Tần số Nyquist và hiện tượng giả danh (Aliasing)

A signal or function is bandlimited if it contains no energy at frequencies higher than some bandlimit or bandwidth B.

A signal that is bandlimited is constrained in how rapidly it changes in time, and therefore how much detail it can convey in an interval of time. The sampling theorem asserts that the uniformly spaced discrete samples are a complete representation of the signal if this bandwidth is less than half the sampling rate.
To formalize these concepts, let x(t) represent a continuous-time signal and X(f) be the continuous Fourier transform of that signal (which exists if x(t) is square-integrable):

The signal x(t) is bandlimited to a one-sided baseband bandwidth B if X(f) for all | f | > B
Then the sufficient condition for exact reconstructability from samples at a uniform sampling rate f_s (in samples per unit time)

or equivalently 2B is called the Nyquist rate and is a property of the bandlimited signal, while f_s is called the Nyquist frequency and is a property of this sampling system.

The time interval between successive samples is referred to as the sampling interval


The sampling theorem leads to a procedure for reconstructing the original x(t) from the samples x[n] and states sufficient conditions for such a reconstruction to be exact.

Aliasing
Hypothetical spectrum of a properly sampled bandlimited signal (blue) and images (green) that do not overlap. A "brick-wall" low-pass filter can remove the images and leave the original spectrum, thus recovering the original signal from the samples.

If the sampling condition is not satisfied, then frequencies will overlap; that is, frequencies above half the sampling rate will be reconstructed as, and appear as, frequencies below half the sampling rate. The resulting distortion is
called aliasing; the reconstructed signal is said to be an alias of the original signal,
in the sense that it has the same set of sample values.

Top: Hypothetical spectrum of an insufficiently sampled bandlimited signal (blue), X(f), where the images (green) overlap. These overlapping edges or "tails" of the images add creating a spectrum unlike the original.
Bottom: Hypothetical spectrum of a marginally sufficiently sampled bandlimited signal (blue), X_A(f), where the images (green) narrowly do not overlap. But the overall sampled spectrum of X_A(f) is identical to the overall inadequately sampled spectrum of X(f) (top) because the sum of baseband and images are the same in both cases. The discrete sampled signals x_A[n] and x[n] are also identical. It is not possible, just from examining the spectra (or the sampled signals), to tell the two situations apart. If this
were an audio signal, x_A[n] and x[n] would sound the same and the presumed "properly" sampled x_A[n] would be the alias of x[n] since the spectrum X_A(f) masquarades as the spectrum X(f).

For a sinusoidal component of exactly half the sampling frequency, the component will in general alias to another sinusoid of the same frequency, but with a different phase and amplitude. To prevent or reduce aliasing, two things can be done:

1. Increase the sampling rate, to above twice some or all of the frequencies that are aliasing.
   
2. Introduce an anti-aliasing filter or make the anti-aliasing filter more stringent.
   

The anti-aliasing filter is to restrict the bandwidth of the signal to satisfy the condition for proper sampling. Such a restriction works in theory, but is not precisely satisfiable in reality, because realizable filters will always allow some leakage of high frequencies. However, the leakage energy can be made small enough so that the aliasing effects are negligible.The reconstruction low-pass filter transition band is between B and f_s-B and
the filter response need not be precisely defined in that region (since there is no non-zero spectrum in that region). However, the worst case is when the bandwidth B is virtually as large as the Nyquist frequency f_s/2 and in that worst case, the reconstruction filter H(f) must be:

where rect[u] is the rectangular function.
With H(f) so defined, it is clear that

Spectrum, X_s(f), of a properly sampled bandlimited signal (blue) and images (green) that do not overlap. A "brick-wall" low-pass filter, H(f), removes the images, leaves the original spectrum, X(f), and recovers the original signal from the samples and the spectrum of the original signal that was sampled, X(f), is recovered from the spectrum of the sampled signal, X_s(f). This means, in the time domain, that the original signal that was sampled, x(t), is recovered from the sampled signal, x_s(t).

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