---
title: Beam-Delay-Doppler (BDD) Domain Analysis
url: https://www.emergentmind.com/topics/beam-delay-doppler-bdd-domain
type: topic
---

# Beam-Delay-Doppler (BDD) Domain Analysis

Searching arXiv for recent 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 [2510.12648][2405.19182].

## 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 [2510.12648].

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 [2510.12648].

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 [2308.01802].

## 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 [2405.19182].

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 [2510.12648].

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 [2510.12648].

## 3. Channel representation, sparsity, and nonstationarity

The DD-domain channel is commonly modeled as a sparse set of multipath components,
\[
h(\tau,\nu)=\sum_{i=1}^{P} h_i\,\delta(\tau-\tau_i)\,\delta(\nu-\nu_i),
\]
or in normalized discrete form through delay indices \(l_i\) and Doppler indices \(k_i\). In BDD, the natural extension is
\[
h[b,l,k]=\sum_{i=1}^{P} h_i\,u_b(\theta_i)\,\delta(l-l_i)\,\delta(k-k_i),
\]
where \(u_b(\theta_i)\) is the beamforming gain of beam \(b\) for path angle \(\theta_i\). This expresses a physically meaningful decomposition: each path occupies a small subset of beams together with a localized delay–Doppler support [2509.25854].

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 [2510.12648].

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 \(10^2\) 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 \(392\) ms, \(172.2\) ms, and \(100.8\) ms for weak, moderate, and strong time-varying conditions, whereas the minimum quasi-invariant intervals in strong conditions can be as small as about \(0.93\) ms [2509.25854]. In a BDD interpretation, this means that beam–delay–Doppler power structure can remain

Source: https://www.emergentmind.com/topics/beam-delay-doppler-bdd-domain