# Carson's Rule

Carson's rule gives the estimation of the bandwidth of an FM system. This rule states that the bandwidth of an FM system is double the sum of the maximum frequency deviation and the highest modulating frequency *f _{m}*. Thus, if B is the bandwidth of the system; then according to Carson's rule:

*B=2( f _{d} + f_{m}) -------------------------(1)*

This rule is based on experimental results that proved that there are limited number of side bands around the carrier frequency, which contain approximately 98 percent of the total power of an FM signal. All the remaining side hands in the frequency spectrum contain only 2 percent of the total power. Thus, it is obvious that the bandwidth of an FM system can be reduced to a practically feasible limit at the expense of 2 percent of the total power.

The loss of 2 percent of total power is a wise trade between the qualities of the received. Signal and the bandwidth, as infinite bandwidth is required to transmit total power of an FM signal.

It is also observed that if only a few side bands are transmitted through limited bandwidth, then, the distortion produced in the recovered baseband signal at the receiver is negligible and the quality of the recovery signal is not severely degraded. Therefore, the bandwidth of an FM system is brought within practically feasible limits by transmitting, only those side bands that contain 98 percent of the total power system.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|

0 | 0.7652 | 0.2239 | -0.2601 | -0.3971 | -0.1776 | 0.1506 | 0.3001 | 0.1717 | -0.09033 | -0.2459 |

1 | 0.4401 | 0.5767 | 0.3391 | -0.06607 | -0.3276 | -0.2767 | 0.004683 | 0.2346 | 0.2453 | 0.04347 |

2 | 0.1149 | 0.3528 | 0.4861 | 0.3641 | 0.04657 | -0.2429 | -0.3014 | -0.1130 | 0.1448 | 0.2546 |

3 | 0.01956 | 0.1289 | 0.3091 | 0.4302 | 0.3648 | 0.1148 | -0.1676 | -0.2911 | -0.1809 | 0.05838 |

4 | 0.002477 | 0.03400 | 0.1320 | 0.2811 | 0.3912 | 0.3576 | 0.1578 | -0.1054 | -0.2655 | -0.2196 |

5 | 0.007040 | 0.04303 | 0.1321 | 0.2611 | 0.3621 | 0.3479 | 0.1858 | -0.05504 | -0.2341 | |

6 | 0.001202 | 0.01139 | 0.04900 | 0.1310 | 0.2458 | 0.3392 | 0.3376 | 0.2043 | -0.01446 | |

7 | 0.002547 | 0.01518 | 0.05338 | 0.1296 | 0.2336 | 0.3206 | 0.3275 | 0.2167 | ||

8 | 0.004029 | 0.01841 | 0.05653 | 0.1280 | 0.2235 | 0.3051 | 0.3179 | |||

9 | 0.005520 | 0.02117 | 0.05892 | 0.1263 | 0.2149 | 0.2919 | ||||

10 | 0.001468 | 0.006964 | 0.02354 | 0.06077 | 0.1247 | 0.2075 | ||||

11 | 0.002048 | 0.008335 | 0.02560 | 0.06222 | 0.1231 | |||||

12 | 0.002656 | 0.009624 | 0.02739 | 0.06337 | ||||||

13 | 0.003275 | 0.01083 | 0.02897 | |||||||

14 | 0.001019 | 0.003895 | 0.01196 | |||||||

15 | 0.001286 | 0.004508 | ||||||||

16 | 0.001567 |

Table gives the values of the various Bessel's functions that appear in the equation of an FM wave. These values are the amplitudes or the side bands present in the modulated signal used to determine the power distribution in an FM It is also found that for a given that for a given value of modulation index m_{f}, 98 percent of the total power of the FM signal is contained in the side bands that lie between the order n = 0 and n = (m_{f} + 1 ). For example, consider m_{f} = 2. According to this, the side bands containing 98 percent of total power are those with the amplitude J_{0}(2), J_{1}(2), J_{2}(2) and J_{3}(2). Similarly, if m_{f} is 5 there will be seven side bands, J_{0}(5) to J_{6}(5) that contain 98 percent of the total power. Therefore. only these side bands are J,(5), that contain 98 percent of total power. Therefore only these sidebands are transmitted and all other side bands are suppressed. This reduces the bandwidth of an FM system from its infinite value.

Note: Consider m_{t} = 1. The side bands containing 98 percent of the total power are J_{0}(1), J_{1}(1) and J_{2}(1) because,

n = m_{t} + 1

n = 1 + 1

n = 2

While deriving the equation of the FM wave, it was assumed for simplicity that the amplitude of a carrier signal is unity. Therefore, E_{c} = 1. In addition, assume that the resistance of the transmitting antenna is 1 ohm. Thus, the powers contained in the frequency components are:

Power in the carrier = [E_{c}^{2}/2(J_{0}^{2}(1))] = ½[J_{2}^{0}(1)]

Power in first order sidebands = J_{1}^{2} (1)

Power in second order sidebands = J_{2}^{2} (1)

Total power in the FM signal = E_{c}^{2}/2 = 1/2

The total power in an FM signal, P_{T}, is obtained as the sum of all the powers, as:

P_{T} = ½[J_{2}^{0}(1) + J_{2}^{1}(1) + J_{2}^{2}(1)]

Substituting the values of J_{0}(1), J_{1}(1), and J_{2}(1) from Table, you get:

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.

Consider the above figure that illustrates the frequency spectrum for m_{f}. There are four significant USBs and LSBs. However, as discussed, only J_{0}(1), J_{1}(1) and J_{2}(1) need to be transmitted. Therefore, only two USBs located at (ω_{c} + ω_{m}) and (ω_{c} + 2ω_{m}) and two LSBs located at (ω_{c} - ω_{m}) and (ω_{c} - 2ω_{m} along with the carrier signal at ω_{c} are to be transmitted. The total bandwidth required to transmit these frequency components can be written as:

B = 2(m_{f} + 1)f_{m} ------------------------------------(2)

or

B = 2(m_{f}f_{m} + f_{m}) ---------------------------------(3)

The modulation index, m_{f} is called as:

m_{f} = f_{d}/f_{m} -------------------------------------(4)

or

f_{d} = m_{f}f_{m} -------------------------------------(5)

Substituting Equation (4) in Equation (3), you get:

B = 2(f_{d} + f_{m}) --------------------------(6)

Equation (4) is the Carson's rule as defined by the Equation (1). This was the method used by Carson to derive the rule for the estimation of the bandwidth of ad FM system. This rule is given in Equations (5) and (1).

Consider Equation (2) to estimate the bandwidth of NBFM and WBFM systems. In NBFM, the modulation index m_{f} is lesser than unity. Therefore, m_{f} can be ignored as compared to unity in Equation (2). Thus, the bandwidth of NBFM obtained from Eqution (2) is:

B_{NBFM} = 2f_{m} ------------------------(7)

In Equation (7), the bandwidth of NBFM is twice the modulating frequency, f_{m}. It should be noted that the bandwidth of an AM signal (DSB-FC and DSB-SC) is also equal to double the modulating frequency. Therefore, the bandwidth of AM and NBFM are the same. If m_{f} is considered large, then a large number of side bands are included in the transmitted signal and the resultant FM is called WBFM. Thus, in WBFM, the value of m_{f} is greater than unity and in Equation (2), unity can he ignored in comparison with m_{f} approximate the bandwidth to:

B_{WBFM} = 2m_{f}f_{m} -----------------------------(8)

Substituting Equation (4) in Equation (5), you get:

B_{WBFM} = 2f_{d} ---------------------------(9)

Therefore, in WBFM, the approximate bandwidth of the system is twice maximum frequency deviation of the FM signal, as given by Equation (9).