# ztrans

## Syntax

`ztrans(f,trans_index,eval_point)`

## Description

`ztrans(f,trans_index,eval_point)` computes the Z-transform of `f` with respect to the transformation index `trans_index` at the point `eval_point`.

## Input Arguments

 `f` Symbolic expression, symbolic function, or vector or matrix of symbolic expressions or functions. `trans_index` Symbolic variable representing the transformation index. This variable is often called the "discrete time variable". Default: The variable `n`. If `f` does not contain `n`, then the default variable is determined by `symvar`. `eval_point` Symbolic variable or expression representing the evaluation point. This variable is often called the "complex frequency variable".Default: The variable `z`. If `z` is the transformation index of `f`, then the default evaluation point is the variable `w`.

## Examples

Compute the Z-transform of this expression with respect to the transformation index `k` at the evaluation point `x`:

```syms k x f = sin(k); ztrans(f, k, x)```
```ans = (x*sin(1))/(x^2 - 2*cos(1)*x + 1)```

Compute the Z-transform of this expression calling the `ztrans` function with one argument. If you do not specify the transformation index, `ztrans` uses the variable `n`.

```syms a n x f = a^n; ztrans(f, x)```
```ans = -x/(a - x)```

If you also do not specify the evaluation point, `ztrans` uses the variable `z`:

`ztrans(f)`
```ans = -z/(a - z)```

Compute the following Z-transforms that involve the Heaviside function and the binomial coefficient:

```syms n z ztrans(heaviside(n - 3), n, z)```
```ans = (1/(z - 1) + 1/2)/z^3```
`ztrans(nchoosek(n, 2)*heaviside(5 - n), n, z)`
```ans = z/(z - 1)^3 + 5/z^5 + (6*z - z^6/(z - 1)^3 + 3*z^2 + z^3)/z^5```

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

```syms f(n) z F = ztrans(f, n, z)```
```F = ztrans(f(n), n, z)```

`iztrans` returns the original expression:

`iztrans(F, z, n)`
```ans = f(n)```

Find the Z-transform of this matrix. Use matrices of the same size to specify the transformation variable and evaluation point.

```syms a b c d w x y z ztrans([exp(x), 1; sin(y), i*z],[w, x; y, z],[a, b; c, d])```
```ans = [ (a*exp(x))/(a - 1), b/(b - 1)] [ (c*sin(1))/(c^2 - 2*cos(1)*c + 1), (d*1i)/(d - 1)^2]```

When the input arguments are nonscalars, `ztrans` acts on them element-wise. If `ztrans` is called with both scalar and nonscalar arguments, then `ztrans` expands the scalar arguments into arrays of the same size as the nonscalar arguments with all elements of the array equal to the scalar.

```syms w x y z a b c d ztrans(x,[x, w; y, z],[a, b; c, d])```
```ans = [ a/(a - 1)^2, (b*x)/(b - 1)] [ (c*x)/(c - 1), (d*x)/(d - 1)]```

Note that nonscalar input arguments must have the same size.

When the first argument is a symbolic function, the second argument must be a scalar.

```syms f1(x) f2(x) a b f1(x) = exp(x); f2(x) = x; ztrans([f1, f2],x,[a, b])```
```ans = [ a/(a - exp(1)), b/(b - 1)^2]```

expand all

### Z-Transform

The Z-transform of the expression f = f(n) is defined as follows:

$F\left(z\right)=\sum _{n=0}^{\infty }\frac{f\left(n\right)}{{z}^{n}}.$

### Tips

• If you call `ztrans` with two arguments, it assumes that the second argument is the evaluation point `eval_point`.

• If `f` is a matrix, `ztrans` acts element-wise on all components of the matrix.

• If `eval_point` is a matrix, `ztrans` acts element-wise on all components of the matrix.

• To compute the inverse Z-transform, use `iztrans`.