Nyquist Theorem -- Sampling Rate Versus Bandwidth
The Nyquist theorem states that a signal must be sampled at least twice as fast as the bandwidth of
the signal to accurately reconstruct the waveform; otherwise, the high-frequency content will alias at a
frequency inside the spectrum of interest (passband). An alias is a false lower frequency component
that appears in sampled data acquired at too low a sampling rate. The following figure shows a 5 MHz
sine wave digitized by a 6 MS/s ADC. The dotted line indicates the aliased signal recorded by the ADC
at that sample rate.
Sine Wave Demonstrating the Nyquist Frequency
The 5 MHz frequency aliases back in the passband, falsely appearing as a 1 MHz sine wave. To
prevent aliasing in the passband, a lowpass filter limits the frequency content of the input signal above
the Nyquist rate.
Filtering
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In the frequency domain, its relatively easy to remove certain frequencies from the
digital signal.
Removing high frequency material is commonplace, since human perception is geared
towards low frequency information.
Multimedia is commonly subject to a low pass filtering operation before sampling.
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
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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 40 kHz cannot be detected, but they can still pollute the
sampled signal through aliasing. Therefore, frequency components above 40 kHz are
removed from the sound signal before sampling by a band-pass or low-pass analog filter.
Practically speaking, the low-pass filter is set at 44 kHz (rather than 40 kHz) in order to
avoid signal contamination from 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
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Under Sampling
When the sampling rate is lower than or equal to the Nyquist rate, a condition defined as
under sampling, it is impossible to rebuild the original signal according to the sampling
theorem.
An example is illustrated below, where the reconstructed signal built from data sampled
at the Nyquist rate is way off from the original signal.
Aliasing
Under sampling causes frequency components that are higher than half of the sampling
frequency to overlap with the lower frequency components. As a result, the higher
frequency components roll into the resconstructed signal and cause distortion of the
signal. This type of signal distortion is called aliasing.
The schematic below repeats the above aliasing argument in the frequency domain.
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Spectrum of Sampled Signal
Sampled at less than the Nyquist rate
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