Consider an FM signal \(s(t) = 10\sin\left(4\pi \times 10^6 t + 9\cos\left(2\pi \times 10^3 t\right)\right)\) the frequency deviation and bandwidth of FM wave are
9 kHz, 20 kHz
Step 1 — match the signal to the standard FM form.
\(s(t)=A_c\sin\left(\omega_c t+\beta\cos\omega_m t\right)\)
Comparing with \(s(t)=10\sin\left(4\pi\times10^{6}t+9\cos\left(2\pi\times10^{3}t\right)\right)\):
Carrier: \(\omega_c=4\pi\times10^{6} \Rightarrow f_c = 2\ \text{MHz}\). Modulating signal: \(\omega_m=2\pi\times10^{3} \Rightarrow f_m = 1\ \text{kHz}\). Modulation index (the coefficient of the phase-modulating term): \(\beta = 9\).
Step 2 — frequency deviation. By definition the modulation index of an FM wave is the ratio of peak deviation to modulating frequency:
\(\beta = \dfrac{\Delta f}{f_m} \Rightarrow \Delta f = \beta f_m\)
\(\Delta f = 9 \times 1\ \text{kHz} = 9\ \text{kHz}\)
Physically, the instantaneous frequency swings 9 kHz above and 9 kHz below the 2 MHz carrier. (You can also get this by differentiating the phase: \(f_i = f_c - \beta f_m \sin\omega_m t\), whose peak excursion is βfm.)
Step 3 — bandwidth by Carson's rule.
\(BW = 2\left(\Delta f + f_m\right) = 2\left(\beta+1\right)f_m\)
\(BW = 2(9+1)\ \text{kHz} = 20\ \text{kHz}\)
Why Carson's rule and not "infinite bandwidth". An FM spectrum strictly contains infinitely many sidebands spaced fm apart, with amplitudes given by Bessel functions Jn(β). Carson's rule keeps the (β + 1) significant pairs that carry about 98 % of the power, which is the accepted practical bandwidth.
Watch the distractors. 10 kHz would come from wrongly reading the deviation as (β + 1)fm; 30 kHz would come from \(2(\Delta f + 2f_m)\) or from using β = 14. Note also that β = 9 ≫ 1 means this is wideband FM, so Carson's rule is the correct tool.
Hence, the frequency deviation is 9 kHz and the bandwidth is 20 kHz.
Similar Questions
In FM
Consider the following :
ST1 : F.M. signal produces more side bands than A.M.
ST2 : The carrier in a F.M. signal can never be dropped to zero amplitude.Which of the following is valid ?
Which of the following is true ?
The FM transmitters have the following blocks as per the following correct sequence :
(A) Crystal Oscillator
(B) Antenna
(C) Frequency multiplier
(D) Phase modulate / Audio source
(E) Power amplifierChoose the most appropriate answer from the options given below :
In case of wideband FM, the modulation index value is :
A high frequency signal is frequency modulated by n number of modulating signals. The ideal number of sidebands in the modulated signal will be :
The approximate rule for transmission of an FM signal generated by a single-tone modulating signal of frequency fm, modulation index β and maximum frequency deviation Δf, is defined as :
Assertion (A) : The FM radio broadcast of analog signals provides higher fidelity.
Reason (R) : FM uses significantly larger channel bandwidth for signal transmission.
Select your answer using the codes given below :
What bandwidth is needed for an FM signal that has a peak deviation of ± 3 KHz and handles audio signals from 200 Hz to 5 KHz ?
Which of the following techniques are used to generate frequency modulated signal :
1. Armstrong
2. Foster-Sealy Discriminator
3. Balanced Modulator
4. Reactance ModulatorWhich one of the following is true ?
Important Questions from Frequency Modulation
Which of the following statements is true for FM?
What is the modulation index in a frequency modulated signal with a modulating frequency of 500 Hz and frequency deviation of 10 kHz?
Which of the following rules states that the bandwidth required to transmit an angle modulated wave is twice the sum of the peak frequency deviation and highest modulating signal frequency?
If f mis modulating frequency and m fis modulation index, then by the Carson's rule, the bandwidth of an FM signal at the input of a conventional discriminator will be:
In FM, "M" stands for
