Message-ID: <002301c4d740$4795e0a0$5e437ad5@jgtdiynm>
From: "James Moritz" <james.moritz@btopenworld.com>
To: rsgb_lf_group@blacksheep.org
Date: Wed, 1 Dec 2004 00:54:13 -0000
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Subject: LF: Variable-PSK theory (warning - goes on a bit!)
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<DIV><FONT face=Arial size=2>Dear LF Group,</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>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.</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>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:</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>forward: theta = (pi/2 +pi.t/T)</FONT></DIV>
<DIV><FONT face=Arial size=2>reverse: theta = (pi/2 - pi.t/T)</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>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) =&nbsp;cosy.sinx + siny.cosx:</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>I component: cos(pi/2 +/- 
pi.t/T).sin(wct)</FONT></DIV>
<DIV><FONT face=Arial size=2>Q component: sin(pi/2 +/- 
pi.t/T).cos(wct)</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>A bit of thought about the&nbsp;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.</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>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&nbsp;envelope meeting the zero line; it will have the fourier 
spectrum of a full-wave rectified sine wave, whatever that is, reflected 
around&nbsp;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&nbsp;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 ( <A 
href="http://www.wireless.org.uk/moritz.htm">http://www.wireless.org.uk/moritz.htm</A>&nbsp;)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.</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>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&nbsp;in no baseband signal component, so this should have no effect on 
the actual demodulation. However, it might have&nbsp;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&nbsp;confirm that 
there is little effect in the "real world".</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>I hope I haven't made any stupid errors in the 
above, but someone has to stick their neck out!&nbsp;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.</FONT></DIV>
<DIV>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>Cheers, Jim Moritz</FONT></DIV>
<DIV><FONT face=Arial size=2>73 de M0BMU</FONT></DIV></BODY></HTML>