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Sampling Theorem

The sampling theorem states that for a limited bandwidth (band-limited) signal with maximum frequency fmax, the equally spaced sampling frequency fs must be greater than twice of the maximum frequency fmax, i.e.,

fs > 2·fmax

in order to have the signal be uniquely reconstructed without aliasing.

The frequency 2·fmax is called the Nyquist sampling rate. Half of this value, fmax, is sometimes called the Nyquist frequency.

The sampling theorem is considered to have been articulated by Nyquist in 1928 and mathematically proven by Shannon in 1949. Some books use the term "Nyquist Sampling Theorem", and others use "Shannon Sampling Theorem". They are in fact the same sampling theorem.
Practical Issues
The sampling theorem clearly states what the sampling rate should be for a given range of frequencies. In practice, however, the range of frequencies needed to faithfully record an analog signal is not always known beforehand. Nevertheless, engineers often can define the frequency range of interest. As a result, analog filters are sometimes used to remove frequency components outside the frequency range of interest before the signal is sampled.

For example, the human ear can detect sound across the frequency range of 20 Hz to 20 kHz. According to the sampling theorem, one should sample sound signals at least at 40 kHz in order for the reconstructed sound signal to be acceptable to the human ear. Components higher than 20 kHz cannot be detected, but they can still pollute the sampled signal through aliasing. Therefore, frequency components above 20 kHz are removed from the sound signal before sampling by a band-pass or low-pass analog filter. Practically speaking, the sampling rate is typically set at 44 kHz (rather than 40 kHz) in order to avoid signal contamination from the filter rolloff.

What if an engineer is interested in sampling a mechanical signal across ALL frequencies? Most mechanical signals have frequencies limited to below 100 kHz. Therefore, using a 200 kHz sampling rate should satisfy most mechanical engineering applications. The price for such a high sampling rate will be the huge amount of sample data to be stored and processed. Note that this limit should NOT be applied to electric engineering, where signals can contain much higher frequencies!

Over Sampling

Graphically, if the sampling rate is sufficiently high, i.e., greater than the Nyquist rate, there will be no overlapped frequency components in the frequency domain. A "cleaner" signal can be obtained to reconstruct the original signal. This argument is shown graphically in the frequency-domain schematic below.

Spectrum of Sampled Signal
Sampled at greater than the Nyquist rate