Definitions#

Let $w$ be a set of $M$ finite $d$-dimensional discrete sequences $w^m$ of length(s) $n_1\times n_2\times \cdots \times n_d$ ($d\in {\mathbb{Z}}_{>0}$, $M\in {\mathbb{Z}}_{>0}$, $n_{\ell }\in {\mathbb{Z}}_{>0}\mathrm{\forall }\ell \in \{1,2,\dots ,d\}$, and $m\in \{0,1,\dots ,M-1\}$). Let $w_{k_1,k_2,\dots ,k_d}^m$ be the entry of multi-index $(k_1,k_2,\dots ,k_d)$ in $w^m$ wherein integer indices $k_{\ell }$ are such that $0\le k_{\ell }<n_{\ell },\mathrm{\forall }\ell \in \{1,2,\dots ,d\}$.

For every $m\in \{0,1,\dots ,M-1\}$, the DFT of sequence $w^m$ is the $d$-dimensional discrete sequence $z^m$ of length(s) $n_1\times n_2\times \cdots \times n_d$ whose entries are defined as

where ${ı}^2=-1$. In (1), $\delta$ determines one of the two “directions” of the DFT: $\delta =-1$ defines the “forward DFT” while $\delta =+1$ defines the “backward DFT”. ${\sigma }_{\delta }$ is a (real) scaling factor associated with either operation.

The domain of input (resp. output) discrete sequences for a forward (resp. backward) DFT is referred to as “forward domain”. Conversely, the domain of output (resp. input) discrete sequences for forward (resp. backward) DFT is referred to as “backward domain”.

oneMath supports single-precision (fp32) and double-precision (fp64) floating-point arithmetic for the calculation of DFTs, using two kinds of forward domains:

• the set of complex $d$-dimensional discrete sequences, referred to as “complex forward domain”;

• the set of real $d$-dimensional discrete sequences, referred to as “real forward domain”.

Similarly, we refer to DFTs of complex (resp. real) forward domain as “complex DFTs” (resp. “real DFTs”). Regardless of the kind of forward domain, the backward domain’s data sequences are always complex.

The calculation of the same DFT for several, i.e., $M>1$, data sets of the same kind of forward domain, using the same precision is referred to as a “batched DFT”.

Elementary range of indices#

In general, all entries of multi-indices $(k_1,\dots ,k_d)$ such that $0\le k_{\ell }<n_{\ell },\mathrm{\forall }\ell \in \{1,\dots ,d\}$ unambiguously determine any relevant $d$-dimensional sequence unambiguously (for any valid $m$). In case of real DFTs, the data sequences in backward domain can be fully determined from a smaller range of indices. Indeed, if all entries of $w$ are real in (1), then the entries of $z$ are complex and, for any valid $m$, ${\left(z_{k_1,k_2,\dots ,k_d}^m\right)}^{\ast }=z_{j_1,j_2,\dots ,j_d}^m$ where $j_{\ell }=(n_{\ell }-k_{\ell })(\mathrm{mod}{n}_{\ell }),\mathrm{\forall }\ell \in \{1,\dots ,d\}$ and ${\lambda }^{\ast }$ represents the conjugate of complex number $\lambda$. This conjugate symmetry relation makes roughly half the data redundant in backward domain: in case of real DFTs, the data sequences in backward domain can be fully determined even if one of the $d$ indices $k_{\ell }$ is limited to the range $0\le k_{\ell }\le \lfloor \frac{n_{\ell }}{2}\rfloor$. In oneMath, the index $k_d$, i.e., the last dimension’s index, is restricted as such to capture an elementary set of non-redundant entries for data sequences belonging to the backward domain of real DFTs.

In other words, oneMath expects and produces a set of $M$ $d$-dimensional data sequences ${(\cdot )}_{k_1,k_2,\dots ,k_d}^m$ with integer indices $m$ and $k_{\ell }(\ell \in \{1,\dots ,d\})$ in the elementary range

• $0\le m<M$;

• $0\le k_j<n_j,\mathrm{\forall }j\in \{1,\dots ,d-1\}$, if $d>1$;

• $0\le k_d<n_d$, except for backward domain’s data sequences of real DFTs;

• $0\le k_d\le \lfloor \frac{n_d}{2}\rfloor$, for backward domain’s data sequences of real DFTs.

Recommended usage#

The desired DFT to be computed is entirely defined by an object desc of a specialization of the oneapi::math::dft::descriptor class template. The desired floating-point format and kind of forward domain are determined by desc’s particular class, i.e., by the specialization values of the template parameters prec and dom of the descriptor class template, respectively. Once desc is created, the length(s) $\{n_1,n_2,\dots ,n_d\}$ (and the dimension $d$) cannot be changed, as they are read-only parameters. Other configuration details for the DFT under consideration may be specified by invoking the appropriate configuration-setting member function(s) of desc for every relevant configuration parameter (e.g., the number $M$ of sequences to consider in case of a batched DFT). Once configured as desired, desc must be initialized for computation by using its committing member function, which requires a sycl::queue object. The successful completion of that operation makes desc ready to compute the desired DFT as configured, for the particular device and context encapsulated by the latter. desc may then be used with user-provided, device-accessible data, in a oneapi::math::dft::compute_forward (resp. oneapi::math::dft::compute_backward) function to enqueue operations relevant to the desired forward (resp. backward) DFT calculations.