Fourier Series and Transform Methods

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Fourier series expresses a periodic function as a sum of sines and cosines. For period 2L:

f(x) = a₀/2 + Σ [aₙ cos(nπx/L) + bₙ sin(nπx/L)]
aₙ = (1/L)∫₀^{2L} f(x)cos(nπx/L)dx
bₙ = (1/L)∫₀^{2L} f(x)sin(nπx/L)dx

⚡ GATE quick wins: Even functions → only aₙ (no sin terms). Odd functions → only bₙ (no cos terms). Half-wave symmetry → only odd harmonics survive (n = 1, 3, 5…).

Half-range expansions: On (0, L), you can expand as sine series only (odd extension) or cosine series only (even extension). GATE gives you a domain (0, π) and asks “write the half-range cosine expansion.”

Fourier transforms:

• Fourier sine transform of f(x): F_s(s) = ∫₀^∞ f(x) sin(sx) dx
• Fourier cosine transform: F_c(s) = ∫₀^∞ f(x) cos(sx) dx

Parseval’s identity: ∫_{-∞}^{∞} |f(x)|² dx = (1/π)∫_{-∞}^{∞} |F(ω)|² dω. GATE uses this to compute total energy or mean square value.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Fourier Series — Full Derivation

Dirichlet Conditions

Any reasonably behaved periodic function can be expressed as a Fourier series. Sufficient conditions:

1. Single-valued, finite discontinuities (piecewise continuous)
2. Finite extrema (finite number of maxima/minima)
3. Absolutely integrable over one period

📌 GATE won’t ask you to verify these — they’re just the “when does this work?” guarantees.

Computing Coefficients

For a function with period 2L:

Coefficient	Formula	Used When
a₀	(1/L)∫₀^{2L} f(x)dx	DC / average value
aₙ	(1/L)∫₀^{2L} f(x)cos(nπx/L)dx	Even part
bₙ	(1/L)∫₀^{2L} f(x)sin(nπx/L)dx	Odd part

Common GATE setup: f(x) = x² on (-π, π) expanded as Fourier series. Compute aₙ via integration by parts twice. Watch for evenness — since x² is even, bₙ = 0.

⚡ Trap alert: Always check the function’s symmetry before integrating. Odd × odd = even (keep in integral), odd × even = odd (integral over symmetric interval = 0).

Half-Range Expansions

When f(x) is defined only on (0, L), extend it:

• Half-range sine series (odd extension to (-L, 0)): aₙ = 0, compute only bₙ
• Half-range cosine series (even extension): bₙ = 0, compute only aₙ

📌 GATE 2017: “Find the half-range cosine expansion of f(x) = x on (0, 2).” Here L = 2, so compute aₙ = (1/2)∫₀^4 x cos(nπx/2) dx. Result has only cosine terms.

Fourier Integral and Transforms

Fourier Integral Theorem

For non-periodic (aperiodic) functions defined on (-∞, ∞):

f(x) = (1/π)∫₀^∞ [A(ω)cos(ωx) + B(ω)sin(ωx)] dω
A(ω) = ∫_{-∞}^{∞} f(x)cos(ωx) dx
B(ω) = ∫_{-∞}^{∞} f(x)sin(ωx) dx

This is the limiting case of Fourier series as L → ∞.

Fourier Sine and Cosine Transforms

Fourier sine transform (for functions on (0, ∞)):

F_s(ω) = ∫₀^∞ f(x) sin(ωx) dx
f(x) = (2/π)∫₀^∞ F_s(ω) sin(ωx) dω

Fourier cosine transform:

F_c(ω) = ∫₀^∞ f(x) cos(ωx) dx
f(x) = (2/π)∫₀^∞ F_c(ω) cos(ωx) dω

⚡ GATE twist: They sometimes give you F_s(ω) or F_c(ω) and ask you to invert. Use the inverse formula above. Or ask for the transform of a known function (e.g., transform of e^{-ax} for a > 0).

Standard transforms to know:

f(x)	F_c(ω)	F_s(ω)
e^{-ax} (a>0)	a/(a²+ω²)	ω/(a²+ω²)
1 (0<x<a), 0 (x>a)	[sin(ωa)]/ω	[1-cos(ωa)]/ω
`e^{-a	x	}`

Parseval’s Identity and Energy

Parseval’s Theorem (Fourier Series)

(1/L)∫₀^{2L} [f(x)]² dx = (a₀/2)² + Σ_{n=1}^∞ (aₙ² + bₙ²)

GATE uses this to find the sum of an infinite series — set up the series, evaluate the integral, equate.

📌 GATE classic: Given Fourier coefficients of f(x), find Σ (1/n⁴). Use Parseval in conjunction with known series sum Σ 1/n⁴ = π⁴/90.

Parseval for Fourier Transforms

∫_{-∞}^{∞} [f(x)]² dx = (1/π)∫_{-∞}^{∞} |F(ω)|² dω

This is the Plancherel theorem — total energy is preserved. Used in signal processing contexts.

Even and Odd Functions — Key Rules

Property	Even (f(-x)=f(x))	Odd (f(-x)=-f(x))
Product even×even	Even	—
Product odd×odd	Even	—
Product even×odd	—	Odd
Integral symmetric	2∫₀^a	0
Fourier series	Only aₙ terms	Only bₙ terms

⚡ Common GATE trap: A function that is neither even nor odd can still be split: f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 → even part + odd part.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Convergence of Fourier Series

Pointwise Convergence

At a point of discontinuity x₀, the Fourier series converges to:

[f(x₀⁺) + f(x₀⁻)] / 2

i.e., the average of left and right limits. GATE has asked this exactly: “At a discontinuity, the Fourier series converges to ___.”

Gibbs Phenomenon

Near a discontinuity, the partial sum overshoots by ~9% — this is Gibbs phenomenon. It’s inherent to Fourier series; you can’t eliminate it by taking more terms.

Differentiation and Integration

• Term-by-term differentiation: Valid if the resulting series converges uniformly AND the original function is continuous with piecewise continuous derivative.
• Integration: Always valid term-by-term — no extra conditions.

⚡ GATE trap: Students often assume they can differentiate Fourier series freely. It’s NOT always valid. GATE asks this as a trick.

Harmonic Analysis and Orthogonality

The set {1, cos(nx), sin(nx)} for n ≥ 1 is orthogonal on (-π, π):

∫_{-π}^{π} sin(mx)cos(nx) dx = 0
∫_{-π}^{π} sin(mx)sin(nx) dx = π δ_mn
∫_{-π}^{π} cos(mx)cos(nx) dx = π δ_mn (m≠0), 2π (m=0)

This orthogonality is the reason we can find coefficients by projecting f(x) onto each basis function — the integrals give the component along each direction.

Fourier Transform Properties

Linearity and Shifting

F{af + bg} = aF{f} + bF{g}           (linearity)
F{f(x-a)} = e^{-iωa}F(ω)             (time shift)
F{f(x)cos(ω₀x)} = [F(ω-ω₀) + F(ω+ω₀)]/2

Fourier Transform of Derivative

F{df/dx} = iωF(ω)
F{d²f/dx²} = -(ω)²F(ω)

This is why ODEs become algebraic in the Fourier domain — the backbone of solving PDEs and signal processing.

Convolution Theorem

F{f * g} = F{f}·F{g}
F{f·g} = (1/2π)[F{f} * F{g}]

In the frequency domain, convolution becomes multiplication. GATE may ask you to compute the transform of a convolution.

Complex Form of Fourier Series

Using Euler’s formula e^{inx} = cos(nx) + i sin(nx):

f(x) = Σ_{n=-∞}^{∞} cₙ e^{inπx/L}
cₙ = (1/2L)∫_{-L}^{L} f(x)e^{-inπx/L} dx

This compact form unifies sines and cosines into complex exponentials. GATE occasionally asks complex Fourier series for functions defined on (-π, π) — cₙ = (1/2π)∫_{-π}^{π} f(x)e^{-inx} dx.

Previous Year GATE Patterns

Year	Topic Tested	Format
2022	Fourier series of f(x)=x(π-x)	Find aₙ, bₙ coefficients
2021	Half-range sine expansion	Expand on (0, π) as sine series
2020	Parseval’s identity	Sum infinite series using Parseval
2019	Fourier transform of e^{-a|x|}	Compute F_c(ω)
2018	Convergence at discontinuity	Value at jump point
2017	Fourier sine transform	Find F_s(ω) for given f(x)

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