Beam-Delay-Doppler (BDD) Domain Analysis

• Beam–Delay–Doppler (BDD) is a joint domain that indexes channel components by beam, delay, and Doppler coordinates.
• It extends delay–Doppler processing with beamspace transforms, enabling sparse, structured representations in high-mobility and mmWave/THz applications.
• BDD leverages discrete transforms to create quasi-static channel models, crucial for advanced MIMO and integrated sensing systems.

Searching arXiv for papers on Beam–Delay–Doppler, delay–Doppler, and related MIMO/beamspace formulations. Beam–Delay–Doppler (BDD) domain denotes a joint representation of wireless propagation and waveform processing in which channel or signal components are indexed by a spatial beam or angle coordinate together with delay and Doppler coordinates. In the cited literature, this representation is often introduced not as a standalone standardized transform, but as a natural extension of delay–Doppler (DD) processing—particularly OTFS/ODDM-type formulations—combined with beamspace processing through spatial DFTs or beamforming. The resulting domain is therefore best understood as a structured angle–delay–velocity description in which the channel can appear sparse and quasi-static over suitable observation intervals, especially in high-mobility, massive-MIMO, mmWave/THz, and ISAC settings (Arous et al., 14 Oct 2025, Kulhandjian et al., 2024).

1. Conceptual status and relation to transform-domain signaling

The BDD domain sits at the intersection of two lines of development. The first is the migration from time–frequency multicarrier design toward DD-domain modulation, where channel variation is represented in terms of path delays and Doppler shifts rather than rapidly changing subcarrier gains. The second is the use of beamspace as a spatial processing domain, usually obtained from an aperture-domain MIMO channel by a DFT over antennas. When these are combined, the natural coordinates become beam, delay, and Doppler (Arous et al., 14 Oct 2025).

In this sense, BDD is not merely time–frequency processing with an added spatial FFT. The cited framework distinguishes between physical domains and “logical” domains: DD is treated as a logical domain reached by structured DFT-based pre- and post-processing, while beamspace is an additional processing domain arising from array geometry and spatial transforms. A BDD representation is therefore a multi-dimensional transform-domain construction, not just a relabeling of time–frequency samples. This also explains why some papers do not explicitly define the term “BDD” even while providing all of its ingredients (Arous et al., 14 Oct 2025).

The DD half of this construction inherits the central motivation of ODDM and related DD-domain multicarrier schemes: doubly selective channels are sparse and approximately invariant in DD over a stationary interval, whereas their time–frequency representation is typically denser and more rapidly varying. ODDM further emphasizes that practical waveform design need only satisfy sufficient (bi)orthogonality on a finite grid, rather than global orthogonality on the entire time–frequency plane, which opens a finite-dimensional design space that is directly relevant once a beam axis is added (Lin et al., 2023).

2. Mathematical construction from delay–Doppler and beamspace

A standard OTFS frame places symbols on a discrete DD grid

$$\Gamma=\left\{\left(m\Delta\tau,n\Delta\nu\right): m=0,\dots,M-1,\; n=0,\dots,N-1\right\},$$

with

$$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$

The DD symbols are mapped to the time–frequency grid by the ISFFT, converted to a continuous-time waveform by the Heisenberg transform, and recovered by cross-ambiguity sampling followed by the SFFT. This TX/RX chain is the basic DD processing block inherited by BDD systems (Kulhandjian et al., 2024).

The spatial extension is introduced by beamspace processing. For a uniform planar array with $N_x\times N_y$ elements, the beamspace transform is a 2D DFT over the aperture: $$\mathbf{H}_{\mathrm{beam}}=\mathbf{F}_{N_x}\,\mathbf{H}_{\mathrm{aperture}}\,\mathbf{F}_{N_y}^{H}.$$ This maps element-domain channels into discrete angular or beam indices. Once each antenna-domain time–frequency channel has been transformed to DD, or equivalently once DD-domain quantities are stacked across antennas and spatially transformed, one obtains a beam–delay–Doppler tensor. A conceptual BDD operator can be written as

$$\mathbf{H}_{\mathrm{BDD}}=\mathcal{T}_{\mathrm{BDD}}\,\mathbf{H}_{\mathrm{TF}},\qquad \mathcal{T}_{\mathrm{BDD}}=\left(\mathbf{F}_{\mathrm{spat}}\otimes \mathbf{F}_N\otimes \mathbf{F}_M\right),$$

with the DD part following the appropriate OTFS sign convention (Arous et al., 14 Oct 2025).

A corresponding discrete BDD resource may therefore be indexed as $(b,m,n)$ or $(b,l,k)$, where $b$ is a beam index, $l$ a delay index, and $k$ a Doppler index. This yields a 3D grid for 1D beamspace and, when both azimuth and elevation are retained, effectively a 4D representation with two angular indices plus delay and Doppler (Arous et al., 14 Oct 2025).

3. Channel representation, sparsity, and nonstationarity

The DD-domain channel is commonly modeled as a sparse set of multipath components,

$$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$0

or in normalized discrete form through delay indices $$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$1 and Doppler indices $$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$2. In BDD, the natural extension is

$$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$3

where $$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$4 is the beamforming gain of beam $$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$5 for path angle $$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$6. This expresses a physically meaningful decomposition: each path occupies a small subset of beams together with a localized delay–Doppler support (Zhou et al., 30 Sep 2025).

The principal structural claim attached to BDD is joint sparsity. DD representations already tend to isolate multipath into a small number of delay–Doppler taps; beamspace adds angular sparsity, especially in mmWave and THz channels. The literature explicitly notes that DD-beamspace sparsity can reduce computational burden and enable parallelization, because within each beam the channel often remains sparse in delay and Doppler (Arous et al., 14 Oct 2025).

Channel measurement results in high-speed railway scenarios sharpen this picture. In that setting, DD-domain statistics are quasi-stationary over intervals on the order of $$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$7 ms, while the quasi-invariant interval of individual DD fading coefficients is only on the order of ms. The reported quasi-stationary intervals are approximately $$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$8 ms, $$\Delta\tau=\frac{1}{M\Delta f},\qquad \Delta\nu=\frac{1}{NT},\qquad \Delta f=\frac{1}{T}.$$9 ms, and $N_x\times N_y$0 ms for weak, moderate, and strong time-varying conditions, whereas the minimum quasi-invariant intervals in strong conditions can be as small as about $N_x\times N_y$1 ms (Zhou et al., 30 Sep 2025). In a BDD interpretation, this means that beam–delay–Doppler power structure can remain