ft_ref3 a table of fourier transforms
PDF · 1 pages · 106.1 KB
Open PDF file
Reference table of Fourier transforms found in the Spectral Theory Book downloads folder, apparently collected from an outside source rather than written by Phil. It lists x(t) against X(f) and X(omega) for the delta function, constants, complex exponentials, cosine and sine, rect and sinc, triangle, one-sided and two-sided exponentials, Gaussian, sgn, unit step, derivative of delta, and the impulse train. The extracted symbols are somewhat garbled.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
Fourier Transform Table
()xt ()Xf () Xω
()tδ 1 1
1 ()fδ 2( )πδω
0 ()ttδ− 0 2jfteπ− 0jteω−
02jfteπ 0 ()ffδ− 0 2( ) πδωω−
0 cos( 2 )ftπ
001() (2ff ff δ δ−+ +) 00 () ( πδωωδωω −+ + )
0 sin(2 )ftπ
001() (2ff ffjδ δ−− +) 00 () ( jπδωωδωω −− −+ )
()rectt sin ()cf sin2cω
π
sin()ct () rectf
2rectω
π
()tΛ 2sin () cf 2sin2cω
π
2sin () ct ()fΛ
2ω
πΛ
(), 0teu tαα−> 1
2jfαπ+ 1
jαω+
(), 0tte utαα−>
()21
2jfαπ+
()21
jαω+
||,0teαα−>
222
(( 2)fα
απ+ 222
(( )α
αω+
2teπ− 2feα− 2feα−
sgn( )t 1
jfπ 2
jω
()ut 11()22fjfδπ+ 1()jπδωω+
()dtdtδ 2jfπ jω
0 ()
ntn Tδ∞
=−∞−∑
001
nnfTTδ∞
=−∞−
∑
0012
nn
TTπδω∞
=−∞−
∑