- 2074 Chaitra · 6 marks
Properties of continuous-time Fourier series
Short answer
Linearity, time shift (c_k e^{-jkω₀t₀}), frequency shift (c_{k-M}), time reversal (c_{-k}), conjugation (c*_{-k}), multiplication ↔ discrete convolution, differentiation (jkω₀ c_k), integration (c_k/(jkω₀) if c₀=0, Parseval Σ|c_k|². Real signals have even magnitude and odd phase; real even signals have real even coefficients.
State the properties of continuous time Fourier series.
Let x(t) be periodic with period T₀ and ω₀ = 2π/T₀, with exponential Fourier coefficients cₖ (x(t) ↔ cₖ). Then:
| Property | Signal | Coefficients |
|---|---|---|
| Linearity | a·x(t) + b·y(t) | a·cₖ + b·dₖ |
| Time shift | x(t - t₀) | cₖ e^(-jkω₀t₀) |
| Frequency shift | e^(jMω₀t) x(t) | c_(k-M) |
| Time reversal | x(-t) | c_(-k) |
| Time scaling | x(αt) | same cₖ, fundamental frequency αω₀ |
| Conjugation | x*(t) | c*_(-k) |
| Multiplication | x(t) y(t) | Σ cₘ d_(k-m) (discrete convolution) |
| Periodic convolution | ∫ x(τ) y(t - τ) dτ over T₀ | T₀ cₖ dₖ |
| Differentiation | dx/dt | jkω₀ cₖ |
| Integration (if c₀ = 0) | ∫ x dt | cₖ / (jkω₀) |
| Parseval's theorem | (1/T₀) ∫|x|² dt | Σ |cₖ|² |
Symmetry properties:
- Real x(t): c_(-k) = cₖ*, so |cₖ| is even and the phase is odd.
- Real and even x(t): cₖ is real and even (only cosine terms).
- Real and odd x(t): cₖ is purely imaginary and odd (only sine terms).
- Half-wave symmetric x(t): only odd harmonics exist.
✓ Answer checked by Claude Opus 5.5 (high effort)
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