Page 290 - Satellite Communications, Fourth Edition

270 Chapter Nine

2
The available carrier power at the input to the FM detector is E c /4R,
and the available noise power at the FM detector input is kT s B N (as
explained in Sec. 12.5), so the input carrier-to-noise ratio, denoted by
C/N,is
2
C E c
(9.5)
N 4RkT B N
s
When a sinusoidal signal of frequency, f , frequency modulates a
m
carrier of frequency, f : The instantaneous frequency is given by
c
f i f c f sin 2 f m t , where f is peak frequency deviation. The output
signal power following the FM detector is
2
P s A f (9.6)
where A is a constant of the detection process.
The thermal noise at the output of a bandpass filter, for which f c >> B N
has a randomly varying amplitude component and a randomly varying
phase component. (It cannot directly frequency modulate the carrier, the
frequency of which is determined at the transmitter, which is at a great
distance from the receiver and may be crystal controlled). When the car-
rier amplitude is very much greater than the noise amplitude the noise
amplitude component can be ignored for FM, and the carrier angle as
t std , where (t) is the noise phase
a function of time is
std 2 f c n n
modulation. Now the instantaneous frequency of a phase modulated
wave in general is given by d
(t)/dt and since 2 f , the equiv-
i
i
i
alent FM resulting from the noise phase modulation is
1 d n std
f (9.7)
f eq.n c
2 dt
What this shows is that the output of the FM detector, which responds
to equivalent FM, is a function of the time rate of change of the phase
change. Now as noted earlier, the available noise power at the input to
the detector is kT B and the noise spectral density, which is the noise
s
N
power per unit bandwidth just kT . A result from Fourier analysis is that
s
the power spectral density of the time derivative of a waveform is (2 f) 2
times the spectral density of the input. Thus the output spectral density
2
as a function of frequency is (2 f) kT . The variation of output spectral
s
noise density as a function of frequency is sketched in Fig. 9.11a. Since
voltage is proportional to the square root of power, the noise voltage spec-
tral density will be proportional to frequency as sketched in Fig. 9.11b.
Figure 9.11a shows that the output power spectrum is not a flat func-
tion of frequency. The available noise output power in a very small band
2
df would be given by s2 fd kT
f . The total average noise output power
s