Theory and Modern Applications
Table 1 Z-transform
From: Investigation of linear difference equations with random effects
x(n), n = 0,1,2,3,… | \(\tilde{ x } ( z ) = \sum_{ i = 0 }^{\infty } x ( i ) z ^{- i } \) |
1 | \(\frac{ z }{ z - 1 } \) |
\(a ^{ n } \) | \(\frac{ z }{ z - a } \) |
\(a ^{ n - 1 } \) | \(\frac{ 1 }{ z - a } \) |
n | \(\frac{ z }{ ( z - 1 )^{ 2 }} \) |
\(n ^{ 2 } \) | \(\frac{ z ( z + 1 )}{ ( z - 1 )^{ 3 }} \) |
\(n ^{ k } \) | \(( - 1 )^{ k } D ^{ k } ( \frac{ z }{ z - 1 } )\); \(D = z \frac{ d }{ dz } \) |
\(n a ^{ n } \) | \(\frac{ az }{ ( z - a )^{ 2 }} \) |
\(n ^{ 2 } a ^{ n } \) | \(\frac{ az ( z + a )}{ ( z - a )^{ 3 }} \) |
\(n ^{ k } a ^{ n } \) | \(( - 1 )^{ k } D ^{ k } ( \frac{ z }{ z - a } )\); \(D = z \frac{ d }{ dz } \) |
x(n − k) | \(z ^{- k } \tilde{ x } ( z ) \) |
x(n + k) | \(z ^{- k } \tilde{ x } ( z ) - \sum_{ r = 0 }^{ k - 1 } x ( r ) z ^{ k - r } \) |