Mr. Integral
Embedded notes/Moving and Processing Data/Analog Signals and Conversion

UNIT 11 · LESSON 1 OF 6

Sampling, Aliasing, and Bandwidth

How can an ADC report a frequency that is not there, and how do you stop it?

INTERACTIVESampling too slowly invents a different signal
A sine wave, its samples and the lower-frequency alias through the same samples2 mssignal 1 kHz, sampled at 10 kHz: Nyquist frequency 5 kHzBelow Nyquist: the samples (dots) describe the 1 kHz signal uniquely.
A sine wave, its samples and the lower-frequency alias through the same samples2 mssignal 1 kHz, sampled at 10 kHz: Nyquist frequency 5kHzBelow Nyquist: the samples (dots) describe the 1 kHzsignal uniquely.

Try this

Sampling rate
1 kHz is below Nyquist.

An ADC sees the signal only at the sampling instants. Any frequency above half the sampling rate (the Nyquist frequency) produces the same samples as a lower one (with suitable phase), its alias, at the distance from the nearest multiple of the sampling rate. After sampling, nothing can tell them apart.

What you will be able to do
  • Explain why a sampled signal is ambiguous and state the Nyquist condition.
  • Compute the alias frequency of a signal above the Nyquist frequency.
  • Explain why aliasing must be prevented before the ADC and cannot be filtered out afterwards.
  • Estimate how much a first-order anti-alias filter attenuates an interferer, in LSBs.
  • Choose a sampling rate and filter for a given signal bandwidth.
Before you start
  • Capacitors and RC time constants (unit 1, lesson 4).
  • Timers and periodic events (unit 8, lessons 2 and 4).
Steps in this lesson
  1. Sampling
  2. Aliasing
  3. Anti-alias filtering
  4. Bandwidth, not just frequency
  5. Worked example: the 10 Hz wobble
  6. Common misconceptions

The puzzle

A logger samples a vibration sensor 100 times a second and shows a slow 10 Hz wobble. The machine has no part turning at 10 Hz. It does have a motor at 90 Hz. How can an ADC report a frequency that is not there, and how do you stop it?

STEP 1

Sampling

An ADC measures the input at discrete instants, the sampling rate f_s times per second. Between samples it sees nothing. The sampling theorem says that a signal containing only frequencies below f_s / 2 is completely described by its samples. That limit is the Nyquist frequency:

fN=fs2f_N = \frac{f_s}{2}

In practice, sample a good margin faster than twice the highest frequency you need, because real filters do not stop sharply at f_N.

STEP 2

Aliasing

Above f_N the samples become ambiguous: a sine at frequency f produces the same samples as a sine (with suitable phase) at |f − k·f_s| for the integer k that makes it smallest. The higher frequency takes on the identity of the lower one, its alias:

falias=∣f−kfs∣,f_{\text{alias}} = \left| f - k f_s \right|, k=round⁡ ⁣(ffs)k = \operatorname{round}\!\left(\frac{f}{f_s}\right)

↑ This step uses the figure at the top of the page.

A signal exactly at f_s is sampled at the same phase every time and looks constant; one slightly off f_s looks like a slow drift. Once the samples are taken, nothing in software can tell the alias from a real signal at that frequency: a digital filter only sees the sequence of numbers.

STEP 3

Anti-alias filtering

The cure is to remove everything above f_N before it reaches the ADC, with an analog anti-alias filter. A single RC low-pass is common because it is cheap, but it falls off only 20 dB per decade above its cut-off f_c:

∣H(f)∣=11+(f/fc)2|H(f)| = \frac{1}{\sqrt{1 + (f/f_c)^2}}

An interferer ten times above f_c is reduced only tenfold. Whether that is enough depends on how large the interferer is compared with one LSB of the converter.

INTERACTIVEAn anti-alias filter in front of the ADC
RC low-pass response with the Nyquist frequency and an interfering tone0-40-80100 Hz1 kHz10 kHz100 kHzNyquist45 kHz interference: -33.1 dB, so 500 mV becomes 11.1 mV = 14 LSB of a 12-bit, 3.3 VADCIt is above Nyquist and folds to 5 kHz.
RC low-pass response with the Nyquist frequency and an interfering tone0-40-80100 Hz1 kHz10 kHz100 kHzNyquist45 kHz interference: -33.1 dB, so 500 mV becomes11.1 mV = 14 LSB of a 12-bit, 3.3 V ADCIt is above Nyquist and folds to 5 kHz.
Filter cut-off
Interference frequency
Sampling rate
45 kHz reduced to 11.1 mV (13.8 LSB), folds to 5 kHz.

Aliasing cannot be undone after sampling, so unwanted high frequencies must be removed before the ADC. A single RC low-pass falls off only 20 dB per decade: an interferer far above the band of interest is reduced, but may still be many LSBs large. The readout compares what is left with one LSB (0.81 mV) of a 12-bit converter on a 3.3 V range.

Sampling much faster than needed (oversampling) relaxes the filter: the Nyquist frequency moves up, a gentle filter has more room to fall, and a digital filter can then reduce the bandwidth and the sample rate afterwards.

STEP 4

Bandwidth, not just frequency

The sampling rate must cover the bandwidth of the signal you need, including fast edges and harmonics, not just its fundamental. A 50 Hz current waveform with harmonics up to the 20th (1 kHz) needs a sampling rate well above 2 kHz; sampling it at 200 Hz gives four samples per cycle, and every harmonic folds onto DC, 50 Hz or 100 Hz, silently corrupting the measured fundamental and RMS value.

STEP 5

Worked example: the 10 Hz wobble

The logger samples at f_s = 100 Hz, so f_N = 50 Hz. The motor’s 90 Hz vibration is above f_N; with k = round(90/100) = 1,

falias=∣90−1×100∣=10 Hzf_{\text{alias}} = |90 - 1 \times 100| = 10\ \text{Hz}

exactly the phantom wobble. The fix is an analog low-pass well below 50 Hz in front of the ADC (if 90 Hz content is unwanted), or sampling fast enough to capture 90 Hz honestly, more than 180 Hz and in practice several hundred.

MYTHS AND FACTS

Common misconceptions

Sampling at twice the frequency is enough

Only for a signal with nothing above f_N, which needs a filter; exactly 2f samples can land on the zero crossings.

A digital filter can remove aliasing

After sampling, an alias is indistinguishable from a real in-band signal.

The ADC’s maximum rate is the rate to use

Sample at what the signal and the filter need; faster wastes memory and power, slower aliases.

Only high-frequency signals alias

Any interference above f_N does, including switching noise and mains harmonics on a slowly sampled sensor.

Check yourself

Answer in your head, then open the card.

A 1.2 kHz tone is sampled at 1 kHz. What frequency does the firmware see?

k = round(1.2) = 1: |1200 − 1000| = 200 Hz.

Why can’t an averaging filter in firmware remove a 60 Hz alias of a 540 Hz vibration sampled at 600 Hz?

The alias is a genuine 60 Hz component of the sampled sequence; a filter that removed it would also remove a real 60 Hz signal. It had to be stopped before sampling.

An RC filter has f_c = 1 kHz. By how much does it attenuate a 20 kHz interferer?

1/√(1 + 20²) ≈ 0.050, about −26 dB, a factor of 20.

The RP2040 samples one input back to back with adc_set_clkdiv(0). What is the Nyquist frequency?

Each conversion takes 96 cycles of 48 MHz, 500 kS/s, so f_N = 250 kHz.

Sources (3)
  1. Raspberry Pi Ltd, pico-sdk 1.5.1, hardware_adc/adc.h — RP2040 ADC: “SAR ADC”, “500 kS/s (Using an independent 48MHz clock)”, “12 bit (8.7 ENOB)”, a 5-input mux, a 4-sample FIFO; adc_set_clkdiv: “Period of samples will be (1 + div) cycles on average. Note it takes 96 cycles to perform a conversion”
  2. Raspberry Pi Ltd, pico-examples, adc/dma_capture/dma_capture.c — captures 1000 samples back to back into a buffer with DMA; the ADC runs free, “timed by the 48 MHz ADC clock” (adc_set_clkdiv(0))
  3. Arm, CMSIS-DSP, Source/FilteringFunctions/arm_fir_f32.c — a digital FIR filter computes “y[n] = b[0] * x[n] + b[1] * x[n-1] + …” from a state buffer of earlier input samples: it operates only on the already-sampled sequence