---
title: Affine Filter Bank Modulation (AFBM)
url: https://www.emergentmind.com/topics/affine-filter-bank-modulation-afbm
type: topic
---

# Affine Filter Bank Modulation (AFBM)

Searching arXiv for recent papers on Affine Filter Bank Modulation to ground the article in the cited literature.
Affine Filter Bank Modulation (AFBM) is a multicarrier waveform that combines filter-bank synthesis with discrete affine Fourier transform (DAFT) precoding to address doubly-dispersive channel conditions, especially in high-mobility and integrated sensing and communications (ISAC) settings. In the cited literature, AFBM is described as a hybrid of affine frequency division multiplexing (AFDM) and filter-bank multicarrier principles, designed to preserve delay–Doppler robustness while reducing peak-to-average power ratio (PAPR) and out-of-band emission (OOBE) through prototype-filter localization and a compensation stage that restores complex orthogonality [2505.03589]. Subsequent work develops low-complexity receivers, analyzes signal-to-interference ratio (SIR) under minimum mean square error (MMSE) equalization, and studies robustness to power-amplifier nonlinearities, thereby positioning AFBM as a communications-and-sensing waveform with explicit architectural, algorithmic, and hardware-level considerations [2506.17010], [2511.21615], [2606.11879].

## 1. Origin, motivation, and relation to prior waveforms

AFBM was introduced as a waveform for high-mobility communications and ISAC in doubly-dispersive channels, where conventional OFDM loses subcarrier orthogonality and suffers severe inter-carrier interference [2505.03589]. The motivating requirements stated in the literature are robustness to delay–Doppler distortion, low OOBE to limit interference outside the occupied band, and low PAPR to ease power-amplifier linearity constraints. AFBM addresses these by combining a filter-bank structure with DAFT precoding and a compensation mechanism for complex orthogonality [2505.03589].

The conceptual lineage is explicit. Filter-bank multicarrier methods contribute well-localized prototype filters and strong spectral containment, while AFDM contributes chirp-domain spreading, quasi-orthogonality in doubly-dispersive channels, and a delay–Doppler channel structure with shifted diagonals [2505.03589]. The resulting design is not merely a superposition of two ideas; it is a structured synthesis in which the filter bank controls spectral localization, the DAFT shapes the subcarriers as chirps, and a compensation vector is selected so that the cascaded filtering and affine spreading approximate a diagonal guard structure [2505.03589].

The early comparative literature frames AFBM against OFDM, AFDM, and FBMC/DFT-s-FBMC. In that comparison, OFDM is characterized as poor in doubly-dispersive robustness, AFDM as excellent in doubly-dispersive robustness but with high OOBE and high PAPR, and AFBM as combining excellent doubly-dispersive robustness with approximately \(8\) dB PAPR and OOBE in the \(\approx-70\) dB to \(\approx-100\) dB range depending on the prototype filter [2505.03589]. This suggests that the defining research question around AFBM is not whether chirp-domain multicarrier works in doubly-dispersive channels—that premise already existed in AFDM—but whether chirp-domain robustness can be retained under a filter-bank architecture with materially improved spectral and amplifier-facing behavior.

## 2. Signal model, DAFT structure, and orthogonality restoration

The core AFBM transmitter model uses \(K\) multicarrier symbols on \(L\) subcarriers, with occupation restricted to the first and last \(L/4\) positions in each block to avoid filter overlap [2505.03589], [2506.17010], [2511.21615]. In the discrete-time formulation, the information vector is inserted through an indexing matrix \(\Xi\), producing a time-frequency grid with guard subcarriers [2506.17010]. The per-block DAFT operator is
\[
\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},
\]
where \(\mathbf F_L\) is the \(L\)-point DFT and \(\Lambda_{c_i,L}\) are chirp-diagonal matrices [2505.03589], [2506.17010], [2511.21615]. A compensation stage then forms
\[
\mathbf C_f=\mathbf W_L\,\mathrm{diag}(\tilde{\mathbf b})
\]
or equivalently \(\mathbf C_f=\mathrm{diag}(\tilde{\mathbf b})\,\mathbf W_L\) depending on the paper’s convention for ordering, but in all cases the role of \(\tilde{\mathbf b}\) is to restore complex orthogonality against the prototype filter [2505.03589], [2506.17010], [2511.21615].

After DAFT spreading, AFBM employs an oversampled or extended IDAFT stage through a pruned transform of length \(P\), with \(L<P<N\), followed by zero-padding and an \(N\)-point filter-bank synthesis stage [2505.03589], [2506.17010], [2511.21615]. The resulting synthesis matrix is combined with a block-Toeplitz filter-bank matrix \(\mathbf G\) built from the prototype filter \(g[n]\). One representative transmit model is
\[
\mathbf s=\mathbf G\,(\mathbf I_K\otimes \mathbf Q_P\mathbf C_f)\,\bm\Xi\,\mathbf x,
\]
with total transmit length \(M=ON+\tfrac{N}{2}(K-1)\) in the formulations that use overlap factor \(O\) [2511.21615], [2506.17010], [2509.05683].

The orthogonality-restoration problem is central to AFBM. Because the prototype-filter and DAFT chain is only approximately orthogonal, the design imposes
\[
\mathbf{C}_f^H \mathbf{Q}_P^H \widetilde{\mathbf G}^T \widetilde{\mathbf G}\mathbf Q_P \mathbf C_f \simeq \mathbf U,
\]
where \(\mathbf U\) is diagonal with ones in the first and last \(L/4\) entries [2505.03589]. Closed-form entries of \(\tilde{\mathbf b}\) are obtained from the diagonal of
\[
\mathbf W_L^H \mathbf Q_P^H \widetilde{\mathbf G}^T \widetilde{\mathbf G}\mathbf Q_P \mathbf W_L,
\]
with zeros in the unused middle subcarriers [2505.03589]. In the SIR analysis, the residual orthogonality approximation error is explicitly written as
\[
\mathbf E_{\rm ortho}
=\mathbf{C}_f\,\mathbf{Q}_P^H\,\tilde{\mathbf G}^T\,\tilde{\mathbf G}\,\mathbf{Q}_P\,\mathbf{C}_f-\mathbf{U},
\]
and this residual later appears in the post-equalizer interference matrices [2511.21615].

A continuous-time expression is also given in later work:
\[
x(t)=\sum_{m=0}^{L-1}\sum_{n=0}^{K-1}d_{m,n}\;
p\bigl(t-n\tfrac{T}{2}\bigr)\,
e^{j2\pi mFt}\,
e^{j\pi \alpha t^2},
\]
where \(p(t)\) is the prototype filter and \(\alpha=c_2\) is the chirp rate [2509.05683]. This formulation emphasizes that AFBM can be interpreted as a chirp-filtered pulse set on a filter-bank lattice, with DAFT parameters inherited from affine-domain modulation theory.

## 3. Effective channel structure and communications receivers

In the original waveform paper, the post-demodulation effective channel is
\[
\mathbf H_{\rm eff}=\mathbf Q_P^H\,\mathbf G^H\,\mathbf H\,\mathbf G\,\mathbf Q_P \in \mathbb C^{L\times L},
\]
and it is reported to exhibit shifted diagonals, one per path, indexed by integer delays \(\ell_r\) and Dopplers \(f_r\) [2505.03589]. The literature describes this diagonal-spread structure as identical in spirit to AFDM’s and as the basis for low-complexity equalization and full diversity [2505.03589]. This suggests that AFBM’s principal structural inheritance from AFDM is not merely chirp spreading, but the preservation of a channel geometry that remains exploitable after filter-bank processing.

A distinct line of work develops a low-complexity Gaussian Belief Propagation (GaBP) receiver for AFBM [2506.17010]. After front-end filtering and demodulation by \((GQ)^H\), the equivalent model is written as
\[
\bar{\mathbf r}=\bar H\,\mathbf x+\bar{\mathbf w},
\]
with known \(\bar H\) [2506.17010]. The receiver constructs a factor graph with variable nodes corresponding to transmitted symbols and observation nodes corresponding to received samples, with channel gains on the edges [2506.17010]. Each GaBP iteration consists of three stages: soft interference cancellation, belief generation via extrinsic Gaussian updates, and soft replica generation through a denoiser [2506.17010]. The update rules are scalar and element-wise, and the per-iteration complexity is stated as \(\mathcal O(\bar N\,\bar M)\), in contrast to \(\mathcal O(\bar M^3)\) for a conventional LMMSE detector [2506.17010].

The specific GaBP equations include the soft interference cancellation residual
\[
\tilde r_{x:\bar n,\bar m}^{(i)}
=\bar r_{\bar n}
-\sum_{e\neq\bar m} h_{\bar n,e}\,\hat x_{\bar n,e}^{(i-1)},
\]
the associated interference-plus-noise variance, extrinsic mean and variance updates, and a QPSK Bayes-optimal denoiser based on hyperbolic tangent nonlinearities [2506.17010]. The formulation is noteworthy because it reinterprets AFBM detection as sparse or dense linear inference on a bipartite graph rather than direct matrix inversion. The paper states that the receiver converges in a few dozen iterations and uses only element-wise scalar operations [2506.17010].

Communication performance results from this receiver study report that, at BER \(10^{-3}\), AFBM with GaBP outperforms AFDM by \(\approx 2\) dB, for example with AFDM at \(16\) dB and AFBM at \(\approx 14\) dB in the cited setup [2506.17010]. Gains are reported to persist for \(P=192,256,320\), and AFBM achieves \(>20\) dB OOBE reduction compared to AFDM’s rectangular chirp pulses [2506.17010]. The same literature reports that Hermite and PHYDYAS filters yield virtually identical BER in the tested setting, indicating that improved OOBE does not necessarily impose a BER penalty in the considered doubly-dispersive regime [2506.17010].

## 4. SIR analysis, MMSE equalization domains, and the filtered time-domain effect

The most detailed analytical treatment of AFBM detection performance appears in the SIR study under MMSE equalization [2511.21615]. That work contrasts two equalization domains: the affine domain, obtained after undoing filter-bank and IDAFT operations, and the filtered time-domain (FTD), where equalization is delayed until after re-filtering the DAFT-spread blocks into the time domain [2511.21615].

In the affine-domain formulation, after passage through a doubly-dispersive channel
\[
\mathbf{H}=\sum_{r=1}^R h_r\,\mathbf{Z}^{f_r}\,\mathbf{\Pi}^{\ell_r},
\]
the observation is
\[
\mathbf y=\mathbf H_{\rm AFB}\,\bm\Xi\,\mathbf x+\mathbf n_{\rm AFB},
\]
with
\[
\mathbf{H}_{\rm AFB}
=(\mathbf{I}_K\otimes\mathbf{C}_f\mathbf{Q}_P)\,
\mathbf{G}\,\mathbf{H}\,\mathbf{G}\,
(\mathbf{I}_K\otimes\mathbf{Q}_P\mathbf{C}_f)
\]
in the paper’s notation [2511.21615]. The affine-domain MMSE equalizer is
\[
\mathbf E_{\rm AFB}
=
\bigl(\mathbf H_{\rm AFB}\mathbf H_{\rm AFB}^H+\sigma_n^2\mathbf I\bigr)^{-1}
\mathbf H_{\rm AFB},
\]
and the estimate takes the form
\[
\hat{\mathbf x}_{\rm AFB}
=\bigl(\mathbf I+\Delta_{\rm AFB}\bigr)\mathbf x
+\bm\Xi\,\mathbf E_{\rm AFB}\,\mathbf n_{\rm AFB},
\]
where \(\Delta_{\rm AFB}\) captures channel-induced inter-symbol interference together with imperfect DAFT-filter orthogonality [2511.21615].

In the FTD formulation, the effective channel is
\[
\bar{\mathbf H}
=\mathbf G\,\mathbf H\,\mathbf G\,
(\mathbf I_K\otimes \mathbf Q_P\mathbf C_f)\,\bm\Xi,
\]
with observation \(\bar{\mathbf r}=\bar{\mathbf H}\mathbf x+\bar{\mathbf n}\), MMSE equalizer
\[
\mathbf E_{\rm FTD}
=\bigl(\bar{\mathbf H}\bar{\mathbf H}^H+\sigma_n^2\mathbf I\bigr)^{-1}\bar{\mathbf H},
\]
and estimate
\[
\hat{\mathbf x}_{\rm FTD}
=\bigl(\mathbf I+\Delta_{\rm FTD}\bigr)\mathbf x+\mathbf E_{\rm FTD}\bar{\mathbf n}
\]
[2511.21615]. The key reported result is that the FTD equalizer “sees” the DAFT, despreading, and prototype filter jointly with the channel, and therefore cancels much of the DAFT-filter orthogonality error mixed into the channel interference; this cancellation does not occur in the purely affine-domain equalizer [2511.21615].

The SIR study ties this directly to the residual orthogonality error. Since \(\mathbf E_{\rm ortho}\) appears in both \(\Delta_{\rm AFB}\) and \(\Delta_{\rm FTD}\), only the FTD domain provides a mechanism to invert the channel and approximation error together [2511.21615]. The paper characterizes this as “an interesting and counter-intuitive cancellation” in the filtered time-domain, absent in the affine domain [2511.21615]. A plausible implication is that the conventional intuition—equalize in the transform domain aligned to channel structure—fails here because the dominant impairment is not solely channel-induced interference but the interaction between channel distortion and approximate orthogonality of the modulation basis.

Under QPSK and interference-limited operation, BER is approximated as
\[
\mathrm{BER}\approx Q\!\Bigl(\sqrt{2\,\mathrm{SIR}}\Bigr),
\]
with the usual Gaussian \(Q\)-function definition given in the paper [2511.21615]. The analysis thereby links equalization-domain choice, residual interference geometry, and observed BER without appealing solely to simulation.

## 5. Performance characteristics: PAPR, OOBE, BER, and sensing metrics

The performance profile of AFBM in the literature is defined by four recurring attributes: low PAPR, strong OOBE suppression, competitive or improved BER in doubly-dispersive channels, and favorable ambiguity or sensing behavior.

On PAPR, the initial AFBM paper reports a complementary cumulative distribution function comparison at \(10^{-3}\): AFDM is \(\simeq 11\) dB, whereas AFBM with a PHYDYAS filter and \(O=4\) is \(\simeq 8\) dB, i.e., an improvement of \(\sim 3\) dB [2505.03589]. Another summary paper states that AFBM’s PAPR is about \(2\) dB lower than regular AFDM and OFDM at typical \(10^{-3}\) CCDF levels [2509.05683]. The nonlinearity-focused paper gives a broader operating characterization, stating that AFBM typically achieves PAPR of the order of \(6\)–\(8\) dB, versus \(10\)–\(12\) dB for plain OFDM [2606.11879]. These statements differ in exact baseline and scenario, but all support the same qualitative result: chirp spreading plus filtering materially reduces PAPR relative to unfiltered chirp-domain or conventional multicarrier baselines.

On OOBE, the original waveform paper reports measured power spectral density far from the occupied band as \(\approx -20\) dB for AFDM with a rectangular window, \(\approx -70\) dB for AFBM with truncated Hermite (\(O=1.5\)), and \(\approx -100\) dB for AFBM with PHYDYAS (\(O=4\)) [2505.03589]. The later 6G-oriented summary gives a relative suppression of \(30\)–\(50\) dB versus AFDM and reports representative out-of-band power levels of \(-60\) dB for Hermite and \(-80\) dB for PHYDYAS [2509.05683]. The low-complexity receiver paper reports \(>20\) dB OOBE reduction compared to AFDM [2506.17010]. These values are not identical because they correspond to different measurement conventions and setups, but together they establish that AFBM’s filter-bank localization is the source of a large and repeatedly observed OOBE advantage.

The following table collects parameterized comparisons stated explicitly in the cited literature.

| Quantity | Reported AFBM result | Context |
|---|---:|---|
| PAPR at CCDF \(10^{-3}\) | \(\simeq 8\) dB | PHYDYAS filter, \(O=4\) [2505.03589] |
| OOBE far from band | \(\approx -70\) dB | Truncated Hermite, \(O=1.5\) [2505.03589] |
| OOBE far from band | \(\approx -100\) dB | PHYDYAS, \(O=4\) [2505.03589] |
| BER gain over AFDM | \(\approx 2\) dB at \(10^{-3}\) BER | GaBP receiver setup [2506.17010] |
| Range RMSE | \(\approx 0.1\) at \(10\) dB SNR | AFBM–PDA in 3-path scenario [2509.05683] |
| Velocity RMSE | \(\approx 0.02\) at \(10\) dB SNR | AFBM–PDA in 3-path scenario [2509.05683] |

For MMSE equalization in fading channels, the SIR analysis paper reports average SIR over \(200\) realizations as follows: AFB domain, PHYDYAS, \(P=192\), \(\sim 12.3\) dB; AFB domain, Hermite, \(P=256\), \(\sim 20.7\) dB; FTD domain, PHYDYAS, \(P=192\), \(\sim 42.4\) dB; FTD domain, Hermite, \(P=256\), \(\sim 45.2\) dB [2511.21615]. The same paper reports waveform-only SIR up to \(30\) dB when \(P=N\) and a Hermite filter is used, falling to \(\sim 15\) dB for PHYDYAS with \(P=192\) [2511.21615]. At BER \(10^{-2}\), FTD detection outperforms AFB-domain detection by \(\sim 5\) dB regardless of filter or \(P\), and the FTD curve is nearly insensitive to both \(P\) and prototype-filter choice [2511.21615].

In sensing-oriented evaluation, a later paper couples AFBM with an expectation maximization-assisted probabilistic data association framework for range and velocity estimation [2509.05683]. Under a 3-path scenario, it reports range RMSE \(\approx 0.1\) and velocity RMSE \(\approx 0.02\) at \(10\) dB SNR, on par with AFDM–PDA and slightly better in range [2509.05683]. The ambiguity function is described as closely matching AFDM’s mainlobe while reducing sidelobes by \(5\)–\(10\) dB in delay and Doppler [2509.05683].

## 6. Sensing behavior and robustness to power-amplifier nonlinearities

AFBM’s relevance to ISAC is treated not only through communication metrics but also through ambiguity-function structure and robustness to front-end nonidealities. The power-amplifier nonlinearity study models the pre-amplifier signal as
\[
\mathbf s=\mathbf M\,\mathbf x,
\qquad
\mathbf M\equiv \bar G = G\,Q\,C\,\Xi,
\]
and applies a Rapp/SSPA amplitude nonlinearity
\[
y(n)=g[s(n)]
=s(n)\Bigl[1+\bigl|s(n)/V_{\rm sat}\bigr|^{2q}\Bigr]^{-1/(2q)},
\]
with smoothness parameter \(q\) and saturation voltage \(V_{\rm sat}\) [2606.11879]. Using a Bussgang decomposition, the output is written
\[
\mathbf y=\kappa\,\mathbf s+\mathbf d=\kappa\,\mathbf M\,\mathbf x+\mathbf d,
\]
where \(\mathbf d\perp \mathbf s\), \(E[\mathbf d]=0\), and \(\mathrm{Var}[\mathbf d]=\sigma_d^2\) [2606.11879].

The undistorted ambiguity function is expressed as
\[
A(l,\nu)=\mathbf s^H D_\nu J_l \mathbf s
=\mathbf x^H\Phi_{l,\nu}\mathbf x,
\qquad
\Phi_{l,\nu}=\mathbf M^H D_\nu J_l \mathbf M,
\]
and, under the nonlinear PA approximation,
\[
A_y(l,\nu)
=(\kappa \mathbf x+\mathbf t)^H \Phi_{l,\nu}(\kappa \mathbf x+\mathbf t),
\qquad
\mathbf t=\mathbf M^H\mathbf d
\]
[2606.11879]. The analysis emphasizes that distortion terms are weighted by traces of \(\Phi\) and the distortion covariance \(R_t\), and that because \(\mathbf M^H\mathbf M\approx I\) for well-localized filters and small overlap, \(\Phi_{l,\nu}\) retains diagonal dominance and low off-diagonal energy [2606.11879]. The stated consequence is that distortion adds only a small pedestal to already low sidelobes [2606.11879].

Simulation results in that study report that the zero-Doppler cut \(A(l,0)\) retains sidelobes at \(-30\) to \(-40\) dB before and after PA nonlinearity, even at input back-off \(1\) dB [2606.11879]. The zero-delay Doppler cut \(A(0,\nu)\) preserves main-lobe width, with only a slight uplift in Doppler-sidelobe valleys and no change in peak sidelobes [2606.11879]. Compared under the same nonlinearity, OFDM and AFDM exhibit larger ambiguity perturbations, in delay and Doppler respectively [2606.11879]. In a 3-path doubly-dispersive channel with a PDA estimator and \(IBO=2\) dB, AFBM has \(<1\) dB RMSE loss across \(0\)–\(20\) dB SNR, whereas AFDM degrades by \(2\)–\(3\) dB in the same regime [2606.11879].

These results sharpen an apparent tension in ISAC waveform design. High sensing performance often favors high transmit power, while high linearity is usually required to avoid ambiguity degradation. The AFBM literature argues that low PAPR and approximate modulation-matrix orthonormality mitigate this contradiction, since reduced instantaneous peaks lessen out-of-band regrowth and the structure of \(\mathbf M\) limits distortion propagation inside the ambiguity function [2606.11879]. This does not imply immunity to nonlinearities, but it does indicate that AFBM’s sensing degradation is comparatively mild in the tested settings.

## 7. Practical trade-offs, misconceptions, and research directions

Several trade-offs recur across the literature. Prototype-filter choice is one. PHYDYAS with \(O=4\) is reported to give the best OOBE, down to \(\approx -100\) dB in the original evaluation, but with longer latency and overlap [2505.03589]. Hermite with \(O=1.5\) offers lower latency and still exceeds \(-70\) dB OOBE in that study [2505.03589]. The original paper further states that overlap factor \(O\le 1.5\) with Hermite guarantees interference limited to the main diagonal, whereas larger \(O\) with PHYDYAS requires careful equalizer design [2505.03589]. Yet the MMSE SIR analysis later reports that FTD detection is nearly insensitive to prototype-filter choice and to \(P\), which suggests that receiver-domain design can absorb much of the modulation-side sensitivity [2511.21615].

A second trade-off concerns transform and oversampling dimensions. Practical parameter sets repeatedly use \(L=128\), \(N=256\), \(K=8\), with \(P\) in \(\{192,256\}\) or comparable values [2505.03589], [2506.17010], [2511.21615], [2509.05683]. The orthogonality and channel-robustness conditions require DAFT parameters to satisfy
\[
2(f_{\max}+\xi)(\ell_{\max}+1)+\ell_{\max}\le P,
\]
with \(P<N\) in the original construction [2505.03589], [2506.17010]. This means that the chirp-domain design is constrained jointly by channel support and synthesis dimensions rather than by a purely spectral design rule.

A common misconception would be to treat AFBM as simply AFDM with a prototype filter appended. The literature does not support that simplification. The compensation vector \(\tilde{\mathbf b}\), the occupancy of only the first and last \(L/4\) subcarriers, the pruned IDAFT structure, and the orthogonality-approximation analysis are all integral rather than auxiliary [2505.03589], [2511.21615]. Another possible misconception is that affine-domain equalization is automatically the natural receiver because the waveform is DAFT-based. The MMSE SIR analysis shows the opposite in the tested settings: filtered time-domain equalization can outperform affine-domain equalization by \(20\)–\(30\) dB in SIR and by about \(4\)–\(6\) dB in BER, precisely because it jointly inverts channel interference and orthogonality-approximation error [2511.21615].

Implementation-wise, the papers point to efficient realization by polyphase network plus IFFT and pointwise chirp multiplications [2505.03589], and later to FPGA/ASIC amenability of the polyphase-network, IDD, and GaBP chain [2506.17010], [2509.05683]. The 6G-oriented summary further identifies communications detection through GaBP and sensing through EM-assisted PDA as two complementary algorithmic layers built on the same hybrid filtered time-domain input–output model [2509.05683].

Across the cited works, the cumulative picture is technically consistent. AFBM is a filter-bank chirp waveform whose defining properties are quasi-orthogonality restored by compensation, shifted-diagonal effective channels in doubly-dispersive propagation, lower PAPR than AFDM, substantially reduced OOBE, and receiver behavior that depends critically on where equalization is performed [2505.03589], [2506.17010], [2511.21615]. Later sensing studies add that these properties persist under power-amplifier nonlinearities to an extent sufficient to preserve ambiguity behavior and induce less than \(1\) dB RMSE loss in the reported regime [2606.11879]. This suggests that the most consequential open technical questions are likely to concern receiver-domain approximations, hardware-constrained implementations, and the interaction between orthogonality-restoration accuracy and equalization strategy rather than the basic viability of the waveform itself.

Source: https://www.emergentmind.com/topics/affine-filter-bank-modulation-afbm