What is a Modulation Index?

Solution:

A carrier wave or signal, after being modulated, if the modulated status is estimated, then such a try is called Modulation Index or also called Modulation Depth.

It states that the level of rearranging the following equation:

s(t) = [A_c+ A_m cos(2πf_mt)]cos(2πf_ct) ———-> eq (1)

We can write it as:

s(t) = A_c[1+ (A_m/A_c) cos(2πf_mt)]cos(2πf_ct)

s(t) = A_c[1+ µ*cos(2πf_mt)]cos(2πf_ct) ————> eq (2)

Where ‘µ’ is called Modulation Index and it is equivalent to the ratio of A_m and A_c. Mathematically we can write it as:

µ = A_m/A_c ———--> eq (3)

Thus we can estimate the value of the Modulation index by employing the above formula when the amplitudes of the message as well as carrier signals or waves are known.

Now we drive one additional formula for Modulation Index by considering the above equation (1).

We can utilize this formula for estimating modulation index values when the maximum (max) and minimum (min) amplitudes of modulated waves are known.

Let A_max and A_min be the maximum amplitude and minimum amplitude of the modulated waves.

We will achieve the maximum (max) amplitude of the modulated wave when cos(2πf_mt) is ‘1’.

A_max = A_c + A_m ———-> eq(4)

and get the minimum (min) amplitude of modulated wave when cos(2πf_mt) is ‘-1’.

A_min = A_c – A_m ———-> eq(5)

Add eq (4) and (5)

A_max + A_min = A_c + A_m + A_c – A_m

A_max + A_min = 2A_c

where

A_c  =  (A_max + A_min)/2 ———-> eq(6)

Subtract eq (5) from eq (4) and we get:

A_max – A_min = A_c + A_m – (A_c – A_m)

A_max – A_min = A_c + A_m – A_c + A_m

A_max – A_min = 2A_m

A_m = (A_max – A_min)/2 ————-> eq (7)

The ratio of eq (7) and eq (6) as following:

A_m/A_c = ((A_max – A_min)/2)/((A_max + A_min)/2)

µ = (A_max – A_min)/(A_max + A_min) ————-> eq (8)

Therefore equation (3) and (8) are two formulas for Modulation Index.

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