## What is Carson's Rule?

Carson's formula is used to calculate the bandwidth (BW) of a single tone wideband FM.
According to carson's rule, the FM bandwidth is given as, twice the sum of frequency deviation and the highest modulating frequency.
But it should be noted here that this rule is just an approximation.

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So Carson's rule can be written mathematically as-

BW = 2(∆w + wm)

But mf = ∆w/wm

Therefore

BW = 2(mfwmwm)

= 2wm (mf + 1)

Now we have two special cases for the carson's rule -

1 - If ∆w << wm and

2- ∆w >> wm

## (adsbygoogle = window.adsbygoogle || []).push({}); Derivation of Carson's Rule for Narrowband FM and Wideband FM

Case 1- If ∆w << wm

Since mf = ∆w/wm

If ∆w << wm

=> mf << 1

=> It is the case for narrowband FM

Since the bandwidth by the carson's rule is given
as-

BW = 2(∆w + wm)

BW = 2(mfwmwm)

= 2wm (mf + 1)

Therefore for m<< 1

BW = 2wm

Note here that this is equivalent to Amplitude Modulation (AM)

Case 2- ∆w >> wm

Since mf = ∆w/wm

Therefore if ∆w >> wm

=> m>> 1   as is the case for wideband FM
Then, since by Carson's rule

BW = 2wm (mf + 1)

Therefore for mf  >> 1

BW = 2wmmf

But wmmf = ∆w

Therefore

BW = 2∆w

Note- For large values of m this BW relationship can be considered accurate for all practical purposes.

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