1. 268 Chapter Nine
Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
results usually give a good indication of what to expect with an arbitrary
signal.
Example 9.2 A test tone of frequency 800 Hz is used to frequency modulate a carrier, the peak
deviation being 200 kHz. Calculate the modulation index and the
bandwidth.
Solution
f = 200
= 250 0.8
B = 2(200 + 0.8) = 401.6 kHz
Carson's rule is widely used in practice, even though it tends to give an
underestimate of the bandwidth required for deviation ratios in the range 2
< D < 10, which is the range most often encountered in practice. For this
range, a better estimate of bandwidth is given by
BIF = 2(AF + 2FM)
(9.4
)
Example 9.3 Recalculate the bandwidths for Examples 9.1 and 9.2.
Solution For the video signal,
Bif = 2(10.75 + 8.4) = 38.3 MHz
For the 800 Hz tone:
Bif = 2(200 + 1.6) = 403.2 kHz
In Examples 9.1 through 9.3 it will be seen that when the deviation ratio (or
modulation index) is large, the bandwidth is determined mainly by the peak
deviation and is given by either Eq. (9.1) or Eq. (9.4). However, for the
video signal, for which the deviation ratio is relatively low, the two
estimates of bandwidth are 29.9 and 38.3 MHz. In practice, the standard
bandwidth of a satellite transponder required to handle this signal is 36
MHz.
The peak frequency deviation of an FM signal is proportional to the peak
amplitude of the baseband signal. Increasing the peak amplitude results in
increased signal power and hence a larger signal-to-noise ratio. At the same
time, AF, and hence the FM signal bandwidth, will increase as shown
previously. Although the noise power at the demodulator input is
proportional to the IF filter bandwidth, the noise power output after the
demodulator is determined by the bandwidth of the baseband filters, and
therefore, an increase in IF filter bandwidth does not increase output noise.
Thus an improvement in signal-to-noise ratio

2. Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
Analog Signals 269
is possible but at the expense of an increase in the IF bandwidth. This is the
large-amplitude signal improvement referred to in Sec. 9.6 and considered
further in the following section.
9.6.3 FM detector noise
and processing gain
At the input to the FM detector, the thermal noise is spread over the IF
bandwidth, as shown in Fig. 9.10a. The noise is represented by the system
noise temperature Ts, as will be described in Sec. 12.5. At the input to the
detector, the quantity of interest is the carrier-to-noise ratio. Since both the
carrier and the noise are amplified equally by the receiver gain following the
antenna input, this gain may be ignored in the carrier- to-noise ratio
calculation, and the input to the detector represented by the voltage source
shown in Fig. 9.10b. The carrier root-mean-square (rms) voltage is shown
as Ec.
SnS£B|F —
H l^Sf
iI| |t.i | f
,
+w
(a I
(b)
Figure 9.10 (a) The predetector noise bandwidth BN is approximately equal to the IF bandwidth BIF. The
LF bandwidth W fixes the equivalent postdetector noise bandwidth at 2W. Sf is an infinitesimally
small noise bandwidth. (b) Receiving system, including antenna represented as a voltage source up to
the FM detector.

3. 270 Chapter Nine
Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark.
The available carrier power at the input to the FM detector is E2
/4R, and
the available noise power at the FM detector input is kTsBN (as explained in
Sec. 12.5), so the input carrier-to-noise ratio, denoted by C/N, is
C E2
C
= -------- c
----
------------- (9.5)
N 4RkTs BN
When a sinusoidal signal of frequency, fm, frequency modulates a carrier
of frequency, fc: The instantaneous frequency is given by fi = fc + Afsin
2wfmt, where Af is peak frequency deviation. The output signal power
following the FM detector is
Ps = AAf2 (9.6)
where A is a constant of the detection process.
The thermal noise at the output of a bandpass filter, for which fc >> BN
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 carrier amplitude is
very much greater than the noise amplitude the noise amplitude component
can be ignored for FM, and the carrier angle as a function of time is 9(t) =
2ft + $n(t), where $n(t) is the noise phase modulation. Now the instantaneous
frequency of a phase modulated wave in general is given by Mi = dQ(t)/dt
and since Mi = 2^fi, the equivalent FM resulting from the noise phase
modulation is
feq.n = fc + ~~~ (9.7)
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
kTsBN and the noise spectral density, which is the noise power per unit
bandwidth just kTs. A result from Fourier analysis is that 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 as a function of
frequency is (2wf)2
kTs. The variation of output spectral 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 spectral density will be
proportional to frequency as sketched in Fig. 9.116.
Figure 9.11a shows that the output power spectrum is not a flat function
of frequency. The available noise output power in a very small band df
would be given by (2wf)2
kTshf. The total average noise output power