Fourier Transform - Formula - Properties

Fourier transform (ft) for a function f(t) is unique i.e no two functions can have same fourier transform.

Consider a continuous time signal x(t). Its fourier transform is defined as

Above expression is called fourier transform formula.

Conditions for existence of fourier transform

Fourier transform of a function x(t) exists if

(1) signal x(t) is absolutely integrable 

(2) signal x(t) is deterministic over any finite interval

i.e  (a) It should have finite number of maxima and minima over    a finite interval.

(b) It should have finite number of discontinuity over a finite interval.

These conditions are sufficient but not necessary. It means that there are signals that violate either one or both conditions , yet possess fourier transform.

Properties of Fourier Transform

The knowledge of properties of fourier transform reduces the labour involved in fourier transform calculations, in certain cases.

How fourier transform properties are useful can be seen by analysing solved examples.

fourier transform of x(t) is represented as

 
   

(1) Linearity :-

(2) Time shifting :-

(3) Frequency shifting :-

(4) Time scaling:-

(5) Time reversal :-

(6) Duality :-

(7) Differentiation in time :-

(8) Integration in time :-

(9) Convolution in time :-

(10) Frequency convolution :-

(11) Frequency differentiation :-

(12) Conjugate property :-

(13) Parseval’s Power theorem :-

(14) Area under x(t) and X(w) :-

Related :-

   (1) Fourier transform solved problems
   (2) Fourier Series | examples-  sawtooth (triangular) and square wave  | Formula