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MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.


ztrans(f, k, z)


ztrans(f, k, z) computes the Z transform of the expression f = f(k) with respect to the index k at the point z.

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


If ztrans cannot find an explicit representation of the transform, it returns an unevaluated function call. See Example 4.

If f is a matrix, ztrans applies the Z transform to all components of the matrix.

To compute the inverse Z transform, use iztrans.


Example 1

Compute the Z transform of these expressions:

ztrans(1/k!, k, z)

ztrans(sin(k), k, z)

Example 2

Compute the Z transform of this expression and then simplify the result:

ztrans(cos(a*k + b), k, z)


Example 3

Compute the Z transform of this expression with respect to the variable k:

F := ztrans(2*k + 3, k, z)

Evaluate the Z transform of the expression at the points z = 2 a + 3 and z = 1 + i. You can evaluate the resulting expression F using | (or its functional form evalAt):

F | z = 2*a + 3

Also, you can evaluate the Z transform at a particular point directly:

ztrans(2*k + 3, k, 1 + I)

Example 4

If ztrans cannot find an explicit representation of the transform, it returns an unevaluated call:

ztrans(f(k), k, z)

iztrans returns the original expression:

iztrans(%, z, k)

Example 5

Compute the following Z transforms that involve Kronecker's Delta function and the Heaviside function:

ztrans(f(k)*kroneckerDelta(k, 1) +
       g(k)*kroneckerDelta(k, -5), k, z)

ztrans(binomial(k, 2)*heaviside(5 - k), k, z)

Simplify the last expression using simplify:


Example 6

Compute the Z transforms of this expression that involves the Heaviside function:

ztrans(heaviside(k - 3), k, z)

Note that MuPAD® uses the value heaviside(0) = 1/2. You can define a different value for heaviside(0):

heaviside(0) := 1:

For better performance, MuPAD remembers the previously computed value of the Z transform. To force the system to recalculate the transform, clear its remember table:

ztrans(Remember, Clear):

For details about the remember mechanism, see Remember Mechanism.

Defining a different value for heaviside(0) produces a different value of the Z transform:

ztrans(heaviside(k - 3), k, z)

For further computations, restore the original value:

heaviside(0):= 1/2:

Example 7

Compute the Z tranforms of these expressions:

ztrans(k*f(k), k, z)

ztrans(f(k + 1), k, z)



Arithmetical expression or matrix of such expressions


Identifier or indexed identifier


Arithmetical expression representing the evaluation point

Return Values

Arithmetical expression or unevaluated function call of type ztrans. An explicit result can be a piecewise object. If the first argument is a matrix, the result is returned as a matrix.

Overloaded By


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