When sampling a function at frequency f s (intervals 1/ f s), the following functions of time (t) yield identical sets of samples: {sin(2π( f+Nf s) t + φ), N = 0, ±1, ±2, ±3,... }. A frequency spectrum of the samples produces equally strong responses at all those frequencies. Without collateral information, the frequency of the original function is ambiguous. So the functions and their frequencies are said to be aliases of each other. Noting the trigonometric identity :