maybe +/-
10 samples. If there is a second pulse of similar amplitude somewhere
within +/- 10 pulses, the correlation will have a significant error,
it is better to use the few data points at the pulse maximum only.
This is a real world experience, the correlator was my first guess.
The explanation is that the significant problem is not the white noise
floor, it is multiple pulses.

Although the pulse shape is known we can not correlate it directly
because the phase with respect to I and Q is unknown, so is the phase
with respect to the sampling clock and also the phase and amplitude
ratio between the two channels (polarisation)

Linrad solves this by working with the pulse power rather than the
pulse amplitude as a first step. This information is already
implicitly contained in "detected by a power level above threshold."

Once the point of maximum is known, the power within a time span
+/- blanker_pulsewidth is collected for the two rx channels together
with the correlation between the channels over this period of time.
The time corresponds to the separation between the -15dB points on
the pulse power.

Having collected powers and correlation (complex) it is possible
to calculate the polarisation. At this point an improved result
would probably be obtained by weighing by S/N which would be the
"true" pulse power time function - but that is not known at this
stage because we do not know how the pulse timing is with respect
to  the sampling clock yet.

Once the polarisation is knowm, the two input channels are subjected
to an orthonormal transformation that brings all the pulse energy into
one channel, leaving noise only in the other channel (hopefully).
The polarisation constants c1,c2 and c2 are stored.

At this point the pulse is represented by a single complex function.
This is where we would start if there is only one receive channel.
The pulse can be expressed as amplitude and phase vs time. Phase
vs time is a slow function and therefore it is independent of the
position of the pulse vs the sampling clock. The known dp/dt function
is used to calculate a version of the pulse that should have a constant
phase angle.

The phase is rotated for the average phase of the three largest
samples to become zero.
The phase angle p0 is stored.

The three largest data points of the real part of the new function
are used to get the pulse position with respect to the sampling
clock by use of a parabolic fit. The phase between the pulse and
the sampling clock, p1, is used as an index to a library of pulse
shapes from which we get the "correct" pulse response for a pulse
with the current position with respect to the sampling clock.
Here is the point where correlation would be useful if the problem
had been white noise. Something else would be optimum if the
limiting factor is another pulse but Linrad simply subtracts the
complex pulse response taken from the library with the amplitude
that makes the largest sample zero. For the subtraction, the
stored parameters are used to transform the library pulse to the
correct phase p0 and polarisation c1,c2,c3.

For the Linrad procedure to work well it is essential that the biggest
pulse is removed first. The oscillations of a larger pulse will be
removed with a high accuracy close to the maximum of a smaller pulse
close in time and therefore the smaller pulse will also be accurately
removed - but not at the location of the maximum point of the big pulse.
The error will be the magnitude of the oscillations belonging to the
weak pulse at this point. It was set to zero by the subtraction
procedure, but it should have been set to the value of the oscillation
of the weaker pulse at the position of the larger pulse.

Multiple pulses are obviously not treated correctly by Linrad, I am
sure it can be done although other errors are most probably much more
important for the blanker/pulse removal performance today.
The reason it works so well is that the pulses are made very short
by the filtering through a filter taylored for that purpose.

The amplitude and phase with respect to the sampling clock, p1, are
determined from the three largest samples only. The transformed
pulse is computed over the entire size of the pulse plus oscillations.
This is where I tried a correlator initially. I intended the information
to be used some day so Linrad still computes it - and then throws it away;)

> Amplitude also can be calculated from correlator value. This is well
> described in any book on 'radar theory'. Does linrad use such a procedure?
In short:  no ;)

I might add that the calibration procedure where the operator selects the
desired frequency response also means that he selects the desired pulse
response. The optimum compromise seems to be a very complicated problem
from a theoretical point of view but it is not very critical in situations
that I have experience with.

73

Leif  /  SM5BSZ

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