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Dear LF Group,
I made a first attempt at trying to analyse the
"variable phase" type of PSK modulation that DL4YHF has experimented with
recently. The following is far from complete, but I think it gives some
interesting results and pointers for further discussion and
development.
I beleive Wolf has used a type of PSK where the
phase of the carrier is varied linearly from 0 to pi radians, or pi to 0
radians, for a 0 to 1 or 1 to 0 bit transition. So during the transition the
signal could be represented as sin(wct + theta) where wc is the carrier
frequency and theta is the phase shift, proportional to the time. There are 2
types of transition, "forward" (0 to 1) where the carrier goes 0 to pi radians,
and "reverse" where it goes pi to 0 radians. For linearly changing phase,
assuming that the transition lasts 1 bit period T, and the mid-point of the
transition is at t=0, theta will be:
forward: theta = (pi/2 +pi.t/T)
reverse: theta = (pi/2 - pi.t/T)
so during the transition period -T/2 to T/2, the
signal will be sin(wct + [pi/2 +/- pi.t/T]), the +/- depending whether the
transition is forward or reverse. Using the standard trig identities (remember
those?), this can be resolved into in-phase (sin wct) and quadrature (cos wct)
components, since sin(x+y) = cosy.sinx + siny.cosx:
I component: cos(pi/2 +/-
pi.t/T).sin(wct)
Q component: sin(pi/2 +/-
pi.t/T).cos(wct)
A bit of thought about the I component shows
it is just exactly the normal envelope-modulated PSK waveform used in PSK31 etc.
- it starts at +/-sin(wct), ramps down to zero, then comes up with the opposite
phase as -/+sin(wct), the signs depending on whether it is a fwd or rev
transition. But in addition to this we have the Q quadrature component - it
turns out with a bit more trigonometric fiddling that sin(pi/2 + pi.t/T) =
sin(pi/2 - pi.t/T), therefore the quadrature component is like the in-phase
component but without the phase reversals - in effect it is a carrier 100%
modulated with a half-sine wave envelope occuring each time a transition
occurs.
So now we can see the effect of variable phase
modulation on the spectrum - we will have the "ideal" envelope-modulated BPSK
spectrum, plus the spectrum of the quadrature component. This will have infinite
sidebands, due to the "sharp corners" of the half-sine-wave
modulation envelope meeting the zero line; it will have the fourier
spectrum of a full-wave rectified sine wave, whatever that is, reflected
around a carrier frequency component - the amplitudes of the sidebands will
be lower than those of "abrupt" phase transitions (rectangular pulse spectrum),
but I guess still significant. This shows that the linear ramping of phase
is not the optimum as far as the spectrum is concerned - I think the solution is
to replace the linear ramp with perhaps a gaussian sort of smoothly curved
ramp, or maybe a cosine-shaped ramp. I guess that this will have the effect of
increasing the lower order sidebands, but reducing the higher order sidebands -
perhaps ending up with a signal that has a bandwidth about twice the bit rate -
for moderate bit rates, this should be very acceptable compared to on-off keyed
CW or normal FSK. Allthough I have not attempted to analyse this properly, this
is actually what my variable -phase modulator ( http://www.wireless.org.uk/moritz.htm )did
- the phase keyed signal is first turned into trapezoidal phase ramps, then low
pass filtered by a bessel response filter to produce a smoothly curved phase
signal. I did this just as an intuitive guess - I can't find the spectra I
obtained with this at the moment, but I recall they were essentially in
agreement with the above, with a visible carrier and somewhat more 2nd order
sideband component.
The other thing of interest is - what effect will
the additional quadrature component have on the demodulation process? The
classic way of demodulating PSK is to multiply the signal with a regenerated
carrier sin(wct), and low-pass filter out the "DC" (actually the baseband
signal). Multiplying the quadrature component by this sin(wct) will
result in no baseband signal component, so this should have no effect on
the actual demodulation. However, it might have an effect on the recovery
of the carrier. From looking at the previous mails on demodulators, I guess that
what is essentially being done is to measure the average phase of the signal
over a period of time - if there are equal 1s and 0s, the phase will be 0 or pi
each 50% of the time, giving an average of pi/2, so adding in the Q component
with a constant pi/2 phase should not change that - but I am on dodgy ground
there, I think. However, Wolf's on-air experiments seem to confirm that
there is little effect in the "real world".
I hope I haven't made any stupid errors in the
above, but someone has to stick their neck out! If I'm wrong, please let me
know. In conclusion, I think the "variable phase" scheme should work OK, both
from the point of view of the signal spectrum (with attention to the shape of
the phase ramp), and also demodulation - so it seems to be worth pursuing
it.
Cheers, Jim Moritz
73 de M0BMU
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