Nyquist–Shannon sampling theorem: Difference between revisions

Revision as of 11:32, 14 October 2023

Revision as of 11:33, 14 October 2023

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==Critical frequency==

To illustrate the necessity of <math>f_s>2B</math>, consider the family of sinusoids generated by different values of <math>\theta</math> in this formula''':'''

:<math display="block">x(t) = \frac{\cos(2 \pi B t + \theta )}{\cos(\theta )}\ = \ \cos(2 \pi B t) - \sin(2 \pi B t)\tan(\theta ), \quad -\pi/2 < \theta < \pi/2.</math>

[[File:CriticalFrequencyAliasing.svg|thumb|right|A family of sinusoids at the critical frequency, all having the same sample sequences of alternating +1 and –1. That is, they all are aliases of each other, even though their frequency is not above half the sample rate.]]

With <math>f_s=2B</math> or equivalently <math>T=1/2B</math>, the samples are given by''':'''

:<math display="block">x(nT) = \cos(\pi n) - \underbrace{\sin(\pi n)}_{0}\tan(\theta ) = (-1)^n</math>

{{em|regardless of the value of <math>\theta</math>}}. That sort of ambiguity is the reason for the ''strict'' inequality of the sampling theorem's condition.