Nyquist rate

When a continuous function, x(t), is sampled at a constant rate, f_s samples/second, there is always an unlimited number of other continuous functions that fit the same set of samples. But only one of them is bandlimited to \tfrac{1}{2}f_s cycles/second (hertz), which means that its Fourier transform, X(f), is 0 for all |f| \ge \tfrac{1}{2}f_s. The mathematical algorithms that are typically used to recreate a continuous function from samples create arbitrarily good approximations to this theoretical, but infinitely long, function. It follows that if the original function, x(t), is bandlimited to \tfrac{1}{2}f_s, which is called the Nyquist criterion, then it is the one unique function the interpolation algorithms are approximating. In terms of a function's own bandwidth (B), as depicted here, the Nyquist criterion is often stated as f_s > 2B. And 2B is called the Nyquist rate for functions with bandwidth B. When the Nyquist criterion is not met {{nowrap|(say, B > \tfrac{1}{2}f_s),}} a condition called aliasing occurs, which results in some inevitable differences between x(t) and a reconstructed function that has less bandwidth. In most cases, the differences are viewed as distortion. Intentional aliasing Figure 3 depicts a type of function called baseband or lowpass, because its positive-frequency range of significant energy is [0, B). When instead, the frequency range is (A, A+B), for some A > B, it is called bandpass, and a common desire (for various reasons) is to convert it to baseband. One way to do that is frequency-mixing (heterodyne) the bandpass function down to the frequency range (0, B). One of the possible reasons is to reduce the Nyquist rate for more efficient storage. And it turns out that one can directly achieve the same result by sampling the bandpass function at a sub-Nyquist sample-rate that is the smallest integer-sub-multiple of frequency A that meets the baseband Nyquist criterion: fs > 2B. For a more general discussion, see bandpass sampling. ==Relative to signaling==