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# ifourier

Inverse Fourier transform

## Syntax

ifourier(F,trans_var,eval_point)

## Description

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.

## Input Arguments

 F Symbolic expression, symbolic function, or vector or matrix of symbolic expressions or functions. trans_var Symbolic variable representing the transformation variable. This variable is often called the "frequency variable". Default: The variable w. If F does not contain w, then the default variable is determined by symvar. eval_point Symbolic variable or expression representing the evaluation point. This variable is often called the "time variable" or the "space variable".Default: The variable x. If x is the transformation variable of F, then the default evaluation point is the variable t.

## Examples

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)```

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`

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(1/(w^2 + 1), w, t)`
```ans =
(pi*exp(-t)*heaviside(t) + pi*heaviside(-t)*exp(t))/(2*pi)```

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)```

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)*i)/2 - (dirac(c + 1)*i)/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)*i, x*dirac(b)]
[     x*dirac(c), x*dirac(d)]```

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

expand all

### 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

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