🚀 go-pugleaf

I am working on an instantaneous frequency problem where
instantaneous frequency is defined as follows:

u(t)		signal of interest
v(t)		Hilbert transform of u(t)
z(t)		analytical signal z(t) = u(t) + i*v(t) (i=sqrt(-1))

a(t)		unwrapped argument of z(t)
f(t)		instantaneous frequency f(t) = d/dt[a(t)]/(2*pi)

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my question concerns a frequency modulated signal (modulated with
a pure tone as follows):

t		time
wc		carrier frequency (radians/sec)
wm		modulation frequency
wd		frequency deviation 

p(t)		phase, p = wc*t + (wd/wm)*sin(wm*t)
u(t) 		modulated signal u(t) = cos(p*t)

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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))

I have not bothered to solve the problem explicitly yet but feel fairly
confident that results will confirm my discreet approximation.
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Am I missing something?

Any comments, notes of experiences will be greatly appreciated.