Affine Filter Bank Modulation (AFBM)

• AFBM is a multicarrier waveform that integrates filter-bank synthesis with DAFT precoding to achieve robustness in doubly-dispersive channels while lowering PAPR and out-of-band emissions.
• It employs a compensation stage to restore approximate complex orthogonality and leverages a hybrid of AFDM and filter-bank techniques for improved spectral localization and effective equalization.
• AFBM demonstrates notable performance gains including reduced BER, enhanced OOBE suppression, and robustness against power-amplifier nonlinearities, making it ideal for high-mobility and ISAC applications.

Searching arXiv for 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 (Senger et al., 6 May 2025). 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 (Ranasinghe et al., 20 Jun 2025, Senger et al., 26 Nov 2025, Gourar et al., 10 Jun 2026).

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 (Senger et al., 6 May 2025). 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 (Senger et al., 6 May 2025).

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 (Senger et al., 6 May 2025). 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 (Senger et al., 6 May 2025).

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 (Senger et al., 6 May 2025). 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 (Senger et al., 6 May 2025, Ranasinghe et al., 20 Jun 2025, Senger et al., 26 Nov 2025). In the discrete-time formulation, the information vector is inserted through an indexing matrix $\Xi$, producing a time-frequency grid with guard subcarriers (Ranasinghe et al., 20 Jun 2025). 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 $\approx-70$0 are chirp-diagonal matrices (Senger et al., 6 May 2025, Ranasinghe et al., 20 Jun 2025, Senger et al., 26 Nov 2025). A compensation stage then forms

$\approx-70$1

or equivalently $\approx-70$2 depending on the paper’s convention for ordering, but in all cases the role of $\approx-70$3 is to restore complex orthogonality against the prototype filter (Senger et al., 6 May 2025, Ranasinghe et al., 20 Jun 2025, Senger et al., 26 Nov 2025).

After DAFT spreading, AFBM employs an oversampled or extended IDAFT stage through a pruned transform of length $\approx-70$4, with $\approx-70$5, followed by zero-padding and an $\approx-70$6-point filter-bank synthesis stage (Senger et al., 6 May 2025, Ranasinghe et al., 20 Jun 2025, Senger et al., 26 Nov 2025). The resulting synthesis matrix is combined with a block-Toeplitz filter-bank matrix $\approx-70$7 built from the prototype filter $\approx-70$8. One representative transmit model is

$\approx-70$9

with total transmit length $\approx-100$0 in the formulations that use overlap factor $\approx-100$1 (Senger et al., 26 Nov 2025, Ranasinghe et al., 20 Jun 2025, Ranasinghe et al., 6 Sep 2025).

The orthogonality-restoration problem is central to AFBM. Because the prototype-filter and DAFT chain is only approximately orthogonal, the design imposes

$\approx-100$2

where $\approx-100$3 is diagonal with ones in the first and last $\approx-100$4 entries (Senger et al., 6 May 2025). Closed-form entries of $\approx-100$5 are obtained from the diagonal of

$\approx-100$6

with zeros in the unused middle subcarriers (Senger et al., 6 May 2025). In the SIR analysis, the residual orthogonality approximation error is explicitly written as

$\approx-100$7

and this residual later appears in the post-equalizer interference matrices (Senger et al., 26 Nov 2025).

A continuous-time expression is also given in later work: $\approx-100$8 where $\approx-100$9 is the prototype filter and $K$0 is the chirp rate (Ranasinghe et al., 6 Sep 2025). 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

$K$1

and it is reported to exhibit shifted diagonals, one per path, indexed by integer delays $K$2 and Dopplers $K$3 (Senger et al., 6 May 2025). 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 (Senger et al., 6 May 2025). 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 (Ranasinghe et al., 20 Jun 2025). After front-end filtering and demodulation by $K$4, the equivalent model is written as

$K$5

with known $K$6 (Ranasinghe et al., 20 Jun 2025). 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 (Ranasinghe et al., 20 Jun 2025). Each GaBP iteration consists of three stages: soft interference cancellation, belief generation via extrinsic Gaussian updates, and soft replica generation through a denoiser (Ranasinghe et al., 20 Jun 2025). The update rules are scalar and element-wise, and the per-iteration complexity is stated as $K$7, in contrast to $K$8 for a conventional LMMSE detector (Ranasinghe et al., 20 Jun 2025).

The specific GaBP equations include the soft interference cancellation residual

$K$9

the associated interference-plus-noise variance, extrinsic mean and variance updates, and a QPSK Bayes-optimal denoiser based on hyperbolic tangent nonlinearities (Ranasinghe et al., 20 Jun 2025). 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 (Ranasinghe et al., 20 Jun 2025).

Communication performance results from this receiver study report that, at BER $L$0, AFBM with GaBP outperforms AFDM by $L$1 dB, for example with AFDM at $L$2 dB and AFBM at $L$3 dB in the cited setup (Ranasinghe et al., 20 Jun 2025). Gains are reported to persist for $L$4, and AFBM achieves $L$5 dB OOBE reduction compared to AFDM’s rectangular chirp pulses (Ranasinghe et al., 20 Jun 2025). 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 (Ranasinghe et al., 20 Jun 2025).

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 (Senger et al., 26 Nov 2025). 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 (Senger et al., 26 Nov 2025).

In the affine-domain formulation, after passage through a doubly-dispersive channel

$L$6

the observation is

$L$7

with

$L$8

in the paper’s notation (Senger et al., 26 Nov 2025). The affine-domain MMSE equalizer is

$L$9

and the estimate takes the form

$L/4$0

where $L/4$1 captures channel-induced inter-symbol interference together with imperfect DAFT-filter orthogonality (Senger et al., 26 Nov 2025).

In the FTD formulation, the effective channel is

$L/4$2

with observation $L/4$3, MMSE equalizer

$L/4$4

and estimate

$L/4$5

(Senger et al., 26 Nov 2025). 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 (Senger et al., 26 Nov 2025).

The SIR study ties this directly to the residual orthogonality error. Since $L/4$6 appears in both $L/4$7 and $L/4$8, only the FTD domain provides a mechanism to invert the channel and approximation error together (Senger et al., 26 Nov 2025). The paper characterizes this as “an interesting and counter-intuitive cancellation” in the filtered time-domain, absent in the affine domain (Senger et al., 26 Nov 2025). 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

$L/4$9

with the usual Gaussian $\Xi$0-function definition given in the paper (Senger et al., 26 Nov 2025). 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 $\Xi$1: AFDM is $\Xi$2 dB, whereas AFBM with a PHYDYAS filter and $\Xi$3 is $\Xi$4 dB, i.e., an improvement of $\Xi$5 dB (Senger et al., 6 May 2025). Another summary paper states that AFBM’s PAPR is about $\Xi$6 dB lower than regular AFDM and OFDM at typical $\Xi$7 CCDF levels (Ranasinghe et al., 6 Sep 2025). The nonlinearity-focused paper gives a broader operating characterization, stating that AFBM typically achieves PAPR of the order of $\Xi$8–$\Xi$9 dB, versus $$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$0–$$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$1 dB for plain OFDM (Gourar et al., 10 Jun 2026). 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 $$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$2 dB for AFDM with a rectangular window, $$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$3 dB for AFBM with truncated Hermite ($$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$4), and $$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$5 dB for AFBM with PHYDYAS ($$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$6) (Senger et al., 6 May 2025). The later 6G-oriented summary gives a relative suppression of $$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$7–$$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$8 dB versus AFDM and reports representative out-of-band power levels of $$\mathbf W_L=\Lambda_{c_1,L}\,\mathbf F_L\,\Lambda_{c_2,L},$$9 dB for Hermite and $\mathbf F_L$0 dB for PHYDYAS (Ranasinghe et al., 6 Sep 2025). The low-complexity receiver paper reports $\mathbf F_L$1 dB OOBE reduction compared to AFDM (Ranasinghe et al., 20 Jun 2025). 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 $\mathbf F_L$2	$\mathbf F_L$3 dB	PHYDYAS filter, $\mathbf F_L$4 (Senger et al., 6 May 2025)
OOBE far from band	$\mathbf F_L$5 dB	Truncated Hermite, $\mathbf F_L$6 (Senger et al., 6 May 2025)
OOBE far from band	$\mathbf F_L$7 dB	PHYDYAS, $\mathbf F_L$8 (Senger et al., 6 May 2025)
BER gain over AFDM	$\mathbf F_L$9 dB at $L$0 BER	GaBP receiver setup (Ranasinghe et al., 20 Jun 2025)
Range RMSE	$L$1 at $L$2 dB SNR	AFBM–PDA in 3-path scenario (Ranasinghe et al., 6 Sep 2025)
Velocity RMSE	$L$3 at $L$4 dB SNR	AFBM–PDA in 3-path scenario (Ranasinghe et al., 6 Sep 2025)

For MMSE equalization in fading channels, the SIR analysis paper reports average SIR over $L$5 realizations as follows: AFB domain, PHYDYAS, $L$6, $L$7 dB; AFB domain, Hermite, $L$8, $L$9 dB; FTD domain, PHYDYAS, $\approx-70$00, $\approx-70$01 dB; FTD domain, Hermite, $\approx-70$02, $\approx-70$03 dB (Senger et al., 26 Nov 2025). The same paper reports waveform-only SIR up to $\approx-70$04 dB when $\approx-70$05 and a Hermite filter is used, falling to $\approx-70$06 dB for PHYDYAS with $\approx-70$07 (Senger et al., 26 Nov 2025). At BER $\approx-70$08, FTD detection outperforms AFB-domain detection by $\approx-70$09 dB regardless of filter or $\approx-70$10, and the FTD curve is nearly insensitive to both $\approx-70$11 and prototype-filter choice (Senger et al., 26 Nov 2025).

In sensing-oriented evaluation, a later paper couples AFBM with an expectation maximization-assisted probabilistic data association framework for range and velocity estimation (Ranasinghe et al., 6 Sep 2025). Under a 3-path scenario, it reports range RMSE $\approx-70$12 and velocity RMSE $\approx-70$13 at $\approx-70$14 dB SNR, on par with AFDM–PDA and slightly better in range (Ranasinghe et al., 6 Sep 2025). The ambiguity function is described as closely matching AFDM’s mainlobe while reducing sidelobes by $\approx-70$15–$\approx-70$16 dB in delay and Doppler (Ranasinghe et al., 6 Sep 2025).

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

$\approx-70$17

and applies a Rapp/SSPA amplitude nonlinearity

$\approx-70$18

with smoothness parameter $\approx-70$19 and saturation voltage $\approx-70$20 (Gourar et al., 10 Jun 2026). Using a Bussgang decomposition, the output is written

$\approx-70$21

where $\approx-70$22, $\approx-70$23, and $\approx-70$24 (Gourar et al., 10 Jun 2026).

The undistorted ambiguity function is expressed as

$\approx-70$25

and, under the nonlinear PA approximation,

$\approx-70$26

(Gourar et al., 10 Jun 2026). The analysis emphasizes that distortion terms are weighted by traces of $\approx-70$27 and the distortion covariance $\approx-70$28, and that because $\approx-70$29 for well-localized filters and small overlap, $\approx-70$30 retains diagonal dominance and low off-diagonal energy (Gourar et al., 10 Jun 2026). The stated consequence is that distortion adds only a small pedestal to already low sidelobes (Gourar et al., 10 Jun 2026).

Simulation results in that study report that the zero-Doppler cut $\approx-70$31 retains sidelobes at $\approx-70$32 to $\approx-70$33 dB before and after PA nonlinearity, even at input back-off $\approx-70$34 dB (Gourar et al., 10 Jun 2026). The zero-delay Doppler cut $\approx-70$35 preserves main-lobe width, with only a slight uplift in Doppler-sidelobe valleys and no change in peak sidelobes (Gourar et al., 10 Jun 2026). Compared under the same nonlinearity, OFDM and AFDM exhibit larger ambiguity perturbations, in delay and Doppler respectively (Gourar et al., 10 Jun 2026). In a 3-path doubly-dispersive channel with a PDA estimator and $\approx-70$36 dB, AFBM has $\approx-70$37 dB RMSE loss across $\approx-70$38–$\approx-70$39 dB SNR, whereas AFDM degrades by $\approx-70$40–$\approx-70$41 dB in the same regime (Gourar et al., 10 Jun 2026).

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 $\approx-70$42 limits distortion propagation inside the ambiguity function (Gourar et al., 10 Jun 2026). 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 $\approx-70$43 is reported to give the best OOBE, down to $\approx-70$44 dB in the original evaluation, but with longer latency and overlap (Senger et al., 6 May 2025). Hermite with $\approx-70$45 offers lower latency and still exceeds $\approx-70$46 dB OOBE in that study (Senger et al., 6 May 2025). The original paper further states that overlap factor $\approx-70$47 with Hermite guarantees interference limited to the main diagonal, whereas larger $\approx-70$48 with PHYDYAS requires careful equalizer design (Senger et al., 6 May 2025). Yet the MMSE SIR analysis later reports that FTD detection is nearly insensitive to prototype-filter choice and to $\approx-70$49, which suggests that receiver-domain design can absorb much of the modulation-side sensitivity (Senger et al., 26 Nov 2025).

A second trade-off concerns transform and oversampling dimensions. Practical parameter sets repeatedly use $\approx-70$50, $\approx-70$51, $\approx-70$52, with $\approx-70$53 in $\approx-70$54 or comparable values (Senger et al., 6 May 2025, Ranasinghe et al., 20 Jun 2025, Senger et al., 26 Nov 2025, Ranasinghe et al., 6 Sep 2025). The orthogonality and channel-robustness conditions require DAFT parameters to satisfy

$\approx-70$55

with $\approx-70$56 in the original construction (Senger et al., 6 May 2025, Ranasinghe et al., 20 Jun 2025). 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 $\approx-70$57, the occupancy of only the first and last $\approx-70$58 subcarriers, the pruned IDAFT structure, and the orthogonality-approximation analysis are all integral rather than auxiliary (Senger et al., 6 May 2025, Senger et al., 26 Nov 2025). 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 $\approx-70$59–$\approx-70$60 dB in SIR and by about $\approx-70$61–$\approx-70$62 dB in BER, precisely because it jointly inverts channel interference and orthogonality-approximation error (Senger et al., 26 Nov 2025).

Implementation-wise, the papers point to efficient realization by polyphase network plus IFFT and pointwise chirp multiplications (Senger et al., 6 May 2025), and later to FPGA/ASIC amenability of the polyphase-network, IDD, and GaBP chain (Ranasinghe et al., 20 Jun 2025, Ranasinghe et al., 6 Sep 2025). 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 (Ranasinghe et al., 6 Sep 2025).

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 (Senger et al., 6 May 2025, Ranasinghe et al., 20 Jun 2025, Senger et al., 26 Nov 2025). Later sensing studies add that these properties persist under power-amplifier nonlinearities to an extent sufficient to preserve ambiguity behavior and induce less than $\approx-70$63 dB RMSE loss in the reported regime (Gourar et al., 10 Jun 2026). 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.