A spectrum is a measurement

What does a finite observation hide?

What does a finite observation hide?

1. Set the fundamental to 3 Hz with harmonics disabled.
2. Change to 3.5 Hz and inspect neighbouring bars.
3. Add the third harmonic and locate its actual frequency.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

A discrete Fourier transform measures projection onto a finite set of sinusoidal basis functions. In this instrument 256 samples cover one second, giving 1 Hz bin spacing and a 128 Hz Nyquist limit. A frequency exactly aligned to a bin has a clean peak. A fractional-bin tone spreads over neighbouring bins because the finite rectangular window cuts it at unequal phases. More interpolation would make a smoother picture without increasing the underlying resolution.

Build the argument

1. Write a prediction before running the experiment.
2. Hold all but one variable constant, collect repeated measurements, and calculate dimensionless ratios.
3. Compare the observations to an analytic limit and document the remaining discrepancy.

Worked example 1

At 3 Hz the third harmonic occurs at 9 Hz.

Worked example 2

Doubling the observation window would halve bin spacing, at fixed sample rate.

Applications

A useful model makes a testable prediction and states where it fails. Reproducible settings and exported measurements turn a visual experiment into evidence.

Common misconception

More decimal places do not guarantee greater accuracy. Distinguish discretisation error, uncertainty in inputs, and limits of the physical model.

A further investigation

Smoothing the plotted curve increases physical resolution?

Check your understanding

Smoothing the plotted curve increases physical resolution?

• No
• Always
• Only in 3D

Model limits

Direct DFT, 256 Hz sampling, 256 samples, rectangular window; only the first 40 Hz are displayed.

6.003 Signals and Systems — Lecture Notes