Exercise set

Sampling, Aliasing and Signal Measurement Exercises

Solved signal measurement exercises for sampling rate, aliasing, FFT resolution, quantization, SNR, ENBW, filters and release checks.

Branch: Engineering Physics
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.1 · reviewed

These exercises practise the measurement side of sampled signals. The goal is to decide whether a measurement system captures the physical signal, rejects unwanted content, preserves timing, and reports noise and frequency content honestly enough for engineering use.

Assume idealized sampling and simple filters unless an exercise states otherwise. Real signal measurement should also check sensor bandwidth, analog front end limits, anti-alias filtering, trigger settings, clock accuracy, windowing, data acquisition range, grounding, shielding, calibration, software scaling and uncertainty.

How to Use These Exercises

For each problem, state the signal band, sample rate, acquisition time, analog filter, quantization range and decision target. Sampling fast enough is necessary, but not sufficient; the analog signal must be conditioned before the sampler.

Release Evidence Notes

A sampled signal is credible only if the analog path and the digital record support the same measurement claim. The record should identify sensor bandwidth, mounting condition, cable path, analog gain, ADC range, anti-alias filter, sample clock, trigger, record length, windowing, scaling, units and software processing. Without those details, a waveform may look clean while still being aliased, clipped, filtered too heavily or reported in the wrong units.

Aliasing is a release blocker because it moves real high-frequency content into a false lower-frequency location. Increasing FFT length after acquisition cannot remove a tone that already folded into the sampled band. The prevention must happen before sampling: adequate sample rate, analog low-pass filtering, known sensor bandwidth and a documented out-of-band environment.

Noise decisions should distinguish sensor noise, amplifier noise, quantization noise, environmental interference and processing bandwidth. A high SNR in a narrow band does not prove that transient peaks, timing, phase or broadband vibration are valid. If the measurement supports protection, diagnosis or acceptance, the release record should include a plausibility check against an independent method or known physical limit.

For troubleshooting, preserve raw data when possible. Filtered, averaged or resampled data can be useful for reporting, but it can also hide clipping, dropouts, clock jitter, intermittent interference and short pulses. A release package should say which data stream is raw, which is processed and which one supports the decision.

Engineering Boundary Notes

These exercises use ideal sampling, simple filters and compact noise models. They do not replace analog front-end design, sensor calibration, electromagnetic compatibility testing, clock-jitter analysis, anti-alias filter validation or domain-specific signal qualification. A numerically adequate sample rate can still fail if the sensor, cable, amplifier or processing chain corrupts the signal before analysis.

Common Release Mistakes

treating the Nyquist minimum as a complete acquisition specification;
increasing FFT length after acquisition to hide poor analog filtering;
quoting SNR without bandwidth, window, averaging and scaling details;
ignoring clipping, ADC range, sensor bandwidth and anti-alias transition band;
validating processed data while discarding raw records needed for diagnosis;
using a clean waveform from one operating mode to approve all speeds, loads or environments.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Sampling and aliasing	1, 2, 3, 4, 5	Nyquist rate, alias frequency and anti-alias margin	Decide whether the acquired signal can represent the physical signal.
Record and spectral resolution	6, 7, 8, 9	record length, frequency bin width, time step and window loss	Decide whether events and spectral peaks can be resolved.
Quantization and noise	10, 11, 12, 13, 14	ADC count, quantization noise, SNR, ENOB and ENBW	Decide whether the signal is above the measurement floor.
Release checks	15, 16, 17, 18	filter delay, RMS conversion, alias rejection and final gate	Accept, retest or redesign the measurement chain.

Validation Package Checklist

signal band, out-of-band environment and decision target are stated;
sensor bandwidth, mounting and analog front-end limits are documented;
anti-alias filter cutoff, slope and transition margin match the sample rate;
ADC range, resolution, clock, trigger and record length are recorded;
FFT window, bin width, scaling and bandwidth basis are explicit;
raw and processed data streams are preserved or distinguished;
final release action names whether to accept, retest, filter redesign or change sample rate.

Exercise 1: Nyquist Minimum Sampling Rate

A vibration measurement must capture content up to $800\ \text{Hz}$ . Find the Nyquist minimum sample rate.

Solution

Nyquist requires:

f_s\ge 2f_{max}

f_s\ge 2(800)=1600\ \text{Hz}

Engineering Comment

The Nyquist minimum is not a good engineering sample rate by itself. Practical systems need filter transition band, timing margin and diagnostic headroom.

Plausibility Check

The sample rate must be at least twice the highest frequency, so $1600\ \text{Hz}$ is expected.

Exercise 2: Sampling Margin

A data acquisition system samples at $5000\ \text{Hz}$ and the highest required signal frequency is $1000\ \text{Hz}$ . Find samples per cycle.

Solution

\displaystyle N=\frac{f_s}{f}= \frac{5000}{1000}=5

Engineering Comment

Five samples per cycle may be adequate for frequency detection but may be weak for waveform shape, peak capture or phase analysis.

Plausibility Check

At $1000\ \text{Hz}$ , each cycle lasts $1\ \text{ms}$ ; at $5000\ \text{Hz}$ , samples are $0.2\ \text{ms}$ apart, giving five samples.

Exercise 3: Alias Frequency

A $1300\ \text{Hz}$ tone is sampled at:

f_s=1000\ \text{Hz}

Find the alias frequency in the first Nyquist band.

Solution

Subtract one sample-rate multiple:

f_a=|1300-1000|=300\ \text{Hz}

Engineering Comment

Aliasing creates a plausible but false low-frequency component. Software cannot reliably repair aliasing after acquisition.

Plausibility Check

The signal is $300\ \text{Hz}$ above the sample rate, so it folds to $300\ \text{Hz}$ .

Exercise 4: Anti-Alias Filter Cutoff

A measurement needs valid data to $400\ \text{Hz}$ and samples at $2000\ \text{Hz}$ . The Nyquist frequency is:

\displaystyle f_N=\frac{f_s}{2}

Find $f_N$ and comment on a $700\ \text{Hz}$ low-pass cutoff.

Solution

\displaystyle f_N=\frac{2000}{2}=1000\ \text{Hz}

A $700\ \text{Hz}$ cutoff is below Nyquist but above the required $400\ \text{Hz}$ band, leaving a transition region.

Engineering Comment

The cutoff alone is not enough. Attenuation at and above Nyquist must be sufficient for the sensor environment.

Plausibility Check

The useful band, cutoff and Nyquist are ordered as $400<700<1000\ \text{Hz}$ , which is sensible.

Exercise 5: Alias Rejection Requirement

An out-of-band tone could be $2.0\ \text{V}$ before filtering. Its allowed aliased amplitude is $0.02\ \text{V}$ . Find required attenuation ratio and dB.

Solution

Amplitude ratio:

\displaystyle A=\frac{0.02}{2.0}=0.01

In dB:

20\log_{10}(0.01)=-40\ \text{dB}

Engineering Comment

Anti-alias filtering should be specified from allowed error, not from a generic filter order.

Plausibility Check

Reducing amplitude by a factor of $100$ corresponds to $40\ \text{dB}$ .

Exercise 6: Frequency Resolution from Record Length

A spectrum is computed from a $4.0\ \text{s}$ record. Find frequency bin spacing.

Solution

\displaystyle \Delta f=\frac{1}{T}=\frac{1}{4.0}=0.25\ \text{Hz}

Engineering Comment

Longer records improve frequency resolution but may hide non-stationary behavior if the signal changes during the record.

Plausibility Check

A multi-second record should resolve sub-hertz spacing, so $0.25\ \text{Hz}$ is credible.

Exercise 7: Samples in a Record

A signal is sampled at $2400\ \text{Hz}$ for $2.5\ \text{s}$ . Find number of samples.

Solution

N=f_sT=2400(2.5)=6000

Engineering Comment

The number of samples controls storage, FFT length and statistical averaging. It does not guarantee signal quality.

Plausibility Check

Thousands of samples are expected for seconds of data at kilohertz rate.

Exercise 8: Time Step

A system samples at $20\ \text{kHz}$ . Find time between samples.

Solution

\displaystyle \Delta t=\frac{1}{f_s}=\frac{1}{20000}=50\ \mu\text{s}

Engineering Comment

Timing resolution matters for transient capture, time delay estimation and phase calculations.

Plausibility Check

$20{,}000$ samples per second means each sample is one twentieth of a millisecond apart.

Exercise 9: Window Amplitude Correction

A window reduces coherent amplitude by factor $0.50$ . A spectral peak reads $0.80\ \text{V}$ . Estimate corrected amplitude.

Solution

\displaystyle A_{corr}=\frac{0.80}{0.50}=1.60\ \text{V}

Engineering Comment

Windowing can reduce leakage but requires amplitude correction when absolute amplitude matters.

Plausibility Check

If the window halves coherent amplitude, the corrected value should double.

Exercise 10: ADC Count Size

A $12$ -bit ADC measures a $0$ to $10\ \text{V}$ range. Find one count.

Solution

Number of levels:

2^{12}=4096

Count size:

\displaystyle q=\frac{10}{4096}=0.00244\ \text{V}

Engineering Comment

ADC resolution at the converter does not include input amplifier noise, offset, gain error or anti-alias filter effects.

Plausibility Check

A 12-bit converter over $10\ \text{V}$ should have millivolt-level counts.

Exercise 11: Quantization RMS Noise

Using count size $q=0.00244\ \text{V}$ , estimate RMS quantization noise:

\displaystyle u_q=\frac{q}{\sqrt{12}}

Solution

\displaystyle u_q=\frac{0.00244}{\sqrt{12}}=0.000704\ \text{V}

Engineering Comment

Quantization noise is only one noise term. If analog noise is larger, increasing ADC bits may not improve the measurement.

Plausibility Check

The RMS quantization noise is smaller than one count, which is expected.

Exercise 12: Signal-to-Noise Ratio

A sensor signal RMS is $1.2\ \text{V}$ and noise RMS is $0.006\ \text{V}$ . Find SNR in dB.

Solution

\displaystyle SNR=20\log_{10}\left(\frac{1.2}{0.006}\right)

SNR=20\log_{10}(200)=46.0\ \text{dB}

Engineering Comment

High SNR supports amplitude measurement, but timing, bandwidth and calibration can still limit the result.

Plausibility Check

A $200:1$ amplitude ratio is a little above $40\ \text{dB}$ , so $46\ \text{dB}$ is plausible.

Exercise 13: ENOB from SNR

Estimate effective number of bits from:

SNR=62\ \text{dB}

using:

\displaystyle ENOB=\frac{SNR-1.76}{6.02}

Solution

\displaystyle ENOB=\frac{62-1.76}{6.02}=10.0

Engineering Comment

ENOB is a system performance measure. It can be much lower than nominal ADC resolution.

Plausibility Check

About $6\ \text{dB}$ per bit means $60\ \text{dB}$ is roughly ten bits.

Exercise 14: Equivalent Noise Bandwidth

A white noise density is:

e_n=20\ \text{nV}/\sqrt{\text{Hz}}

and equivalent noise bandwidth is $10{,}000\ \text{Hz}$ . Estimate RMS noise.

Solution

e_{rms}=e_n\sqrt{ENBW}

e_{rms}=20\sqrt{10000}=2000\ \text{nV}=2.0\ \mu\text{V}

Engineering Comment

Noise increases with square root of bandwidth. Filtering can improve noise if it does not remove required signal content.

Plausibility Check

The square root of $10{,}000$ is $100$ , so the density multiplies by $100$ .

Exercise 15: First-Order Filter Delay Screen

A first-order low-pass filter has cutoff:

f_c=20\ \text{Hz}

Estimate time constant:

\displaystyle \tau=\frac{1}{2\pi f_c}

Solution

\displaystyle \tau=\frac{1}{2\pi(20)}=0.00796\ \text{s}

So:

\tau=8.0\ \text{ms}

Engineering Comment

Filtering reduces noise but adds lag. That lag may matter for transient timing, control loops or protection evidence.

Plausibility Check

A cutoff of tens of hertz should have a time constant of milliseconds.

Exercise 16: RMS to Peak for a Sine

A vibration channel reports:

v_{rms}=3.0\ \text{mm/s}

Assuming a sine wave, find peak value.

Solution

v_{peak}=\sqrt{2}v_{rms}=1.414(3.0)=4.24\ \text{mm/s}

Engineering Comment

RMS, peak and peak-to-peak values are not interchangeable. The waveform assumption must be stated.

Plausibility Check

For a sine wave, peak should be about $41\%$ higher than RMS.

Exercise 17: Missed Pulse Screen

A pulse width is $0.30\ \text{ms}$ . A system samples every $0.50\ \text{ms}$ . Is one sample guaranteed inside every pulse?

Solution

The sample interval is longer than the pulse width:

0.50>0.30\ \text{ms}

One sample is not guaranteed inside every pulse.

Engineering Comment

Pulse detection often needs hardware capture, faster sampling, pulse stretching or synchronized triggering. Average sample rate is not enough.

Plausibility Check

A pulse can occur entirely between two samples if it is shorter than the sample interval.

Exercise 18: Signal Measurement Release Gate

A vibration measurement has these checks:

Check	Result	Gate
Highest required frequency	$800\ \text{Hz}$	captured
Sample rate	$2000\ \text{Hz}$	$\ge 5f_{max}$
Anti-alias attenuation	$28\ \text{dB}$	$\ge 40\ \text{dB}$
SNR	$42\ \text{dB}$	$\ge 40\ \text{dB}$
Record length	$1.0\ \text{s}$	$\ge 2.0\ \text{s}$

Decide whether to release the measurement setup.

Solution

Sample-rate gate:

2000<5(800)=4000\ \text{Hz}

fails.

Anti-alias attenuation fails:

28<40\ \text{dB}

SNR passes, but record length fails:

1.0<2.0\ \text{s}

The setup should not be released.

Engineering Comment

The SNR is acceptable, but aliasing and spectral resolution are not. Clean-looking data can still be invalid.

Plausibility Check

Three gates fail, including anti-aliasing, so rejection is consistent.

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