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 :
\(2\Delta f\left(1+\dfrac{1}{\beta}\right)\)
This is Carson's rule, and it is easiest to recognise in its plain form first :
\(B=2\left(\Delta f+f_{m}\right)\)
— twice the sum of the peak deviation and the highest modulating frequency. Every version in the options is this same statement rewritten using the modulation index
\(\beta=\dfrac{\Delta f}{f_{m}}\qquad\Rightarrow\qquad f_{m}=\dfrac{\Delta f}{\beta}\)
Substitute and factorise :
\(B=2\left(\Delta f+\dfrac{\Delta f}{\beta}\right)=2\Delta f\left(1+\dfrac{1}{\beta}\right)\)
which is option 1. Factorising the other way gives the equally valid \(B=2f_{m}(1+\beta)\) — note that options 2 and 4 are exactly these two correct forms with \(\Delta f\) and \(f_{m}\) swapped, which is the trap.
| Form | Expression | Correct? |
|---|---|---|
| Plain | \(2(\Delta f+f_{m})\) | ✓ |
| In terms of Δf | \(2\Delta f\left(1+\tfrac{1}{\beta}\right)\) | ✓ option 1 |
| In terms of fm | \(2f_{m}(1+\beta)\) | ✓ |
| Option 2 | \(2\Delta f(1+\beta)\) | ✗ mixes the two |
Why an approximation is needed at all. Strictly, an FM signal has infinitely many sidebands, spaced fm apart, with amplitudes given by the Bessel functions \(J_{n}(\beta)\). Carson's rule is the working compromise: it counts only the sidebands carrying about 98 % of the total power and discards the rest, which is why it is called approximate.
The two limits are worth checking. For narrowband FM, \(\beta\ll1\), the rule gives \(B\approx2f_{m}\) — the same as AM, since only the first sideband pair survives. For wideband FM, \(\beta\gg1\), it gives \(B\approx2\Delta f\) — the bandwidth is set by the deviation alone.
A worked example : commercial FM broadcasting uses \(\Delta f=75\ \text{kHz}\) and \(f_{m}=15\ \text{kHz}\), so \(\beta=5\) and \(B=2(75+15)=180\ \text{kHz}\) — comfortably inside the 200 kHz channel allocation.
Hence, the required bandwidth is 2Δf(1 + 1/β).
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 :
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
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 :
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
