Designing analog filters to shape signals before digitization, and configuring DAQ systems to correctly capture those signals without aliasing artifacts.
Active filters use op-amps with RC feedback networks to achieve filtering without inductors. Advantages over passive RC filters: no signal attenuation in the passband, higher input impedance, lower output impedance, and ability to cascade stages without loading.
The -3 dB frequency: where output power drops to half (-3 dB = 0.707 of input voltage). The boundary between passband and transition band.
Determines the roll-off slope. Each pole adds -20 dB/decade. A 2nd-order filter rolls off at -40 dB/decade beyond fc.
Maximally flat passband (no ripple). Moderate roll-off. Best choice when flat frequency response matters more than transition sharpness.
Steeper roll-off than Butterworth at the cost of passband ripple. Used when sharp transition between pass and stop bands is critical.
f_c = 1 / (2π · R · C) (for equal R, equal C design)
For Butterworth Q = 0.707:
Select R1 = R2 = R, C1 = 2C, C2 = C (or use tables)
Roll-off above f_c: -40 dB/decade (2nd order)
To faithfully reconstruct a continuous signal from its digital samples, the sampling rate must be at least twice the highest frequency component present in the signal. Sampling below this rate causes aliasing.
f_s ≥ 2 · f_max (Nyquist criterion)
f_Nyquist = f_s / 2 (highest representable frequency)
In practice: use f_s = 5 to 10 × f_max to leave margin
for realistic (imperfect) anti-aliasing filter roll-off.
When a signal component above f_Nyquist is sampled, it "folds" back into the spectrum and appears as a false low-frequency component. Aliased frequencies cannot be distinguished from real signals after digitization. The only cure is an anti-aliasing filter applied before the ADC.
A complete DAQ chain: Anti-aliasing filter → Sample-and-hold (S/H) → Multiplexer (MUX) → Analog-to-Digital Converter (ADC) → Digital bus.
The sample-and-hold circuit freezes the analog value during the ADC conversion time (aperture time). The multiplexer sequentially connects multiple channels to a single ADC — note that multiplexed channels are not simultaneously sampled.
Quantization step size: Δ = V_FSR / 2^N
Quantization noise (RMS): e_q = Δ / √12
ENOB = (SNR_measured - 1.76) / 6.02
where SNR is in dB. Ideal N-bit ADC: SNR = 6.02N + 1.76 dB
ENOB (Effective Number of Bits) reflects actual ADC performance degraded by noise, nonlinearity, and distortion. A 16-bit ADC may only achieve 14 ENOB in practice.
Nyquist frequency = f_s / 2 = 5 kHz. An ideal filter would cut off exactly at 5 kHz to prevent aliasing.
However, a 2nd-order Butterworth has a gradual roll-off. At 5 kHz a signal component must be attenuated to below quantization noise. Setting f_c = 3 kHz (equal to signal bandwidth) leaves 1 octave (3 to 5 kHz) for the filter to attenuate. At 2 octaves above f_c, a 2nd-order filter attenuates by about 40 dB × 2 = 80 dB, which is typically sufficient.
Ideal brick-wall: f_c = 5 kHz. Real 2nd-order Butterworth: f_c ≈ 3 kHz. Increasing sampling rate to ≥30 kHz is preferred in practice.
Δ = V_FSR / 2^N = 5.0 / 2^10 = 5.0 / 1024 ≈ 4.88 mV
e_q (RMS) = Δ / √12 = 4.88 / 3.464 ≈ 1.41 mV
Step size ≈ 4.88 mV, RMS quantization noise ≈ 1.41 mV
Nyquist = 500/2 = 250 Hz. The 400 Hz signal is above Nyquist. Alias frequency = |f_signal - f_s| = |400 - 500| = 100 Hz. Alternatively: alias = f_s - f_signal = 500 - 400 = 100 Hz.
Alias appears at 100 Hz. The 400 Hz component is indistinguishable from a 100 Hz signal in the digitized data.