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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
δ⁢t=0 δ t 0 for t≠0t0, ∫−∞∞δ⁢tdt=1 t δ t 1 δ⁢n={1  if  n=00  otherwise   δ n 1 n0 0
   
Unit step function Unit step function
u⁢t={1  if  t≥00  otherwise   u t 1 t0 0 u⁢n={1  if  n≥00  otherwise   u n 1 n0 0
   
Angular frequency Angular frequency
Ω=2⁢π⁢F Ω 2 F ω=2⁢π⁢f ω 2 f
   
Energy Ea=∫−∞∞|x⁢t|2dt Ea t x t 2 Energy Ed=∑n=−∞∞|x⁢n|2 Ed n x n 2
   
Power Pa=1T0⁢∫−T02T02|x⁢t|2dt Pa 1 T0 t T0 2 T0 2 x t 2 Power Pd=1N⁢∑n=N1N1+N−1|x⁢n|2 Pd 1 N n N1 N1 N 1 x n 2
   
Convol. y⁢t=∫−∞∞x⁢τ⁢h⁢t−τdτ y t τ x τ h t τ Convol. y⁢n=∑k=−∞∞x⁢k⁢h⁢n−k y n k x k h n k
   
Fourier Transformation Discrete Time Fourier Transform
X⁢i⁢Ω=∫−∞∞x⁢t⁢e−(i⁢Ω⁢t)dt X Ω t x t Ω t X⁢ei⁢ω=∑n=−∞∞x⁢n⁢e−(i⁢ω⁢n) X ω n x n ω n
   
Inverse Fourier Transform Inverse DTFT
x⁢t=12⁢π⁢∫−∞∞X⁢i⁢ω⁢ei⁢Ω⁢tdΩ x t 1 2 Ω X ω Ω t x⁢n=12⁢π⁢∫−ππX⁢ei⁢ω⁢ei⁢ω⁢ndω x n 1 2 ω X ω ω n
   
Fourier coeffecients Discrete Fourier Transform
αk=1T0⁢∫0T0x⁢t⁢e−(i⁢k⁢Ω0⁢t)dt αk 1 T0 t 0 T0 x t k Ω0 t X⁢k=∑n=0N−1x⁢n⁢e−(i⁢2⁢πN⁢k⁢n) X k n 0 N1 x n 2 N k n
   
Series expansion Inverse DFT
x⁢t=∑k=−∞∞αk⁢ei⁢k⁢Ω0⁢t x t k αk k Ω0 t x⁢n=1N⁢∑k=0N−1X⁢k⁢ei⁢2⁢πN⁢k⁢n x n 1 N k 0 N1 X k 2 N k n
   
Parseval Parseval
∫T1T0|x⁢t|2dt=∑k=1∞|αk|2 t T1 T0 x t 2 k 1 αk 2 1N⁢∑k=0N−1|X⁢k|2=∑n=0N−1|x⁢n|2 1N k 0 N1 X k 2 n 0 N1 x n 2

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