Frequency modulation (FM) can be mathematically expressed as the process of varying the instantaneous frequency of a carrier signal with the amplitude of a modulating signal. Let’s consider a carrier wave with a frequency of fc(t) and a modulating signal with an amplitude of m(t). The modulated FM signal can be expressed as:

s(t) = Ac cos[2πfc(t)t + 2πkf ∫m(τ)dτ]

where Ac is the amplitude of the carrier wave, kf is the frequency deviation constant, and τ is the integration variable. The term 2πkf ∫m(τ)dτ represents the frequency deviation of the carrier wave due to the modulating signal, which is integrated over time.

To understand the equation, let’s break it down into its components. The first part of the equation, Ac cos[2πfc(t)t], represents the unmodulated carrier wave with a frequency of fc(t) and amplitude of Ac. The second part of the equation, 2πkf ∫m(τ)dτ, represents the modulation component, which varies the frequency of the carrier wave in proportion to the amplitude of the modulating signal.

The modulation index, which is the ratio of the amplitude of the modulating signal to the frequency deviation constant, can be expressed as:

β = Am / kf

where Am is the peak amplitude of the modulating signal. The modulation index determines the extent to which the carrier wave is frequency modulated, with higher values resulting in more significant frequency deviation and wider sidebands.

In summary, FM can be mathematically expressed as the process of varying the instantaneous frequency of a carrier signal with the amplitude of a modulating signal. The frequency deviation of the carrier wave is proportional to the amplitude of the modulating signal, with the modulation index determining the extent of the frequency deviation.