To obtain the frequency-domain representation of Wide Band FM signal for the condition one must express the FM signal in complex representation (or) Phasor Notation (or) in the exponential form
i.e, Single-tone FM signal is
Now by expressing the above signal in terms of Phasor notation ( , None of the terms can be neglected)
Let is the complex envelope of FM signal.
is a periodic function with period . This can be expressed in it’s Complex Fourier Series expansion.
i.e, this approximation is valid over . Now the Fourier Coefficient
let implies
as and
let as order Bessel Function of first kind then .
Continuous Fourier Series expansion of
Now substituting this in the Equation (I)
The Frequency spectrum can be obtained by taking Fourier Transform
n value | wide Band FM signal |
0 | |
1 | |
-1 | |
… | …. |
From the above Equation it is clear that
- FM signal has infinite number of side bands at frequencies for n values changing from to .
- The relative amplitudes of all the side bands depends on the value of .
- The number of significant side bands depends on the modulation index .
- The average power of FM wave is Watts.