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# iztrans

Inverse Z-transform

## Syntax

``iztrans(F)``
``iztrans(F,transVar)``
``iztrans(F,var,transVar)``

## Description

example

````iztrans(F)` returns the Inverse Z-Transform of `F`. By default, the independent variable is `z` and the transformation variable is `n`. If `F` does not contain `z`, `iztrans` uses the function `symvar`.```

example

````iztrans(F,transVar)` uses the transformation variable `transVar` instead of `n`.```

example

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

## Examples

### Inverse Z-Transform of Symbolic Expression

Compute the inverse Z-transform of `2*z/(z-2)^2`. By default, the inverse transform is in terms of `n`.

```syms z F = 2*z/(z-2)^2; iztrans(F)```
```ans = 2^n + 2^n*(n - 1)```

### Specify Independent Variable and Transformation Variable

Compute the inverse Z-transform of `1/(a*z)`. By default, the independent and transformation variables are `z` and `n`, respectively.

```syms z a F = 1/(a*z); iztrans(F)```
```ans = kroneckerDelta(n - 1, 0)/a```

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

```syms m iztrans(F,m)```
```ans = kroneckerDelta(m - 1, 0)/a```

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

`iztrans(F,a,m)`
```ans = kroneckerDelta(m - 1, 0)/z```

### Inverse Z-Transforms Involving Kronecker Delta Function

Compute the inverse Z-transforms of these expressions. The results involve the Kronecker Delta function.

```syms n z iztrans(1/z,z,n)```
```ans = kroneckerDelta(n - 1, 0)```
```f = (z^3 + 3*z^2)/z^5; iztrans(f,z,n)```
```ans = kroneckerDelta(n - 2, 0) + 3*kroneckerDelta(n - 3, 0)```

### Inverse Z-Transform of Array Inputs

Find the inverse 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, `iztrans` 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]; iztrans(M,vars,transVars)```
```ans = [ exp(x)*kroneckerDelta(a, 0), kroneckerDelta(b, 0)] [ iztrans(sin(y), y, c), iztrans(z, z, d)*1i]```

If `iztrans` 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 iztrans(x,vars,transVars)```
```ans = [ x*kroneckerDelta(a, 0), iztrans(x, x, b)] [ x*kroneckerDelta(c, 0), x*kroneckerDelta(d, 0)]```

### Inverse Z-Transform of Symbolic Function

Compute the Inverse 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; iztrans([f1, f2],x,[a, b])```
```ans = [ iztrans(exp(x), x, a), iztrans(x, x, b)]```

### If Inverse Z-Transform Cannot Be Found

If `iztrans` cannot compute the inverse transform, it returns an unevaluated call.

```syms F(z) n F(z) = exp(z); f = iztrans(F,z,n)```
```f = iztrans(exp(z), z, n)```

Return the original expression by using `ztrans`.

`ztrans(f,n,z)`
```ans = exp(z)```

## Input Arguments

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

Independent variable, specified as a symbolic variable, expression, vector, or matrix. This variable is often called the "complex frequency variable." If you do not specify the variable, then `iztrans` uses `z`. If `F` does not contain `z`, then `iztrans` uses the function `symvar`.

Transformation variable, specified as a symbolic variable, expression, vector, or matrix. It is often called the"time variable" or "space variable." By default, `iztrans` uses `n`. If `n` is the independent variable of `F`, then `iztrans` uses `k`.

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### Inverse Z-Transform

Where R is a positive number, such that the function F = F(z) is analytic on and outside the circle |z| = R, the inverse Z-transform is

`$f\left(n\right)=\frac{1}{2\pi i}\underset{|z|=R}{\oint }F\left(z\right){z}^{n-1}dz,\text{ }n=0,1,2...$`

## Tips

• If any argument is an array, then `iztrans` 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 direct Z-transform, use `ztrans`.