##
__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.

**Watch the Complete Video Here**So Carson's rule can be written mathematically as-

BW = 2(∆w + wm)

But m

**f**= ∆w/wm

Therefore

BW = 2(m

**f**wm+ wm)

= 2wm (m

**f**+ 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__

__Derivation of Carson's Rule for Narrowband FM and Wideband FM__

**Case 1- If ∆w << wm**Since m

**f =**∆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(m

**f**wm+ wm)

= 2wm (m

**f**+ 1)

Therefore for mf << 1

BW = 2wm

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

**Case 2- ∆w >> wm**Since m

**f**= ∆w/wm

Therefore if ∆w >> wm

=> mf >> 1 as is the case for wideband FM

Then, since by Carson's rule

BW = 2wm (m

**f**+ 1)

Therefore for mf >> 1

BW = 2wmm

**f**

**But wmm**

**f =**∆w

Therefore

BW = 2∆w

Note- For large values of m

**f**this BW relationship can be considered accurate for all practical purposes.

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