Editing Nyquist rate (section)

==Relative to sampling==
[[Image:Bandlimited.svg|thumb|Fig 2: Fourier transform of a bandlimited function (amplitude vs frequency)|right|240px]]
When a continuous function, <math>x(t),</math> is sampled at a constant rate, <math>f_s</math> ''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 [[Bandlimiting|bandlimited]] to <math>\tfrac{1}{2}f_s</math> ''cycles/second'' ([[hertz]]),{{efn-ua
|The factor of <math>\tfrac{1}{2}</math> has the units ''cycles/sample'' (see [[Sampling (signal processing)#Theory|Sampling]] and [[Sampling theorem]]).
}} which means that its [[Fourier transform]], <math>X(f),</math> is <math>0</math> for all <math>|f| \ge \tfrac{1}{2}f_s.</math>&nbsp; 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, <math>x(t),</math> is bandlimited to <math>\tfrac{1}{2}f_s,</math> 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 (signal processing)|bandwidth]] <math>(B), </math> as depicted here, the '''Nyquist criterion''' is often stated as <math>f_s > 2B.</math>&nbsp; And <math>2B</math> is called the '''Nyquist rate''' for functions with bandwidth <math>B.</math>  When the Nyquist criterion is not met {{nowrap|<math>(</math>say, <math>B > \tfrac{1}{2}f_s),</math>}} a condition called [[aliasing]] occurs, which results in some inevitable differences between <math>x(t)</math> and a reconstructed function that has less bandwidth.  In most cases, the differences are viewed as distortion.

[[File:Bandpass sampling depiction.svg|thumb|right|300px|Fig 3: The top 2 graphs depict Fourier transforms of 2 different functions that produce the same results when sampled at a particular rate.  The baseband function is sampled faster than its Nyquist rate, and the bandpass function is undersampled, effectively converting it to baseband.  The lower graphs indicate how identical spectral results are created by the aliases of the sampling process.]]

===Intentional aliasing===
{{main|Undersampling}}
Figure 3 depicts a type of function called [[Baseband|baseband or lowpass]], because its positive-frequency range of significant energy is [0,&nbsp;''B'').  When instead, the frequency range is (''A'',&nbsp;''A''+''B''), for some ''A''&nbsp;>&nbsp;''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,&nbsp;''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:&nbsp; f<sub>s</sub>&nbsp;>&nbsp;2''B''.  For a more general discussion, see [[Undersampling|bandpass sampling]].