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The main difference is in the distribution of sidebands. The sidebands are described by Bessel functions . See https://www.tpub.com/neets/book12/49e.htm for a nice explanation.
Although phase and freq. modulation are two types of angle modulation schemes they differ from each other in terms of their respective transmission bandwidth.
Frequency Modulation (FM):
The carrier frequency 'fc' is linearly deviated by the message (modulating) signal 'm(t)', and the linear constant 'fm' is known as the Frequency Modulation Index.
Phase Modulation (PM):
The carrier phase is linearly deviated by the message signal, and the linear constant 'km' is known as the Phase Modulation Index.
This means that the frequency of a PM signal is deviated by the Rate-Of-Change of the message signal, not by the amplitude of the message signal in the case of FM.
I hope this helps.
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Background Information:
When we talk about Frequency and Phase, we are talking about Periodic signals.
A Periodic signal repeats itself over and over again.
Each repetition is called a Cycle.
The Frequency of a periodic signal is typically measured in Cycles/Second (Hertz).
Phase is the time integral of frequency and is simply measured in Cycles.
The Sinusoid is the most fundamental form of periodic signal.
If we trace the circumference of a circle, centered at the origin of an x-y coordinate system, at a constant rate, and plot the y-value over time, we get a Sine Wave.
If we plot the x-value over time, we get a Cosine Wave, which is identical to a Sine wave but phase shifted by 1/4 of a cycle ahead.
In the case of a sinusoid we can define phase not just in cycles, but in angles or in relative distance along the circumference.
1 cycle = 360 degrees = 2*pi radians
Fourier theory shows that all complex periodic waveforms can be represented as a sum of different sinusoids (frequency components).
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