Fourier Transform Table - Tables of Transform Pairs 2005 by...

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Tables of Transform Pairs 2005 by Marc Stoecklin — marc a stoecklin.net — http://www.stoecklin.net/ — December 8, 2005 — version 1.5 Students and engineers in communications and mathematics are confronted with transforma- tions such as the z-Transform, the Fourier transform, or the Laplace transform. Often it is quite hard to quickly find the appropriate transform in a book or the Internet, much less to have a good overview of transformation pairs and corresponding properties. In this document I present a handy collection of the most common transform pairs and properties of the continuous-time frequency Fourier transform (2 πf ), continuous-time pulsation Fourier transform ( ω ), z-Transform , discrete-time Fourier transform DTFT , and Laplace transform arranged in a table and ordered by subject. The properties of each transformation are indicated in the first part of each topic whereas specific transform pairs are listed afterwards. Please note that, before including a transformation pair in the table, I verified their cor- rectness. However, it is still possible that there might be some mistakes due to typos. I’d be grateful to everyone for dropping me a line and indicating me erroneous formulas. Some useful conventions and formulas Sinc function sinc ( x ) sin( x ) x Convolution f * g ( t ) = + -∞ f ( τ ) g * ( t - τ ) Parseval theorem + -∞ f ( t ) g * ( t ) dt = + -∞ F ( f ) G * ( f ) df + -∞ | f ( t ) | 2 dt = + -∞ | F ( f ) | 2 df Real part e { f ( t ) } = 1 2 [ f ( t ) + f * ( t )] Imaginary part m { f ( t ) } = 1 2 [ f ( t ) - f * ( t )] Sine / Cosine sin ( x ) = e jx - e - jx 2 j cos ( x ) = e jx + e - jx 2 Geometric sequences k =0 x k = 1 1 - x n k =0 x k = 1 - x n +1 1 - x General case : n k = m x k = x m - x n +1 1 - x 1
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Marc Stoecklin : TABLES OF TRANSFORM PAIRS 2 Table of Continuous-time Frequency Fourier Transform Pairs f ( t ) = F - 1 { F ( f ) } = R + -∞ f ( t ) e j 2 πft df F == F ( f ) = F { f ( t ) } = R + -∞ f ( t ) e - j 2 πft dt f ( t ) F == F ( f ) f ( - t ) F == F ( - f ) f * ( t ) F == F * ( - f ) f ( t ) is purely real F == F ( f ) = F * ( - f ) even/symmetry f ( t ) is purely imaginary F == F ( f ) = - F * ( - f ) odd/antisymmetry even/symmetry f ( t ) = f * ( - t ) F == F ( f ) is purely real odd/antisymmetry f ( t ) = - f * ( - t ) F == F ( f ) is purely imaginary time shifting f ( t - t 0 ) F == F ( f ) e - j 2 πft 0 f ( t ) e j 2 πf 0 t F == F ( f - f 0 ) frequency shifting time scaling f ( af ) F == 1 | a | F f a 1 | a | f f a F == F ( af ) frequency scaling af ( t ) + bg ( t ) F == aF ( f ) + bG ( t ) f ( t ) g ( t ) F == F ( f ) * G ( f ) f ( t ) * g ( t ) F == F ( f ) G ( f ) δ ( t ) F == 1 δ ( t - t 0 ) F == e - j 2 πft 0 1 F == δ ( f ) e j 2 πf 0 t F == δ ( f - f 0 ) e - a | t | a > 0 F == 2 a a 2 +4 π 2 f 2 e - πt
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