Fixed DFTs documentations. (#11762)

docs/ops/signals/DFT_7.md

@@ -32,16 +32,16 @@
 
 **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}]`.
+For simplicity, assume that an input tensor `A` has the shape `[B_0, ..., B_{k-1}, M_0, ..., M_{r-1}, 2]`, `axes=[k,...,k+r-1]`, 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.
+\f[X[j_0,\dots,j_{k-1},j_k,\dots,j_{k+r-1}]=D[j_0,\dots,j_{k-1},j_k,\dots,j_{k+r-1},0]+iD[j_0,\dots,j_{k-1},j_k,\dots,j_{k+r-1},1]\f]
+for all indices `j_0,...,j_{k+r-1}`, 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]
+\f[Y[n_0,\dots,n_{k-1},m_0,\dots,m_{r-1}]=\sum\limits_{j_0=0}^{S_0-1}\cdots\sum\limits_{j_{r-1}=0}^{S_{r-1}-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_q}\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]

docs/ops/signals/IDFT_7.md

@@ -13,7 +13,7 @@
 **Inputs**
 
 *   **1**: `data` - Input tensor of type *T* with data for the IDFT 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**: **2**: `axes` - 1D tensor of type *T_IND* specifying dimension indices where IDFT 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.**
+*   **2**: `axes` - 1D tensor of type *T_IND* specifying dimension indices where IDFT 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 IDFT 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, if `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, then the size of `signal_size` must be the same as the size of `axes`.
@@ -32,16 +32,16 @@
 
 **Detailed description**: *IDFT* performs the discrete Fourier transformation of 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_{r-1}, 2]`, `axes=[k+1,...,k+r]`, and `signal_size=[S_0,...,S_{r-1}]`.
+For simplicity, assume that an input tensor `A` has the shape `[B_0, ..., B_{k-1}, M_0, ..., M_{r-1}, 2]`, `axes=[k,...,k+r-1]`, 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.
+\f[X[j_0,\dots,j_{k-1},j_k,\dots,j_{k+r-1}]=D[j_0,\dots,j_{k-1},j_k,\dots,j_{k+r-1},0]+iD[j_0,\dots,j_{k-1},j_k,\dots,j_{k+r-1},1]\f]
+for all indices `j_0,...,j_{k+r-1}`, where `i` is an imaginary unit, that is `X` is a complex tensor.
 
 Then the inverse discrete Fourier transform is the tensor `Y` of the same shape as the tensors `X`, such that
-\f[Y[n_0,\dots,n_{k-1},m_0,\dots,m_{r-1}]=\frac{1}{\prod\limits_{j=0}^{r-1}S_j}\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]
+\f[Y[n_0,\dots,n_{k-1},m_0,\dots,m_{r-1}]=\frac{1}{\prod\limits_{q=0}^{r-1}S_q}\sum\limits_{j_0=0}^{S_0-1}\cdots\sum\limits_{j_{r-1}=0}^{S_{r-1}-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_q}\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]

docs/ops/signals/IRDFT_9.md

@@ -54,7 +54,7 @@ Define the complex tensor `F` with the shape `[B_0, ..., B_{k-1}, 2 * (M_0 - 1),
 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
-\f[Y[n_0,\dots,n_{k-1},m_0,\dots,m_{q-1}]=\frac{1}{\prod\limits_{j=0}^{q-1}S_j}\sum\limits_{p_0=0}^{S_0}\cdots\sum\limits_{p_{q-1}=0}^{S_{q-1}}X[n_0,\dots,n_{k-1},j_0,\dots,j_{q-1}]\exp\left(2\pi i\sum\limits_{b=0}^{q-1}\frac{m_bj_b}{S_b}\right)\f]
+\f[Y[n_0,\dots,n_{k-1},m_0,\dots,m_{q-1}]=\frac{1}{\prod\limits_{b=0}^{q-1}S_b}\sum\limits_{j_0=0}^{S_0-1}\cdots\sum\limits_{j_{q-1}=0}^{S_{q-1}-1}X[n_0,\dots,n_{k-1},j_0,\dots,j_{q-1}]\exp\left(2\pi i\sum\limits_{b=0}^{q-1}\frac{m_bj_b}{S_b}\right)\f]
 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`.

docs/ops/signals/RDFT_9.md

@@ -39,16 +39,16 @@
 
 **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_{1-1}]`.
+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_{1-1}]`.
+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
 \f[X=X[j_0,\dots,j_{k-1},j_k,\dots,j_{k+q-1}]\f]
 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
-\f[Y[n_0,\dots,n_{k-1},m_0,\dots,m_{q-1}]=\sum\limits_{p_0=0}^{S_0}\cdots\sum\limits_{p_{q-1}=0}^{S_{q-1}}X[n_0,\dots,n_{k-1},j_0,\dots,j_{q-1}]\exp\left(-2\pi i\sum\limits_{b=0}^{q-1}\frac{m_bj_b}{S_b}\right)\f]
+\f[Y[n_0,\dots,n_{k-1},m_0,\dots,m_{q-1}]=\sum\limits_{j_0=0}^{S_0-1}\cdots\sum\limits_{j_{q-1}=0}^{S_{q-1}-1}X[n_0,\dots,n_{k-1},j_0,\dots,j_{q-1}]\exp\left(-2\pi i\sum\limits_{b=0}^{q-1}\frac{m_bj_b}{S_b}\right)\f]
 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.