🚀 go-pugleaf

In <1993Jan22.153135.17150@exlog.com> johnk@exlog.com  (John Kingman) writes:

[abbreviated]
> u(t)		signal of interest
> v(t)		Hilbert transform of u(t)
> p(t)		phase, p = wc*t + (wd/wm)*sin(wm*t)
> u(t) 		modulated signal u(t) = cos(p*t)
>
> In playing with this using discreet time series', I have found that in
> general, for deviation ratios (wd/wm) >= 1, the unwrapped argument of the 
> analytic  function, a(t), is NOT the same as the phase p(t)!  Another way of 
> putting this is:
>
>		v(t) .ne. sin(p(t))
>

There is no reason for the two be equal. The fact that the Hilbert transform of
cos(wt) = sin(wt) for *constant* 'w' really doesn't imply anything about the 
general case of arbitrary argument (such as p(t) = wc*t + (wd/wm)*sin(wm*t) in
your example). There are many ways to see this; look at the integral definition
of the Hilbert transform for example and notice how nontrivial it is to get
closed form Hilbert transforms. The usual interprestation of the Hilbert
transform as a "90 degree phase shifter" might be a little misleading here since
it shifts every frequency *component* by 90 degrees; the overall trasnformed
signal could be fairly different from the original. If the argument (p(t)) 
is "more or less constant" that is if (p(t) - wc*t) is a low frequency
signal then to some measure you can use the approximation: 
v(t) .approx. sin(p(t)). This approximation will of course get worse and
worse as your deviation ratio gets larger.


Jacob Hirbawi
JcbHrb@CERF.net