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Table of Formulas

Module by: Anders Gjendemsjø. E-mail the author

Summary: Lists important formulas, both analog and time discrete.

Table 1
Analog Time Discrete
Delta function Unit sample
δt=0 δ t 0 for t0t0, δtdt=1 t δ t 1 δn={1  if  n=00  otherwise   δ n 1 n0 0
   
Unit step function Unit step function
ut={1  if  t00  otherwise   u t 1 t0 0 un={1  if  n00  otherwise   u n 1 n0 0
   
Angular frequency Angular frequency
Ω=2πF Ω 2 F ω=2πf ω 2 f
   
Energy Ea=|xt|2dt Ea t x t 2 Energy Ed=n=|xn|2 Ed n x n 2
   
Power Pa=1T0T02T02|xt|2dt Pa 1 T0 t T0 2 T0 2 x t 2 Power Pd=1Nn=N1N1+N1|xn|2 Pd 1 N n N1 N1 N 1 x n 2
   
Convol. yt=xτhtτdτ y t τ x τ h t τ Convol. yn=k=xkhnk y n k x k h n k
   
Fourier Transformation Discrete Time Fourier Transform
XjΩ=xte(jΩt)dt X Ω t x t Ω t Xejω=n=xne(jωn) X ω n x n ω n
   
Inverse Fourier Transform Inverse DTFT
xt=12πXjωejΩtdΩ x t 1 2 Ω X ω Ω t xn=12πππXejωejωndω x n 1 2 ω X ω ω n
   
Fourier coeffecients Discrete Fourier Transform
αk=1T00T0xte(jkΩ0t)dt αk 1 T0 t 0 T0 x t k Ω0 t Xk=n=0N1xne(j2πNkn) X k n 0 N1 x n 2 N k n
   
Series expansion Inverse DFT
xt=k=αkejkΩ0t x t k αk k Ω0 t xn=1Nk=0N1Xkej2πNkn x n 1 N k 0 N1 X k 2 N k n
   
Parseval Parseval
T1T0|xt|2dt=k=1|αk|2 t T1 T0 x t 2 k 1 αk 2 1Nk=0N1|Xk|2=n=0N1|xn|2 1N k 0 N1 X k 2 n 0 N1 x n 2

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