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This package is an implementation of the Z-transform of a sequence. This is the discrete analogue of the Laplace Transform.
Authors: Wolfram Koepf and Lisa Temme.
The Z-Transform of a sequence {fn} is the discrete analogue of the Laplace Transform, and
This series converges in the region outside the circle |z| = |z0| = limsupn→∞.
The calculation of the Laurent coefficients of a regular function results in the following
inverse formula for the Z-Transform:
If F(z) is a regular function in the region |z| > ρ then ∃ a sequence {fn} with
𝒵{fn} = F(z) given by
where k,λ ∈ N−{0} and A,B are fractions or variables (B > 0) and α,β, & ϕ are angles in radians.
Solution of difference equations
In the same way that a Laplace Transform can be used to solve differential equations, so
Z-Transforms can be used to solve difference equations.
Given a linear difference equation of k-th order
![]() | (16.99) |
with initial conditions f0 = h0, f1 = h1, …, fk−1 = hk−1 (where hj are given), it is
possible to solve it in the following way. If the coefficients a1,…,ak are constants, then
the Z-Transform of (16.99) can be calculated using the shift equation, and results in a
solvable linear equation for 𝒵{fn}. Application of the Inverse Z-Transform then results
in the solution of (16.99).
If the coefficients a1,…,ak are polynomials in n then the Z-Transform of (16.99)
constitutes a differential equation for 𝒵{fn}. If this differential equation can be solved
then the Inverse Z-Transform once again yields the solution of (16.99). Some examples
of these methods of solution can be found in §16.83.6.
Here are some examples for the Z-Transform
Here are some examples for the Inverse Z-Transform
Examples: Solutions of Difference Equations
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