# SJSU EE 112 - Fourier Transform Properties (28 pages)

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**View the full content.**## Fourier Transform Properties

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## Fourier Transform Properties

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- Pages:
- 28
- School:
- San Jose State University
- Course:
- Ee 112 - Introduction to Signal Processing

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Si Signals l and dS Systems t EE 112 Lecture 15 Fourier Transform properties Khosrow Ghadiri Electrical Engineering Department San Jose State University Khosrow Ghadiri Jean Baptiste Charles Fourier Jean Baptiste Charles Fourier 3 21 1768 5 16 1830 3 21 1768 5 16 1830 Khosrow Ghadiri Signal and System EE Dept SJSU 2 Outline Definition Linearity Symmetry Time scaling Time shifting Multiplication Frequency shifting Signal modulation Ti Time diff differentiation ti ti Frequency differentiation Time integration Conjugation of time and frequency functions Time convolution Frequency convolution Area under Area under Parseval s Theorem Khosrow Ghadiri Signal and System EE Dept SJSU 3 Fourier transform properties Linearity If F f t F F g t G And a and b are any y real or complex p scalars then Linearity Proof evaluating g the Fourier transform of a f t b g t a f t b g t a F b G F a f t b g t a f t b g t e j t dt By y linearity y of integration g F a f t b g t a F Khosrow Ghadiri f t e dt b g t e j t dt j t a f t b g t a F b G Signal and System EE Dept SJSU 4 Example Application of Linearity Consider the below signal g Find its Fourier transform f t F f t Let s write f t as a linear combination of t and t Then f t t t f t t t F t F t sinc f sinc f Khosrow Ghadiri 2 Signal and System EE Dept SJSU 5 Fourier transform properties Symmetry y y If F is the Fourier transform of f t then the symmetry property of the Fourier transform states that FT F t 2 F That is if in F we replace with t we get the Fourier transform pair Proof Since 1 f t Then 2 F e j t d 2 f t F e j t d Interchanging t and we get 2 f F t e Khosrow Ghadiri j t d Signal and System EE Dept SJSU FT f t 6 Fourier transform properties Time scaling g If F f is the Fourier transform of f t and a is a real constant then F 1 f f at F a a That is the time scaling property of the Fourier transform states that if we replace the variable f in the frequency d domain i by b f a and d divide di id F f a by b the th absolute b l t

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