# ifourier

Inverse Fourier transform

## Syntax

``ifourier(F)``
``ifourier(F,transVar)``
``ifourier(F,var,transVar)``

## Description

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````ifourier(F)` returns the Inverse Fourier Transform of `F`. By default, the independent variable is `w` and the transformation variable is `x`. If `F` does not contain `w`, `ifourier` uses the function `symvar`.```

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````ifourier(F,transVar)` uses the transformation variable `transVar` instead of `x`.```

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````ifourier(F,var,transVar)` uses the independent variable `var` and the transformation variable `transVar` instead of `w` and `x`, respectively.```

## Examples

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Compute the inverse Fourier transform of `exp(-w^2/4)`. By default, the inverse transform is in terms of `x`.

```syms w F = exp(-w^2/4); ifourier(F)```
```ans = exp(-x^2)/pi^(1/2)```

Compute the inverse Fourier transform of `exp(-w^2-a^2)`. By default, the independent and transformation variables are `w` and `x`, respectively.

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

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

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

Compute the inverse Fourier transform of expressions in terms of Dirac and Heaviside functions.

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

Specify parameters of the inverse Fourier transform.

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

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

Change the Fourier parameters to `c = 1`, ```s = 1``` by using `sympref`, and compute the transform again. The sign of the result changes.

```sympref('FourierParameters',[1 1]); ifourier(f,w,t)```
```ans = -t*exp(-t^2)```

Change the Fourier parameters to `c = 1/(2*pi)`, `s = 1`. The result changes.

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

Preferences set by `sympref` persist through your current and future MATLAB® sessions. Restore the default values of `c` and `s` by setting `FourierParameters` to `'default'`.

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

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

If `ifourier` 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.

`ifourier(x,vars,transVars)`
```ans = [ x*dirac(a), -dirac(1, b)*1i] [ x*dirac(c), x*dirac(d)]```

If `ifourier` cannot transform the input, then it returns an unevaluated call to `fourier`.

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

## 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 "frequency variable." If you do not specify the variable, then `ifourier` uses `w`. If `F` does not contain `w`, then `ifourier` uses the function `symvar` to determine the independent variable.

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

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

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

## Tips

• If any argument is an array, then `ifourier` 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.

• The toolbox computes the inverse Fourier transform via the 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, then it returns results in terms of the Fourier transform.

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

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

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