I would like to know why would someone choose to use Music algorithm from frequency estimation when one can simply use DFT/FFT which are much simpler. What is the intuitive way for understanding Music algorithm?
So both of MUSIC and DFTs are different tools that can be used to accomplish similar goals. DFTs lean on the solid background of Fourier analysis, whereas iterative methods like MUSIC and similar sub-space/AR methods try for better resolution than a DFT is capable of.
With a DFT, the spacing of the frequency bins is prescribed by the number of samples you have, which means that you’ll have spaces “between” your frequency bins that will be ambiguous/smeared between two bins. With MUSIC and similar algorithms, we search over an arbitrary grid of frequencies and can get as fine of resolution as we’d like. In this sense, people will often say that MUSIC is a “super resolution technique”, as it can better resolve closely spaced sinusoids that a DFT would not reveal as being two separate signals.
MUSIC generally outperforms a DFT when trying to detect sinusoids and you know the number of signals a priori. Of course, the major downside here is that you need to tell the algorithm how many signals you care about are present in your measurement. If you don’t know this a priori, it can be difficult to get optimal results.
You should give the Wikipedia page for the MUSIC algorithm a read if you haven’t already, it does a great job comparing the method to the DFT.