Fourier Transform —
Aperiodic Signal Analysis
- 1. From Fourier Series to Fourier Transform
- 2. Fourier Transform — Definition & Inverse
- 3. Standard Fourier Transform Pairs
- 4. Properties of Fourier Transform
- 5. Convolution Theorem & LTI Systems
- 6. Energy Spectral Density (ESD)
- 7. Parseval's Theorem for FT
- 8. LTI System Frequency Response H(jω)
- 9. GATE Questions (12)
For an aperiodic signal, the spectrum is continuous (not a line spectrum like periodic signals). The FT maps time-domain signal ↔ frequency-domain spectrum. For real-valued x(t): amplitude spectrum |X(jω)| is even, phase spectrum ∠X(jω) is odd.
1. From Fourier Series to Fourier Transform
The Fourier Series represents periodic signals as a discrete sum of harmonics. To handle aperiodic signals, we let the period T₀ → ∞ — the discrete harmonics merge into a continuum, and the sum becomes an integral. This limiting process yields the Fourier Transform (FT).
Fourier Series: Periodic signals → discrete line spectrum (Cₙ at nω₀)
Fourier Transform: Aperiodic signals → continuous spectrum X(jω)
The FT can also represent periodic signals using impulse functions δ(ω).
2. Fourier Transform — Definition & Inverse
Existence Conditions for FT
The FT of x(t) exists if any of these is satisfied:
3. Standard Fourier Transform Pairs
These pairs must be memorized for GATE — especially the exponential, sinc, rect, Gaussian, and signum functions.
| x(t) | X(jω) | Notes / Conditions |
|---|---|---|
| δ(t) | 1 | Impulse → flat spectrum (all frequencies equally) |
| 1 (DC) | 2πδ(ω) | Constant → impulse at DC |
| u(t) (unit step) | πδ(ω) + 1/(jω) | Non-absolutely integrable, distributional FT |
| e^(−at)u(t) | 1/(a + jω) | a > 0; |X| = 1/√(a²+ω²) |
| e^(−a|t|) | 2a/(a² + ω²) | a > 0; even, real, Lorentzian spectrum |
| te^(−at)u(t) | 1/(a + jω)² | a > 0 |
| e^(jω₀t) | 2πδ(ω − ω₀) | Complex exponential → impulse at ω₀ |
| cos(ω₀t) | π[δ(ω−ω₀) + δ(ω+ω₀)] | Two impulses at ±ω₀ |
| sin(ω₀t) | (π/j)[δ(ω−ω₀) − δ(ω+ω₀)] | Two impulses, imaginary |
| rect(t/τ) | τ·sinc(ωτ/2) = τ·sin(ωτ/2)/(ωτ/2) | rect pulse → sinc spectrum |
| sinc(Wt) = sin(Wt)/(Wt) | (π/W)·rect(ω/2W) | sinc in time → rect in freq (by duality) |
| sgn(t) = ±1 | 2/(jω) | Signum function |
| e^(−t²/2σ²) | σ√(2π)·e^(−σ²ω²/2) | Gaussian → Gaussian (self-dual) |
| δ(t − t₀) | e^(−jωt₀) | Time shift → phase shift in frequency |
Time-Bandwidth product: Narrower pulse (small τ) → wider spectrum; wider pulse (large τ) → narrower spectrum. This is the time-frequency uncertainty principle.
4. Properties of Fourier Transform
All 10 properties below are GATE-critical. Master these to solve FT problems without computing integrals from scratch.
| Property | Time Domain | Frequency Domain |
|---|---|---|
| Linearity | ax(t) + by(t) | aX(jω) + bY(jω) |
| Time Shifting | x(t − t₀) | e^(−jωt₀) · X(jω) [magnitude unchanged, phase changes] |
| Frequency Shifting (Modulation) | x(t) · e^(jω₀t) | X(j(ω − ω₀)) [spectrum shifted by ω₀] |
| Time Scaling | x(at), a ≠ 0 | (1/|a|) X(jω/a) [compress time → expand freq] |
| Time Reversal | x(−t) | X(−jω) = X*(jω) [for real x(t)] |
| Duality | X(jt) [replace ω by t] | 2π x(−ω) [key: swap time and freq] |
| Differentiation (time) | d^n x(t)/dt^n | (jω)^n X(jω) [nth derivative → multiply by (jω)ⁿ] |
| Integration | ∫₋∞^t x(τ)dτ | X(jω)/(jω) + π X(0)δ(ω) [divides by jω + DC term] |
| Convolution in time | x(t) * h(t) | X(jω) · H(jω) [convolution → multiplication] |
| Multiplication (modulation) | x(t) · g(t) | (1/2π) X(jω) * G(jω) [multiplication → convolution/2π] |
Time delay x(t−t₀) adds phase −ωt₀ to X(jω): the magnitude |X(jω)| is unchanged, only the phase angle changes by −ωt₀. Many GATE questions check if you know the magnitude spectrum is invariant under time shifts.
Known: rect(t/τ) ↔ τ·sinc(ωτ/2)
Apply duality (X(jt) ↔ 2πx(−ω)):
τ·sinc(tτ/2) ↔ 2π·rect(−ω/τ) = 2π·rect(ω/τ) [since rect is even]
→ sinc(Wt) ↔ (π/W)·rect(ω/2W) where W = τ/2
4.1 Symmetry Properties (for real x(t))
5. Convolution Theorem & LTI Systems
The convolution theorem is the most important result in signal processing — it converts a time-domain convolution (complex integral) to simple multiplication in the frequency domain.
The convolution theorem is the power of the Fourier Transform. Instead of computing ∫x(τ)h(t−τ)dτ, simply multiply X(jω)·H(jω) and take the inverse FT.
6. Energy Spectral Density (ESD)
For energy signals (finite energy), the ESD describes how energy is distributed across frequencies.
ESD of e^(−at)u(t)
7. Parseval's Theorem for FT
∫₀^∞ [sin(ωt)/(ωt)]² dt — use Parseval's theorem. x(t) = sinc(t/π) whose FT is π·rect(ω/2). Then E = (1/2π)∫|π·rect|² dω = (1/2π)·π²·(2ω_c) where ω_c=1, giving E = π/2. Parseval's is consistently tested as a shortcut to avoid direct integration.
8. LTI System Frequency Response H(jω)
Ideal Filters
| Filter | H(jω) = 1 for | H(jω) = 0 for | h(t) |
|---|---|---|---|
| Ideal LPF (cutoff ωc) | |ω| ≤ ωc | |ω| > ωc | ωc/π · sinc(ωct/π) — non-causal, unrealizable |
| Ideal HPF | |ω| > ωc | |ω| ≤ ωc | δ(t) − LPF impulse response |
| Ideal BPF (ω₁ to ω₂) | ω₁ ≤ |ω| ≤ ω₂ | Otherwise | LPF(ω₂) − LPF(ω₁) impulse |
| Ideal BSF (band-stop) | |ω| < ω₁ or |ω| > ω₂ | ω₁ ≤ |ω| ≤ ω₂ | δ(t) − BPF impulse response |
Ideal LPF: h(t) = (ωc/π)sinc(ωct/π) — non-zero for t < 0, hence non-causal. Real filters (Butterworth, Chebyshev) approximate this with causal, realizable transfer functions.
GATE Questions — Fourier Transform (12)
x(t) = e^(−2|t|) is a two-sided decaying exponential.
Using the FT pair: e^(−a|t|) ↔ 2a/(a² + ω²) with a = 2:
Answer: X(jω) = 4/(4 + ω²)
Time shifting property: x(t − t₀) ↔ e^(−jωt₀) X(jω)
With t₀ = 3: FT{x(t−3)} = e^(−j3ω) X(jω)
Note: magnitude spectrum unchanged, phase gets −3ω added.
Answer: (B) X(jω) e^(−j3ω)
Time scaling property: x(at) ↔ (1/|a|)·X(jω/a)
With a = 2: FT{x(2t)} = (1/2)·X(jω/2)
Interpretation: compressing time (a=2 > 1) → stretching frequency by 2, amplitude scaled by 1/2.
Answer: (B) (1/2)·X(jω/2)
Method 1 (time domain): E = ∫₀^∞ |e^(−3t)|² dt = ∫₀^∞ e^(−6t) dt = 1/6
Method 2 (Parseval's): X(jω) = 1/(3+jω), |X|² = 1/(9+ω²)
Answer: (A) 1/6
rect(t) centered at 0: FT = sinc(ω/2π) [using normalized sinc]
Time shift by +0.5 → multiply by e^(−jω·0.5) = e^(−jω/2)
Magnitude |X(jω)| = |sinc(ω/2π)| [unchanged by shift], phase gains −ω/2.
Answer: (B) e^(−jω/2)·sinc(ω/2π)
Duality: If x(t) ↔ X(jω), then X(jt) ↔ 2πx(−ω)
Known pair: rect(t) ↔ sinc(ω/2) [derived from rect↔sinc definition with proper normalization]
By duality from sinc(t) ↔ rect(ω/2): applying duality to this pair gives rect(t) ↔ 2π·sinc(−ω/2)/2π = sinc(ω/2) [since sinc is even]
Answer: (B) sinc(ω/2)
X(jω) = 1/(3 + jω) = 1/|3 + jω| · e^(−j∠(3+jω))
Answer: (A) 1/√(9 + ω²)
By convolution theorem: Y(jω) = X(jω)·H(jω)
Note: y(t) = e^(−t)u(t) * e^(−2t)u(t) = (e^(−t) − e^(−2t))u(t) [by partial fractions in time]
Answer: (A) 1/[(1+jω)(2+jω)]
Known FT pair: e^(jω₀t) ↔ 2πδ(ω − ω₀)
With ω₀ = 5: X(jω) = 2πδ(ω − 5) → x(t) = e^(j5t)
Intuition: a complex exponential at frequency ω₀ has all its energy at exactly ω₀.
Answer: (B) e^(j5t)
x(t) = e^(−|t|) → X(jω) = 2/(1 + ω²) [standard pair with a=1]
Differentiation property: FT{dx/dt} = jω·X(jω)
Note: x(t) = e^(−|t|) has a kink at t=0, derivative is a sgn(t) correction: x'(t) = −sgn(t)e^(−|t|). The FT result = 2jω/(1+ω²). Answer choice B shows −2jω/(1+ω²) — check sign conventions used in the original exam. Most editions give 2jω/(1+ω²).
Answer: FT{x'(t)} = 2jω/(1+ω²) = jω·X(jω) [differentiation property]
For real x(t): Hermitian symmetry holds → X(−jω) = X*(jω)
This means: |X(−jω)| = |X*(jω)| = |X(jω)| [magnitude is EVEN]
And: ∠X(−jω) = ∠X*(jω) = −∠X(jω) [phase is ODD]
Answer: (C)
sinc(t) = sin(πt)/(πt) has FT: X(jω) = rect(ω/2π) [width 2π, height 1, using normalized sinc convention]
Equivalently: sinc(t) ↔ rect(f) in Hz, and ∫|sinc(t)|² dt = ∫|rect(f)|² df
rect(f)=1 for |f| ≤ 1/2 and 0 elsewhere, so the integral is just 1 × (width 1) = 1.
Answer: (B) 1