# Discrete Fourier Transformation (DFT) {#openvino_docs_ops_signals_DFT_7}

**Versioned name**: *DFT-7*

**Category**: *Signal processing*

**Short description**: *DFT* operation performs the discrete complex-to-complex Fourier transformation of input tensor by specified dimensions.

**Attributes**:

    No attributes available.

**Inputs**

*   **1**: `data` - Input tensor of type *T* with data for the DFT transformation. Type of elements is any supported floating-point type. The last dimension of the input tensor must be equal to 2, that is the input tensor shape must have the form `[D_0, D_1, ..., D_{N-1}, 2]`, representing the real and imaginary components of complex numbers in `[:, ..., :, 0]` and in `[:, ..., :, 1]` correspondingly. **Required.**
*   **2**: `axes` - 1D tensor of type *T_IND* specifying dimension indices where DFT is applied, and `axes` is any unordered list of indices of different dimensions of input tensor, for example, `[0, 4]`, `[4, 0]`, `[4, 2, 1]`, `[1, 2, 3]`, `[-3, 0, -2]`. These indices should be integers from `-(r - 1)` to `(r - 2)` inclusively, where `r = rank(data)`. A negative axis `a` is interpreted as an axis `r - 1 + a`. Other dimensions do not change. The order of elements in `axes` attribute matters, and is mapped directly to elements in the third input `signal_size`. **Required.**
*   **NOTE**: The following constraint must be satisfied: `rank(data) >= len(axes) + 1 and input_shape[-1] == 2 and (rank(data) - 1) not in axes and (-1) not in axes`.
*   **3**: `signal_size` - 1D tensor of type *T_SIZE* describing signal size with respect to axes from the input `axes`. If `signal_size[i] == -1`, then DFT is calculated for full size of the axis `axes[i]`. If `signal_size[i] > input_shape[: r - 1][axes[i]]`, then input data are zero-padded with respect to the axis `axes[i]` at the end. Finally, `signal_size[i] < input_shape[: r - 1][axes[i]]`, then input data are trimmed with respect to the axis `axes[i]`. More precisely, if `signal_size[i] < input_shape[: r - 1][axes[i]]`, the slice `0: signal_size[i]` of the axis `axes[i]` is considered. Optional, with default value `[input_shape[: r - 1][a] for a in axes]`.
*   **NOTE**: If the input `signal_size` is specified, the size of `signal_size` must be the same as the size of `axes`.

**Outputs**

*   **1**: Resulting tensor with elements of the same type as input `data` tensor. The shape of the output is calculated as follows: if the input `signal_size` is not specified, the shape of output is the same as the shape of `data`. Otherwise, `output_shape[axis] = input_shape[axis]` for `axis not in axes`, and if `signal_size[i] == -1`, then `output_shape[: r - 1][axes[i]] = input_shape[: r - 1][axes[i]]`, else `output_shape[: r - 1][axes[i]] = signal_size[i]`.

**Types**

* *T*: floating-point type.

* *T_IND*: `int64` or `int32`.

* *T_SIZE*: `int64` or `int32`.

**Detailed description**: *DFT* performs the discrete Fourier transformation of input tensor with respect to specified axes. Calculations are performed according to the following rules.

For simplicity, assume that an input tensor `A` has the shape `[B_0, ..., B_{k-1}, M_0, ..., M_{r-1}, 2]`, `axes=[k+1,...,k+r]`, and `signal_size=[S_0,...,S_{r-1}]`.

Let `D` be an input tensor `A`, taking into account the `signal_size`, and, hence, `D` has the shape `[B_0, ..., B_{k-1}, S_0, ..., S_{r-1}, 2]`.

Next, put
\f[X[j_0,\dots,j_{k-1},j_k,\dots,j_{k+r}]=D[j_0,\dots,j_{k-1},j_k,\dots,j_{k+r},0]+iD[j_0,\dots,j_{k-1},j_k,\dots,j_{k+r},1]\f]
for all indices `j_0,...,j_{k+r}`, where `i` is an imaginary unit, that is `X` is a complex tensor.

Then the discrete Fourier transform is the tensor `Y` of the same shape as the tensor `X`, such that
\f[Y[n_0,\dots,n_{k-1},m_0,\dots,m_{r-1}]=\sum\limits_{p_0=0}^{S_0}\cdots\sum\limits_{p_{r-1}=0}^{S_{r-1}}X[n_0,\dots,n_{k-1},j_0,\dots,j_{r-1}]\exp\left(-2\pi i\sum\limits_{q=0}^{r-1}\frac{m_qj_q}{S_s}\right)\f]
for all indices `n_0,...,n_{k-1}`, `m_0,...,m_{r-1}`, and the result of the operation is the real tensor `Z` with the shape `[B_0, ..., B_{k-1}, S_0, ..., S_{r-1}, 2]` and such that
\f[Z[n_0,\dots,n_{k-1},m_0,\dots,m_{r-1}, 0]=Re Y[n_0,\dots,n_{k-1},m_0,\dots,m_{r-1}],\f]
\f[Z[n_0,\dots,n_{k-1},m_0,\dots,m_{r-1}, 1]=Im Y[n_0,\dots,n_{k-1},m_0,\dots,m_{r-1}].\f]

Calculations for the generic case of axes and signal sizes are similar.

**Example**:

There is no `signal_size` input (4D input tensor):
```xml
<layer ... type="DFT" ... >
    <input>
        <port id="0">
            <dim>1</dim>
            <dim>320</dim>
            <dim>320</dim>
            <dim>2</dim>
        </port>
        <port id="1">
            <dim>2</dim> <!-- axes input contains [1, 2] -->
        </port>
    <output>
        <port id="2">
            <dim>1</dim>
            <dim>320</dim>
            <dim>320</dim>
            <dim>2</dim>
        </port>
    </output>
</layer>
```

There is no `signal_size` input (3D input tensor):
```xml
<layer ... type="DFT" ... >
    <input>
        <port id="0">
            <dim>320</dim>
            <dim>320</dim>
            <dim>2</dim>
        </port>
        <port id="1">
            <dim>2</dim> <!-- axes input contains [0, 1] -->
        </port>
    <output>
        <port id="2">
            <dim>320</dim>
            <dim>320</dim>
            <dim>2</dim>
        </port>
    </output>
</layer>
```


There is `signal_size` input (4D input tensor):
```xml
<layer ... type="DFT" ... >
    <input>
        <port id="0">
            <dim>1</dim>
            <dim>320</dim>
            <dim>320</dim>
            <dim>2</dim>
        </port>
        <port id="1">
            <dim>2</dim> <!-- axes input contains [1, 2] -->
        </port>
        <port id="2">
            <dim>2</dim> <!-- signal_size input contains [512, 100] -->
        </port>
    <output>
        <port id="3">
            <dim>1</dim>
            <dim>512</dim>
            <dim>100</dim>
            <dim>2</dim>
        </port>
    </output>
</layer>
```


There is `signal_size` input (3D input tensor):
```xml
<layer ... type="DFT" ... >
    <input>
        <port id="0">
            <dim>320</dim>
            <dim>320</dim>
            <dim>2</dim>
        </port>
        <port id="1">
            <dim>2</dim> <!-- axes input contains [0, 1] -->
        </port>
        <port id="2">
            <dim>2</dim> <!-- signal_size input contains [512, 100] -->
        </port>
    <output>
        <port id="3">
            <dim>512</dim>
            <dim>100</dim>
            <dim>2</dim>
        </port>
    </output>
</layer>
```


There is `signal_size` input (5D input tensor, `-1` in `signal_size`, unsorted axes):
```xml
<layer ... type="DFT" ... >
    <input>
        <port id="0">
            <dim>16</dim>
            <dim>768</dim>
            <dim>580</dim>
            <dim>320</dim>
            <dim>2</dim>
        </port>
        <port id="1">
            <dim>3</dim> <!-- axes input contains  [3, 1, 2] -->
        </port>
        <port id="2">
            <dim>3</dim> <!-- signal_size input contains [170, -1, 1024] -->
        </port>
    <output>
        <port id="3">
            <dim>16</dim>
            <dim>768</dim>
            <dim>1024</dim>
            <dim>170</dim>
            <dim>2</dim>
        </port>
    </output>
</layer>
```


There is `signal_size` input (5D input tensor, `-1` in `signal_size`, unsorted axes, the second example):
```xml
<layer ... type="DFT" ... >
    <input>
        <port id="0">
            <dim>16</dim>
            <dim>768</dim>
            <dim>580</dim>
            <dim>320</dim>
            <dim>2</dim>
        </port>
        <port id="1">
            <dim>3</dim> <!-- axes input contains  [3, 0, 2] -->
        </port>
        <port id="2">
            <dim>3</dim> <!-- signal_size input contains [258, -1, 2056] -->
        </port>
    <output>
        <port id="3">
            <dim>16</dim>
            <dim>768</dim>
            <dim>2056</dim>
            <dim>258</dim>
            <dim>2</dim>
        </port>
    </output>
</layer>
```
