Sampling and Anti-Aliasing in a Vibration Monitoring Channel
A condition-monitoring system samples an accelerometer signal to detect bearing defects. The analyst wants to capture defect frequencies up to 8 kHz, and the hardware provides an anti-aliasing filter with a selectable cutoff. The current design samples at 20 kS/s and uses a 10 kHz analogue low-pass filter of second order, with a stopband attenuation of only 12 dB per octave. The analyst notices that a 14 kHz component, which is a known gear-mesh harmonic, appears in the spectrum at an apparent frequency of 6 kHz and is being mistaken for a bearing defect. The engineer proposes simply raising the sampling rate to 50 kS/s to 'get rid of the problem', while the data-storage budget allows only 20 kS/s per channel.
1. Explain why the 14 kHz component appears at 6 kHz, and evaluate whether raising the sampling rate to 50 kS/s solves the problem.
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With a sampling rate of 20 kS/s the Nyquist frequency is 10 kHz. A component at 14 kHz is above Nyquist, so it aliases to |14 - 20| = 6 kHz, exactly where it was observed. The analogue filter, being second order with 12 dB per octave, gives only about 12 x log2(14/10) = 5.8 dB of attenuation at 14 kHz, which is nowhere near enough to suppress it.
Raising the sampling rate to 50 kS/s moves Nyquist to 25 kHz, so 14 kHz is no longer aliased and the problem disappears for that component. But it does not solve the general problem: any component above 25 kHz will now alias, and the same second-order filter gives even less attenuation there. The real defect is an inadequate anti-aliasing filter, not an inadequate sampling rate. The correct fix is a steeper analogue filter, for example a fourth- or sixth-order Butterworth or an elliptic filter with a cutoff near 8-9 kHz and at least 60 dB attenuation at the sampling frequency, combined with oversampling and digital decimation. That keeps the 20 kS/s storage budget while providing genuine protection.
2. Design a practical acquisition chain that meets the 8 kHz analysis bandwidth and the 20 kS/s storage limit, and state the filter order needed for 60 dB attenuation at Nyquist.
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Chain: analogue anti-aliasing filter, then an ADC running at a higher rate (for example 80 or 100 kS/s), then digital low-pass filtering and decimation to 20 kS/s for storage. The analogue filter handles the frequencies that the digital stage cannot; the digital stage provides the sharp cutoff and linear phase.
Filter order: a Butterworth response of order n gives attenuation 20n log10(f/fs) dB. To get 60 dB at the first alias frequency, which for a 20 kS/s output is 20 kHz, with a cutoff at 8 kHz: 60 = 20n log10(20/8) = 20n x 0.398, so n = 7.5, i.e. an eighth-order filter. If the ADC runs at 100 kS/s and the analogue filter only needs to suppress above 50 kHz, the requirement is far easier: 60 = 20n log10(50/8), so n = 2.4, i.e. a third-order filter suffices. This is the practical reason to oversample. Then a digital decimation filter with a sharp transition band at 8-9 kHz provides the final protection. Specify the accelerometer bandwidth, the filter's passband ripple and the ADC's effective number of bits, and verify with a swept-sine test injecting a 14 kHz tone.