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Table of Formulas

Module by: Anders Gjendemsjø

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

Analog Time Discrete
Delta function Unit sample
δt=0 δ t 0 for t≠0t0, ∫-∞∞δtdt=1 t δ t 1 δn=1ifn=00otherwise δ n 1 n0 0
   
Unit step function Unit step function
ut=1ift≥00otherwise u t 1 t0 0 un=1ifn≥00otherwise 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=1T0∫-T02T02|xt|2dt Pa 1 T0 t T0 2 T0 2 x t 2 Power Pd=1N∑n=N1N1+N-1|xn|2 Pd 1 N n N1 N1 N 1 x n 2
   
Convol. yt=∫-∞∞xτht-τdτ y t τ x τ h t τ Convol. yn=∑k=-∞∞xkhn-k y n k x k h n k
   
Fourier Transformation Discrete Time Fourier Transform
XⅈΩ=∫-∞∞xtⅇ-ⅈΩtdt X Ω t x t Ω t Xⅇⅈω=∑n=-∞∞xnⅇ-ⅈωn X ω n x n ω n
   
Inverse Fourier Transform Inverse DTFT
xt=12π∫-∞∞Xⅈωⅇ+ⅈΩtdΩ x t 1 2 Ω X ω Ω t xn=12π∫-ππXⅇⅈωⅇ+ⅈωndω x n 1 2 ω X ω ω n
   
Fourier coeffecients Discrete Fourier Transform
αk=1T0∫0T0xtⅇ-ⅈkΩ0tdt αk 1 T0 t 0 T0 x t k Ω0 t Xk=∑n=0N-1xnⅇ-ⅈ2πNkn X k n 0 N1 x n 2 N k n
   
Series expansion Inverse DFT
xt=∑k=-∞∞αkⅇⅈkΩ0t x t k αk k Ω0 t xn=1N∑k=0N-1Xkⅇⅈ2πNkn 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 1N∑k=0N-1|Xk|2=∑n=0N-1|xn|2 1N k 0 N1 X k 2 n 0 N1 x n 2

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