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## Friday, 28 September 2012

### Nyquist rate relative to sampling

The Nyquist rate is the minimum sampling rate required to avoid aliasing, equal to twice the highest frequency contained within the signal. $f_N \ \stackrel{\mathrm{def}}{=}\ 2 B\,$
where $B\,$ is the highest frequency at which the signal can have nonzero energy.
To avoid aliasing, the sampling rate must exceed the Nyquist rate: $f_S > f_N.\,$
A signal whose positive-frequency range of significant energy is (0, B), as depicted above, is called baseband or lowpass. But when the frequency range is (AB), for some A > B − A, it is called bandpass. In that case, aliasing is not necessarily detrimental, and sampling below the Nyquist rate, called bandpass sampling, is sometimes done. With careful design, a rate as low as 2(B − A) may be achievable, and it is equivalent to mixing (heterodyne) the signal into the frequency range (0, B − A), whose Nyquist rate is 2(B − A). An even lower Nyquist rate can be achieved for a bandpass signal, such as amplitude modulation, whose energy distribution in (AB) is symmetrical. In that case, the homodyne mixer translates the signal frequencies by (A + B)/2, with a synchronized phase, which moves the highest component to (B − A)/2 and the Nyquist rate to just (B − A).