Fourier Transform

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Fourier Transform is a mathematical model which helps to transform the signals between two different domains, such as transforming signal from frequency domain to time domain or vice versa. Fourier transform has many applications in Engineering and Physics, such as signal processing, RADAR, and so on. In this article, we are going to discuss the formula of Fourier transform, properties, tables, Fourier cosine transform, Fourier sine transform with complete explanations.

What is Fourier Transform?

The generalisation of the complex Fourier series is known as the Fourier transform. The term “Fourier transform” can be used in the mathematical function, and it is also used in the representation of the frequency domain. The Fourier transform helps to extend the Fourier series to the non-periodic functions, which helps us to view any functions in terms of the sum of simple sinusoids.

Fourier Transform Formula

As discussed above, the Fourier transform is considered to be a generalisation of the complex Fourier series in the limit L→∞. Also, convert discrete An to the continuous F(k)dk and let n/L→k. Finally, convert the sum to an integral.

Thus, the Fourier transform of a function f(x) is given by:

\(\beginf(x) =\int_<-\infty >^<\infty >F(k)e^<2\pi ikx>dk \end \) \(\beginF(k)= \int_<-\infty >^<\infty >f(x)e^<-2\pi ikx>dx\end \)

Forward and Inverse Fourier Transform

From the Fourier transform formula, we can derive the forward and inverse Fourier transform.

\(\beginF(k)= F_[f(x)](k) = \int_<-\infty >^<\infty >f(x)e^<-2\pi ikx>dx\end \) \(\beginf(x) = F^_[F(k)](x)= \int_<-\infty >^<\infty >F(k)e^<2\pi ikx>dk\end \) \(\beginForward\ Fourier\ Transform\ : \hat(k)\end \) \(\beginInverse\ Fourier\ Transform\ : \check(x)\end \)

Fourier Transform Properties

The following are the important properties of Fourier transform:

  • Duality – If h(t) has a Fourier transform H(f), then the Fourier transform of H(t) is H(-f).
  • Linear transform – Fourier transform is a linear transform. Let h(t) and g(t) be two Fourier transforms, which are denoted by H(f) and G(f), respectively. In this case, we can easily calculate the Fourier transform of the linear combination of g and h.
  • Modulation property – According to the modulation property, a function is modulated by the other function, if it is multiplied in time.

Fourier Transform in Two Dimensions

Fourier transform in two-dimensions is given as follows:

\(\beginF(x, y)= \int_<-\infty >^<\infty >\int_<-\infty >^<\infty >f(k_, k_)e^<-2\pi i(k_x + k_y)>dk_dk_\end \) \(\beginf(k_, k_)= \int_<-\infty >^<\infty >\int_<-\infty >^<\infty >F(x, y)e^<2\pi i(k_x + k_y)>dx dy\end \)

Fourier Transform Table

The following table presents the Fourier transform for different functions:

Function