Measurements · Module 2 of 10
Signal Characteristics
Before you can measure a signal well, you must describe it: its average, its size, and how fast it changes. This module covers the mean, the RMS, and the frequency content of a measured signal.
Readiness check
This module describes signals. Tick only what you can do closed-notes.
- Recall that a sine wave has an amplitude and a period.
- Divide one by a period to get frequency.
- Multiply a frequency by 2π.
- Divide a number by the square root of two.
- Recall that a mean is an average.
The core idea
A signal is described by its mean, its amplitude, and its frequency content. A periodic signal has a period T and a frequency f = 1/T, with angular frequency ω = 2πf. Its effective magnitude is the root-mean-square, and for a pure sinusoid the RMS is the amplitude divided by the square root of two.
frequency f = 1/Tangular frequency ω = 2πfRMS of a sinusoid = A / √2Every measured quantity arrives as a signal that varies, and describing it well is the first step to measuring it correctly. Signals divide into static, essentially constant over the measurement, and dynamic, changing in time; and into analog, continuous, and discrete, sampled. The simplest useful descriptors are the mean, the average or DC level, and the amplitude, the size of the variation about that mean. For a periodic signal the period T is the time of one cycle, the frequency f = 1/T counts cycles per second in hertz, and the angular frequency ω = 2πf expresses the same rate in radians per second, the form that appears in the sine function. The single most useful magnitude is the root-mean-square, or RMS, which captures the effective value and relates directly to signal power; for a pure sinusoid of amplitude A it is A divided by the square root of two, about 0.707 A. Finally, real signals are rarely one sine, so the Fourier idea decomposes them into a sum of sinusoids at different frequencies, the frequency spectrum, which tells you which frequencies your instrument must faithfully capture. This vocabulary drives the dynamic-response and sampling modules that follow.
The skills, taught in order
Five skills describe any measured signal.
2.1 Signal classes
Signals are static or dynamic, analog or discrete, and deterministic or random. Classifying a signal first decides how you sample and analyze it: a slow static reading and a fast vibration need very different treatment.
2.2 Mean, amplitude, RMS
The mean is the average or DC level; the amplitude is the size of the variation; the RMS is the effective magnitude that relates to power. Reporting all three characterises the signal far better than any one alone.
2.3 Period, frequency, angular frequency
The period T is one cycle's time, the frequency f = 1/T is cycles per second, and ω = 2πf is the angular frequency in rad/s. The three are the same information; the angular form is what the sine function uses.
| Descriptor | Formula | Tells you |
|---|---|---|
| Frequency | f = 1/T | how fast it cycles |
| Angular frequency | ω = 2πf | rate in rad/s |
| RMS (sinusoid) | A/√2 | effective magnitude |
The core descriptors of a periodic signal. RMS, not amplitude, is what most meters report.
2.4 RMS and power
The root-mean-square is defined as the square root of the average of the square, and it is the value that determines power. For a sinusoid it works out to A/√2, which is why an AC meter reading differs from the peak.
2.5 The frequency spectrum
The Fourier transform decomposes a signal into sinusoids at different frequencies, its spectrum. The highest important frequency in that spectrum sets how fast the instrument must respond and how fast you must sample.
Engineering connection: a vibration signal's spectrum reveals the machine's running frequency and its harmonics, telling you the bandwidth the accelerometer and sampler must cover.
Worked example 1: RMS of a sinusoid
A sinusoidal voltage has an amplitude of 10 V. Find its root-mean-square value.
- ProblemFind the RMS of the sinusoid in Figure 1.
- Given / findAmplitude A = 10 V. Find the RMS.
- AssumptionsA pure sinusoid with zero mean.
- ModelFor a sinusoid, RMS = A / √2.
- EquationsRMS = 10 / √2
- SolveRMS = 10 / 1.414 = 7.07 V.
- Check0.707 × 10 = 7.07 V, the standard sinusoid factor, so the result is consistent.
- ConclusionThe effective value is 7.07 V, which is what an RMS-reading meter would display, not the 10 V peak.
Worked example 2: period and angular frequency
A measured signal oscillates at 60 Hz. Find its period and its angular frequency.
- ProblemFind the period and angular frequency for the signal in Figure 2.
- Given / findFrequency f = 60 Hz. Find T and ω.
- AssumptionsA single-frequency periodic signal.
- ModelT = 1/f; ω = 2πf.
- EquationsT = 1/60ω = 2π(60)
- SolveT = 0.01667 s = 16.67 ms; ω = 377 rad/s.
- Checkω = 2π/T = 2π/0.01667 = 377 rad/s, matching the direct computation.
- ConclusionThe 60 Hz signal has a 16.67 ms period and a 377 rad/s angular frequency, the mains-power rate familiar from electrical noise.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Mean confused with RMS | Zero-mean AC reported as zero magnitude | "Average, or effective value?" | RMS captures magnitude even when the mean is zero. |
| RMS equals peak | Magnitude overstated by 41 percent | "Did I divide by √2?" | For a sinusoid, RMS = A/√2. |
| Dropping 2π | Angular frequency far too small | "Is this f or ω?" | ω = 2πf, in rad/s. |
| Period and frequency swapped | Milliseconds and hertz confused | "Cycles per second, or seconds per cycle?" | f = 1/T; they are reciprocals. |
Practice ladder
A sinusoid has an amplitude of 5 V. Find its RMS.
Show answer
RMS = 5 / √2 = 3.54 V.
A signal oscillates at 50 Hz. Find its period and angular frequency.
Show answer
T = 1/50 = 20 ms; ω = 2π(50) = 314.16 rad/s.
A signal is y = 2 + 4 sin(ωt) volts. Find its mean and the RMS of its oscillating part.
Show answer
Mean = 2 V (the DC term); RMS of the AC part = 4 / √2 = 2.83 V.
You record a vibration and see a dominant peak at 25 Hz in its spectrum. State the period, angular frequency, and the minimum instrument bandwidth you would want.
What good work looks like
T = 1/25 = 40 ms; ω = 2π(25) = 157 rad/s. To capture it and a few harmonics faithfully, the instrument bandwidth should reach several times 25 Hz, say 100 to 250 Hz. A good answer converts the rate three ways and reasons about capturing harmonics.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Take one real measured signal and report its mean, RMS, dominant frequency, period, and angular frequency.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. Write frequency from period.
f = 1/T.
2. Write angular frequency.
ω = 2πf.
3. RMS of a sinusoid?
Amplitude divided by the square root of two.
4. Mean versus RMS?
Mean is the average level; RMS is the effective magnitude tied to power.
5. What does a spectrum show?
Which frequencies make up a signal, from the Fourier decomposition.
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 |
|---|---|
| Signal analysis and descriptors | Figliola and Beasley, Section 2.3, Signal Analysis |
| Amplitude and frequency | Figliola and Beasley, Section 2.4, Signal Amplitude and Frequency |
| Fourier transform and spectrum | Figliola and Beasley, Section 2.5, Fourier Transform and the Frequency Spectrum |
Section numbers refer to Figliola and Beasley, 5th edition. Any edition with the same chapter titles is equivalent for study.