Discrete Fourier Transformation for real-valued input (RDFT)#
Versioned name: RDFT-9
Category: Signal processing
Short description: RDFT operation performs the discrete real-to-complex Fourier transformation of the input tensor by specified dimensions.
Attributes:
No attributes available.
Inputs
1:
data
- Input tensor of type T with data for the RDFT transformation. Required.2:
axes
- 1D tensor of type T_IND specifying dimension indices where RDFT is applied, andaxes
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
tor - 1
inclusively, wherer = rank(data)
. A negative axisa
is interpreted as an axisr + a
. Other dimensions do not change. The order of elements inaxes
attribute matters, and is mapped directly to elements in the third inputsignal_size
. Required.3:
signal_size
- 1D tensor of type T_SIZE describing signal size with respect to axes from the inputaxes
. Ifsignal_size[i] == -1
, then RDFT is calculated for full size of the axisaxes[i]
. Ifsignal_size[i] > data_shape[axes[i]]
, then input data is zero-padded with respect to the axisaxes[i]
at the end. Finally,signal_size[i] < data_shape[axes[i]]
, then input data is trimmed with respect to the axisaxes[i]
. More precisely, ifsignal_size[i] < data_shape[axes[i]]
, the slice0: signal_size[i]
of the axisaxes[i]
is considered. Optionally, with default value[data_shape[a] for a in axes]
.NOTE: If the input
signal_size
is specified, the size ofsignal_size
must be the same as the size ofaxes
.
Outputs
1: Resulting tensor with elements of the same type as input
data
tensor and with rankr + 1
, wherer = rank(data)
. The shape of the output has the form[S_0, S_1, ..., S_{r-1}, 2]
, where allS_a
are calculated as follows:
Calculate
normalized_axes
, where eachnormalized_axes[i] = axes[i]
, ifaxes[i] >= 0
, andnormalized_axes[i] = axes[i] + r
otherwise.If
a not in normalized_axes
, thenS_a = data_shape[a]
.If
a in normalized_axes
, thena = normalized_axes[i]
for somei
.When
i != len(normalized_axes) - 1
,S_a
is calculated asS_a = data_shape[a]
if thesignal_size
input is not specified, or, if it is specified,signal_size[i] = -1
; andS_a = signal_size[a]
otherwise.When
i = len(normalized_axes) - 1
,S_a
is calculated asS_a = data_shape[a] // 2 + 1
if thesignal_size
input is not specified, or, if it is specified,signal_size[i] = -1
; andS_a = signal_size[a] // 2 + 1
otherwise.
Types
T: any supported floating-point type.
T_IND:
int64
orint32
.T_SIZE:
int64
orint32
.
Detailed description: RDFT performs the discrete Fourier transformation of real-valued 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_{q-1}]
, axes=[k,...,k+q-1]
, and signal_size=[S_0,...,S_{q-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_{q-1}]
.
Next, let
for all indices j_0,...,j_{k+q-1}
, be a real-valued input tensor.
Then the transformation RDFT of the tensor X
is the tensor Y
of the shape [B_0, ..., B_{k-1}, S_0 // 2 + 1, ..., S_{r-1} // 2 + 1]
, such that
for all indices n_0,...,n_{k-1}
, m_0,...,m_{q-1}
.
Calculations for the generic case of axes and signal sizes are similar.
Example:
There is no signal_size
input (3D input tensor):
<layer ... type="RDFT" ... >
<input>
<port id="0">
<dim>1</dim>
<dim>320</dim>
<dim>320</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>161</dim>
<dim>2</dim>
</port>
</output>
</layer>
There is no signal_size
input (2D input tensor):
<layer ... type="RDFT" ... >
<input>
<port id="0">
<dim>320</dim>
<dim>320</dim>
</port>
<port id="1">
<dim>2</dim> <!-- axes input contains [0, 1] -->
</port>
<output>
<port id="2">
<dim>320</dim>
<dim>161</dim>
<dim>2</dim>
</port>
</output>
</layer>
There is signal_size
input (3D input tensor):
<layer ... type="RDFT" ... >
<input>
<port id="0">
<dim>1</dim>
<dim>320</dim>
<dim>320</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>51</dim>
<dim>2</dim>
</port>
</output>
</layer>
There is signal_size
input (2D input tensor):
<layer ... type="RDFT" ... >
<input>
<port id="0">
<dim>320</dim>
<dim>320</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>51</dim>
<dim>2</dim>
</port>
</output>
</layer>
There is signal_size
input (4D input tensor, -1
in signal_size
, unsorted axes):
<layer ... type="RDFT" ... >
<input>
<port id="0">
<dim>16</dim>
<dim>768</dim>
<dim>580</dim>
<dim>320</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>513</dim>
<dim>170</dim>
<dim>2</dim>
</port>
</output>
</layer>
There is signal_size
input (4D input tensor, -1
in signal_size
, unsorted axes, the second example):
<layer ... type="RDFT" ... >
<input>
<port id="0">
<dim>16</dim>
<dim>768</dim>
<dim>580</dim>
<dim>320</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>1029</dim>
<dim>258</dim>
<dim>2</dim>
</port>
</output>
</layer>