When can an analog signal be reconstructed from its sample values?

Answer 1

An analog signal can be reconstructed from its sample values when the samples were taken often enough.

Exactly how often is "often enough" depends on how accurately you want to reconstruct the original. The crucial question is "what is the highest frequency of interest in the original signal?". The usual rule of thumb is to sample at a minimum of twice that frequency.

Beware. If you want to sample a piano playing a concert pitch A (440 Hz), your sample rate will depend on why you want to do it. If you just want to know that the piano played A rather than, say, middle C, a sample rate of 1KHz would do. However, if you want to be able to identify the instrument as a piano, a 1 KHz sample rate won't do at all. When the orchestra tunes up, and the Oboe gives an A, it doesn't sound anything like a piano. The fundamental note is 440 Hz just like the piano, but every musical instrument produces multiples of the fundamental, called harmonics. It is the number of harmonics and their relative amplititudes which gives the instrument its distinctive sound. These are the "highest frequency of interest".