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# ztrans

## Syntax

``ztrans(f)``
``ztrans(f,transVar)``
``ztrans(f,var,transVar)``

## Description

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````ztrans(f)` finds the Z-Transform of `f`. By default, the independent variable is `n` and the transformation variable is `z`. If `f` does not contain `n`, `ztrans` uses `symvar`.```

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````ztrans(f,transVar)` uses the transformation variable `transVar` instead of `z`.```

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````ztrans(f,var,transVar)` uses the independent variable `var` and transformation variable `transVar` instead of `n` and `z`, respectively.```

## Examples

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Compute the Z-transform of `sin(n)`. By default, the transform is in terms of `z`.

```syms n f = sin(n); ztrans(f)```
```ans = (z*sin(1))/(z^2 - 2*cos(1)*z + 1)```

Compute the Z-transform of `exp(m+n)`. By default, the independent variable is `n` and the transformation variable is `z`.

```syms m n f = exp(m+n); ztrans(f)```
```ans = (z*exp(m))/(z - exp(1))```

Specify the transformation variable as `y`. If you specify only one variable, that variable is the transformation variable. The independent variable is still `n`.

```syms y ztrans(f,y)```
```ans = (y*exp(m))/(y - exp(1))```

Specify both the independent and transformation variables as `m` and `y` in the second and third arguments, respectively.

`ztrans(f,m,y)`
```ans = (y*exp(n))/(y - exp(1))```

Compute the Z-transform of 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))`
```ans = z/(z - 1)^3```

Find the Z-transform of the matrix `M`. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. When the arguments are nonscalars, `ztrans` acts on them element-wise.

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

If `ztrans` is called with both scalar and nonscalar arguments, then it expands the scalars to match the nonscalars by using scalar expansion. Nonscalar arguments must be the same size.

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

Compute the Z-transform of symbolic functions. When the first argument contains symbolic functions, then 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]```

If `ztrans` cannot transform the input then it returns an unevaluated call.

```syms f(n) f(n) = 1/n; F = ztrans(f,n,z)```
```F = ztrans(1/n, n, z)```

Return the original expression by using `iztrans`.

`iztrans(F,z,n)`
```ans = 1/n```

## Input Arguments

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Input, specified as a symbolic expression, function, vector, or matrix.

Independent variable, specified as a symbolic variable. This variable is often called the "discrete time variable". If you do not specify the variable, then `ztrans` uses `n`. If `f` does not contain `n`, then `ztrans` uses the function `symvar`.

Transformation variable, specified as a symbolic variable, expression, vector, or matrix. This variable is often called the "complex frequency variable." By default, `ztrans` uses `z`. If `z` is the independent variable of `f`, then `ztrans` uses `w`.

## More About

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### Z-Transform

The Z-transform F = F(z) of the expression f = f(n) with respect to the variable `n` at the point `z` is

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

## Tips

• If any argument is an array, then `ztrans` acts element-wise on all elements of the array.

• If the first argument contains a symbolic function, then the second argument must be a scalar.

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

## See Also

Introduced before R2006a

## Support

#### Mathematical Modeling with Symbolic Math Toolbox

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