Carson’s Rule for Bandwidth Calculation

Last Updated : 13 Mar, 2024

Carson’s Bandwidth Rule is an important concept in Electronics and Communication Engineering. This rule is used for the calculation of FM modulation bandwidth or occupancy for the FM signal. Let’s say you are designing an FM system on satellite, you will need to make sure that your signal is not cross-talking into other signals on the system.

FM creates modulation sidebands that in theory extend to infinite bandwidth. These sidebands consist of Bessel Functions of any order. However, practically the band occupancy of an FM-modulated carrier solely needs to count the Bessel Function sidebands of necessary amplitude. Carson’s Rule is used to calculate this bandwidth.

This article will incorporate definitions and basic principles of Carson’s Rule followed by the important applications, merits, demerits, and at last a solved example to enhance our understanding of the rule.

Table of Content

Carson’s Rule
Working Principle
Carson’s Rule Table
Derivation
Special Cases
Applications
Solved Example
Advantages and Disadvantages

What is Carson’s Rule?

Carson’s Bandwidth Rule is used for the calculation of FM modulation bandwidth or occupancy for the FM signal which is an important concept in Electronics and Communication Engineering. Given bellow is the statement for the Carson’s Rule.

Statement

Carson’s Rule states that the bandwidth (B) required for an amplitude-modulated signal can be approximated by doubling the sum of the highest modulating frequency (f_m) and the maximum frequency deviation (f_d), then according to Carson’s rule:

B=2( f_d + f_m) (1)

Formula of Carson’s Rule

The formula represents Carson’s Rule is

B = 2(f_m+ f_d)

where,

B is Bandwidth

f_mis the highest modulating frequency

f_d is the maximum frequency deviation

Working Principle of Carson’s Bandwidth Rule

This rule follows the experimental conclusion that established that there is a finite number of side bands around the carrier frequency, of which about 98% contain the total power of an FM signal, Rest of the Side Bands in the frequency spectrum contain about 2%. Thus, 2% of the total power can be sacrificed to reduce the bandwidth of an FM system to a practically feasible limit. The 2% power loss is acceptable in exchange for the received enhancement. Infinite bandwidth is required for Signal and its total power to be transmitted.

It is also to be noticed that if only a few sidebands are transmitted through limited bandwidth, then, the distortion produced in the recovered baseband signal at the receiver is almost insignificant and the quality of the recovery signal is not that compromised. To bring the bandwidth within the practically feasible parameters, we use only those sidebands which contain the 98% of the total power.

Carson’s Rule Table

The table provides values for Bessel’s functions used in the equation of an FM wave. The values in the above table give the values for Bessel’s functions used in the equation of an FM wave, They represent the sideband’s amplitude present in the modulated signal, which helps to determine the distribution of power in FM.

Let’s say we have a modulation index mf, 98% of the total power of the FM signal is contained in the sidebands between the orders n = 0 and n = (mf + 1). For example, if mf = 2, the sidebands containing 98 percent of the total power are J₀(2), J₁(2), J₂(2), and J₃(2).

Similarly, for mf = 5, seven sidebands from J₀(5) to J₆(5) contain 98 percent of the total power. As we discussed above, Only these sidebands are transmitted while others are suppressed because these sidebands i.e J₀(5) to J₆(5) contain most of the total power.

Note: Let’s say that m_t equals 1. The sidebands with 98% of the total power are J₀(1), J1(1), and J₂(1)

because,

n = 1 + 1 n = 2 and n = mt + 1

Derivation of Carson’s Bandwidth Rule

In deriving the equation for an FM wave, it was assumed, for the sake of simplicity, that the carrier signal’s amplitude is one (E_c = 1).The powers contained in the frequency components, assuming the resistance of the transmitting antenna is 1 ohm, are calculated as:

Carrier power = [Ec²/2(J₀²(1))] = ½[J₀²(1)]

First-order sideband power = J₁²(1)

J₂²(1) = Power in second-order sidebands

The FM signal’s total power is equal to E_c²/2 = 1/2.

The sum of these powers is the total power (PT) of an FM broadcast.

PT = ½[J₂⁰(1) + J₂¹(1) + J₂²(1)]

By changing J₀(1), J₁(1), and J₂(1) values from the table as follows:

P_T = ½[(0.7652)² + (0.4401)² + (0.1149)²]

P_T = 0.2927 + 0.1936 + 0.0132

P_T = 0.4995

Thus, the total power contained in the side bands up to n = 2 is and the carrier signal is 0.4495. The total power in an FM signal is 1/2 or 0.5. Therefore, the total power in the side bands covered up to n = 2, is 99.9 percent, and all the remaining side bands in the spectrum of the signal contain only 0.1 percent of the power.

So Carson’s rule can be written mathematically as,

B = 2(f_m + f_d)

But m_f = f_d/f_m

Therefore

B = 2(m_ff_m+ f_m)

B = 2f_m(m_f + 1)

Now we have two special cases for the carson’s rule.

Special Cases of Carson’s Bandwidth Rule

Case 1. fd << fm (NBFM)

Since m_f = f_d/f_m

If f_d << f_m

=> m_f << 1

hence, It is the case for narrowband FM

Since the bandwidth by the carson’s rule is given as B = 2(fd + fm)

B = 2(m_ff_m+ f_m)

= 2f_m(m_f + 1)

Therefore for m_f << 1

B = 2f_m

Case 2. f_d >> f_m (WBFM)

Since m_f= f_d/f_m

Therefore if f_d >> f_m

=> m_f >> 1

hence, It is the case for wideband FM

Then, since by Carson’s rule

B = 2f_m (m_f + 1)

Therefore for m_f >> 1

B = 2f_mm_f

But f_mm_f = f_d

Therefore

B = 2f_d

Note: This Bandwidth relationship is considered accurate for all practical purposes when mf is large.

Applications of Carson’s Rule

Radio Broadcasting: Carson’s Rule figures out how much bandwidth amplitude-modulated (AM) radio stations need. The best possible transmission quality and spectrum use can be guaranteed by broadcasters by precisely calculating the bandwidth.
Wireless Communication Systems: The determination of how much bandwidth is needed to send signals over different wireless channels in wireless communication is helped by Carson’s Rule. This determination enables engineers to design and develop cost-efficient Wireless Communication Systems.
Radar Systems: In Radar Systems, We employ Carson’s rule to determine the required bandwidth as the transmission and reception of signals in radar systems depend on exact bandwidth estimations.
Television Broadcasting: Broadcasters may provide viewers with high-quality television programs by precisely predicting the bandwidth required using Carson’s rule.
Satellite Communication: Carson’s Rule provides the efficient distribution of bandwidth for data transmission between ground stations and satellites. This increases satellite efficiency.

Solved Example on Carson’s Rule for Bandwidth Calculation

A broadcasting station has a highest modulating frequency f_m = 10kHz and a maximum frequency deviation f_d =4 kHz. Calculate the bandwidth required using Carson’s Rule.

Using Carson’s Rule formula:

B = 2(f_m + f_d)

B = 2(10+4)

B = 2×14

B = 28kHz

Therefore, the required bandwidth for the broadcasting station is 28kHz.

Advantages and Disadvantages of Carson’s Rule for Bandwidth Calculation

There are some Advantages and Disadvantages of Carson’s Rule for Bandwidth Calculation given below :

Advantages

Provides a speedy and precise bandwidth estimation.
Easy formula with real-world applications.
Aids in effective use of the spectrum.
Extensively employed while designing communication systems.
Aids in the transmission resource optimization process.

Disadvantages

Assumes idealized circumstances; in complex contexts, it could not be accurate.
Applicable only to amplitude-modulated communications; not suitable for other forms of modulation.
Disregards elements that impact performance in the actual world, such as interference and noise.
Sometimes overestimates bandwidth, which results in inefficiencies.
Requires modulation parameter information, which is not always straightforward to get.

Conclusion

Carson’s rule for bandwidth computation describes the estimated bandwidth needs of communication system components for a carrier signal that is frequency modulated across a continuous or large frequency range rather than a single frequency. We can use it different areas like for radio and television broadcasting, radar system, satellite communications, etc.