Sampling (signal Processing) - Complex Sampling

Complex sampling refers to the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as complex numbers. Usually one waveform is the Hilbert transform of the other waveform and the complex-valued function, 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 ≥ B can be reduced to just B (complex samples/sec), instead of 2B (real samples/sec). More apparently, the equivalent baseband waveform, also has a Nyquist rate of B, because all of its non-zero frequency content is shifted into the interval [-B/2, B/2).

Although complex-valued samples can be obtained as described above, they are much more commonly created by manipulating samples of a real-valued waveform. For instance, the equivalent baseband waveform can be created without explicitly computing by processing the product sequence through a digital lowpass filter whose cutoff frequency is B/2. 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 s(t) waveform can be recovered, if necessary.