Fourier Transform

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 A_n 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:

\begin{array}{l} f (x) = \int_{- \infty}^{\infty} F (k) e^{2 π i k x} d k \end{array}

\begin{array}{l} F (k) = \int_{- \infty}^{\infty} f (x) e^{- 2 π i k x} d x \end{array}

Forward and Inverse Fourier Transform

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

$\begin{array}{l} F (k) = F_{x} [f (x)] (k) = \int_{- \infty}^{\infty} f (x) e^{- 2 π i k x} d x \end{array}$
is known as the forward Fourier transform or simply Fourier transform.
$\begin{array}{l} f (x) = F_{k}^{- 1} [F (k)] (x) = \int_{- \infty}^{\infty} F (k) e^{2 π i k x} d k \end{array}$
is known as the inverse Fourier transform.
The symbols used to denote the forward and inverse Fourier transform are given as follows:
- $\begin{array}{l} F o r w a r d F o u r i e r T r a n s f o r m : \hat{f} (k) \end{array}$
- $\begin{array}{l} I n v e r s e F o u r i e r T r a n s f o r m : \overset{ˇ}{f} (x) \end{array}$

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:

\begin{array}{l} F (x, y) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f (k_{x}, k_{y}) e^{- 2 π i (k_{x} x + k_{y} y)} d k_{x} d k_{y} \end{array}

\begin{array}{l} f (k_{x}, k_{y}) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} F (x, y) e^{2 π i (k_{x} x + k_{y} y)} d x d y \end{array}

Fourier Transform Table

The following table presents the Fourier transform for different functions:

Function	f(x)	F(k) = F_x [f(x)]
Fourier Transform: 1	1	$\begin{array}{l} δ (k) \end{array}$
Fourier Transform: Sine Function	$\begin{array}{l} s i n (2 π k_{0} x) \end{array}$	$\begin{array}{l} \frac{1}{2} i [δ (k + k_{0}) - δ (k - k_{0})] \end{array}$
Fourier Transform: Cosine Function	$\begin{array}{l} c o s (2 π k_{0} x) \end{array}$	$\begin{array}{l} \frac{1}{2} [δ (k - k_{0}) + δ (k + k_{0})] \end{array}$
Fourier Transform: Inverse Function	$\begin{array}{l} - P V \frac{1}{π x} \end{array}$	$\begin{array}{l} i [1 - 2 H (- k)] \end{array}$
Fourier Transform: Exponential Function	$\begin{array}{l} e^{- 2 π k_{0} \| x \|} \end{array}$	$\begin{array}{l} \frac{1}{π} \frac{k_{0}}{k^{2} + k_{0}^{2}} \end{array}$
Fourier Transform: Gaussian Function	$\begin{array}{l} e^{- a x^{2}} \end{array}$	$\begin{array}{l} \sqrt{\frac{π}{a}} e^{- π^{2} k^{2} / a} \end{array}$

Applications of Fourier Transform

Fourier transform is used in a wide range of applications, such as:

Image Compression
Image Analysis
Image Filtering
Image Reconstruction

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Fourier Sine Transform

The Fourier sine transform is defined as the imaginary part of full complex Fourier transform, and it is given by:

\begin{array}{l} F_{x}^{(s)} [f (x)] (k) = I [F_{x} [f (x)] (k)] \end{array}

\begin{array}{l} F_{x}^{(s)} [f (x)] (k) = \int_{- \infty}^{\infty} s i n (2 π k x) f (x) d x \end{array}

Fourier Cosine Transform

The Fourier transform for cosines of a real function is defined as the real part of a full complex Fourier transform.

\begin{array}{l} F_{x}^{(c)} [f (x)] (k) = R [F_{x} [f (x)] (k)] \end{array}

\begin{array}{l} F_{x}^{(c)} [f (x)] (k) = \int_{- \infty}^{\infty} c o s (2 π k x) f (x) d x \end{array}

Frequently Asked Questions on Fourier Transform

Is Fourier transform a generalised form of the Fourier series?

Yes, Fourier transform is the generalised form of a complex Fourier series.

Why do we use Fourier transform?

Fourier transform is one of the important concepts used in image processing, which helps to decompose the image into the sine and cosine components.

Give one comparison between the Laplace transform and the Fourier transform.

The Laplace transform is used to analyse the unstable system, and has a convergence factor. Whereas, the Fourier transform cannot be used to analyse the unstable systems, and it does not have any convergence factor.

What are the properties of Fourier transform?

The properties of Fourier transform are:
Linearity property
Frequency shifting property
Time reversal property
Time-shifting property, and so on.

Is the Fourier transform linear?

Yes, the Fourier transform is linear.

What is the use of Fourier transform?

The Fourier transform is used in the transition of signal from the time spectrum to the frequency spectrum.

MATHS Related Links
Precision	Area Of Kite
Surface Area Of A Cone	Linear Equations
Linear Equations In One Variable	Reflection Symmetry
Integral Calculus	LCM And HCF
Right Angled Triangle	Volume Of Cube

MATHS Related Links

Precision

Area Of Kite

Surface Area Of A Cone

Linear Equations

Linear Equations In One Variable

Reflection Symmetry

Integral Calculus

LCM And HCF

Right Angled Triangle

Volume Of Cube