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• Examples and How To

# ztrans

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```ztrans(f, k, z)
```

## Description

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.

## Examples

### 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)`

`Simplify(%)`

### 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:

`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):

```unprotect(heaviside):
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:
protect(heaviside):```

### Example 7

Compute the Z tranforms of these expressions:

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

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

## Parameters

 f Arithmetical expression or matrix of such expressions k z 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.