Inverse Discrete complex-to-real Fourier Transformation (IRDFT)#
Versioned name: IRDFT-9
Category: Signal processing
Short description: IRDFT operation performs the inverse complex-to-real discrete 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 IRDFT transformation. 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 IRDFT is applied, andaxes
is any unordered list of indices of different dimensions of the 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, wherer = rank(data)
. A negative axisa
is interpreted as an axisr - 1 + a
. Other dimensions do not change. The order of elements in theaxes
attribute matters, and is mapped directly to elements in the third inputsignal_size
. Required.Note
The following constraint must be satisfied:
rank(data) >= len(axes) + 1 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 inputaxes
. Ifsignal_size[i] == -1
, then IRDFT is calculated for full size of the axisaxes[i]
. Ifsignal_size[i] > data_shape[: r - 1][axes[i]]
, then input data is zero-padded with respect to the axisaxes[i]
at the end. Finally, ifsignal_size[i] < data_shape[: r - 1][axes[i]]
, then input data is trimmed with respect to the axisaxes[i]
. More precisely, ifsignal_size[i] < data_shape[: r - 1][axes[i]]
, the slice0: signal_size[i]
of the axisaxes[i]
is considered. Optionally, with default value[data_shape[: r - 1][a] for a in axes]
.Note
If the input
signal_size
is specified, then 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-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 - 1
otherwise.If
a not in normalized_axes
, thenS_a = data_shape[a]
.If
a in normalized_axes
, thena = normalized_axes[i]
for somei
. In such case,S_a = 2 * (data_shape[a] - 1)
if thesignal_size
input is not specified, or, if it is specified,signal_size[i] = -1
; andS_a = signal_size[a]
otherwise. + Wheni != 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. + Wheni = len(normalized_axes) - 1
,S_a
is calculated asS_a = 2 * (data_shape[a] - 1)
if thesignal_size
input is not specified, or, if it is specified,signal_size[i] = -1
; andS_a = signal_size[a]
otherwise.
Types
T: any supported floating-point type.
T_IND:
int64
orint32
.T_SIZE:
int64
orint32
.
Detailed description: IRDFT performs the discrete Fourier transformation of the input tensor, 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}, 2]
, axes=[k,...,k + q - 1]
, and signal_size=[S_0,...,S_{q-1}]
.
Let D
be a value of the input tensor A
.
Next, put
for all indices j_0,...,j_{k+q-1}
, where i
is an imaginary unit, that is X
is a complex tensor.
Define the complex tensor F
with the shape [B_0, ..., B_{k-1}, 2 * (M_0 - 1), ..., 2 * (M_{q-1} - 1)]
using the formula
Construct the complex tensor G
with the shape [B_0, ..., B_{k-1}, S_0, ..., S_{q-1}]
by the following way. If S_a > 2 * (M_a - 1)
, then the axis k + a
of F
will be padded by zeros; if S_a < 2 * (M_a - 1)
, then the axis k + a
of F
will be trimmed, that is, we will consider only the slice 0: S_a
of this axis; finally, if S_a = 2 * (M_a - 1)
, then we consider the full axis k + a
of F
.
Let Y
be a complex tensor with the shape [B_0, ..., B_{k-1}, S_0, ..., S_{q-1}]
such that
for all indices n_0,...,n_{k-1}
, m_0,...,m_{q-1}
.
Finally, the result of the inverse discrete complex-to-real Fourier transform is a real part of the tensor Y.
Calculations for the generic case of axes and signal sizes are similar.
Example:
There is no signal_size
input (4D input tensor):
<layer ... type="IRDFT" ... >
<input>
<port id="0">
<dim>1</dim>
<dim>161</dim>
<dim>161</dim>
<dim>2</dim>
</port>
<port id="1">
<dim>2</dim> <!-- [1, 2] -->
</port>
<output>
<port id="2">
<dim>1</dim>
<dim>161</dim>
<dim>320</dim>
</port>
</output>
</layer>
There is no signal_size
input (3D input tensor):
<layer ... type="IRDFT" ... >
<input>
<port id="0">
<dim>161</dim>
<dim>161</dim>
<dim>2</dim>
</port>
<port id="1">
<dim>2</dim> <!-- [0, 1] -->
</port>
<output>
<port id="2">
<dim>161</dim>
<dim>320</dim>
</port>
</output>
</layer>
There is signal_size
input (4D input tensor):
<layer ... type="IRDFT" ... >
<input>
<port id="0">
<dim>1</dim>
<dim>161</dim>
<dim>161</dim>
<dim>2</dim>
</port>
<port id="1">
<dim>2</dim> <!-- [1, 2] -->
</port>
<port id="2">
<dim>2</dim> <!-- [512, 100] -->
</port>
<output>
<port id="3">
<dim>1</dim>
<dim>512</dim>
<dim>100</dim>
</port>
</output>
</layer>
There is signal_size
input (3D input tensor):
<layer ... type="IRDFT" ... >
<input>
<port id="0">
<dim>161</dim>
<dim>161</dim>
<dim>2</dim>
</port>
<port id="1">
<dim>2</dim> <!-- [0, 1] -->
</port>
<port id="2">
<dim>2</dim> <!-- [512, 100] -->
</port>
<output>
<port id="3">
<dim>512</dim>
<dim>100</dim>
</port>
</output>
</layer>
There is signal_size
input (5D input tensor, -1
in signal_size
, unsorted axes):
<layer ... type="IRDFT" ... >
<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>
</port>
</output>
</layer>
There is signal_size
input (5D input tensor, -1
in signal_size
, unsorted axes, the second example):
<layer ... type="IRDFT" ... >
<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>
</port>
</output>
</layer>