Signals and Systems – Time-Shifting Property of Fourier Transform

Statement – The time shifting property of Fourier transform states that if a signal 𝑥(𝑡) is shifted by 𝑡₀ in time domain, then the frequency spectrum is modified by a linear phase shift of slope (−𝜔𝑡₀). Therefore, if,

Then, according to the time-shifting property of Fourier transform,

Proof

From the definition of Fourier transform, we have

Or, it can also be represented as,

The time shifting property of Fourier transform has a very important implication. That is,

$$\mathrm> X\left ( \omega \right )\right |=\left | X\left ( \omega \right ) \right |>$$

$$\mathrm>X\left ( \omega \right )=e^<-j\omega t_>+\angle X\left ( \omega \right )=\angle \left ( -\omega t_ \right )+\angle X\left ( \omega \right )>$$

From this, it is clear that the shifting of a function by 𝑡₀ in time domain results in multiplying its Fourier transform by 𝑒 −𝑗𝜔𝑡₀ . Hence, there is no change in the magnitude spectrum but the phase spectrum is linearly shifted.

Numerical Example

Using time-shifting property of Fourier transform, find the Fourier transform of signal [𝑒 −𝑎|𝑡−2| ].

Solution

Since the Fourier transform of two-sided exponential signal is defined as,

Now, by using time-shifting property $\mathrm < \left [i.e.\: x\left ( t-t_\right )\overset<\leftrightarrow>e^<-j\omega t_>X\left ( \omega \right ) \right ]>$ of the Fourier transform, we have,

Or, it can also be written as,