# ifourier

Inverse Fourier transform

## Syntax

• ``ifourier(F,trans_var,eval_point)``
example

## Description

example

````ifourier(F,trans_var,eval_point)` computes the inverse Fourier transform of `F` with respect to the transformation variable `trans_var` at the point `eval_point`.```

## Examples

### Inverse Fourier Transform of Symbolic Expression

Compute the inverse Fourier transform of this expression with respect to the variable `y` at the evaluation point `x`.

```syms x y F = sqrt(sym(pi))*exp(-y^2/4); ifourier(F, y, x)```
```ans = exp(-x^2)```

### Default Transformation Variable and Evaluation Point

Compute the inverse Fourier transform of this expression calling the `ifourier` function with one argument. If you do not specify the transformation variable, `ifourier` uses the variable `w`.

```syms a w t real F = exp(-w^2/(4*a^2)); ifourier(F, t)```
```ans = exp(-a^2*t^2)/(2*pi^(1/2)*(1/(4*a^2))^(1/2))```

If you also do not specify the evaluation point, `ifourier` uses the variable `x`:

`ifourier(F)`
```ans = exp(-a^2*x^2)/(2*pi^(1/2)*(1/(4*a^2))^(1/2))```

For further computations, remove the assumptions:

`syms a w t clear`

### Inverse Fourier Transforms Involving Dirac and Heaviside Functions

Compute the following inverse Fourier transforms that involve the Dirac and Heaviside functions.

```syms t w ifourier(dirac(w), w, t)```
```ans = 1/(2*pi)```
`ifourier(2*exp(-abs(w)) - 1, w, t)`
```ans = -(2*pi*dirac(t) - 4/(t^2 + 1))/(2*pi)```
`ifourier(exp(-w)*heaviside(w), w, t)`
```ans = -1/(2*pi*(- 1 + t*1i))```

### Parameters of Inverse Fourier Transform

Specify parameters of the inverse Fourier transform.

Compute the inverse Fourier transform of this expression using the default values `c = 1`, `s = -1` of the Fourier parameters. (For details, see Inverse Fourier Transform.)

```syms t w ifourier(-(sqrt(sym(pi))*w*exp(-w^2/4)*i)/2, w, t)```
```ans = t*exp(-t^2)```

Change the values of the Fourier parameters to `c = 1``s = 1` by using `sympref`. Then compute the inverse Fourier transform of the same expression again.

```sympref('FourierParameters', [1, 1]); ifourier(-(sqrt(sym(pi))*w*exp(-w^2/4)*i)/2, w, t)```
```ans = -t*exp(-t^2)```

Change the values of the Fourier parameters to ```c = 1/2π````s = 1` by using `sympref`. Compute the inverse Fourier transform using these values.

```sympref('FourierParameters', [1/(2*sym(pi)), 1]); ifourier(-(sqrt(sym(pi))*w*exp(-w^2/4)*i)/2, w, t)```
```ans = -2*t*pi*exp(-t^2)```

The preferences set by `sympref` persist through your current and future MATLAB® sessions. To restore the default values of `c` and `s`, set `sympref` to `'default'`.

`sympref('FourierParameters','default');`

### Inverse Fourier Transform of Matrix

Find the inverse Fourier 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 ifourier([exp(x), 1; sin(y), i*z],[w, x; y, z],[a, b; c, d])```
```ans = [ exp(x)*dirac(a), dirac(b)] [ (dirac(c - 1)*1i)/2 - (dirac(c + 1)*1i)/2, dirac(1, d)]```

When the input arguments are nonscalars, `ifourier` acts on them element-wise. If `ifourier` is called with both scalar and nonscalar arguments, then `ifourier` 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 ifourier(x,[x, w; y, z],[a, b; c, d])```
```ans = [ -dirac(1, a)*1i, x*dirac(b)] [ x*dirac(c), x*dirac(d)]```

Note that nonscalar input arguments must have the same size.

### Inverse Fourier Transform of Vector of Symbolic Functions

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; ifourier([f1, f2],x,[a, b])```
```ans = [ fourier(exp(x), x, -a)/(2*pi), -dirac(1, b)*1i]```

### If Inverse Fourier Transform Cannot be Found

If `ifourier` cannot find an explicit representation of the transform, it returns results in terms of the direct Fourier transform.

```syms F(w) t f = ifourier(F, w, t)```
```f = fourier(F(w), w, -t)/(2*pi)```

## Input Arguments

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### `F` — Input functionsymbolic expression | symbolic function | vector of symbolic expressions or functions | matrix of symbolic expressions or functions

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

### `trans_var` — Transformation variable`w` (default) | symbolic variable

Transformation variable, specified as a symbolic variable. This variable is often called the "frequency variable".

If you do not specify the transformation variable, `ifourier` uses the variable `w` by default. If `F` does not contain `w`, then the default variable is determined by `symvar`.

### `eval_point` — Evaluation point`x` (default) | `t` | symbolic variable | symbolic expression | vector of symbolic variables or expressions | matrix of symbolic variables or expressions

Evaluation point, specified as a symbolic variable, expression, or vector or matrix of symbolic variables or expressions. This is often called the "time variable" or the "space variable".

If you do not specify the evaluation point, `ifourier` uses the variable `x` by default. If `x` is the transformation variable of `F`, then the default evaluation point is the variable `t`.

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### Inverse Fourier Transform

The inverse Fourier transform of the expression F = F(w) with respect to the variable w at the point x is defined as follows:

$f\left(x\right)=\frac{|s|}{2\pi c}\underset{-\infty }{\overset{\infty }{\int }}F\left(w\right){e}^{-iswx}dw.$

Here, c and s are parameters of the inverse Fourier transform. The `ifourier` function uses c = 1, s = –1.

### Tips

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

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

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

• The toolbox computes the inverse Fourier transform via the direct Fourier transform:

$ifourier\left(F,w,t\right)=\frac{1}{2\pi }fourier\left(F,w,-t\right)$

If `ifourier` cannot find an explicit representation of the inverse Fourier transform, it returns results in terms of the direct Fourier transform.

• To compute the direct Fourier transform, use `fourier`.

## References

[1] Oberhettinger, F. "Tables of Fourier Transforms and Fourier Transforms of Distributions", Springer, 1990.