Sampling, aliasing and reconstruction

Every robot controller is a computer, and computers see the continuous world only through samples. An ADC reads a current sensor 20,000 times a second, an IMU delivers 1,000 readings a second, a joint controller updates its command every millisecond. Between samples the computer is blind, and after computing a command it holds that value until the next update. Both steps change the signals in ways that matter. Sample too slowly and high frequencies masquerade as low ones, an effect called aliasing that can make a vibrating machine appear to drift or turn a 50 Hz electrical hum into a slow wobble that no later filter can remove. Hold commands for too long and the loop acquires extra delay that erodes stability. This lesson explains the sampling theorem, aliasing and how to prevent it with anti-aliasing filters, quantization noise, and the zero-order hold that turns samples back into continuous signals.

Ideal sampling

Sampling a continuous signal x(t) every T seconds produces the sequence \( x[n] = x(nT) \). The sampling rate is \( f_s = 1/T \) samples per second (hertz), or \( \omega_s = 2\pi/T \) in radians per second. A useful mental model writes the sampled signal as the continuous signal multiplied by a train of impulses spaced T apart. In the frequency domain, multiplication becomes convolution, and the spectrum of an impulse train is another impulse train, spaced \( f_s \) apart. The result is the single most important fact about sampling: the spectrum of a sampled signal is the original spectrum repeated, or replicated, at every multiple of the sampling rate:

If the original spectrum is confined to frequencies below \( f_s/2 \), the copies do not overlap and the original can be recovered exactly. If it extends beyond \( f_s/2 \), the copies overlap and the information is scrambled irreversibly.

The sampling theorem

The Nyquist-Shannon sampling theorem, developed by Harry Nyquist (1928), Claude Shannon (1949) and earlier by Vladimir Kotelnikov (1933), states: a signal containing no frequencies at or above B hertz is completely determined by samples taken at a rate \( f_s > 2B \). The frequency \( f_s/2 \) is called the Nyquist frequency, and 2B the Nyquist rate. Below the Nyquist frequency, sampling loses nothing. In practice, engineers sample several times faster than 2B, for three reasons: real anti-aliasing filters do not cut off sharply, control loops need many samples per response time to keep the delay of sampling small, and reconstruction with a simple hold is crude unless samples are close together. Common rules of thumb are 5 to 10 times the highest frequency of interest for measurement and 10 to 30 times the closed-loop bandwidth for digital control.

Aliasing

What happens to a frequency above the Nyquist frequency? It appears, after sampling, at a lower alias frequency. A sinusoid at frequency f produces exactly the same samples as one at \( |f - kf_s| \) for any integer k, and the observed frequency is the one that falls between 0 and \( f_s/2 \):

The figure shows the classic case: a 9 Hz sinusoid sampled at 10 Hz yields samples indistinguishable from a 1 Hz sinusoid. The same effect makes wagon wheels in films appear to turn slowly backwards, and makes a spinning propeller filmed with a rolling-shutter camera look bent. In a robot, aliasing can turn high-frequency vibration, PWM ripple or electrical hum into apparent slow motion that the controller then tries to correct, causing real low-frequency movement. Once aliased, the false component lies in the same band as the true signal and no digital filter can separate them. Aliasing must be prevented before sampling.

Anti-aliasing filters

The defence against aliasing is an analogue low-pass filter in front of the ADC that removes content above the Nyquist frequency, or at least attenuates it below the resolution of the converter. Because no real filter has a vertical cutoff, the filter's passband edge must sit well below \( f_s/2 \), and the gap between them, the transition band, determines how steep, and therefore how complex, the filter must be. A first-order RC filter rolls off at only 20 dB per decade; reaching the 72 dB that matches a 12-bit ADC's resolution would take more than three decades of margin. Higher-order active filters (Butterworth, Bessel) roll off at 20n dB per decade for order n. The modern alternative is oversampling: sample an inexpensive sigma-delta ADC at a very high rate, where a simple RC filter suffices, then filter and decimate digitally to the rate actually needed. Most MEMS IMUs and audio converters work this way, with configurable digital low-pass filters, which is why IMU datasheets list an output data rate and a separate bandwidth setting (ROB-220).

Quantization

Sampling discretizes time; quantization discretizes amplitude. An N-bit ADC with full-scale range V resolves steps of \( \Delta = V/2^N \) (ROB-200). The rounding error is at most \( \Delta/2 \) and, for busy signals, behaves like uniform random noise with standard deviation

The resulting signal-to-quantization-noise ratio for a full-scale sinusoid is about \( 6.02N + 1.76 \) dB: 74 dB for 12 bits, 98 dB for 16 bits. Each extra bit buys 6 dB. Oversampling also helps here: averaging four samples halves the noise, which is worth one extra bit, so oversampling by \( 4^k \) gains k bits, provided the noise is genuinely random. Quantization in encoders works the same way: an encoder of resolution q radians adds noise of \( q/\sqrt{12} \) to every angle reading, which matters when angles are differenced to get velocity (lesson 11).

Reconstruction and the zero-order hold

Going the other way, a controller's discrete commands must become a continuous voltage or current. The ideal reconstruction implied by the sampling theorem uses a perfect low-pass filter, which would need an infinite, non-causal impulse response, impossible in real time. Instead, digital-to-analogue converters and PWM stages use a zero-order hold (ZOH): each command is held constant until the next one arrives, producing a staircase. The ZOH has impulse response equal to a rectangle of width T, and its transfer function is

Its frequency response has a magnitude shaped like \( \sin(\omega T/2)/(\omega T/2) \) and, most importantly, a phase lag of \( \omega T/2 \): on average the staircase lags the smooth signal by half a sample. In a control loop this is an effective delay of T/2 on top of computation and sensor delays. At a loop crossover frequency \( \omega_c \), the lost phase margin is \( \omega_c T/2 \) radians. This is the quantitative reason digital controllers must sample well above their bandwidth.

import numpy as np
fs = 1000.0
for f in [50, 450, 550, 950, 1050, 1950]:
    n = np.arange(2000)
    x = np.cos(2*np.pi*f*n/fs)                      # sampled cosine
    X = np.abs(np.fft.rfft(x*np.hanning(n.size)))
    f_obs = np.fft.rfftfreq(n.size, 1/fs)[np.argmax(X)]
    k = round(f/fs); f_pred = abs(f - k*fs)
    print(f"true {f:5.0f} Hz -> observed {f_obs:6.1f} Hz  (predicted alias {f_pred:5.0f} Hz)")

true    50 Hz -> observed   50.0 Hz  (predicted alias    50 Hz)
true   450 Hz -> observed  450.0 Hz  (predicted alias   450 Hz)
true   550 Hz -> observed  450.0 Hz  (predicted alias   450 Hz)
true   950 Hz -> observed   50.0 Hz  (predicted alias    50 Hz)
true  1050 Hz -> observed   50.0 Hz  (predicted alias    50 Hz)
true  1950 Hz -> observed   50.0 Hz  (predicted alias    50 Hz)

Frequencies below 500 Hz appear where they should; every frequency above it folds back into the 0 to 500 Hz band exactly as the alias formula predicts. A 950 Hz and a 1,050 Hz component both land at 50 Hz, indistinguishable from a genuine 50 Hz signal.