04 · The frequencies inside a signal

How can simple waves build a complex signal?

Where will a third harmonic create a spectral peak?

Select signal and spectrum.
Set the fundamental to 2 Hz.
Change the third-harmonic amplitude; compare the time trace and DFT.

Measurement

Sample rate f_s, N, bin spacing Δf=f_s/N, Nyquist limit f_s/2, and peak frequencies.

The DFT projects samples onto sinusoidal basis functions. The spectrum is calculated from the synthetic signal, not decoration. A one-second observation resolves bins 1 Hz apart. Noninteger cycles cause spectral leakage. Frequencies beyond half the sample rate can alias into lower ones; practical electronics use antialias filters.

f₀ in Hz; A₃ is third-harmonic amplitude; n=0…N−1.

s(t)=sin(2πf₀t)+A₃sin(6πf₀t); X_k=Σₙ sₙ exp(−i2πkn/N)

Worked example 1

For f₀=2 Hz and A₃=0.3, peaks occur at 2 and 6 Hz with amplitudes 1 and 0.3.

Worked example 2

For f_s=256 Hz and N=256, Δf=1 Hz and Nyquist is 128 Hz. A 150 Hz component aliases to 106 Hz.

At fixed f_s, how does doubling observation time change Δf?

Assumptions

Analytic linear educational model, not a numerical fluid solver. No damping, dispersion or nonlinear coupling.

6.003 Signals and Systems — Lecture Notes