Editing Sampling (signal processing) (section)

== Complex sampling {{anchor|Complex}} ==
'''Complex sampling''' (or '''I/Q sampling''') is the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as [[complex numbers]].{{efn-ua|
Sample pairs are also sometimes viewed as points on a [[constellation diagram]].
}} When one waveform, <math>\hat s(t)</math>, is the [[Hilbert transform]] of the other waveform, <math>s(t)</math>, the complex-valued function, <math>s_a(t)\triangleq s(t)+i\cdot\hat s(t)</math>, is called an [[analytic signal]], whose Fourier transform is zero for all negative values of frequency. In that case, the [[Nyquist rate]] for a waveform with no frequencies ≥&nbsp;''B'' can be reduced to just ''B'' (complex samples/sec), instead of <math>2B</math> (real samples/sec).{{efn-ua|
When the complex sample-rate is ''B'', a frequency component at 0.6&nbsp;''B'', for instance, will have an alias at −0.4&nbsp;''B'', which is unambiguous because of the constraint that the pre-sampled signal was analytic.  Also see {{slink|Aliasing|Complex sinusoids}}.
}} More apparently, the [[Baseband#Equivalent baseband signal|equivalent baseband waveform]], <math>s_a(t)\cdot e^{-i2\pi\frac{B}{2}t}</math>, also has a Nyquist rate of <math>B</math>, because all of its non-zero frequency content is shifted into the interval <math>[-B/2,B/2]</math>.

Although complex-valued samples can be obtained as described above, they are also created by manipulating samples of a real-valued waveform. For instance, the equivalent baseband waveform can be created without explicitly computing <math>\hat s(t)</math>, by processing the product sequence, <math>\left[s(nT)\cdot e^{-i2\pi\frac{B}{2}Tn}\right]</math>,{{efn-ua|
When ''s''(''t'') is sampled at the Nyquist frequency (1/''T'' {{=}} 2''B''), the product sequence simplifies to <math>\left [s(nT)\cdot (-i)^n\right ].</math>
}} through a digital low-pass filter whose cutoff frequency is <math>B/2</math>.{{efn-ua|
The sequence of complex numbers is convolved with the impulse response of a filter with real-valued coefficients.  That is equivalent to separately filtering the sequences of real parts and imaginary parts and reforming complex pairs at the outputs.
}} Computing only every other sample of the output sequence reduces the sample rate commensurate with the reduced Nyquist rate. The result is half as many complex-valued samples as the original number of real samples. No information is lost, and the original <math>s(t)</math> waveform can be recovered, if necessary.