Assume $$\omega_i(\tau)$$ is the frequency in radians per second and $$f_i(\tau)$$ is the frequency in Hz. In case of frequency modulation, the instantaneous value of that angle is the integral of instantaneous frequency of the modulated signal. It varies with time depending on the message signal.
When the carrier is angle modulated, the instantaneous value of $$\theta$$ depends on the message signal. The angle modulated wave has the information content available in the angle part ($$\theta_c$$) of the carrier. It may be represented as $$A_c square(\theta_c)$$. If we want the carrier to be a square wave as shown in the figure, the built-in function would not help. But this function modulates the given message signal with a sinusoidal carrier of specified frequency. Gnu-octave has built-in function fmmod() available in octave-communications package for implementing frequency modulation. Sinusoidal frequency modulation of square wave carrier It is a kind of more generic technique called angle modulation. That is the information regarding the message is available in the frequency of the carrier. Here the instantaneous frequency of the carrier wave is varied with the instantaneous amplitude of the message signal.
Frequency modulation is a common analog modulation technique.
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