# How phase is modulated according to our message signal?

My question: let message signal m(t) be sin(w1t).For frequency modulation (instantaneous phase deviation=phi(t)) d[phi(t)]/dt=kf m(t) so i get that at 90° or π/2 frequency will be max because the amplitude of sinw1t is max. So same results should be given by when phi(t)=kf*integral[m(t)]. {indefinite integral of m(t) over the limit 0 to t will be 1-cosw1t}. Does that mean there will be a maximum lead according to maximum amplitude of 1-cosw1t and that phase leading must be making the frequencies closer? But according to this result(as i have shown) frequencies should be much closer at 180° not at 90°. This same problem is arising when i do phase modulation, i get that if phi(t)=kp m(t) therefore d[phi(t)]/dt =kp dm/dt (dm/dt=cosine wave), so frequencies are higher or shrinked near 0°. But if i apply the same theory on phase lead or lag, then phase lead must be max at 90°and hence the frequencies must be higher there. So the whole problem is arising when i try to understand it from a instantaneous phase deviation view.

• Why are you wanting to integrate the modulating signal? Commented Jun 7, 2023 at 18:29
• I am not integrating the modulating signal. I am integrating the message signal, because its integration will be proportional to instantaneous phase deviation and give us the angle modulated signaI want to know what exactly is meant be phase being proportional to it. Commented Jun 8, 2023 at 5:17
• modulating signal is not the modulated signal. Commented Jun 8, 2023 at 8:13
• Can you tell me how the instantaneous phase deviation is actually happening in the signal Commented Jun 9, 2023 at 13:46