How many samples per cycle(time period) is actually enough to reproduce the sine wave sound properly? I have been working on Active Noise Cancellation in real time for Air vent in kitchen.
Initially for the demonstration purpose, I am using sinusoidal sound ( 40k Hz Fs,1000 samples, 2 kHz frequency, 1 Amplitude). I am using simple LMS Algorithm initially to test this, I could produce the anti noise with the same frequency and amplitude but because it is a real time application, I could be able to produce the sine wave with 30-31 samples per each cycle of sine wave.   

The problem is the sound coming through the noise cancelling speakers is not good compared to the source noise, there is some buzz noise coming in addition (kind of background noise). This could mainly be because of distortions in speaker, I don't have any idea how to control this !!
But I am thinking this may be could be because of very less number of samples that I am using for Cancelling speaker.May be the reproduction of the noise is not so good because of less number of samples.    

So my question is how many samples per cycle are ideally required for good reproduction of sine wave sound??
It depends on how good your interpolation between samples is.  If you have really good interpolation, anything more than 2 samples per cycle will suffice.  This is not just a theoretical fact, it is my experience in practice using polyphase interpolation that combines 64 adjacent table-lookup samples.

If you're using linear interpolation (which combines 2 adjacent table-lookup samples in the interpolation), I would use no less than 1024 samples per cycle for a high-quality sine wave (about 120 dB S/N).