# Properties of fourier transform

Student
4. Jul 2016
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### Properties of fourier transform

• 1. Prepared By:- Nisarg Amin Topic:- Properties Of Fourier Transform
• 2. Properties Of Fourier Transform •There are 11 properties of Fourier Transform: i. Linearity Superposition ii. Time Scaling iii. Time Shifting iv. Duality Or Symmetry v. Area Under x(t) vi. Area Under X(f) vii. Frequency Shifting viii. Differentiation In Time Domain ix. Integration In Time Domain x. Multiplication In Time Domain xi. Convolution In Time Domain
• 3. Linearity Superposition • If and Then • This follows directly from the definition of the Fourier Transform (as the integral operator is linear). It is easily extended to a linear combination of an arbitrary number of signals )()( 11 fXtx F  )()( 12 fXtx F  )()()()( 212211 fXafXatxatxa F 
• 4. Time Scaling • Let x(t) and X(f) be Fourier Transform pairs and let ‘α’ be a constant. Then time scaling property states that • x(αt) represents a time scaled signal and X(f/ α) represents frequency scaled signal. • For α<1, x(αt) represents compressed signal but X(f/ α) represents expanded version of X(f). • For α>1, x(αt) will be an expanded signal in the time domain. But its Fourier Transform X(f/ α) represents version of X(f).          f Xtx || 1 )( F
• 5. Time Shifting • The time shifting property states that if x(t) and X(f) form a Fourier transform pair then, • Here the signal is time shifted signal. • It is the same signal x(t) only shifted in time.  fXettx dftj d 2 )(   F )( dttx 
• 6. Duality Or Symmetry • This property states that, if then • The duality theorem tells us that the shape of the signal in the time domain and the shape of the spectrum an be interchanged. )()( fXtx F  )()( fxtX F 
• 7. Area Under x(t) • This property states that the area under the curve x(t) equals the value of its Fourier Transform at f=0 i.e. if Then x(t)=X(0) )()( fXtx F 
• 8. Area Under X(f) • This property states that the area under the curve X(f) equals the value of signal x(t) at t=0. i.e. if Then x(0)=X(f) )()( fXtx F 
• 9. Frequency Shifting • The frequency shifting characteristics states that if x(t) and X(f) form a Fourier Transform pair then, • Fc is a real constant. )()(2 c Ftfj ffXtxe c 
• 10. Differentiation In Time Domain • This property is applicable if and only if the derivative of x(t) is Fourier Transformable. )(2)( ffXjtx dt d F 
• 11. Integration In Time Domain • Integration in time domain is equivalent to dividing the Fourier Transform by (j2πf). • If and provided that X(0)=0)()( fXtx F  )( 2 1 )( fX fj dx t F   
• 12. Multiplication In Time Domain • The multiplication theorem states that: If and are the two Fourier Transform pairs then, • This means that multiplication of two signals in time domain gets transformed into convolution of their Fourier Transform . )()( 11 fXtx F  )()( 12 fXtx F   dfXXtxtx F )()()()( 2121     )()()()( 2121 fXfXtxtx F 
• 13. Convolution In Time Domain • This property states that the convolution of signals in the time domain will be transformed into the multiplication of their Fourier Transform in the frequency domain. i.e. )()()()( 2121 fXfXtxtx F 