Comparative Analysis of Filter Bank Waveforms for ISAC Systems: 2D-FFT FB and AFBM

I. INTRODUCTION

• ISAC integrates sensing and communication over shared resources, with the stated aims of improving spectral efficiency and reducing hardware redundancy.The introduction presents this as a response to the inefficient use of hardware and spectrum when the functions are implemented separately.Source: §I · p. 1
• Severe delay and Doppler spreads in doubly-dispersive vehicular channels cause channel variation and performance degradation.The spreads stem from relative motion between transmitters, receivers, and surrounding objects.Source: §I · p. 1
• The cited waveform alternatives have distinct trade-offs: OFDM suffers inter-carrier interference under strong Doppler, while FBMC improves spectral localization and reduces OOBE at increased transceiver complexity.OTFS is described as robust to doubly dispersive channels but retaining considerable OOBE from its OFDM-based structure.Source: §I · p. 1
• The paper compares 2D-FFT FB and AFBM under a common framework, with OTFS and AFDM as OFDM-based baselines.Because the two filter-bank waveforms share a filter-bank core and differ only in precoding, the comparison isolates the effect of the precoder.Source: §I · p. 1
• The evaluation covers BER, PAPR, OOBE, ambiguity-function delay and Doppler cuts, and computational complexity under doubly-dispersive channel conditions.The stated aim is to assess waveform trade-offs for high-mobility ISAC.Source: §I · p. 1

II. WAVEFORM TRANSMISSION

• All considered waveform transmission schemes originate from the exact same data payload to ensure a consistent comparison.The schemes are shown in Fig. 1.Source: §II · p. 2
• The common input x contains N_d transmitted complex symbols drawn from the QAM alphabet D.Both schemes use L signal dimensions per frame, with K = 2N_d/L time symbols.Source: §II · p. 2
• The formulation uses the same frame dimensions, L and K, for both schemes.K is defined as 2N_d/L.Source: §II · p. 2

A. The 2D-FFT FB Transmitter in Brief

• DFT-based precoding and filter compensation spread L/2 complex symbols across L subcarriers, restoring complex orthogonality without losing information rate.The scheme replaces OTFS’s OFDM kernel with a filter-bank multicarrier structure.Source: A. The 2D-FFT FB Transmitter in Brief · p. 2
• The resulting grid has L/2 active delay-domain bins and L/2 zero-valued guard bins to mitigate filter-bank interference.Its second axis contains K bins.Source: A. The 2D-FFT FB Transmitter in Brief · p. 2
• An extended N-point IDFT converts the time-frequency matrix X into the time-domain matrix D.Frequency-domain zero padding uses N > L, expanding the signal in time.Source: A. The 2D-FFT FB Transmitter in Brief · p. 2
• The prototype filter has length ON, where O is its overlap factor, and is applied to each time-domain sequence.The filtering operation is modeled using diagonal matrices arranged along a block Toeplitz matrix.Source: A. The 2D-FFT FB Transmitter in Brief · p. 2
• The output filter-bank signal is represented as s_2D, formed using time-shifted versions of the prototype filter.Each block in the filtering structure corresponds to a time-shifted prototype filter.Source: A. The 2D-FFT FB Transmitter in Brief · p. 2

B. The AFBM Transmitter in Brief

• AFBM extends AFDM by replacing conventional chirp-based subcarriers with filtered chirp subcarriers generated through a filter bank.The method combines affine-domain spreading with filter-bank multicarrier processing.Source: B. The AFBM Transmitter in Brief · p. 2
• AFBM uses only L/2 of its L subcarriers for data transmission to preserve near complex orthogonality.The passage states this is configured similarly to 2D-FFT FB.Source: B. The AFBM Transmitter in Brief · p. 2
• After symbol allocation, filter compensation and affine-domain spreading produce the DAFT-spread transmit signal.The spreading uses the L-point DAFT matrix and chirp modulation matrices parameterized by c1 and c2.Source: B. The AFBM Transmitter in Brief · p. 2
• The chirp parameters are derived from an orthogonality condition using maximum normalized Doppler shift, maximum delay, and guard width.The guard width indicates the number of additional reserved elements for Doppler-interference leakage.Source: B. The AFBM Transmitter in Brief · pp. 2–3
• The affine-precoded symbols undergo an extended inverse DAFT operation with frequency-domain zero padding before filtering.The R-point transform is larger than the L-dimensional affine-spread block, with L < R < N.Source: B. The AFBM Transmitter in Brief · p. 3
• Prototype filtering produces the transmitted AFBM signal, and the filter bank confines channel-induced interference so no cyclic prefix is required.The channel is modeled as a linear convolution.Source: B. The AFBM Transmitter in Brief · p. 3

III. CHANNEL MODEL

• The doubly dispersive channel models multipath and relative motion as multiple delayed and Doppler-shifted copies of the transmitted signal.The model is used to capture time-frequency selectivity in high-mobility channels.Source: §III · p. 3
• Each of the P propagation paths is characterized by a complex gain, a delay, and a Doppler shift.The time-varying impulse response is expressed using these path parameters.Source: §III · p. 3
• The discrete-time model defines normalized integer delay ℓp and digital Doppler shift fp, and represents the channel as a matrix.The delay uses the sampling period Ts; the Doppler shift is normalized by sampling frequency fs.Source: §III · p. 3
• The matrix uses a linear shift operator for delay and a diagonal matrix for Doppler-induced phase shifts; this same channel model is applied to every evaluated waveform.The passage also defines t as time and τ as the delay variable.Source: §III · p. 3

IV. WAVEFORM RECEPTION

• The received signal vector is expressed after propagation through the channel.Source: §IV · p. 3
• The noise is modeled as a complex additive white Gaussian noise (AWGN) vector.Source: §IV · p. 3
• The AWGN vector’s dimension corresponds to the length of the transmitted signal.Source: §IV · p. 3

A. The 2D-FFT FB Receiver in Brief

• The receiver first demodulates the received signal with an analysis filter bank, then splits it into K columns and removes frequency-domain zero padding after DFT processing.This recovers the intermediate time-frequency matrix.Source: A. The 2D-FFT FB Receiver in Brief · p. 3
• The reception-chain diagram places the 2D-FFT filter-bank receiver in the upper branch and the AFBM receiver in the lower branch.The lower branch includes analysis filtering, affine-domain processing, compensation, and QAM-symbol recovery.Source: Figure 2 · p. 4
• The receiver reverses the transmitter’s DFT-based frequency spreading and time-axis IDFT using the corresponding unitary transforms.The L-point DFT despreads symbols across subcarriers, while the K-point IDFT is undone along the time axis.Source: A. The 2D-FFT FB Receiver in Brief · p. 4
• The diagonal compensation is removed by its inverse to yield the estimated data matrix.Source: A. The 2D-FFT FB Receiver in Brief · p. 4

B. The AFBM Receiver in Brief

• The AFBM receiver first processes the received signal through an analysis filter bank and affine-domain demodulator.Source: B. The AFBM Receiver in Brief · p. 4
• The receiver then obtains the estimated allocated-grid vector from the vector received before affine-domain despreading and compensation.The received vector is denoted z in the passage.Source: B. The AFBM Receiver in Brief · p. 4
• The estimated data-symbol matrix is recovered by reversing subcarrier allocation, and parallel-to-serial conversion yields the estimated transmitted symbol vector.Source: B. The AFBM Receiver in Brief · p. 4

• The discrete AF assesses each waveform’s sensing capability from its transmitted signal.Source: §V · p. 4
• Delay and Doppler indices are normalized to τnorm = ℓ/M ∈[−1, 1] and νnorm = f/M ∈[−0.5, 0.5] for a system-agnostic representation.Source: §V · p. 4
• Delay and Doppler resolution are assessed from the zero-Doppler and zero-delay cuts, respectively, each normalized to unit peak.The cuts are Aτ[ℓ] = A[ℓ, 0] and Aν[f] = A[0, f].Source: §V · p. 4

• At a CCDF level of 10^-3, AFBM attains approximately 8 dB PAPR, about 1.7 dB, 2.3 dB, and 3 dB below OTFS, 2D-FFT FB, and AFDM, respectively.The passage attributes this to AFBM’s DAFT-based spreading.Source: §VI · p. 4, Figure 4 · p. 5
• Filter bank-based modulations show about 40 dB OOBE gain over OTFS and AFDM.Fig. 5 compares PSD versus normalized frequency; its markers distinguish PHYDYAS-filtered from Hermite-filtered waveforms.Source: §VI · p. 4, Figure 5 · p. 5
• At 500 km/h in a Vehicular A channel, all four waveforms achieve similar BER, with differences below about 1 dB across the simulated SNR range.The BER-versus-SNR comparison is shown in Fig. 3.Source: §VI · p. 4, Figure 3 · p. 5
• With deterministic all-ones sensing symbols, OTFS and 2D-FFT FB have a clean delay cut, while AFBM has the highest delay ambiguity and AFDM repeats its sharp peak in lobes.AFDM’s repeated lobes reach approximately −25 dB; AFBM’s dense ridge approaches −5 dB near the origin.Source: §VI · p. 5, Figure 6 · p. 5
• In the deterministic Doppler cut, AFBM and AFDM have one-bin main lobes, whereas OTFS and 2D-FFT FB have broader lobes with decaying sidelobes.Source: §VI · p. 5, Figure 6 · p. 5
• With random QAM sensing symbols, the four waveforms’ AF responses largely coincide, and AFBM reaches the lowest Doppler-cut floor.The random-symbol cuts are shown in Fig. 7.Source: §VI · p. 5, Figure 7 · p. 6
• The filter-bank schemes’ improved characteristics come with up to three times higher transmitter complexity than OTFS and AFDM.Both require prototype-filter processing and multiple transform stages; AFBM’s additional operations include DAFT processing.Source: §VI · pp. 5–6

• All four waveforms achieved BER within about 1 dB, while the filter bank core yielded about 40 dB lower OOBE than OTFS and AFDM.Source: §VII · p. 6
• AFBM had the lowest PAPR, 1.7 to 3 dB below the other waveforms, and the sharpest Doppler cut, while 2D-FFT FB kept a clean delay cut under deterministic sensing symbols.AFBM had higher delay sidelobes under deterministic sensing symbols.Source: §VII · p. 6
• With random QAM sensing symbols, AF differences largely vanished and AFBM offered the best compromise, at up to three times the transmitter complexity of OTFS and AFDM.Source: §VII · p. 6
• Future work will address hardware impairments in communications and sensing, together with channel estimation.Source: §VII · p. 6