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Final_Tables

# Final_Tables - Table of Fourier Transform Pairs Signal Name...

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Table of Fourier Transform Pairs Signal Name Time-Domain: x.t/ Frequency-Domain: X.j!/ Right-sided exponential e at u.t/ .a > 0/ 1 a C j! Left-sided exponential e bt u. t/ .b > 0/ 1 b j! Square pulse OEu.t C T=2/ u.t T=2/ sin .!T=2/ !=2 “sinc” function sin .! 0 t/ t OEu.! C ! 0 / u.! ! 0 / Impulse ı.t/ 1 Shifted impulse ı.t t 0 / e j!t 0 Complex exponential e j! 0 t 2 ı.! ! 0 / General cosine A cos .! 0 t C / Ae j ı.! ! 0 / C Ae j ı.! C ! 0 / Cosine cos .! 0 t/ ı.! ! 0 / C ı.! C ! 0 / Sine sin .! 0 t/ j ı.! ! 0 / C j ı.! C ! 0 / General periodic signal 1 X k D 1 a k e jk! 0 t 1 X k D 1 2 a k ı.! k! 0 / Impulse Train 1 X n D 1 ı.t nT / 2 T 1 X k D 1 ı.! 2 k=T / Table of Fourier Transform Properties Property Name Time-Domain x.t/ Frequency-Domain X.j!/ Linearity ax 1 .t/ C bx 2 .t/ aX 1 .j!/ C bX 2 .j!/ Conjugation x .t/ X . j!/ Time-Reversal x. t/ X. j!/ Scaling x.at/ 1 j a j X.j.!=a// Delay x.t t d / e j!t d X.j!/ Modulation x.t/e j! 0 t X.j.! ! 0 // Modulation x.t/ cos .! 0 t/ 1 2 X.j.! ! 0 // C 1 2 X.j.! C ! 0 // Differentiation d k x.t/ dt k .j!/ k X.j!/

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