Spectrum Measurements: The most important part of your analysis system

TECHNICAL BRIEF: While much attention is paid to the transducers and the software and firmware that performs the analysis the most important part of your spectrum analysis system is the digitization of the real-world signals. We take deeper dive on this aspect.

Aliasing - The problem described

The common failure is not a wrong answer. It is something that looks to the system like a good one. Whether you know it or not everyone reading this article has witnessed aliasing. If you ever have watched video of a car going down the road and wondered why the wheels seemed to be turning slower or even backward relative to the speed and direction of the car you have witnessed the effects of aliasing. This happens when the frequency of wheel rotation is greater than the video frame capture frequency on the video equipment. While some of the concepts are the same for a spectrum measurement system, I am using this primarily as illustrative of the effects.

A spectrum computed from a badly specified acquisition chain can look exactly like a spectrum computed from a good one — and there is nothing in the output that flags the difference. In the example of the car, what if control decisions were solely made based upon the inputs from the video that falsely represented the signals? Bad things will happen.

Spectrum measurements have a wide variety of applications; the common thread is that the signal frequency is the core aspect of the measurement that matters. Applications include machinery condition monitoring, acoustic analysis, resonance non-destructive testing, transient shock effects and others. Much of this article is dedicated to specifying spectrum measurement systems that perform well and reduce the risks of aliasing.

The measurement path

In most typical condition monitoring applications vibration measurements play a key role in determining the state of the machinery. There are other applications as well that fall into the same spectrum measurement dilemma such as shock and acoustical analysis for various applications. the concepts addressed here apply there as well but with some parametric differences between them.

Between a physical quantity and a usable spectrum sit at least ten stages, each with its own transfer function, its own noise contribution, and its own characteristic failure:

mechanical interface → transducer → cabling → conditioning and excitation → analog anti-alias filter → gain and range → ADC → digital decimation → block framing and windowing → transform → scaling and units.

The system bandwidth is the narrowest of these, and it is frequently set by the earlier stages up to digitization by the ADC. Once the signal has been digitized no recovery protocol can salvage an aliased signal.

Sampling: The Necessary and the Sufficient

Nyquist gives a necessary condition — the sample rate must exceed by 2 times the highest frequency present in the signal reaching the converter — and it is routinely misapplied in two ways. First, the condition applies to the frequency content present, not the frequency content of interest; energy the analyst does not care about still aliases. Second, satisfying it exactly leaves no room for a real filter. The Nyquist frequency is important because it defines the lower boundary under which aliasing is guaranteed.

The better number - Why 2.56 rather than 2?

An analog anti-alias filter cannot transition from passband to full stopband attenuation at a single frequency. It needs a transition band, and the sample rate has to accommodate it. The industry convention in vibration and acoustic analysis is a factor of 2.56, which places the usable maximum analysis frequency at fs/2.56 = 0.39·fs and leaves the band from 0.39·fs to 0.5·fs as filter transition. The number is not arbitrary: a 1024-sample block transformed and truncated at that ratio yields exactly 400 usable spectral lines, which is why analyser line counts are 400, 800, 1600, 3200 rather than powers of two.

If an instrument advertises a 51.2 kS/s sample rate, its honest maximum analysis frequency is 20 kHz, not 25.6 kHz. Anything quoted between those two numbers is inside the filter skirt, where amplitude accuracy is unspecified and alias rejection is incomplete.

Stopband attenuation must exceed converter dynamic range

The anti-alias filter's stopband attenuation at the Nyquist frequency must exceed the dynamic range of the converter, otherwise out-of-band energy folds in above the noise floor and the converter's resolution is wasted. A 24-bit delta-sigma channel delivering 110 dB of usable dynamic range needs better than 110 dB of attenuation at fs/2. This is achievable with a modest analog filter only because the converter oversamples heavily; it is not achievable with a second-order RC network in front of a SAR converter running at the output rate, which is the most common architectural mistake in low-cost designs.

Aliasing: The Only Unrecoverable Error

Every other error in the chain is either correctable or at least detectable after the fact. Gain error can be calibrated out. Offset can be removed. Noise can be averaged down. Aliasing cannot be undone, because the information required to undo it was destroyed at the instant of sampling. An aliased component is mathematically indistinguishable from a real component at the folded frequency. The fold-back mapping is simple and worth carrying in the head. For an input at frequency f sampled at fs, the observed frequency is the distance from f to the nearest integer multiple of fs:

f_observed = | f − k · fs |,  where k is the nearest integer to f / fs.

So a 1.8 kHz component sampled at 2 kS/s appears at 200 Hz, with no amplitude penalty and no other signature. In machinery this is the dangerous case, because the frequencies most likely to be present and unmeasured — switching-supply noise, gear mesh harmonics, bearing resonance, blade-pass content — fold into the 0 to 1 kHz band where imbalance, misalignment and looseness live. A phantom line at a plausible order of running speed is a diagnosis waiting to be made.

Digital anti-aliasing does not remove the analog requirement

A delta-sigma converter contains a digital decimation filter that provides excellent alias rejection around the output rate, and this leads to the assumption that no analog filter is needed. It is wrong. The modulator itself samples at OSR times the output rate, and energy near that modulator rate — or near its multiples — aliases into the passband before the digital filter ever sees it. The analog requirement is relaxed, not removed: a simple single- or two-pole filter placed well above the band of interest is usually sufficient, but its absence leaves a wideband hole through which RF pickup, switching noise, and ultrasonic content enter unattenuated.

Where the requirement is absolute

For transient and shock measurement the analog filter is not negotiable at any sample rate, because the source spectrum is unbounded and unknown. Pyrotechnic and ballistic events carry meaningful energy above 100 kHz. There is no sample rate that outruns the problem, so the band must be limited in the analog domain before digitization.

Resolution Is Time, Not Sample Rate

The single most consequential misunderstanding in applied spectral measurement is the belief that a faster converter yields a finer spectrum. It does not. Frequency resolution is fixed entirely by the length of the acquired record:

Δf = 1 / T,  where T = N / fs is the record duration in seconds.

Doubling the sample rate at a fixed block size halves the record duration and therefore coarsens the resolution by a factor of two, while extending the upper frequency limit by the same factor. The two are traded against one another; neither is free. A 6400-line spectrum spanning 0 to 20 kHz has Δf = 3.125 Hz and requires a 0.32 s record. The same line count spanning 0 to 500 Hz has Δf = 0.078 Hz and requires 12.8 seconds — during which the machine must remain in a steady operating state, or the result is a smear. How this manifests in hardware is a need for longer block sizes to gain increasing resolution.

This is what makes certain diagnoses expensive in a way that has nothing to do with hardware. Detecting broken rotor bars in an induction motor requires resolving sidebands offset from line frequency by twice the slip frequency. At light load, slip may be under one percent, putting the sidebands within 0.5 Hz of a component tens of decibels larger. Resolving that requires Δf of roughly 0.05 Hz, which requires a record of about twenty seconds at genuinely constant load. No converter selection changes this arithmetic.

Objective	f-max	Δf needed	Implied record length and constraint
Overall vibration severity, ISO 20816 band	1 kHz	1–2 Hz	0.5–1 s. Undemanding; the usual default and adequate for trending only.
Imbalance, misalignment, looseness	10× running speed	0.25–0.5 Hz	2–4 s. Needs enough resolution to separate 1× from nearby structural lines.
Rolling-element bearing, spectral	10–20× running speed	0.25 Hz	4 s. Defect orders are non-integer; adjacent running-speed harmonics must resolve.
Rolling-element bearing, envelope	Carrier band 1–20 kHz; envelope span 0–500 Hz	0.25–0.5 Hz	2–4 s of envelope record. Carrier band selection matters more than converter choice.
Gear mesh and sidebands	3× GMF, often 10–50 kHz	0.1–0.25 Hz	4–10 s. Sidebands at ±1× shaft demand fine resolution at high f-max — the most expensive combination.
Induction motor rotor bar	≤ 200 Hz	0.02–0.05 Hz	20–50 s at constant load. Load stability, not instrumentation, is the binding constraint.
Blade pass, cavitation, flow noise	20 kHz+	1–2 Hz	0.5–1 s, heavily averaged. Broadband and stochastic; PSD rather than line spectrum.
Acoustic emission	100 kHz – 1 MHz	n/a	Different discipline. Hit-based parameters, not FFT. Requires dedicated front end.

Transducer Classes and What Each Imposes

The transducer sets more of the achievable measurement than the digitizer does. The digitizer can only resolve what the host transducer can deliver. Each class carries constraints that propagate into the acquisition design, and several carry a settling or warm-up behavior that silently corrupts the first records after power-up.

Class	Coupling	Practical band	What it imposes on the chain
IEPE / ICP accelerometer	AC, 2–20 mA constant current at 18–30 V compliance	0.5 Hz – 10 kHz typical	High-pass corner set by both sensor and input stage; the two interact. Requires seconds of settling after current is applied — records taken during settling show a decaying DC ramp that destroys any integration.
Charge-mode piezoelectric	Charge amplifier	0.1 Hz – 15 kHz, high temperature capable	Cable capacitance is part of the measurement. Triboelectric noise from cable flexing appears as low-frequency content. Cable must be fixed and specified.
MEMS capacitive accelerometer	DC-coupled, ratiometric or digital	DC – 2 to 10 kHz	True DC response, so it measures tilt and can be zeroed. Lower dynamic range, typically 60–80 dB, which limits small-defect detection against a large 1× component.
Moving-coil velocity	Self-generating	10 Hz – 1 kHz	Internal resonance and phase shift near the low corner. Convenient units, but the phase response makes it a poor choice for cross-channel work.
Eddy-current proximity probe	DC, driver required	DC – 10 kHz	Measures relative shaft displacement, not casing motion. Mechanical and electrical runout must be characterised and subtracted. Essential for fluid-film bearings, where casing measurement sees very little.
Microphone / ultrasonic	Polarised or prepolarised	20 Hz – 20 kHz, or 20–100 kHz	Field-dependent; measurement geometry is part of the calibration. Ultrasonic bands need a converter chain to match, not an audio one.
Strain / bridge	DC, regulated excitation	DC – bandwidth of conditioning	Excitation stability enters directly as measurement error. Lead resistance and temperature must be compensated.

The mounting limit

Often overlooked for accelerometers, the mechanical interface usually determines the real upper bandwidth, and it is far below what the datasheet implies. Stud mounting into a prepared flat face preserves 30 kHz or more. Adhesive mounting typically holds to 10 to 20 kHz. A magnet base resonates somewhere between 2 and 7 kHz depending on mass and surface, and a handheld probe is unreliable above roughly 1 kHz. Above the mount resonance the reading is dominated by the mount and bears no fixed relationship to the surface motion — it is not attenuated, it is wrong, and it is frequently amplified. Any specification of f-max above the mount resonance is fiction regardless of the converter behind it.

Post Digitization - Windowing and Averaging

The transform assumes the block is one period of a periodic signal. It never is, so the discontinuity at the block boundary produces leakage — energy from a real component smeared across neighboring bins. Windowing trades amplitude accuracy against leakage suppression, and the choice is dictated by what is being measured rather than by convention.

Window	Noise bandwidth	Worst-case amplitude error	Use
Rectangular (uniform)	1.00 bins	3.92 dB (36%)	Transients wholly contained in the block, and impact testing where the signal decays to zero inside the record. Never for continuous machinery.
Hanning	1.50 bins	1.42 dB (16%)	The default for continuous rotating machinery. Good leakage suppression; amplitude error is acceptable because diagnosis is usually comparative rather than absolute.
Hamming	1.36 bins	1.78 dB	Marginally better main-lobe resolution than Hanning with worse far-field suppression. Rarely the right answer in this domain.
Flat-top	≈ 3.77 bins	< 0.01 dB	Calibration and any measurement where absolute amplitude matters. The wide main lobe destroys resolution, so it is a verification tool rather than a diagnostic one.
Exponential	Signal-dependent	n/a	Response channel in impact testing, to force decay to zero within the block. Adds artificial damping which must be removed from the extracted modal parameters.

Overlap processing recovers what windowing discards at the block edges. With a Hanning window, fifty percent overlap restores approximately uniform weighting across the record and roughly doubles the number of averages available from a given acquisition; sixty-seven and seventy-five percent are used where averaging count matters more than computation. Overlap does not improve resolution and does not add information beyond the record length.

Three kinds of averaging, doing three different things

–   Linear (RMS) spectral averaging reduces the variance of the estimate. Random content falls as the square root of the number of averages while deterministic content remains; it does not remove noise, it stabilises the estimate of it.

–   Time-synchronous averaging requires a once-per-revolution reference and averages the time record itself. Content not synchronous with the reference cancels toward zero. This is the strongest tool available for gear diagnosis, because it isolates one shaft from everything else in the box — and it is unavailable without a tachometer, which is why the tach input is not an optional accessory.

–   Peak hold is not averaging; it retains the maximum in each bin across blocks. Appropriate for run-up and coast-down, misleading anywhere else.

Note: Order tracking deserves a separate callout. On variable-speed machines, a spectrum computed at fixed sample rate smears every shaft-related component across bins as speed drifts. The correct approach is angular resampling — resampling the record onto a constant-samples-per-revolution grid using a tachometer — after which shaft-related content collapses into sharp orders and non-synchronous content smears instead. Doing this properly requires the tachometer signal to be acquired on the same clock as the vibration channels. This is noted here as a reference, but a deep dive is not in scope for this article

Converter Architecture and Selection

Converter selection follows from the measurement, not the other way round. Three architectures cover essentially all of this domain, and they are not interchangeable.

Architecture	Strengths	Costs	Where it belongs
Delta-sigma	20–24 bit; 100–120 dB usable dynamic range; heavy oversampling relaxes the analogue filter; excellent linearity	Substantial group delay; long settling after any configuration change; poor fit for multiplexing; fixed relationship between clock and output rate	Vibration, acoustic, and any measurement where dynamic range matters more than latency. The default for condition monitoring.
SAR	Low latency; no pipeline delay; multiplexes cleanly; sample-and-hold is easy to reason about	Requires a real analogue anti-alias filter with steep rolloff; typically 16–18 bit; alias rejection is entirely the designer's problem	Control loops, mixed-signal I/O, and applications where deterministic latency is a requirement.
Pipeline	Very high sample rate at moderate resolution	12–16 bit; higher power; latency of several clocks	Ultrasonic, acoustic emission, partial discharge, and RF-adjacent measurement.

Nominal bits are not resolution

A converter's nominal word length says nothing useful about achievable measurement quality. The ideal signal-to-noise ratio of an N-bit converter is 6.02·N + 1.76 dB, so 16 bits gives 98 dB and 24 bits gives 146 dB — a figure no 24-bit converter in this class achieves. The honest figure is the effective number of bits derived from measured SINAD:

ENOB = (SINAD − 1.76) / 6.02

A good 24-bit delta-sigma channel delivers 18 to 20 effective bits in practice. That still matters, because the dynamic range requirement in vibration work is severe: an early-stage bearing defect may sit 50 to 60 dB below the running-speed component in the same record, and the input range must accommodate the large component without clipping while the small one remains above the noise floor. Sixteen bits leaves very little margin once real-world SINAD and crest factor are accounted for.

Nexus Engineering Partners has extensive experience in spectrum measurement and analysis systems for condition monitoring systems in power generation, pulp and paper, petrochemicals, as well as transient shock analysis (explosives). We can help architect your edge measurement system to ensure that it delivers the results you expect.