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Properties of Continuous-Time Fourier Transform (CTFT)
Fourier Transform
The Fourier transform of a continuous-time function $x(t)$ is defined as,
$$\mathrm{X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}dt}$$
Inverse Fourier Transform
The inverse Fourier transform of a continuous-time function is defined as,
$$\mathrm{x(t)=\frac{1}{2\pi}\int_{−\infty}^{\infty}X(\omega)e^{j\omega t}d\omega}$$
Properties of Fourier Transform
The continuous-time Fourier transform (CTFT) has a number of important properties. These properties are useful for driving Fourier transform pairs and also for deducing general frequency domain relationships. These properties also help to find the effect of various time domain operations on the frequency domain. Some of the important properties of continuous time Fourier transform are given in the table as −
| Property of CTFT | Time Domain x(t) | Frequency Domain X(ω) |
|---|---|---|
| Linearity Property | $ax_{1}(t)+bx_{2}(t)$ | $aX_{1}(\omega)+bX_{2}(\omega)$ |
| Time Shifting Property | $x(t ± t_{0})$ | $e^{± j\omega t_{0}}X(\omega)$ |
| Frequency Shifting Property | $ e^{± j\omega_{0} t}x(t)$ | $X(\omega ∓ \omega_{0})$ |
| Time Reversal Property | x(-t) | $x(-\omega)$ |
| Time Scaling Property | x(at) | $\frac{1}{|a|} X(\frac{\omega}{a})$ |
| Time Differentiation Property | $\frac{d}{dt} x(t)$ | $j \omega X(\omega)$ |
| Frequency Derivative Property | $t.x(t)$ | $j\frac{d}{d\omega}X(\omega)$ |
| Time Integration Property | $\int_{−\infty}^{\infty} x(t) d τ$ | $\frac{X(\omega)}{j\omega}$ |
| Convolution Property | $x_{1}(t)*x_{2}(t)$ | $X_{1}(\omega)X_{2}(\omega)$ |
| Multiplication Property | $x_{1}(t)x_{2}(t)$ | $\frac{1}{2\pi}[X_{1}(\omega)*X_{2}(\omega)]$ |
| Duality or Symmetry Property | X(t) | $2\pi x(-\omega)$ |
| Modulation Property | $x(t)\:cos\:\omega_{0}t$ | $\frac{1}{2}[X(\omega-\omega_{0})+X(\omega+\omega_{0})]$ |
| $x(t)\:sin\:\omega_{0}t$ | $\frac{1}{2j}[X(\omega-\omega_{0})-X(\omega+\omega_{0})]$ | |
| Conjugation Property | x*(t) | $x*(-\omega)$ |
| Autocorrelation Property | R(τ) | $|X(-\omega)|^{2}$ |
| Parseval’s Theorem | $\int_{−\infty}^{\infty} x_{1}(t)x_{2}^*(t)dt$ | $\frac{1}{2\pi}\int_{−\infty}^{\infty}X_{1}(\omega)x_{2}^*(\omega)d\omega$ |
| Parseval’s Identity | $\int_{−\infty}^{\infty}|x(t)|^{2} dt$ | $\frac{1}{2\pi}\int_{−\infty}^{\infty}|X(\omega)|^{2}d\omega$ |
| Area Under the Curve Property | $\int_{−\infty}^{\infty}x(t)dt$ | $\frac{1}{2\pi}X(0)$ |
| x(0) | $\int_{−\infty}^{\infty}X(\omega)d\omega$ |
Updated on: 03-Dec-2021
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