Complex sampling refers to the accompanying sampling of two different, but related, waveforms, consistent in pairs of samples that are after advised as circuitous numbers. Usually one waveform, \hat s(t), is the Hilbert transform of the added waveform, s(t),\, and the complex-valued function, s_a(t)\ \stackrel{\text{def}}{=}\ s(t) + j\cdot \hat s(t), is alleged an analytic signal, whose Fourier transform is aught for all abrogating ethics of frequency. In that case, the Nyquist amount for a waveform with no frequencies ≥ B can be bargain to just B (complex samples/sec), instead of 2B (real samples/sec).note 1 Added apparently, the agnate baseband waveform, s_a(t)\cdot e^{-j 2\pi \frac{B}{2} t}, aswell has a Nyquist amount of B, because all of its non-zero abundance agreeable is confused into the breach -B/2, B/2).
Although complex-valued samples can be acquired as declared above, they are abundant added frequently created by manipulating samples of a real-valued waveform. For instance, the agnate baseband waveform can be created after absolutely accretion \hat s(t), by processing the artefact sequence, \left s(nT)\cdot e^{-j 2 \pi \frac{B}{2}Tn}\right ,note 2 through a agenda lowpass clarify whose blow abundance is B/2.note 3 Accretion alone every added sample of the achievement arrangement reduces the sample-rate adequate with the bargain Nyquist rate. The aftereffect is bisected as abounding complex-valued samples as the aboriginal amount of absolute samples. No advice is lost, and the aboriginal s(t) waveform can be recovered, if necessary.
Although complex-valued samples can be acquired as declared above, they are abundant added frequently created by manipulating samples of a real-valued waveform. For instance, the agnate baseband waveform can be created after absolutely accretion \hat s(t), by processing the artefact sequence, \left s(nT)\cdot e^{-j 2 \pi \frac{B}{2}Tn}\right ,note 2 through a agenda lowpass clarify whose blow abundance is B/2.note 3 Accretion alone every added sample of the achievement arrangement reduces the sample-rate adequate with the bargain Nyquist rate. The aftereffect is bisected as abounding complex-valued samples as the aboriginal amount of absolute samples. No advice is lost, and the aboriginal s(t) waveform can be recovered, if necessary.
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