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MIMO-AFDM Outperforms MIMO-OFDM in the Face of Hardware Impairments

Zeping Sui, Zilong Liu, Leila Musavian, Yong Liang Guan, Lie-Liang Yang, Lajos Hanzo

arXiv:2601.00502v2eess.SPcs.IT

TL;DR

MIMO-AFDM is studied under practical multiplicative and additive hardware impairments, which can degrade communication performance. The paper derives BER analyses for ML and LMMSE detection and finds that AFDM retains full diversity while generally outperforming MIMO-OFDM under hardware impairments.

  • Problem

    Practical hardware impairments and imperfect channel information can cause severe performance loss in AFDM communication systems.

  • Method

    The paper derives an asymptotic BER upper bound for small-scale MIMO-AFDM with ML detection and an approximate BER expression for large-scale systems with LMMSE detection.

  • Results

    MIMO-AFDM retains full diversity under hardware impairments, while its analytical BER results align with simulated ML and LMMSE performance.

  • Takeaways & Limitations

    Compared with MIMO-OFDM, MIMO-AFDM is more resilient to phase noise, carrier frequency offset, and additive hardware impairments because of chirp signaling and DAFT spreading.

Abstract

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The impact of both multiplicative and additive hardware impairments (HWIs) on multiple-input multiple-output affine frequency division multiplexing (MIMO-AFDM) systems is investigated. For small-scale MIMO-AFDM systems, a tight bit error rate (BER) upper bound associated with the maximum likelihood (ML) detector is derived. By contrast, for large-scale systems, a closed-form BER approximation associated with the linear minimum mean squared error (LMMSE) detector is presented, including realistic imperfect channel estimation scenarios. Our first key observation is that the full diversity order of a hardware-impaired AFDM system remains unaffected, which is a unique advantage. Furthermore, our analysis shows that 1) the BER results derived accurately predict the simulated ML performance in moderate-to-high signal-to-noise ratios (SNRs), while the theoretical BER curve of the LMMSE detector closely matches that of the Monte-Carlo based one. 2) MIMO-AFDM is more resilient to multiplicative distortions, such as phase noise and carrier frequency offset, compared to its orthogonal frequency division multiplexing (OFDM) counterparts. This is attributed to its inherent chirp signal characteristics; 3) MIMO-AFDM consistently achieves superior BER performance compared to conventional MIMO-OFDM systems under the same additive HWI conditions, as well as different velocity values. The latter is because MIMO-AFDM is also resilient to the additional inter-carrier interference (ICI) imposed by the nonlinear distortions of additive HWIs. In a nutshell, compared to OFDM, AFDM demonstrates stronger ICI resilience and achieves the maximum full diversity attainable gain even under HWIs, thanks to its intrinsic chirp signalling structure as well as to the beneficial spreading effect of the discrete affine Fourier transform.

I. INTRODUCTION

The paper analyzes MIMO-AFDM under realistic hardware impairments and imperfect CSI, addressing modeling, performance characterization, and comparisons with MIMO-OFDM. It derives detector-specific BER results and reports resilience from AFDM’s chirp signaling and DAFT spreading.

  • Motivation: The study targets MIMO-AFDM performance under oscillator, converter, mixer, amplifier, and other hardware impairments that can degrade BER and spectral efficiency.The impairments include PN, CFO, low-resolution DACs, IQI, NPA, and DCO, modeled as multiplicative and additive distortions.
  • Contributions: The paper derives a DAFT-domain MIMO-AFDM input-output model incorporating both multiplicative and additive hardware-distortion terms.The model includes additional mirror, DC, and nonlinear-distortion interference terms.
  • Contributions: It derives an ML BER upper bound for small-scale systems and an LMMSE BER approximation for large-scale systems with channel-estimation error.The analysis also derives the SINR of each chirp subcarrier under hardware impairments.
  • Results: MIMO-AFDM is reported to outperform MIMO-OFDM under multiplicative and additive impairments through chirp-based resilience, stronger ICI resistance, and DAFT spreading.The paper specifically investigates resilience to CFO and PN and comparisons under additive HWI settings.
  • Results: The ML BER upper bound becomes tight as SNR grows, while simulated LMMSE BER closely matches the derived approximation.These results support the accuracy of the paper’s detector-specific analytical characterizations.

II. HWI-FREE MIMO-AFDM SYSTEM MODEL

MIMO-AFDM maps QAM symbols onto chirp subcarriers through the DAFT, then models multipath delay-Doppler channels in the DAFT domain. The resulting MIMO channel is sparse, with CPP-based transmission supporting the input-output formulation.

  • System configuration: MIMO-AFDM uses M transmit antennas, J receive antennas, and N chirp subcarriers to carry QAM symbols.
  • AFDM modulation: The inverse DAFT kernel is parameterized by c1 and c2, and the model reduces to OFDM when c1 = c2 = 0.
  • AFDM modulation: A chirp-periodic prefix of length LCPP is employed to harness circulant convolution and alleviate inter-symbol interference.
  • Channel model: Each channel path is characterized by a gain, delay shift, and Doppler shift, producing a delay-Doppler channel response between each antenna pair.
  • DAFT-domain channel: The DAFT-domain MIMO channel is assembled from per-path matrices involving chirp transforms, cyclic delays, Doppler shifts, and CPP effects.
  • DAFT-domain channel: Each row and column of the path matrix has (2kν + 1) non-zero elements, yielding a sparse end-to-end DAFT-domain channel structure.

III. HARDWARE IMPAIRMENTS

The paper models MIMO-AFDM with multiple practical hardware impairments affecting transmitters, receivers, or both. These impairments are summarized according to their effects on the received signal.

  • Impairment model: The impairment model includes low-resolution DACs, IQI, DCO, NPA, CFO, and PN across the transmitter and receiver hardware.
  • Impairment model: The CFO is modeled at the receiver, while low-resolution DACs, IQI, DCO, and NPA are modeled at the transmitter.
  • Impairment assumptions: PN affects both transmitters and receivers, and all impairments except PN are assumed identical across antennas.

A. Multiplicative Distortions

Multiplicative distortions arise from oscillator phase noise and carrier frequency offset in practical MIMO-AFDM transceivers. The analysis considers Wiener phase noise under both common and separate local-oscillator configurations.

  • Phase noise: Transmitter and receiver phase noise is modeled as additive terms generated during signal up-conversion and down-conversion under a Wiener process.
  • Phase noise: The phase-noise variances are associated with transmitter and receiver oscillators and are parameterized through oscillator constants.
  • Phase-noise configurations: The paper considers common local oscillators with identical phase-noise matrices and separate local oscillators with independent transmit phase-noise components.

2) Carrier frequency offset:

The paper models CFO and several additive transmitter impairments, including DAC quantization, IQI, NPA, and DCO. These mechanisms introduce scaling changes, nonlinear distortion, mirror interference, or leakage into the transmitted and received signals.

  • Carrier frequency offset: CFO is represented by a diagonal phase-rotation matrix parameterized by the normalized CFO factor φCFO.
  • Low-resolution DACs: The AQNM and Bussgang decomposition model DAC output as a scaled input plus uncorrelated quantization noise.
  • Low-resolution DACs: The DAC scaling factor η depends on quantizer resolution b, with tabulated values for b ≤5 and an approximation for b > 5.
  • In-phase and quadrature imbalance: IQI creates a desired signal and conjugate mirror component, contaminating subcarrier k with a component from subcarrier N − k and causing ICI.
  • Nonlinear power amplifier distortion: NPA is modeled with a Bussgang-based nonlinear power-amplifier representation whose distortion variance depends on the soft-envelope limiter and clipping level.

4) Direct current offset:

The section develops the hardware-impaired MIMO-AFDM input-output relationship, including transmit- and receiver-side impairments, and decomposes the received signal into desired, mirror-interference, and nonlinear-distortion terms.

  • 4) Direct current offset:: The end-to-end time-domain model incorporates the transmit-side hardware impairments before forming the received signal.The transmit signal is modified by hardware effects, and the resulting time-domain channel relationship is represented before receiver-side CFO and phase-noise processing.
  • 4) Direct current offset:: The DAFT-domain model combines transmit and receive phase noise, carrier frequency offset, and additive distortion through an effective channel representation.The resulting relationship is derived from the time-domain model and the block-diagonal CFO and phase-noise matrices.
  • 4) Direct current offset:: The received signal separates into a desired signal, mirror interference, and nonlinear-distortion interference components.The model also identifies a DC-interference term associated with DCO, nonlinear power amplification, phase noise, and carrier frequency offset.
  • 4) Direct current offset:: The effective hardware-impaired channel retains the sparse structure of the hardware-impairment-free channel, supporting sparse channel-estimation and detection algorithms.Its nonzero structure is preserved despite multiplication by hardware factors, diagonal phase-noise matrices, and diagonal CFO matrices.
  • 4) Direct current offset:: Mirror interference can cause significant SINR loss because its effective channel matrices overlap fully with those of the desired signal.The desired-to-mirror-interference power difference depends only on ρ1 and ρ2.
  • 4) Direct current offset:: The input-output relationship remains valid when any hardware-impairment type is removed, enabling separate error-rate analysis for ML and LMMSE detectors.The formulation is used for subsequent detector analysis under the stated impairment combinations.

V. ERROR RATE ANALYSIS

The error-rate analysis evaluates hardware-impaired MIMO-AFDM with ML detection for small-scale systems and LMMSE detection for large-scale systems under both ideal and imperfect CSI.

  • V. ERROR RATE ANALYSIS: BER performance is analyzed with ML detection in small-scale systems and LMMSE detection in large-scale systems.The analysis covers both detector classes within the hardware-impaired MIMO-AFDM framework.
  • V. ERROR RATE ANALYSIS: Both ideal CSI and imperfect CSI scenarios are considered in the BER analysis.This includes realistic channel-estimation imperfections for the detector analysis.

1) Perfect CSI Conditions:

Under perfect CSI, the section derives pairwise-error and BER expressions for ML detection, then characterizes diversity order through channel and codeword-distance ranks.

  • 1) Perfect CSI Conditions:: The received signal is expressed using effective codeword matrices, channel vectors, and additive noise for ML detection.The ML detector selects the candidate minimizing the received-signal mismatch under the resulting model.
  • 1) Perfect CSI Conditions:: Pairwise error analysis uses the codeword difference matrix and its associated distance matrix to formulate the conditional error probability.The error event compares transmitted and erroneous symbol vectors through their effective Euclidean distance.
  • 1) Perfect CSI Conditions:: The unconditional pairwise error probability is approximated as 1/12 det(I_L + γ1ΓΩ) + 1/4 det(I_L + γ2ΓΩ).The expression follows from averaging the conditional error probability over the channel distribution.
  • 1) Perfect CSI Conditions:: The average BER is obtained from the pairwise-error analysis using a union bound over error events.The Hamming distance between bit sequences weights the corresponding symbol-vector error events.
  • 1) Perfect CSI Conditions:: The diversity order is VD = min∀e{r, rank(Ω)}, and phase-noise and CFO matrices do not change rank(Ω).With independently generated channel coefficients, r = PJ, while the minimum codeword-distance rank is P.

2) Imperfect CSI Conditions:

Under imperfect CSI, channel estimation error is incorporated into the effective noise and channel model, yielding modified pairwise-error and BER expressions.

  • 2) Imperfect CSI Conditions:: The estimated channel is modeled as ˇh = h + eh, with complex Gaussian channel-estimation error.The estimation error variance contributes to the imperfect-CSI analysis.
  • 2) Imperfect CSI Conditions:: With imperfect CSI, the effective noise becomes ˇw = −Ξ(x)eh + ¯w.The estimation error therefore enters the received-signal disturbance together with the original noise.
  • 2) Imperfect CSI Conditions:: Diversity order describes the BER-curve slope versus SNR, whereas degrees of freedom describe parallel high-SNR information streams.The section distinguishes these quantities and notes their fundamental trade-off.
  • 2) Imperfect CSI Conditions:: The imperfect-CSI unconditional pairwise error probability is approximated as 1/12 det(I_L + κ1ˇΓΩ) + 1/4 det(I_L + κ2ˇΓΩ).The coefficients κ1 and κ2 depend on the channel and channel-estimation error variances.
  • 2) Imperfect CSI Conditions:: The imperfect-CSI BER upper bound is obtained from the corresponding pairwise-error expression.The derivation uses the same BER construction after incorporating the channel-estimation imperfections.

B. Performance Analysis of LMMSE Detection

The LMMSE analysis models imperfect channel estimation and hardware-induced interference in the effective DAFT-domain system, then derives output SINR and BER expressions. A Jensen-based lower bound is obtained, with equality when the effective matrix has equal diagonal entries.

  • LMMSE system model: The LMMSE detector produces an estimated symbol vector through a Hermitian weighting matrix built from the estimated channel and interference covariance.The detector output is represented as ˆx = Gy, with G determined through the matrix W and its inverse.
  • LMMSE system model: The imperfect-CSI input-output model combines the estimated channel, estimation error, and effective DAFT-domain noise before LMMSE detection.The effective noise includes mirror, distortion, nonlinear, and receiver-noise terms.
  • SINR analysis: The cth detected symbol is decomposed into a desired term, residual interference, and noise, yielding a corresponding output SINR expression.The decomposition uses T = G ˆHeff and ˜v = G ˜Heffx + Gv.
  • BER analysis: For an |A|-ary QAM constellation, the approximate BER is converted into a lower bound using the convexity of Q(√u2x) and Jensen’s inequality.The modulation order is represented by |A|, and the bound follows by substituting the SINR expression into the BER formulation.
  • BER analysis: The BER lower bound is achieved when every diagonal element of T equals tr(T)/(NM).This condition makes the diagonal entries of the effective matrix uniform.

VI. SIMULATION RESULTS

The simulations evaluate MIMO-AFDM under hardware impairments, imperfect CSI, mobility, and realistic channel settings. Across the tested impairments, AFDM generally preserves full diversity and achieves stronger BER or SINR performance than OFDM, while additive distortions can produce saturation.

  • CFO and phase noise: Under high mobility and ideal hardware, MIMO-AFDM provides about 9 dB SNR gain over MIMO-OFDM at BER 10^-5.With CFO, AFDM performance remains nearly identical to its ideal-hardware case, whereas OFDM incurs SNR losses of 8 dB at φCFO = 0.08 and 1 dB at φCFO = 0.04.
  • CFO and phase noise: At BER 3 × 10^-5, MIMO-AFDM with CLO has a 15 dB SNR advantage over MIMO-OFDM with CLO.AFDM maintains similar BER under ideal and imperfect local oscillators, while OFDM is more affected by common phase noise.
  • Additive hardware impairments: With 5-bit DACs at BER 10^-4, MIMO-AFDM loses about 1 dB SNR from ideal hardware, compared with about 8 dB for MIMO-OFDM.AFDM’s chirp spreading allows quantization noise to be averaged through the DAFT process, reducing inter-carrier leakage.
  • Additive hardware impairments: For IQI parameters λ = 0.05 and β = 1°, MIMO-AFDM gains 8 dB SNR over MIMO-OFDM at BER 2 × 10^-4.Under stronger IQI settings, AFDM can still exhibit an SNR loss or a BER floor caused by mirror interference.
  • Additive hardware impairments: AFDM consistently outperforms OFDM at the tested nonlinear PA clipping levels because chirp signalling and DAFT spreading improve ICI resilience.For νclip = 4 dB, AFDM remains resilient to the minimal PA-induced ICI, while OFDM shows an 8 dB SNR gap at BER 3 × 10^-5.
  • Diversity and channel structure: The AFDM effective channel matrix retains separated path-related diagonal and sub-diagonal lines, supporting full diversity, whereas the OFDM matrix exhibits only one dominant diagonal line.The comparison uses M = J = 2, N = 32, P = 4, and maximum normalized Doppler shift kmax = 1.
  • SINR and channel effects: AFDM achieves higher average LMMSE output SINR than OFDM under both ideal and HWI conditions, although both saturate above 30 dB input SNR in the HWI case.With imperfect CSI, channel-estimation error can dominate the high-SNR behavior; under perfect CSI, DCO can reduce average output SINR by up to 39.6%.
  • Realistic channel validation: For SISO EVA channels with nine paths, HWI BER curves saturate above 40 dB at about 9 × 10^-3 for OFDM and 2 × 10^-5 for AFDM.Under ideal hardware, AFDM and OFDM show a 14 dB SNR gap at BER 10^-4.

VII. CONCLUSIONS

The paper develops BER and SINR analyses for MIMO-AFDM under multiplicative and additive hardware impairments, including imperfect CSI. Its results show preserved full diversity, accurate analytical predictions, and superior performance to MIMO-OFDM under matched impairment conditions.

  • Contributions: The study models both multiplicative and additive hardware impairments and derives the corresponding MIMO-AFDM input-output relationship.The considered impairments include phase noise, CFO, low-resolution DACs, IQI, PA nonlinearity, and DCO.
  • Analytical results: The ML BER upper bound becomes tight at high SNR, while the LMMSE BER approximation aligns with simulated BER results.The analyses include imperfect CSI for large-scale systems.
  • Analytical results: MIMO-AFDM retains full diversity under hardware impairments because the multiplicative HWI-related matrices are diagonal.The paper also derives an LMMSE SINR and approximate BER while accounting for channel-estimation error.
  • Conclusions: AFDM is resilient to CFO and phase noise but remains relatively sensitive to additive hardware impairments.The reported comparisons attribute AFDM’s advantage over OFDM to its chirp subcarriers and full time-frequency diversity.
  • Practical implications: Practical design must balance DAC resolution, clipping level, and phase-noise variance against receiver complexity, hardware cost, and power consumption.Higher DAC resolution and clipping thresholds reduce distortion, while lower phase-noise variance preserves phase coherence.
  • Future directions: Future work should address synchronization, HWI mitigation, channel estimation, and data detection with forward-error-correction-aided mitigation.The paper also identifies fractional-delay and Doppler analysis, including time-offset effects, as open directions.
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