I am trying to observe the differences of phase and frequency modulation. On the figure below, the left scope shows a frequency modulation and demodulation whereas the right scope shows a phase modulation and demodulation. Shouldn't be the modulated signal for PM and FM look similar? However, in my results, the FM modulated signal has more visible phase differences compared to the PM modulated signal. Why does the FM demodulated signal is more distorted compared to the PM demodulated signal? Also for the frequency modulation, like the amplitude modulation, does the Nyquist theorem also apply where the carrier frequency should be atleast twice or more than the highest frequency component, meaning the carrier frequency cannot be the same as the message frequency? One last thing, for both PM and FM, when I try to increase the Hilbert transform modulation order of the demodulator blocks to get a better resolution demodulated signal, the signal gets distorted after passing the order of 250 (last figure). I assumed this is because as the order increases more calculations are done so due to complexity the signal gets distorted, or is there other explanation? Thanks in advance.
The question 'are FM and PM the same or different?' has two answers ...
They are the same
They are both angle modulations of the carrier. If you see an angle-modulated carrier, then you have no way of knowing, without other information, whether you are looking at an 'FM' or 'PM' system.
In principle, any carrier produced by one system can also be produced by the other, subject to the practical limitations below.
They are different
They were invented at different times to serve different purposes, so a 'PM' system will carry with it a different set of assumptions to an 'FM' system, which will usually make a typical signal from either look different.
There are at least two ways to get angle modulations of the carrier. We can generate a fixed carrier and then phase modulate it. Or we can generate a carrier with the correct average frequency, while frequency modulating it.
These two methods tend to produce systems with a different mix of characteristics.
Generally a phase modulator can only be economically built to produce modulations up to a radian or so. With the modulation thus limited, the resulting carrier would fit well into a narrow-band low quality analogue communication network. It's not possible to produce many cycles of FM modulation, at least not directly. As angle is mulitplied when the frequency is multiplied, you could use a +/- 1 radian phase modulator to produce +/- 10 radians by passing the multiplied signal through a frequency multiplier.
A frequency modulator can easily produce angle modulations of many radians, many cycles even, by using a voltage controlled oscillator (VCO). This improves the SNR, and wide deviation FM is used for high quality analogue music transmission, albeit with very wide channels. Uncertain control of DC drift to the VCO means that it would not be possible to return to exactly the same phase of carrier, which a PM system could do, without extra control by a synthesiser.
A significant difference is the 1/s difference in modulation response, as phase is the integral of frequency. This can be mimicked by pre- or de-emphasis on the modulation signal into or out of the modulator or demodulator respectively, so it's not a fundamental difference between the systems.
You can construct a universal angle modulator, where a fixed carrier is IQ modulated with a complex baseband signal. With suitable generation of I and Q, usually in an FPGA or ASIC, the output can have definite phase, many cycles of modulation, and produce signals identical to any FM or PM system. However, this is never done for real FM or PM systems; once you have this level of control, you might as well use the amplitude component as well, to carry more information, and control the channel width better.