Z-Transforms

The Z-transform of the function F(z) is defined as follows:

.

If R is a positive number, such that the function F(Z) is analytic on and outside the circle |z| = R, then the inverse Z-transform is defined as follows:

You can consider the Z-transform as a discrete equivalent of the Laplace transform.

To compute the Z-transform of an arithmetical expression, use the ztrans function. For example, compute the Z-transform of the following expression:

S := ztrans(sinh(n), n, z)

If you know the Z-transform of an expression, you can find the original expression or a mathematically equivalent form by computing the inverse Z-transform. To compute the inverse Z-transform, use the iztrans function. For example, compute the inverse Z-transform of the expression S:

iztrans(S, z, n)

Suppose, you compute the Z-transform of an expression, and then compute the inverse Z-transform of the result. In this case, MuPAD® can return an expression that is mathematically equivalent to the original one, but presented in a different form. For example, compute the Z-transform of the following expression:

C := ztrans(exp(n), n, z)

Now, compute the inverse Z-transform of the resulting expression C. The result differs from the original expression:

invC := iztrans(C, z, n)

Simplifying the resulting expression invC gives the original expression:

simplify(invC)

Besides arithmetical expressions, the ztrans and iztrans functions also accept matrices of arithmetical expressions. For example, compute the Z-transform of the following matrix:

A := matrix(2, 2, [1, n, n + 1, 2*n + 1]):
ZA := ztrans(A, n, z)

Computing the inverse Z-transform of ZA gives the original matrix A:

iztrans(ZA, z, n)

The ztrans and iztrans functions let you evaluate the transforms of an expression or a matrix at a particular point. For example, evaluate the Z-transform of the following expression for the value z = 2:

ztrans(1/n!, n, 2)

Evaluate the inverse Z-transform of the following expression for the value n = 10:

iztrans(z/(z - exp(x)), z, 10)

If MuPAD cannot compute the Z-transform or the inverse Z-transform of an expression, it returns an unresolved transform:

ztrans(f(n), n, z)

iztrans(F(z), z, n)

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