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A few examples together with the above properties will enable one to solve and understand a wide variety of problems. These use the unit stepfunction to remove the negative time part of the signal. This function is defined as
and several bilateral z-transforms are given by
Notice that these are similar to but not the same as a term of a partial fraction expansion.
The z-transform can be inverted in three ways. The first two have similar procedures with Laplace transformations and the third has no counter part.
For example
which is $u\left(n\right)\phantom{\rule{0.166667em}{0ex}}{a}^{n}$ as used in the examples above.
We must understand the role of the ROC in the convergence and inversion of the z-transform. We must also see the difference between the one-sided andtwo-sided transform.
The z-transform can be used to convert a difference equation into an algebraic equation in the same manner that the Laplace converts a differential equation in to an algebraic equation. The one-sided transform isparticularly well suited for solving initial condition problems. The two unilateral shift properties explicitly use the initial values of theunknown variable.
A difference equation DE contains the unknown function $x\left(n\right)$ and shifted versions of it such as $x(n-1)$ or $x(n+3)$ . The solution of the equation is the determination of $x\left(t\right)$ . A linear DE has only simple linear combinations of $x\left(n\right)$ and its shifts. An example of a linear second order DE is
A time invariant or index invariant DE requires the coefficients not be a function of $n$ and the linearity requires that they not be a function of $x\left(n\right)$ . Therefore, the coefficients are constants.
This equation can be analyzed using classical methods completely analogous to those used with differential equations. A solution of the form $x\left(n\right)=K{\lambda}^{n}$ is substituted into the homogeneous difference equation resulting in a second order characteristic equation whose two roots givea solution of the form ${x}_{h}\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{K}_{1}{\lambda}_{1}^{n}+{K}_{2}{\lambda}_{2}^{n}$ . A particular solution of a form determined by $f\left(n\right)$ is found by the method of undetermined coefficients, convolution or some other means. Thetotal solution is the particular solution plus the solution of the homogeneous equation and the three unknown constants ${K}_{i}$ are determined from three initial conditions on $x\left(n\right)$ .
It is possible to solve this difference equation using z-transforms in a similar way to the solving of a differential equation by use of theLaplace transform. The z-transform converts the difference equation into an algebraic equation. Taking the ZT of both sides of the DE gives
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