- Asked 2/20
- 2076 Asoj · 4 marks
- 2073 Chaitra · 6 marks
Properties of the Fourier transform
Short answer
Key properties include linearity, time shift (e^{-jωt0}X(ω)), frequency shift (X(ω-ω0)), scaling (1/|a| X(ω/a)), convolution ↔ multiplication, differentiation ↔ jωX(ω), and Parseval's theorem. For real signals, magnitude is even and phase odd; a real even signal has real even spectrum.
What are the properties of the Fourier transform?
Let x(t) ↔ X(ω) denote a Fourier transform pair, with X(ω) = ∫ x(t) e^(-jωt) dt. The main properties are:
| Property | Time domain | Frequency domain |
|---|---|---|
| Linearity | a·x₁(t) + b·x₂(t) | a·X₁(ω) + b·X₂(ω) |
| Time shifting | x(t - t₀) | e^(-jωt₀) X(ω) |
| Frequency shifting (modulation) | e^(jω₀t) x(t) | X(ω - ω₀) |
| Time scaling | x(at) | (1/|a|) X(ω/a) |
| Time reversal | x(-t) | X(-ω) |
| Conjugation | x*(t) | X*(-ω) |
| Duality | X(t) | 2π x(-ω) |
| Time differentiation | dx/dt | jω X(ω) |
| Time integration | ∫ x(τ) dτ | X(ω)/(jω) + π X(0) δ(ω) |
| Frequency differentiation | t·x(t) | j dX/dω |
| Convolution in time | x₁(t) * x₂(t) | X₁(ω) X₂(ω) |
| Multiplication in time | x₁(t) x₂(t) | (1/2π) X₁(ω) * X₂(ω) |
| Parseval's theorem | ∫|x(t)|² dt | (1/2π) ∫|X(ω)|² dω |
Additional points:
- For a real x(t), X(-ω) = X*(ω): the magnitude is even and the phase is odd.
- If x(t) is real and even, X(ω) is real and even; if x(t) is real and odd, X(ω) is purely imaginary and odd.
- Time scaling shows that compression in time gives expansion in frequency (a short pulse has a wide bandwidth).
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Why the Fourier transform is needed · Data Communication, chapter 3 · asked 2×
- Fourier transform of e^(-at)u(t) and spectra · Data Communication, chapter 3 · asked 2×
- Fourier transform of a DC signal · Data Communication, chapter 3
- Fourier transform of e^(-a|t|) · Data Communication, chapter 3
- Fourier transform of a ramp pulse · Data Communication, chapter 3
- Fourier series vs Fourier transform · Data Communication, chapter 3
- Fourier transform properties; Rayleigh energy theorem · Communication Systems, chapter 1
- Hilbert vs Fourier transform; properties of HT · Communication Systems, chapter 2
- Autocorrelation and ESD as Fourier transform pair · Communication Systems, chapter 1
- Fourier transform of e^(−a|t|) · Signals and Systems, chapter 3 · asked 5×