# Bandwidth of FM Signals: Carson’s Rule

A satellite telemetry link operating in S-band uses frequency modulation to transmit the value of an analog voltage on the satellite to a receiving earth station. The voltage has a range from -1.0 volts to +1.0 volts, and a maximum frequency of 1000Hz.The FM modulator on the satellite has a constant of 10,000 Hz per volt. At the receiving earth station the C/N ratio of the signal is 10 dB measured in the Carson’s rule bandwidth, and is 3 dB above the FM threshold of the FM demodulator.

a. What is the Carson’s rule bandwidth for the FM signal?
b. What is the baseband S/N ratio at the earth station receiver output for the recovered analog signal?

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Carson’s rule bandwidth for the FM signal

Question:
A satellite telemetry link operating in S-band uses frequency modulation to transmit the value of an analog voltage on the satellite to a receiving earth station. The voltage has a range from -1.0 volts to +1.0 volts, and a maximum frequency of 1000Hz.The FM modulator on the satellite has a constant of 10,000 Hz per volt. At the receiving earth station the C/N ratio of the signal is 10 dB measured in the Carson’s rule bandwidth, and is 3 dB above the FM threshold of the FM demodulator.

a. What is the Carson’s rule bandwidth for the FM signal?
b. What is the baseband S/N ratio at the earth station receiver output for the recovered analog signal?

Solution:
a.Carson’s rule states that the bandwidth required to transmit an FM signal is given by

B = 2(Δfpk+ fmax) Hz
Where,
Δfpk = the peak frequency deviation
fmax = the highest frequency present in the modulating signal

1. The frequency deviation of the carrier is directly proportional to the modulating signal voltage.

2. The bandwidth required to transmit an FM signal is found from Carson’s rule.

From here don’t know how to calculate the Δfpk ,the peak frequency deviation.

b. (S/N)out = C/N + 10 log10(BRF/fmax) + 20 log10(Δfpeak/fmax) + 1.8 dB (5.11)

Where

BRF = IF bandwidth of receiver = RF B/W of FM signal from Carson’s rule

fpeak = peak frequency deviation at transmitter
fmax = maximum frequency of baseband signal = receiver baseband bandwidth

And the factor of 1.8 dB is equivalent to the numerical ratio 3/2.

Same as here don’t know how to calculate the Δfpk ,the peak frequency deviation.

Khandaker Mosharraf Arafen
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