Fourier Transform – Representation and Condition for Existence

Fourier Transform

The Fourier transform is defined as a transformation technique which transforms signals from the continuous-time domain to the corresponding frequency domain and vice-versa. In other words, the Fourier transform is a mathematical technique that transforms a function of time $x(t)$ to a function of frequency X(ω) and vice-versa.

For a continuous-time function $x(t)$, the Fourier transform of $x(t)$ can be defined as

$$\mathrm{X(ω)=\int_{−\infty}^{\infty}x(t)\:e^{-j\omega t}dt}$$

Points about Fourier Transform

• The Fourier transform can be applied for both periodic as well as aperiodic signals.

• The Fourier transform is extensively used in the analysis of LTI (linear time invariant) systems, cryptography, signal processing, signal analysis, etc.

• Fourier transform has several application ranging from RADAR to spread spectrum communication.

Magnitude and Phase Representation of Fourier Transform

The magnitude and phase representation of Fourier transform is the tool that is used to analysed the transformed function X(ω). The function X(ω)is a complex valued function of frequency $\omega$. Therefore, it can be written as −

$$\mathrm{X(ω)=X_{real}(\omega)+X_{img}(\omega)… (2)}$$

Where,

• $X_{real}(\omega)$ is the real part of the function $X(\omega)$, and

• $X_{img}(\omega)$ is the imaginary part of the function $X(\omega)$.

Therefore, the magnitude of the function $X(\omega)$ is given by,

$$\mathrm{|X(\omega)|=\sqrt{X_{real}^{2}(\omega)}+X_{img}^{2}(\omega)… (3)}$$

And the phase of the function $X(\omega)$ is given by,

$$\mathrm{\angle\:X(\omega)=\tan^{-1}\left (\frac{X_{real}(\omega)}{X_{img}(\omega)} \right )… (4)}$$

Note:

• The graph plotted between the magnitude of the function $(|X(\omega)|)$ and the frequency (ω) is known as amplitude spectrum of the function.

• The graph plotted between the phase of the function $\angle\:X(\omega)$ and the frequency is called the phase spectrum of the function.

• The amplitude spectrum and phase spectrum together are known as frequency spectrum of the function.

Condition for Existence of Fourier Transform

The Fourier transform does not exist for all non-periodic signal. Hence, for a function $x(t)$ to have Fourier transform, the following conditions (called Dirichlet’s conditions) should be satisfied −

• The function $x(t)$ is absolutely integrable over the time interval $(-\infty\:to\:\infty)$i.e.,

$$\mathrm{\int_{−\infty}^{\infty}|X(t)|dt\:<\infty}$$

• The function $x(t)$ has a finite number of maxima and minima in every finite interval of time.

• The function $x(t)$ has a finite number of discontinuities in every finite interval of time. Also, each of these discontinuities must be finite.

The Dirichlet’s conditions are the sufficient conditions but not necessary conditions, which means, Fourier transform will definitely exist for the functions which satisfy these conditions.