2.1 Analogue and Digital Signals, Sampling and Quantisation

2.1 Analogue and Digital Signals, Sampling and Quantisation

Aircraft sensors produce physical quantities that vary smoothly: pitot and static pressure, total air temperature, and the analogue voltages from linear and rotary variable differential transformers (LVDTs and RVDTs) that report flight-control surface or actuator position. Digital computers, electronic instrument systems, and aircraft data buses cannot store those continuous waveforms as analogue voltages. They store binary words. Commission Implementing Regulation (EU) 2023/989 sets topic 5.3, Data conversion, at level 2 for B2/B2L and level 1 for B1. Its current Appendix I does not spell out the component concepts; the former detailed Appendix I description supplied analogue data, digital data, and converter operation, which this guide retains as historical study scope. This section develops the continuous-versus-discrete distinction, the sampling theorem, aliasing, quantisation, and the anti-alias filters that protect air-data and engine-sensor chains.

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Continuous Analogue Data versus Discrete Digital Data

An analogue signal is continuous in both time and amplitude. At every instant the instantaneous value may occupy any real number within the transducer range. A platinum resistance temperature detector, a capacitive fuel-quantity probe, and an LVDT secondary voltage are analogue sources: their outputs change smoothly as temperature, fuel height, or ram displacement change. There is no inherent “step” in a correctly functioning analogue chain until a later converter imposes one.

A digital representation is discrete in time and discrete in amplitude. Time is sliced into sampling instants separated by the sampling period $T_s = 1/f_s$. Amplitude is restricted to a finite set of allowed levels. Between samples the digital system holds a number, not the original waveform. Once quantised, the sample is encoded as a binary word of $n$ bits and may be transmitted on a digital bus or processed by a flight-guidance or engine computer.

The two domains are complementary rather than competing. Analogue circuits remain the natural interface to physics. Digital processing gives noise immunity on multi-receiver buses, exact repeatability of computed parameters, and straightforward Built-In Test Equipment (BITE) patterns. Every modern air-data computer, full-authority digital engine-control (FADEC) channel, and electronic flight instrument computer therefore contains an analogue front end, a sampler, a quantiser, and a digital processor. A technician who treats “digital” as automatically more accurate than “analogue” will misdiagnose offset, aliasing, and reference errors that originate entirely in the analogue path.

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Sampling Rate and the Sampling Period

Sampling measures the analogue waveform at discrete instants and, with a sample-and-hold, freezes that voltage while the analogue-to-digital converter finishes conversion. The sampling frequency $f_s$ must be chosen from the highest frequency of interest in the sensor bandwidth, not from a convenient leftover clock divider.

If a total-air-temperature signal is known to change no faster than a few hertz after probe thermal lag, a few tens of samples per second can capture the useful information. If a wide-band vibration accelerometer on an engine gearbox carries energy to several kilohertz, $f_s$ must sit well above that band. Undersampling a fast signal is not a mild accuracy loss; it creates a false low-frequency image that the rest of the avionics will treat as a valid parameter.

The sampling period and the converter’s conversion time must also fit together. If conversion occupies most of $T_s$, the analogue value must already be held stable. That is why a sample-and-hold (treated in detail in Section 2.3) sits between the anti-alias filter and the quantiser on almost every aircraft data-acquisition channel.

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The Nyquist Criterion

Harry Nyquist’s sampling theorem states that a band-limited signal containing no energy above $f_{\max}$ can be reconstructed from uniformly spaced samples if

$f_s > 2 f_{\max}$

The quantity $2 f_{\max}$ is the Nyquist rate. Half the sampling frequency, $f_s/2$, is the Nyquist frequency (also called the folding frequency). In some textbooks the inequality is written $f_s \ge 2 f_{\max}$; practical avionics designs use a comfortable margin because real anti-alias filters do not cut off infinitely sharply and because reconstruction filters are not ideal brick walls.

A worked air-data illustration: suppose the highest useful frequency in a filtered pitot-static channel is $20\text{ Hz}$. The theoretical minimum sampling rate is $40\text{ samples/s}$. An air-data computer will typically sample far faster — often hundreds of hertz — so that digital filtering, BITE, and latency requirements are met and so that a practical analogue low-pass filter can begin rolling off well below $f_s/2$. The extra samples are not wasted; they buy filter transition-band width and reduce group-delay compromise.

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Aliasing

When $f_s < 2 f_{\max}$, frequency components above the Nyquist frequency fold back into the baseband. A component at frequency $f$ appears at

$f_{\text{alias}} = \lvert f - k f_s \rvert$

where the integer $k$ is chosen so the result lies between $0$ and $f_s/2$. The aliased tone is indistinguishable from a genuine low-frequency signal. No later digital filter can remove it, because the damage has already occurred at the sampling instant.

On an aircraft this is not an academic curiosity. Engine vibration, $400\text{ Hz}$ alternating-current ripple on a poorly filtered direct-current rail, and switching edges from nearby power electronics all inject energy above the intended sensor bandwidth. If that energy reaches the sampler, a slowly varying airspeed or temperature word can acquire a wandering error that looks like a valid aerodynamic or thermal change. A line-maintenance crew comparing a digital airspeed readout with a mechanical standby instrument may then chase the wrong unit.

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Anti-Alias Filters

The only robust protection is an analogue anti-aliasing low-pass filter placed before the sample-and-hold. Its pass-band covers the frequencies the system must measure. Its stop-band has already attenuated everything above $f_s/2$ to below one least-significant bit (or to a documented noise floor) before the sampling switch closes.

A single-pole RC filter rolls off at only $20\text{ dB/decade}$, so the gap between $f_{\max}$ and $f_s/2$ must be wide, or a multi-pole active filter (Butterworth, Bessel, or elliptic) must be used. Bessel responses are favoured when waveform shape and group delay matter, as in some servo-command and vibration channels. After digitisation, additional digital low-pass filtering can refine the displayed or computed bandwidth, but it cannot repair aliases.

[!WARNING] Do not omit or bypass the analogue anti-alias filter during troubleshooting. Substituting a raw sensor into a converter input “to see the true signal” can inject $400\text{ Hz}$ aircraft-bus ripple and high-frequency electromagnetic interference that alias into the displayed parameter. The indication then disagrees with a calibrated analogue gauge even though the digital box is working as designed.

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Quantisation, Resolution and Bit Depth

After sampling, each held voltage is mapped onto the nearest of $2^n$ discrete levels. An $n$-bit converter therefore provides

$\text{number of levels} = 2^n$

The quantisation step, or least-significant-bit (LSB) voltage, for a unipolar converter is

$V_{\text{LSB}} = \frac{\text{FSR}}{2^n}$

where FSR is the full-scale range. A bipolar converter spanning $\pm V_{\text{ref}}$ uses $2V_{\text{ref}}$ as the full-scale span. The table below lists the bit depths that appear constantly in Module 05 teaching and in aircraft data-acquisition hardware.

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Quantisation Error and Quantisation Noise

Because the true analogue voltage may lie anywhere inside a step, the maximum quantisation error is $\pm \tfrac{1}{2}\text{ LSB}$. That error is not a random hardware fault; it is the irreducible price of representing a continuum with a finite code set. Over many samples of a busy signal the error behaves like additive noise — quantisation noise — with an RMS value of $V_{\text{LSB}}/\sqrt{12}$ for an ideal uniform quantiser.

For a full-scale sine wave the theoretical signal-to-quantisation-noise ratio is approximately

$\text{SQNR} \approx 6.02n + 1.76\text{ dB}$

Each extra bit therefore improves ideal SQNR by about $6\text{ dB}$. Real converters fall short because of differential non-linearity, aperture jitter, and reference noise. Maintenance implications are direct: a 12-bit air-data channel cannot be turned into 16-bit performance by software scaling alone. The hardware resolution, the analogue front-end noise, and the anti-alias filter together set the usable precision.

[!NOTE] Resolution is not the same as accuracy. A 16-bit word can resolve $153\text{ }\mu\text{V}$ on a $10\text{ V}$ range and still be wrong by tens of millivolts if the voltage reference has drifted, the probe excitation is in error, or the analogue input amplifier has an offset. Resolution describes the fineness of the steps; accuracy describes closeness to the true physical quantity.

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Aircraft Signal Paths into Converters

Typical analogue sources that enter analogue-to-digital converters on a transport aeroplane include:

1. Air-data sensors. Pitot and static pressures (often via vibrating-cylinder or solid-state transducers), angle-of-attack vanes, and total-air-temperature probes feed the air-data computer. After analogue conditioning they are sampled, quantised, and computed into computed airspeed, Mach number, pressure altitude, and true airspeed.
2. LVDT and RVDT position sensors. Flight-control surface position, throttle-lever angle, and many actuator rams use alternating-current-excited differential transformers. The analogue secondary voltages are demodulated to direct current (or processed as alternating-current amplitudes) and then digitised so flight-control and engine computers can close digital loops.
3. Temperature probes. Resistance temperature detectors and thermocouples on engines, hydraulic systems, and environmental-control systems produce millivolt or resistance changes that must be amplified, filtered, and converted before they appear on crew alerting displays or enter a FADEC limit schedule.

In each path the same sequence applies: analogue sensing, anti-alias filtering, sampling, quantisation to $2^n$ levels, and digital transmission. Category B2 staff must be able to explain why a “noisy” digital parameter may be an aliasing or resolution problem rather than a failed probe, and why replacing a sensor without confirming the analogue interface and reference voltages can leave the digital word still incorrect.

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Worked Numerical Example: 12-bit Air-Data Channel

A unipolar analogue-to-digital converter in an air-data computer has a full-scale range of $0\text{ V}$ to $10.00\text{ V}$ and a resolution of $12\text{ bits}$. A conditioned pitot-transducer voltage of $3.250\text{ V}$ is presented to the sampler.

Step 1 — Number of levels and LSB size

$2^{12} = 4096 \text{ levels}, \quad V_{\text{LSB}} = \frac{10.00\text{ V}}{4096} = 2.441\text{ mV}$

Step 2 — Ideal code

$\text{code} = \operatorname{round}\left(\frac{3.250\text{ V}}{2.441\text{ mV}}\right) = \operatorname{round}(1331.3) = 1331$

Step 3 — Quantisation error bound

Maximum error $= \pm 1.221\text{ mV}$, which is $\pm 0.012%$ of full scale. Whether that error is acceptable depends on the pressure-to-voltage scale factor and the airspeed accuracy specification, not on the bit count alone.

Step 4 — Nyquist check

If the anti-alias filter cuts off at $25\text{ Hz}$ and $f_s = 200\text{ Hz}$, then $f_s/2 = 100\text{ Hz}$, giving a factor-of-four guard band above $25\text{ Hz}$. Energy at $400\text{ Hz}$ from the alternating-current bus must already be attenuated by the analogue filter; otherwise it aliases toward a near-direct-current offset or a low-frequency wander in the airspeed word.