# Properties of Continuous-Time Fourier Transform (CTFT)

Signals and SystemsElectronics & ElectricalDigital Electronics

## 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 CTFTTime Domain x(t)Frequency Domain X(ω)
Linearity Property$ax_{1}(t)+bx_{2}(t)$$aX_{1}(\omega)+bX_{2}(\omega) Time Shifting Propertyx(t ± t_{0})$$e^{± j\omega t_{0}}X(\omega)$
Frequency Shifting Property$e^{± j\omega_{0} t}x(t)$$X(\omega ∓ \omega_{0}) Time Reversal Propertyx(-t)x(-\omega) Time Scaling Propertyx(at)\frac{1}{|a|} X(\frac{\omega}{a}) Time Differentiation Property\frac{d}{dt} x(t)$$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)$$X_{1}(\omega)X_{2}(\omega) Multiplication Propertyx_{1}(t)x_{2}(t)$$\frac{1}{2\pi}[X_{1}(\omega)*X_{2}(\omega)]$
Duality or Symmetry PropertyX(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 Propertyx*(t)$x*(-\omega)$
Autocorrelation PropertyR(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)$\int_{−\infty}^{\infty}X(\omega)d\omega$