Sampling Theorem
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The sampling theorem states that, if the sampling rate in any pulse modulation system exceeds twice the maximum signal frequency, the original signal can be reconstructed in the receiver with minimal distortion. The sampling theorem is used in practice to determine minimum sampling speeds. Consider pulse modulation used for speech. Transmission is generally over standard telephone channels, so that the audio frequency range is 300 to 3400 Hz. For this application, a sampling rate 8000 samples per second is almost a worldwide standard. This pulse rate is, as can be seen, comfortably more than twice the highest audio frequency. The sampling theorem is satisfied, and the resulting system is free from sampling error.
According to the Nyquist criterion, the sampling frequency, Fs, must be at least twice the maximum frequency component in the signal. If this criterion is violated, a phenomenon known as aliasing occurs. The figure below shows an adequately sampled signal and an under sampled signal. In the under sampled case, the result is an aliased signal that appears to be at a lower frequency than the actual signal.
When the Nyquist criterion is violated, frequency components above half the sampling frequency appear as frequency components below half the sampling frequency, resulting in an erroneous representation of the signal. For example, a component at frequency
appears as the frequency Fs - f0.
The following figure shows the alias frequencies that appear when the signal with real components at 25, 70, 160, and 510 Hz is sampled at 100 Hz. Alias frequencies appear at 10, 30, and 40 Hz.
Before a signal is digitized, you can prevent aliasing by using anti aliasing filters to attenuate the frequency components at and above half the sampling frequency to a level below the dynamic range of the analog-to-digital converter(ADC). For example, if the digitizer has a full-scale range of 80 dB, frequency components at and above half the sampling frequency must be attenuated to 80 dB below full scale.
These higher frequency components do not interfere with the measurement. If you know that the frequency bandwidth of the signal being measured is lower than half the sampling frequency, you can choose not to use an anti-aliasing filter.
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