What I am asking is if it is possible to transmit a frequency modulated wave in AM range(550-1650 Khz) and similarly to transmit an Amplitude modulated wave in FM band (88-108 Mhz). I would like to know about the technical aspects of the problem like what problems we may face if try to do so.
AM now supports Quadrature Stereo AM but still fits within 10Khz channel spacing with some guardband.
FM cannot fit into the AM band , wwhile AM is even used in the ISM band at 928MHz with 6kHz channel spacing, so there is no reason why AM cannot be used at any frequency.
FM has higher SNR and signal bandwidth that prevents application in the lower AM bands.
However one can use AM or FM for cable communication at any frequency. Can you see how to fit the above baseband FM signal in the 10KHz channel spacing of the AM band? If you do it's worth a ton of money.
You could do AM on the FM band .You will be able to squeeze in many more channels but you wont get the SNR advantages of FM .If you tried to do wideband FM on the standard MW AM band the number of channels would be very low .Also the type of Atmospheric fading on the MW band would make the stereo music sound terrible .
Technically any mode of transmission is feasible on any frequency if bandwidth is available. The reason that AM mode is used on the lower broadcast band and FM on the higher broadcast band is by reason of government regulation as well as signal propagation issues. The broadcast FM band in the '40's was originally around 40 Mhz the got moved to 88-108 Mhz band. Depending on the application a multitude of modes are used.
There is no technical reason you couldn't do what you describe. In fact, there is at least one system for AM stereo that is based on modulating the carrier simultaneously with AM and FM.
However, depending on what jurisdiction you live in, there may be regulatory issues.
AM and FM waveforms: -
In case 1 you move the amplitude up and down a bit. In case 2 you wobble the frequency a bit. This can occur at any point in the radio spectrum.
Both can occupy the same amount of spectral bandwidth. Narrow band FM is scattered all over the spectrum and, for low bandwidth speech signals is perfectly OK. Wideband FM used in broadcasts of stereo music occupies a large chunk of bandwidth hence it can't really be used in the traditional AM broadcast region of the spectrum.