If the sampling rate is too low, information is lost or, more seriously, components that do not exist appear in the record. If it is too high, the volume of data simply grows without adding any new information.
A data acquisition system does not record a signal continuously: it takes a sample of its value, waits for a fixed interval and then takes the next one. The number of samples per second is the sampling rate or sampling frequency (fs). It is expressed in hertz: 1 kHz corresponds to one thousand samples per second.
How many samples are needed? There is no single answer. It depends on how quickly the quantity to be measured changes, that is, on the bandwidth of the phenomenon.
Sampling: from a continuous signal to discrete points
The panel below shows a continuous signal, in grey, and the samples that a data acquisition system would record. Moving the slider shows how the signal reconstructed from those points changes.
Sampling a signal
The bandwidth of a physical phenomenon
Any signal, however complex it may appear, can be decomposed into a sum of sinusoidal oscillations of different frequencies.
The bandwidth is the range of frequencies in which the signal has content relevant to the measurement. Its upper limit is the maximum frequency, fmax.
Slow, smooth variations are concentrated at low frequencies. Abrupt changes, such as an impact or a fracture, contain high-frequency components. For this reason, measuring the temperature of a furnace and measuring the impact of a hammer call for very different acquisition equipment.
Frequency range of different phenomena
The Nyquist-Shannon sampling theorem
The criterion linking these two quantities stems from the work of Harry Nyquist (1928) and was formalised by Claude Shannon in 1949. According to the sampling theorem, a signal with no content above fmax can be reconstructed exactly from its samples, provided that:
In other words, more than two samples are required for each cycle of the fastest component. Half the sampling rate, fs/2, is known as the Nyquist frequency and marks the upper limit of what the system can record.
When the criterion is not met: aliasing
Components above that limit do not disappear. They enter the record with a lower apparent frequency and, once sampled, cannot be distinguished from a real signal. This effect is known as aliasing.
Film offers a well-known example: the wheels of a carriage that appear to turn slowly, stop or even turn backwards. The camera records a fixed number of frames per second, and the wheel turns too fast for that rate.
The wheel and the camera
Exactly the same happens with an electrical signal. In the next panel the system samples at 10 Hz, so its Nyquist frequency is 5 Hz. If the signal frequency exceeds that value, the same samples describe a slower wave.
Aliasing with fs = 10 Hz
In practice, twice is not enough
The theorem sets a theoretical minimum. On a real test bench an additional margin is advisable, for three reasons:
- The anti-aliasing filter. The only effective protection against aliasing is to remove high frequencies before sampling, using an analogue filter placed ahead of the converter. Once the signal has been digitised, the error can no longer be corrected. This filter does not cut off abruptly but has a transition band, which is why many vibration analysers work with fs = 2.56 · fmax.
- The waveform. The Nyquist criterion guarantees that the frequency information is in the data, but not that the curve is faithfully displayed on screen. If the peak value of an impact must be read directly from the record, ten or more samples per cycle are advisable. With ten, the maximum peak error is around 5 %.
- Excess also has a cost. Sampling a hundred times faster than necessary multiplies file sizes. Moreover, if the filter bandwidth is widened in the same proportion, the recorded noise increases without providing useful information.
Sensors also have a bandwidth
Acquiring at 50 kHz serves no purpose if the sensor cannot follow the phenomenon. Each element of the measurement chain has its own bandwidth, and the final result is limited by the most restrictive one.
- Phenomenonfmax
- Sensorfrequency response
- Signal conditioneranti-aliasing filter
- A/D converterfs
- Analysistime and FFT
Accelerometers
A piezoelectric accelerometer is essentially a seismic mass resting on a piezoelectric crystal. It responds very well to fast vibrations but cannot measure a constant acceleration: the charge generated by the crystal gradually leaks away.
Its upper limit is set by the resonance of the sensor itself. Close to it, the reading is amplified and no longer reliable. In practice, therefore, it is used only up to a fraction of the resonant frequency: between one third and one fifth, depending on the accepted tolerance.
Frequency response of a piezoelectric accelerometer
Capacitive MEMS accelerometers behave the other way round: they measure from 0 Hz, so they detect gravity and are suitable for measuring inclination or slow movements. In exchange, their upper frequency limit is lower.
Load cells
A strain gauge load cell measures from 0 Hz, which makes it ideal for weighing and slow tests.
Its upper limit is mechanical. The cell behaves as a spring and, together with the mass it supports, has a natural frequency: the heavier the fixture mounted on it, the lower that frequency. In addition, the signal conditioner often includes a filter of a few hertz to stabilise the reading.
A few samples per second are sufficient to weigh a tank. Recording the brittle fracture of a specimen, which occurs within milliseconds, requires rates in the kilohertz range and a stiff load cell or, alternatively, a piezoelectric force sensor.
| Sensor | Typical bandwidth | Remarks |
|---|---|---|
| Piezoelectric accelerometer | 0.5 Hz – 10 kHz | Does not measure constant acceleration |
| Capacitive MEMS accelerometer | 0 Hz – 1 kHz | Measures gravity and inclination |
| Strain gauge load cell | 0 Hz – hundreds of Hz | Depends on the mounted mass |
| Piezoelectric force sensor | ≈ 0.01 Hz – tens of kHz | Suited to impacts |
| Sheathed thermocouple | < 1 Hz | Response time of the order of seconds, depending on diameter |
The FFT: analysing a signal in the frequency domain
So far, signals have been represented as an oscilloscope would show them: amplitude versus time. They can also be analysed in another way, by determining the amplitude of the signal at each frequency.
That representation is the spectrum. The tool that calculates it from the samples is the FFT (Fast Fourier Transform).
Its principle goes back to Joseph Fourier: a periodic signal can be expressed as a sum of sines and cosines. The FFT is a highly efficient algorithm for calculating the amplitude of each of those components. What looks like a tangled signal in the time domain becomes, in the spectrum, a few clearly separated peaks.
From the time domain to the frequency domain
Two parameters determine what can be obtained from an FFT, and both depend on how the acquisition was carried out:
The sampling rate sets how far the spectrum extends, again according to the Nyquist criterion.
The record length, T, sets the level of detail. To distinguish two peaks 0.1 Hz apart, at least 10 seconds must be recorded, whatever the sampling rate; in practice, somewhat longer if a window is applied.
For this reason, the FFT is the reference tool in vibration analysis. In a rotating machine, a peak at the rotational frequency usually indicates unbalance; peaks at higher frequencies may point to gear or bearing defects.
Two common techniques are worth knowing from the outset. The first is windowing, for example with the Hann window (often called Hanning), which tapers the ends of the record to limit spectral leakage. The second is averaging several spectra, which reduces noise.
Practical criteria
- First estimate the bandwidth of the phenomenon: the fastest variation that needs to be measured.
- Sample at more than twice that frequency. As a starting point, 2.56 times if the spectrum is to be analysed and 10 times if the waveform is to be evaluated.
- Filter before sampling: aliasing cannot be corrected afterwards.
- Check that the sensor and the signal conditioner reach that frequency and, if the phenomenon is slow, that they also respond at sufficiently low frequencies.
- In the FFT, fs sets the maximum frequency that can be analysed and the record length sets the resolution.
What is not measured cannot be improved
Data acquisition tailored to each test
Nor can what is measured at the wrong rate. At Adecua we help define the sampling rate, sensors and filtering that each test requires, and we integrate them into a complete measurement system.
