Frequency Modulation Questions and Answers

Answer: A

The deviation follows the message amplitude, so a louder passage swings the carrier further from its rest frequency while a higher-pitched passage swings it more often. Proportionality to message frequency would describe no useful scheme. Proportionality to the integral is phase modulation, which is why an integrator ahead of a phase modulator produces FM. And a square-law dependence would distort badly. The distinction between which message property sets deviation and which sets its rate is the foundation of everything else in the topic.

Answer: B

Deviation is the peak swing away from the unmodulated carrier frequency, quoted as plus or minus so many kilohertz, and it is set by the message amplitude alone. The spread of message frequencies is the baseband bandwidth. The occupied bandwidth is larger than the deviation and is given by Carson's rule. And the rate of sweep is the message frequency. Confusing deviation with bandwidth is the most common error in the topic, and keeping the four quantities distinct settles most of it.

Answer: C

The index is \( \beta = \Delta f/f_m \), so it grows with deviation and shrinks as the message frequency rises. Inverting the ratio would make a loud low note narrowband, which is the reverse of the truth. Dividing by the carrier frequency gives a figure of no use, since the carrier plays no part in the spectrum's width. And a product has the wrong dimensions. Because the index depends on the message frequency, a single FM signal has a different index for every tone it carries.

Answer: D

Dividing deviation by message frequency:

\[ \beta = \frac{75}{15} = 5 \]

so 5, which is the deviation ratio of FM broadcasting at its highest audio frequency. 15 is the message frequency itself. 0.2 inverts the ratio and would describe narrowband FM. And 75 is the deviation quoted back. Because the index is largest for the lowest audio frequency, the worst-case bandwidth is computed at the highest audio frequency where the index is smallest, which is the point Carson's rule captures.

Answer: A

The deviation ratio is peak deviation divided by the highest message frequency, so it is the modulation index evaluated at the worst case for bandwidth. For FM broadcasting it is 75 over 15, that is 5. Dividing by the carrier is meaningless here. The index at the lowest message frequency would be enormous and is not what the term denotes. And bandwidth over deviation is a derived ratio with no standard name. The deviation ratio is the single number that characterises a whole system rather than one tone.

Answer: B

Carson's rule is \( B = 2(\Delta f + f_m) \), capturing about ninety-eight per cent of the power. Twice the deviation alone underestimates it and fails badly for narrowband FM, where the message frequency dominates. Twice the message frequency is the AM answer and the narrowband FM limit. And the last combination is not the rule. Carson's rule is an engineering approximation rather than an exact result, which matters because the true FM spectrum extends indefinitely.

Answer: C

Applying the rule:

\[ B = 2(75 + 15) = 180 \ \mathrm{kHz} \]

so 180 kHz, which is why FM broadcast channels are spaced 200 kHz apart with a guard band. 150 kHz is twice the deviation, omitting the message term. 90 kHz is the sum without the factor of two. And 30 kHz is twice the message frequency, the AM figure. The comparison with AM's 30 kHz for the same audio shows exactly what FM spends to buy its noise performance.

Answer: D

Substituting equal values:

\[ B = 2(3+3) = 12 \ \mathrm{kHz} \]

so 12 kHz, and with an index of one this sits between the narrowband and wideband regimes. 6 kHz is twice the message frequency, the true narrowband limit which applies only when the index is much less than one. 3 kHz is the deviation. And 24 kHz doubles the correct answer. Recognising which regime a signal is in decides whether the simple limit or the full rule should be used.

Answer: A

In the narrowband limit the bandwidth tends to \( 2f_m \), exactly as for AM, because only the first pair of sidebands carries appreciable power. That is why narrowband FM occupies a channel comparable with AM's and why it is used for voice radio. Twice the deviation is the wideband approximation. The deviation alone is never the bandwidth. And half the message frequency is less than the message needs. The two limits of Carson's rule are worth knowing separately, since they bracket every practical case.

Answer: B

The Bessel expansion gives infinitely many sideband pairs, at every multiple of the message frequency from the carrier, with amplitudes that eventually decay. That is why FM bandwidth is a matter of convention rather than of exact arithmetic, and why Carson's rule exists. A single pair is the narrowband approximation. Two pairs is an arbitrary truncation. And the index sets how many pairs are significant rather than how many exist. The infinite spectrum is the fundamental difference from AM.