3. Sampling theory fundamentals for channel budgeting

Every channel in a data acquisition system is a chain of compromises, and the sampling stage is where those compromises become irreversible. Once an analog signal is digitized, no downstream filter, averaging routine, or calibration table can recover information that was aliased or clipped at the converter. Channel budgeting therefore has to begin with sampling theory, not with the ADC datasheet's headline resolution. This section establishes the three quantities that drive nearly every sampling decision — the Nyquist criterion, the anti-aliasing filter transition band, and the oversampling ratio — and shows how they interact when many channels share one converter or one filter bank.

The Nyquist criterion and what it actually constrains

For a signal band-limited to f_max, sampling at f_s > 2·f_max preserves all information in the band. The factor of two is the familiar statement, but the engineering consequence is subtler: the criterion guarantees reconstruction only if the input contains no energy above f_s/2. Real sensors, cables, and switching power supplies do not respect that assumption. A 10 kHz signal of interest sampled at 25 kSPS satisfies Nyquist comfortably, yet a 30 kHz switching spur on the same wire folds to |30 − 25| = 5 kHz and lands directly inside the passband, indistinguishable from genuine signal content.

This is why channel budgeting treats Nyquist as a floor, not a target. The practical sampling rate is set by the highest frequency that can reach the converter, not by the bandwidth the application cares about. Two numbers must be tracked per channel: the signal bandwidth f_sig that must be preserved, and the interference bandwidth f_int that must be rejected. The gap between them determines the filter, and the filter determines the achievable sample rate.

Anti-aliasing filter requirements

An anti-aliasing filter must attenuate every frequency that would fold into the passband down to a level below the converter's noise floor or the required measurement resolution — whichever is stricter. The design question is not "what cutoff frequency?" but "what order and what transition band?" The transition band is the interval between the passband edge f_p (where attenuation must still be near zero) and the stopband edge f_stop (where attenuation must reach the target). For a first-order RC filter, the stopband edge is fixed by the sampling rate: any component above f_s − f_p can fold into the passband, so f_stop ≤ f_s − f_p.

Consider a concrete budget. A channel must resolve 1 mV on a ±5 V range with a 16-bit converter, giving an LSB of about 153 µV. To keep aliased energy below half an LSB, the filter must provide roughly 20·log₁₀(5 V / 76 µV) ≈ 96 dB of attenuation at the stopband edge. If the signal band extends to 1 kHz and the converter runs at 10 kSPS, the stopband edge is 9 kHz — a transition ratio of 9:1. A single-pole RC filter rolls off at 20 dB/decade and would need about five decades to reach 96 dB, which is impossible within a 9:1 band. The required order is approximately 96 / (20·log₁₀(9)) ≈ 50, which is absurd for a passive network. The realistic responses are to raise the sample rate, lower the passband edge, or accept a lower resolution — and that trade is exactly what channel budgeting exists to make explicit.

Oversampling and its multi-channel trade-offs

Oversampling — running the converter well above 2·f_sig — relaxes the analog filter because the stopband edge moves outward, widening the transition band. It also spreads quantization noise over a wider bandwidth, so a digital decimation filter that removes the out-of-band noise recovers effective resolution: each doubling of the oversampling ratio OSR buys about 3 dB (half a bit) for a white-noise converter, or more for a noise-shaping converter. The cost is throughput. In a multiplexed multi-channel system, the aggregate sample rate is shared: if one ADC runs at 100 kSPS and serves 16 channels, each channel sees only 6.25 kSPS. Oversampling one channel by 8× consumes the budget of eight channels.

This is the central tension in multi-channel budgeting. A single high-order analog filter per channel is expensive in board area and component count; a shared filter before the multiplexer is cheap but forces every channel to accept the same cutoff, and the multiplexer's own settling and charge injection add error. Oversampling shifts the burden to the digital domain, where per-channel decimation filters are nearly free in an FPGA or DSP — but only if the converter has the raw throughput to spare. The budget must therefore be solved jointly: choose f_s per channel from the filter order you can afford, multiply by channel count, and check against the converter's maximum rate and the multiplexer's settling time.

Putting the numbers together: a worked channel budget

The following example is illustrative, not measured. A 16-channel strain-gauge system must resolve 1 µV on a ±10 mV bridge output, with signal content up to 500 Hz and a known 25 kHz switching spur on the excitation supply. The converter is a 24-bit delta-sigma with a maximum output rate of 20 kSPS per channel and a 200 kSPS aggregate limit.

1. Resolution target. 1 µV on ±10 mV is 1 part in 20,000, or about 14.3 effective bits. A 24-bit converter has ample headroom, so the budget is set by noise and aliasing, not by quantization.
2. Stopband edge. The spur at 25 kHz must be attenuated. With f_s = 4 kSPS, the stopband edge is 4 − 0.5 = 3.5 kHz, well below the spur — the spur folds to |25 − 4·6| = 1 kHz, inside the passband. With f_s = 50 kSPS, the stopband edge is 49.5 kHz, above the spur, so a modest filter suffices.
3. Aggregate check. 16 channels × 50 kSPS = 800 kSPS, exceeding the 200 kSPS aggregate limit. The system cannot run all channels at that rate simultaneously.
4. Resolution of the conflict. Two viable budgets: (a) run at 4 kSPS per channel with a sharp digital decimation filter and a 4th-order analog anti-aliasing filter ahead of the multiplexer, accepting that the 25 kHz spur must be attenuated by the analog stage before it reaches the converter; or (b) run at 12.5 kSPS per channel (200 kSPS / 16), placing the stopband edge at 12 kHz and requiring a 6th-order filter to push the 25 kHz spur below the noise floor.

Option (b) is usually preferable because the higher sample rate widens the transition band and reduces the analog filter order, at the cost of a faster multiplexer and more digital throughput. The point of the exercise is that the choice is visible and quantified before any hardware is committed — which is precisely what a channel budget is for.

Practical rules for the sampling plan

• State f_sig and f_int explicitly for every channel; they are different numbers and both belong in the budget.
• Compute the required stopband attenuation from the resolution target and the converter's full-scale range, not from the ADC's bit count.
• Estimate filter order from the transition ratio before selecting a topology; if the order exceeds four, revisit the sample rate first.
• Multiply per-channel rate by channel count and compare against the aggregate converter and multiplexer limits early — this constraint often dominates.
• Treat oversampling as a system resource: it trades throughput for filter relaxation and effective resolution, and in a multiplexed system that throughput is shared.

With these three quantities — Nyquist floor, filter transition band, and oversampling ratio — the sampling plan becomes a set of explicit, checkable numbers rather than a default inherited from a reference design. The next step is to carry those numbers into the per-channel error budget, where converter noise, reference drift, and multiplexer settling are added to the aliasing allowance established here.