Measurements · Module 7 of 10
Sampling and Data Acquisition
Computers store numbers, not signals, so an analog measurement must be sampled in time and quantized in amplitude. Get the sample rate wrong and a fast signal masquerades as a slow one.
Readiness check
This module digitizes a signal. Tick only what you can do closed-notes.
- Double a frequency.
- Evaluate a power of two such as 212.
- Divide a voltage range into equal steps.
- Recall that a signal has a highest frequency.
- Recall the anti-alias filter idea.
The core idea
Sampling reads a signal at a fixed rate; the Nyquist criterion says that rate must exceed twice the highest frequency present, or high frequencies alias into false low ones. Quantization then rounds each sample to one of 2N levels, so the resolution is the full-scale range divided by 2N.
Nyquist: fs > 2 fmaxresolution Q = FS / 2Nalias appears at |f − fs|Bringing a measurement into a computer discretises it two ways. Sampling discretises time: the signal is read at a fixed sample rate fs, turning a continuous curve into a list of numbers. The Nyquist sampling criterion is the rule that governs it: the sample rate must be more than twice the highest frequency in the signal, fs > 2 fmax. Sample too slowly and aliasing occurs, where a high-frequency component is indistinguishable from a lower one and appears in the data at a false frequency near |f − fs|, a distortion no later processing can undo, which is why an anti-alias low-pass filter is placed before the converter to remove anything above half the sample rate. Quantization discretises amplitude: an N-bit analog-to-digital converter divides its full-scale range FS into 2N levels, so the smallest resolvable step is Q = FS/2N, and each sample is rounded to the nearest level. More bits mean finer resolution; a faster sample rate means more faithful timing. A data-acquisition system chains a multiplexer, a sample-and-hold, and the converter to do this across channels, and choosing fs and N correctly is the difference between clean data and nonsense.
The skills, taught in order
Five skills bring a signal into the computer without corrupting it.
7.1 Sampling and the sample rate
Sampling reads the signal every 1/fs seconds. A higher rate captures faster detail but produces more data. The rate must be set from the signal's highest meaningful frequency, not guessed.
7.2 Nyquist and aliasing
The Nyquist criterion requires fs > 2 fmax. Violating it aliases high frequencies into false low ones, an irreversible error, so an anti-alias filter must remove content above half the sample rate before conversion.
7.3 Bits and quantization
An N-bit converter has 2N levels across its full-scale range, so its resolution is Q = FS/2N. Each added bit halves the step. Resolution is set jointly by the bit count and the range.
| Bits N | Levels 2N | Q at FS = 10 V |
|---|---|---|
| 8 | 256 | 39.1 mV |
| 12 | 4096 | 2.44 mV |
| 16 | 65536 | 0.153 mV |
Each extra bit halves the quantization step. Choosing bits is choosing amplitude resolution.
7.4 Data-acquisition components
A real system chains a multiplexer to share one converter across channels, a sample-and-hold to freeze each value during conversion, and the ADC itself. Understanding the chain explains its timing and throughput limits.
7.5 Practical acquisition
Record length and sample rate together set how long you capture and the frequency resolution of any spectrum, fs/N. Throughput, channels times per-channel rate, must not exceed the hardware's capacity.
Engineering connection: capturing a 2 kHz vibration cleanly means sampling well above 4 kHz, filtering above 2 kHz, and choosing enough bits to resolve the smallest amplitude of interest.
Worked example 1: the minimum sample rate
A signal contains frequencies up to 500 Hz. What is the minimum sample rate that avoids aliasing?
- ProblemFind the minimum sample rate for the signal in Figure 1.
- Given / findHighest frequency fmax = 500 Hz. Find the minimum fs.
- AssumptionsAll content above 500 Hz is removed by an anti-alias filter.
- ModelNyquist: fs > 2 fmax.
- Equationsfs > 2 × 500
- Solvefs > 1000 Hz (samples per second).
- CheckAt exactly 1000 Hz the 500 Hz component is only just captured; practice uses five to ten times fmax for a faithful shape.
- ConclusionSample above 1000 Hz, and filter out anything above 500 Hz first, to avoid aliasing.
Worked example 2: ADC resolution
A 12-bit analog-to-digital converter has a full-scale range of 10 V. Find its resolution.
- ProblemFind the resolution of the converter in Figure 2.
- Given / findN = 12 bits, FS = 10 V. Find Q.
- AssumptionsUniform quantization over the full-scale range.
- ModelQ = FS / 2N.
- Equations212 = 4096Q = 10 / 4096
- SolveQ = 2.44 mV.
- CheckEight bits gave 39.1 mV; adding four bits divides by 16, and 39.1/16 ≈ 2.44 mV, matching.
- ConclusionThe converter resolves about 2.44 mV, so any signal detail smaller than that is lost to quantization.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Sampling at exactly fmax | Signal poorly reconstructed | "Is fs above twice fmax?" | Sample above 2 fmax, ideally several times. |
| No anti-alias filter | False low-frequency content | "Is content above fs/2 removed?" | Filter before sampling; aliasing is irreversible. |
| Using N instead of 2N | Resolution wildly wrong | "Levels or bits?" | Levels are 2N. |
| Ignoring quantization | Trusting detail below Q | "Is this smaller than one step?" | Signal below Q is lost to rounding. |
Practice ladder
A signal has content up to 2 kHz. Find the minimum sample rate.
Show answer
fs > 2 × 2000 = 4000 Hz (4 kHz).
A 16-bit ADC has a 5 V full-scale range. Find its resolution.
Show answer
Q = 5 / 216 = 5 / 65536 = 76.3 µV.
A 1200 Hz signal is sampled at 1000 Hz. At what false frequency does it alias?
Show answer
The alias appears at |1200 − 1000| = 200 Hz, a false low-frequency component.
Specify a sample rate and bit count to capture a vibration up to 1 kHz with amplitude detail of about 1 mV over a 10 V range.
What good work looks like
Sample well above 2 kHz, say 5 to 10 kHz, with an anti-alias filter near 1 kHz. For 1 mV over 10 V you need Q ≤ 1 mV, so 2N ≥ 10000, giving N ≥ 14 bits. A good answer sets fs from Nyquist with margin and N from the resolution target.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
For one measurement, choose a sample rate and bit count and justify both from the signal's bandwidth and required resolution.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. State the Nyquist criterion.
Sample faster than twice the highest frequency present.
2. What is aliasing?
A too-slow sample rate making a high frequency appear as a false low one.
3. Write the ADC resolution.
Q = FS / 2N.
4. Why an anti-alias filter?
To remove content above half the sample rate before conversion.
5. Effect of one more bit?
It halves the quantization step.
Textbook mapping
This module follows Figliola and Beasley, Theory and Design for Mechanical Measurements, 5th edition. Use these references to read further.
| Topic in this module | Where to read more |
|---|---|
| Sampling and aliasing | Figliola and Beasley, Section 7.2, Sampling Concepts |
| Bits, words, and quantization | Figliola and Beasley, Section 7.3, Digital Devices: Bits and Words |
| Data-acquisition systems | Figliola and Beasley, Section 7.6, Data-Acquisition Systems |
Section numbers refer to Figliola and Beasley, 5th edition. Any edition with the same chapter titles is equivalent for study.