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(mfwm+ wm)
= 2wm (mf + 1)
Now we have two special cases for the carson's rule -
1 - If ∆w << wm and
2- ∆w >> wm
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(mfwm+ wm)
= 2wm (mf + 1)
Therefore for mf << 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
=> mf >> 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 mf this BW relationship can be considered accurate for all practical purposes.
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