---
title: Affine-Doppler Division Multiplexing (ADDM)
url: https://www.emergentmind.com/topics/affine-doppler-division-multiplexing-addm
type: topic
---

# Affine-Doppler Division Multiplexing (ADDM)

Affine-Doppler Division Multiplexing (ADDM) is an orthogonal multicarrier waveform proposed for high-mobility wireless communications systems. In its defining formulation, ADDM modulates information symbols in the Affine-Doppler (A-D) domain by means of a two-dimensional transform, is presented as a generic framework that subsumes both Orthogonal Time-Frequency Space (OTFS) and Affine Frequency Division Multiplexing (AFDM) as particular cases, and is claimed to combine AFDM-like unambiguous Doppler and Doppler-resolution properties with a two-dimensional cyclic-shift structure similar to cyclic-prefix OTFS. Numerical results in the proposing paper show BER comparable to AFDM and better than OTFS in the tested high-mobility setting [2509.02116].

## 1. Terminology, origin, and relation to AFDM

ADDM entered the literature explicitly in 2025 as a new waveform for high-mobility wireless communications [2509.02116]. Its appearance followed several years of closely related work under the name AFDM, a DAFT-based chirp multicarrier waveform designed for doubly dispersive channels. In the AFDM literature, the central design objective is to choose affine parameters so that channel paths with distinct delays or Doppler shifts do not overlap in the DAFT domain, thereby yielding a full delay-Doppler representation and full diversity in linear time-varying channels [2104.11331].

The ADDM paper positions its contribution against two earlier families. OTFS is treated as the canonical two-dimensional delay-Doppler waveform, but one with unambiguous Doppler limited by the subcarrier spacing; AFDM is treated as the waveform with stronger Doppler behavior, but without direct compatibility with the large body of OTFS-oriented estimation and detection methods [2509.02116]. This suggests that ADDM is intended less as a rejection of AFDM than as a two-dimensional reformulation that preserves affine processing while restoring a block structure closer to OTFS.

The broader AFDM literature provides the immediate conceptual background. AFDM is repeatedly described as a chirp-based multicarrier waveform built on the discrete affine Fourier transform (DAFT), with OFDM recovered when \((c_1,c_2)=(0,0)\) and OCDM recovered at a fixed chirp setting \((c_1,c_2)=\left(\frac{1}{2N},\frac{1}{2N}\right)\) [2510.27192]. ADDM inherits this affine/chirp lineage, but relocates it into a genuine two-dimensional modulation architecture [2509.02116].

## 2. Transform-domain construction and modulation architecture

The defining ADDM data object is an \(N\times M\) symbol block \(\mathbf{X}\) in the Affine-Doppler domain, with indices \(n=0,\ldots,N-1\) and \(m=0,\ldots,M-1\) [2509.02116]. The waveform uses both the DFT and the DAFT. In matrix form, the DAFT is
\[
\mathbf{p}_{\mathrm{AT}}=\mathbf{A}\mathbf{s}, \qquad \mathbf{s}=\mathbf{A}^{\mathrm H}\mathbf{p}_{\mathrm{AT}},
\]
with
\[
\mathbf{A}=\mathbf{\Lambda}_{c_2}\mathbf{F}_{N_0}\mathbf{\Lambda}_{c_1},
\qquad
\mathbf{\Lambda}_{c}=\mathrm{diag}\!\left(e^{-2\pi c n^2},n=0,1,\ldots,N_0-1\right),
\]
where \(\mathbf{F}_{N_0}\) is the \(N_0\)-point DFT matrix and \(c_1,c_2\) are the affine parameters [2509.02116].

ADDM modulation proceeds in two stages. First, an IDFT is applied row-wise,
\[
\mathbf{P}=\mathbf{X}\mathbf{F}_M^{\mathrm H}.
\]
Second, an IDAFT is applied column-wise,
\[
\mathbf{S}=\mathbf{A}^{\mathrm H}\mathbf{P}
       =\mathbf{A}^{\mathrm H}\mathbf{X}\mathbf{F}_M^{\mathrm H}.
\]
Thus the A-D domain is mapped to a time-delay domain through a mixed DFT/DAFT synthesis operator [2509.02116].

Because DAFT induces chirp periodicity rather than ordinary periodicity, ADDM uses a chirp-periodic prefix (CPP). The prefixed samples satisfy
\[
{\bf{S}_{\mathrm{cp}}[n,k] = {\bf{S}[N + n,k]e^{ - j2\pi c_1\left( N^2 + 2Nn \right)},
\]
for \(n=-N_{\mathrm{cp}},\ldots,-1\), and the transmit matrix becomes
\[
\widetilde{\mathbf S}
=
\mathbf T_{\mathrm{cp}}^{N_{\mathrm{cp}},c_1}\mathbf S
=
\mathbf T_{\mathrm{cp}}^{N_{\mathrm{cp}},c_1}\mathbf A^{\mathrm H}\mathbf X\mathbf F_M^{\mathrm H}
\]
before serialization [2509.02116]. The ADDM paper also states that if \(c_1=c_2=0\), the A-D domain reduces to the frequency-Doppler domain, making the purely Fourier case an explicit special point in the framework [2509.02116].

## 3. Channel model and the A-D domain input-output relation

ADDM is formulated for doubly selective channels. In time domain, the received signal is modeled as
\[
\mathbf{r}[n]=\sum\limits_{i=1}^{P} h_i\,\mathbf{s}[n-l_i]e^{j2\pi f_i n}+\mathbf{w}[n],
\]
where \(P\) is the number of paths, \(h_i\) is the complex path gain, \(l_i\) is the delay in samples, \(f_i\) is the Doppler shift in digital frequency, and \(\mathbf w[n]\) is noise [2509.02116]. After serial-to-parallel conversion and CPP removal, the received time-delay matrix is written as
\[
{\bf{R} = \sum\limits_{i = 1}^P \widetilde h_i{\bf{\Gamma }_{\mathrm{cpp}_i}{\bf{\Delta }_{f_{1i}}{\bf{\Pi }^{l_i}{\bf{S}{\bf{\Delta }_{f_{2i}}  + {\bf{W},
\]
where \(\mathbf{\Gamma}_{\mathrm{cpp}_i}\) is the prefix-induced diagonal correction, \(\mathbf{\Pi}\) is the cyclic shift matrix, and \(\mathbf{\Delta}_{f_{1i}},\mathbf{\Delta}_{f_{2i}}\) are Doppler-related diagonal matrices [2509.02116].

Receiver processing is the inverse of the transmitter: a column-wise DAFT produces \(\mathbf Y\), then a row-wise DFT produces the A-D domain observation \(\mathbf Z\). The resulting block input-output relation is
\[
{\bf{Z} = \sum\limits_{i = 1}^P \widetilde h_i{\bf{H}_{\mathrm{A},i}{\bf{X} {\bf{H}_{\mathrm{D},i} + {\bf{\tilde W},
\]
with
\[
{\bf{H}_{\mathrm{A},i} \triangleq {\bf{A}{\bf{\Gamma }_{\mathrm{cpp}_i}{\bf{\Delta } _{f_{1i}}{\bf{\Pi }^{l_i}{\bf{A}^\mathrm{H},
\qquad
{\bf{H}_{\mathrm{D},i} \triangleq {\bf{F}_M^\mathrm{H}{\bf{\Delta } _{f_{2i}}{\bf{F}_M}.
\]
After vectorization,
\[
{\rm vec}({\bf{Z}) = \sum\limits_{i=1}^{P}\widetilde h_i\,{\bf H}_{\rm eff,i}\,{\rm vec}(\mathbf X) + {\rm vec}(\widetilde{\mathbf W}),
\qquad
\mathbf{H}_{\mathrm{eff},i}=\mathbf{H}_{\mathrm{D},i}^T\otimes\mathbf{H}_{\mathrm{A},i}^T.
\]
This is the core ADDM channel law: each path acts separably across an affine dimension and a Doppler dimension [2509.02116].

The paper then specializes to normalized Doppler variables
\[
\nu_i \triangleq Nf_i = \alpha_i + a_i,
\qquad
\nu_i' \triangleq N_s f_i = \beta_i + b_i,
\]
and assumes \(2Nc_1l_i\in\mathbb Z\) for exact cyclic behavior [2509.02116]. Under integer-aligned conditions, the channel reduces to a two-dimensional cyclic shift. In the Doppler dimension,
\[
{\bf{Q}_i}[p,q] = \begin{cases} M,& p = \langle q + \lfloor Mb_i \rceil \rangle_M,\\
0,& \text{otherwise},\end{cases}
\]
and in the affine dimension,
\[
{\bf{K}_{i}[m',m] = \begin{cases} N,& m = \langle m' + 2Nc_1l_i - \alpha_i \rangle_N,\\
0,& \text{otherwise}.\end{cases}
\]
Accordingly, the A-D block undergoes a path-dependent two-dimensional shift plus a phase factor,
\[
{\bf{Z}[m',q] = \sum\limits_{i = 1}^P \widetilde h_i e^{j\frac{2\pi }{N}\left[ Nc_1l_i^2 - ml_i  + Nc_2\left( m^2 - m'^2 \right) \right]} {\bf{X}[m,p]} + {\bf{\tilde W}[m',q],
\]
with
\[
m = \langle m' + 2Nc_1l_i - \alpha_i \rangle_N,
\qquad
p = \langle q + \lfloor Mb_i \rceil \rangle_M.
\]
When the normalized Doppler quantities are fractional, the exact one-bin shift is replaced by localized spreading around the same shift locations [2509.02116]. In the proposing paper, this two-dimensional cyclic-shift property is the main reason ADDM is presented as structurally closer to CP-OTFS than AFDM is.

## 4. Generic framework and reductions to AFDM, OTFS, and OFDM

A central claim of ADDM is that it is not merely another waveform but a framework that subsumes established ones [2509.02116]. The following specializations are explicitly stated.

| Waveform | Specialization in the ADDM framework | Interpretation |
|---|---|---|
| **CP-AFDM** | \(\mathbf{T}_{\mathrm F}=\mathbf{A}_{c_1,c_2}^{\mathrm H}\), \(\mathbf{T}_{\mathrm B}=\mathbf{F}_1^{\mathrm H}\) | AFDM appears as the single-column case |
| **CP-OTFS** | \(\mathbf{T}_{\mathrm F}=\mathbf{I}_N\), \(\mathbf{T}_{\mathrm B}=\mathbf{F}_M^{\mathrm H}\), \(c_1=0\) in the prefix | OTFS appears as the non-affine 2D case |
| **CP-OFDM** | \(\mathbf{T}_{\mathrm F}=\mathbf{A}_{0,0}^{\mathrm H}\), \(\mathbf{T}_{\mathrm B}=\mathbf{F}_1^{\mathrm H}\) | OFDM is the purely Fourier one-dimensional special case |

Within this taxonomy, AFDM is effectively ADDM with \(M=1\), while OTFS is ADDM with no affine transform in the \(N\)-dimension [2509.02116]. This is the paper’s formal answer to the incompatibility problem: AFDM and OTFS are re-expressed as members of a common transform family rather than treated as unrelated designs.

The broader AFDM literature clarifies why this unification matters. AFDM’s DAFT-domain path locations depend on a linear combination of delay and Doppler, and proper choice of \(c_1\) yields a full delay-Doppler representation [2204.12798]. OTFS, by contrast, exposes a two-dimensional delay-Doppler grid directly. ADDM combines these two perspectives: it keeps AFDM’s affine processing in one dimension, but restores a two-dimensional block model so that state-of-the-art methods designed for OTFS and AFDM are, in the words of the proposing paper, potentially directly applicable to ADDM [2509.02116].

## 5. High-mobility behavior, reported performance, and stated limitations

The proposing paper evaluates ADDM against AFDM and OTFS under ideal CSI with MMSE equalization. The reported setup uses \(f_c=24\) GHz, bandwidth \(B=7.68\) MHz, QPSK, time interval \(T=0.2667\) ms, prefix length \(N_{\mathrm{cp}}=4\), and \(c_1=0.1211\) for ADDM and AFDM. ADDM and OTFS use \(N=128\) and \(M=16\), whereas AFDM is the \(M=1\) special case with \(N=2048\). The simulated channel has \(P=3\) paths and maximum integer normalized Doppler \(\alpha_{\max}=2\) [2509.02116].

Two cases are highlighted. In **Case I**, the delays are different,
\[
l=[0,1,2],
\]
and ADDM, AFDM, and OTFS exhibit comparable BER. In **Case II**, all paths share the same delay,
\[
l=[1,1,1],
\]
and ADDM and AFDM achieve almost the same BER while both outperform OTFS [2509.02116]. The paper attributes this difference to OTFS’s unambiguous Doppler being limited by the subcarrier spacing, so that equal-delay paths with sufficiently separated Dopplers overlap in the delay-Doppler domain. ADDM and AFDM are said to avoid this because their unambiguous Doppler range is several times larger than the subcarrier spacing in the tested setting [2509.02116].

The conclusion of the ADDM paper states that the derived A-D input-output relation reveals full diversity order in doubly selective channels [2509.02116]. The included text, however, does not reproduce a separate diversity theorem or proof in the style of the earlier AFDM diversity analyses. Likewise, the paper argues that OTFS and AFDM methods may transfer to ADDM, but it does not instantiate a new detection or estimation algorithm beyond MMSE under ideal CSI. It also does not provide a detailed complexity analysis, so the main established result is structural and numerical rather than algorithmic [2509.02116].

## 6. Broader research implications and related directions

The immediate ADDM paper is narrow: it establishes the waveform definition, the transform relations, the A-D channel law, and a first BER comparison [2509.02116]. The broader AFDM literature indicates where ADDM may develop next, although these remain extensions rather than established ADDM results.

First, the AFDM literature already shows that affine/chirp waveforms are attractive for integrated sensing and communications. AFDM-based ISAC has been developed in both sensing-centric and communications-centric forms, including monostatic and bistatic architectures, dechirping-based self-interference cancellation, sub-Nyquist sampling, and one-pilot sensing that achieves almost the same sensing performance as using the entire frame [2511.04471][2402.16468]. Because ADDM preserves a two-dimensional block structure while retaining an affine dimension, this suggests a natural route toward A-D-domain sensing methods that combine AFDM-like Doppler handling with OTFS-like sparse two-dimensional processing.

Second, AFDM has already accumulated a nontrivial toolbox for channel estimation and detection. That toolbox includes pilot-aided channel estimation with single and multiple pilots, message-passing detection on sparse affine-domain factor graphs, and MB-UAMP detection for fractional delay-Doppler dispersion [2203.05781][2307.16109][2410.11421]. ADDM’s two-dimensional cyclic-shift structure suggests that such methods may be portable, and perhaps easier to hybridize with OTFS algorithms than in one-dimensional AFDM.

Third, parameter design in affine waveforms has implications beyond reliability. Secure AFDM parameter design has shown that \(c_1\), \(c_2\), and guard-interval choices affect anti-eavesdropping performance, including a bounded admissible range for \(c_1\), effective periodicity \(c_2\in[0,1]\), and a security risk interval induced by excessive delay guard provisioning [2503.19364]. AFDM has also been proposed for integrated channel sounding and communication, where its full delay-Doppler representation supports extraction of PDP, DPS, RMS delay spread, and RMS Doppler spread [2509.16643]. This suggests that corresponding parameter-design, security, and sounding questions are likely to become central in ADDM once explicit A-D-domain estimation algorithms are developed.

In that sense, ADDM currently stands as a unifying waveform proposal with a clearly defined mathematical structure and an explicitly stated compatibility ambition. Its long-term significance will depend on whether the surrounding AFDM and OTFS ecosystems—channel estimation, sparse equalization, sensing, coexistence, security, and implementation—can be carried into the A-D domain without losing the Doppler advantages that motivated the waveform in the first place [2509.02116].

Source: https://www.emergentmind.com/topics/affine-doppler-division-multiplexing-addm