- 2069 Asar · 5 marks
Fourier series vs Fourier transform
Short answer
Fourier series represents a periodic signal by a discrete line spectrum (coefficients c_k), while the Fourier transform gives a continuous spectral density X(ω) for aperiodic signals. The transform is the limit of the series as period tends to infinity, merging lines into a continuum.
Differentiate between Fourier series and Fourier transform with suitable examples.
The Fourier series represents a periodic signal as a sum of harmonically related sinusoids. The Fourier transform represents a non-periodic signal as a continuous integral of complex exponentials over all frequencies.
| Point | Fourier series | Fourier transform |
|---|---|---|
| Applies to | Periodic signals | Non-periodic (aperiodic) signals (periodic ones too, with impulses) |
| Spectrum | Discrete (lines at kω₀) | Continuous in ω |
| Formula | x(t) = Σ cₖ e^(jkω₀t), cₖ = (1/T₀)∫x e^(-jkω₀t) dt | X(ω) = ∫ x(t) e^(-jωt) dt |
| Output | Coefficients cₖ (amplitude) | Spectral density X(ω) |
| Signal type | Power signal | Energy signal (absolutely integrable) |
| Parseval | Σ |cₖ|² = average power | ∫|x|² dt = (1/2π)∫|X|² dω = energy |
| Example | Square wave of amplitude ±1: |cₖ| = 2/(kπ) for odd k, 0 for even k | Rectangular pulse of width τ: X(ω) = τ sin(ωτ/2)/(ωτ/2) |
The Fourier transform is the limit of the Fourier series as the period T₀ → ∞: the line spacing ω₀ → 0 and the lines merge into a continuous curve.
✓ Answer checked by Claude Opus 5.5 (high effort)
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