Read the mean, fundamental and harmonic terms
A periodic signal repeats after a period T. Its fundamental angular frequency is ω₀ = 2π/T. A real Fourier series is a₀/2 + Σ[aₙcos(nω₀t) + bₙsin(nω₀t)]. The index n identifies the harmonic. The average over one complete period is the constant term a₀/2, because the sine and cosine terms average to zero. Signal units remain the same in the coefficients.
Worked practice example
A periodic displacement in millimetres is x(t) = 4 + 3cos(2t) − 2sin(4t). Identify its Fourier coefficients, mean and period.
| Step | Working |
|---|---|
| Fundamental | The nonzero angular frequencies are 2 and 4 rad/s; the fundamental is 2 rad/s, so T = π s. |
| Constant term | a₀/2 = 4, hence a₀ = 8 mm. |
| First harmonic | a₁ = 3 mm and b₁ = 0. |
| Second harmonic | a₂ = 0 and b₂ = −2 mm; all higher coefficients vanish. |
Check your method
The presence of the 2 rad/s term makes π seconds the smallest positive period. A coefficient’s sign describes phase, not a negative amplitude magnitude. For this signal the second harmonic is twice the fundamental frequency, rather than twice its period.
Try it yourself
For f(t) = 5 + 2cos(6t), state the mean, fundamental period and a₀.
Show the answer and method
The mean is 5, the angular frequency is 6 rad/s, the period is 2π/6 = π/3 s and a₀ = 10. The first cosine coefficient is 2.
This written practice example was prepared for Jemshah revision. It is additional practice, not an official examination question or a transcript of the video.