Fourier Series —
Periodic Signal Representation
1. Introduction to Fourier Series
Any periodic signal satisfying Dirichlet conditions can be decomposed into a sum of sinusoids (harmonics) — this is the Fourier Series representation. The idea, due to Jean-Baptiste Joseph Fourier (1822), is fundamental to all signal processing.
A periodic signal with fundamental period T₀ has fundamental frequency f₀ = 1/T₀ (Hz) and fundamental angular frequency ω₀ = 2π/T₀ = 2πf₀ (rad/s). The Fourier series expresses x(t) as a sum of harmonics at frequencies 0, f₀, 2f₀, 3f₀, ... (the DC, fundamental, 2nd harmonic, 3rd harmonic, ...).
Square wave with amplitude A: only odd harmonics exist. Even harmonics are zero due to half-wave symmetry. Spectrum is discrete (line spectrum) — characteristic of all periodic signals.
2. Dirichlet Conditions
A periodic signal x(t) has a Fourier series representation if and only if it satisfies the Dirichlet conditions:
At a discontinuity, the partial sum of Fourier series shows an overshoot of ~9% (≈ 1.09 times the jump) that does not decrease as more terms are added — only the location of the overshoot narrows. This is the Gibbs phenomenon.
3. Trigonometric Fourier Series
Alternative compact trigonometric form using amplitude and phase:
Worked Example: Square Wave
4. Exponential Fourier Series
For any real signal x(t):
• |C₋ₙ| = |Cₙ| → Magnitude spectrum is even
• ∠C₋ₙ = −∠Cₙ → Phase spectrum is odd
• C₋ₙ = Cₙ* (complex conjugate)
This means the two-sided spectrum has mirror symmetry — you only need n ≥ 0 to fully describe a real signal.
Worked Example: Full-Wave Rectified Sine
5. Fourier Spectrum & Power Spectrum
The Fourier spectrum is the plot of Fourier coefficients vs frequency — it is a discrete (line) spectrum for periodic signals.
| Spectrum | Plot | Symmetry for real x(t) |
|---|---|---|
| Amplitude Spectrum | |Cₙ| vs nω₀ | Even: |C₋ₙ| = |Cₙ| |
| Phase Spectrum | ∠Cₙ vs nω₀ | Odd: ∠C₋ₙ = −∠Cₙ |
| Power Spectrum | |Cₙ|² vs nω₀ | Even: always |
6. Symmetry Conditions
Symmetry conditions reduce integration work by telling which Fourier coefficients are zero before computing them:
| Symmetry | Condition | Effect on FS |
|---|---|---|
| Even Symmetry | x(−t) = x(t) | bₙ = 0 (no sine terms); aₙ = (4/T₀)∫₀^(T₀/2) x(t)cos(nω₀t)dt |
| Odd Symmetry | x(−t) = −x(t) | a₀ = aₙ = 0 (no cosine terms); bₙ = (4/T₀)∫₀^(T₀/2) x(t)sin(nω₀t)dt |
| Half-Wave Symmetry | x(t) = −x(t±T₀/2) | a₀ = 0; aₙ = bₙ = 0 for n even (only odd harmonics present) |
| Quarter-Wave Symmetry (even) | Even + Half-wave | Only aₙ for odd n; bₙ = 0 |
| Quarter-Wave Symmetry (odd) | Odd + Half-wave | Only bₙ for odd n; aₙ = 0 |
• Square wave (odd symmetry + half-wave) → only odd sine harmonics
• Triangular wave (even symmetry + half-wave) → only odd cosine harmonics
• Full-wave rectified sine (even symmetry) → only cosine terms, all harmonics
• Sawtooth wave (odd symmetry, no half-wave) → all sine harmonics
7. Properties of Fourier Series
| Property | Signal | FS Coefficients |
|---|---|---|
| Linearity | ax(t) + by(t) | a·Cₙˣ + b·Cₙʸ |
| Time Shifting | x(t − t₀) | Cₙ · e^(−jnω₀t₀) (magnitude unchanged, phase shifted) |
| Time Reversal | x(−t) | C₋ₙ = Cₙ* |
| Time Scaling | x(at), period T₀/a | Same Cₙ but ω₀ → aω₀ |
| Conjugation | x*(t) | C₋ₙ* |
| Multiplication | x(t)·y(t) | Convolution: Σₖ Cₖˣ · C(n−k)ʸ |
| Convolution | x(t)*y(t) (periodic) | T₀ · Cₙˣ · Cₙʸ |
| Differentiation | dx/dt | jnω₀ · Cₙ |
| Integration | ∫x(τ)dτ | Cₙ/(jnω₀) for n≠0; C₀·t term if n=0 |
8. Parseval's Theorem for Fourier Series
Worked Example — Power of Square Wave
Σₙ₌₁,₃,₅,... (1/n²) = 1 + 1/9 + 1/25 + ... = π²/8
Σₙ₌₁^∞ (1/n²) = π²/6
These identities are frequently needed to verify Parseval's theorem results.
GATE Questions — Unit 2: Fourier Series
Answer: (A)
For even signals x(−t) = x(t): bₙ = (2/T₀)∫x(t)sin(nω₀t)dt = 0 because x(t) is even and sin(nω₀t) is odd — their product is odd, and the integral of an odd function over a symmetric interval is zero. Only cosine terms (and DC) remain.
Answer: (B)
Note: magnitude unchanged |Dₙ| = |Cₙ|; only phase shifts by −nπ/2 per harmonic.
Answer: (A) 16.5 W
Answer: (B)
|cos(πt)| is an even function (even symmetry → no sine terms, bₙ = 0). Its period T₀ = 1 (half the period of cos(πt)), so ω₀ = 2π. Fundamental frequency doubles. The FS contains DC (a₀ = 2/π) and cosine terms at multiples of 2π — these are the even harmonics of the original signal. Hence: DC and even cosine harmonics.
Answer: (B)
Half-wave symmetry: x(t + T₀/2) = −x(t). This implies a₀ = 0 (zero average) and aₙ = bₙ = 0 for all even n. Only odd harmonics (n = 1, 3, 5, ...) appear in the Fourier series.
Answer: (A)
Check: C₋₁ = C₁* = 1 + j1.5. Magnitude |C₁| = √(1² + 1.5²) = √3.25.
Answer: (A)
This means differentiation emphasizes high frequencies (large n) — it acts as a high-pass operation.
Answer: (A) π²/8
Answer: (A)
Each non-zero coefficient Cₙˣ produces contributions at n−1 and n+1. So all original harmonics shift by ±1.
Answer: (C)
This follows directly from the Dirichlet conditions. At a jump discontinuity, the Fourier series converges to the arithmetic mean of the left-hand and right-hand limits. For example, the square wave at t = 0 (where it jumps from −A to A) has FS value = (A + (−A))/2 = 0.