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# Efficient filter-bank pilot structures for multi-user massive MIMO systems

Fatima Hamdar, Jeremy Nadal, Charbel Abdel Nour, Amer Baghdadi

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Fatima Hamdar, Jeremy Nadal, Charbel Abdel Nour, Amer Baghdadi. Efficient filter-bank pilot structures for multi-user massive MIMO systems. IEEE Transactions on Communications, In press, 10.1109/TCOMM.2025.3597802. hal-05226339

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# Efficient filter-bank pilot structures for multi-user massive MIMO systems

Fatima Hamdar, Jeremy Nadal, Charbel Abdel Nour, and Amer Baghdadi

***Abstract***—This work proposes a novel pilot structure (PS) to improve channel estimation (CE) in massive multiple-input multiple-output filter bank multi-carrier (mMIMO-FBMC) communications using offset quadrature amplitude modulation (OQAM). Accurate CE is essential to fully benefit from the corresponding spectral efficiency (SE) advantages and robustness against dispersive channels. Unlike orthogonal frequency division multiplexing (OFDM), FBMC/OQAM shows only real-field orthogonality. Therefore, corresponding signals suffer from intrinsic interference, motivating the use of alternative CE techniques. Furthermore, typical CE methods for mMIMO-FBMC systems require large training overheads, particularly for a large number of users or for flat-fading channel conditions over each subcarrier band. To address these issues, this paper proposes a PS that reduces the training overhead by interleaving user pilots in frequency while using conventional OFDM CE techniques, particularly the least squares (LS) method in the context of single-user (SU) and multi-user (MU) scenarios. Analytical expressions for the mean square error (MSE) and the Cramer-Rao lower bound (CRLB) are derived for the estimation technique considering the proposed PSs. Furthermore, performed simulations over the 5G QuaDRiga channel show that the proposed PS improves SE while delivering optimal performance in both preamble-based MU-mMIMO and SU systems.

***Index Terms***— Channel estimation, Beyond 5G, massive MIMO, filtered multi-carrier waveforms, FBMC/OQAM, Overlap-Save based FBMC receivers, OFDM, QuaDRiGa channel, Intrinsic interference.

## I. INTRODUCTION

Filter bank multicarrier (FBMC)-offset quadrature amplitude modulation (OQAM) has emerged in recent years as a promising alternative to orthogonal frequency division multiplexing (OFDM), gaining significant attention for its potential advantages. Notable advancements have been proposed for FBMC/OQAM in single-input-single-output (SISO) systems [1], [2]. On the transmitter side, an innovative short prototype filter (PF), referred to as near perfect reconstruction 1 (NPR1), was introduced, offering the same duration as an OFDM symbol. This design substantially reduces latency, hardware complexity, and energy consumption [1]. On the receiver side, advanced techniques tailored for short filters were also developed [2], significantly enhancing robustness against doubly-dispersive channels and enabling asynchronous communication. These advancements were further extended to multi-user massive MIMO (MU-mMIMO) systems in [3],

demonstrating their applicability and benefits. However, perfect channel estimation (CE) was assumed in these studies, and the full realization of these improvements relies critically on accurate CE.

FBMC-OQAM, while striving to establish itself as a primary waveform for future applications, confronts intrinsic interference [4] that complicates CE by contaminating training pilots with adjacent data symbols. Indeed, this specific problem has to be solved either for preamble-based approaches [5]–[9], or scattered pilot-based approaches [4], [10]–[12]. These methods generally either focus on mitigating unwanted interference or leveraging it to enhance estimation performance. A major issue about scattered pilots is that they complicate the use of peak-to-average power ratio (PAPR) reduction precoding techniques, a critical factor in multi-carrier systems. Therefore, this paper focuses on preamble-based pilots.

To mitigate this interference in preamble-based approaches, guard FBMC intervals are often inserted between pilots and the remainder of the frame. However, this penalizes spectral efficiency (SE), especially when using long prototype filters. In multi-user (MU) scenarios, additional guard intervals (GI) are required to isolate users pilots, further degrading SE. Therefore, it is essential to explore alternative strategies that can minimize or eliminate the reliance on GIs.

Several preamble-based CE solutions have been proposed in the literature. The frequency domain interference approximation method (IAM) introduced in [5] represents the widely used CE technique for FBMC/OQAM. Researchers have delved deeper into the IAM approach in [6], [13], where complex pilot preamble structures were explored. Unfortunately, the IAM technique suffers from an extremely high PAPR that makes its usage impractical in a real system, as it can lead to signal distortion when the peak power surpasses the linear operating range of the power amplifier.

As an alternative research direction, CE methods based on the time-domain models were explored [14]–[17]. The terminology used to distinguish between time-domain and frequency-domain channel estimators depends solely on the domain of the input pilot symbols. Specifically, if the input to the estimator consists of time-domain signals, the estimator is referred to as a time-domain estimator and vise versa. This classification holds regardless of the format of the output, which may be expressed either as the channel impulse response (CIR) or the channel frequency response (CFR). In [14], a general time-domain CE technique was introduced

<footnote>IMT Atlantique, Lab-STICC, UMR CNRS 6285, F-29238 Brest, France, CORRESPONDING AUTHOR: F. HAMDAR (e-mail: fatima.hamdar@imt-atlantique.fr).</footnote>
for single-user (SU) systems, which requires that the number of pilots is equal to the number of active subcarriers. However, this approach leads to a high loss in SE, particularly when extended to MU-massive multiple-input multiple-output (mMIMO), where additional GIs are needed to distinguish the training signals of different users. Similarly, the method in [15] also relies on GIs. In [17], the authors adopt the general technique from [14] while optimizing preamble design for both OFDM and FBMC/OQAM systems. The proposed method proved beneficial for OFDM, but its performance gain in FBMC/OQAM systems was found to be negligible. An alternative time domain technique that applies for its mMIMO extension was proposed in [18] and explored further in [16]. This approach involves transmitting pilots for each user on all subcarriers, with user pilots being multiplexed in the code domain to allow sharing of the same time-frequency resources for CE. One major drawback resides in the computational load needed at the receiver to demultiplex the channel responses of different users based on their code sequences. Unfortunately, the aforementioned techniques are either bound to be used for short delay spread or set to suffer a significant SE loss, particularly in MU scenarios.

The limitations of the aforementioned state-of-the-art CE techniques naturally lead to a critical question: Can conventional OFDM CE schemes be effectively employed in FBMC/OQAM systems without compromising real orthogonality?

In this paper, we provide a compelling answer to this question by proposing an innovative pilot structure (PS) design, which when integrated with conventional CE techniques, significantly enhances error rate performance while maintaining SE and minimizing computational complexity. The proposed approach is particularly advantageous in the realm of MU-MIMO systems, where efficient CE is essential to maximize system performance. The specific contributions presented in this article can be summarized as follows:

1) Proposal of a spectrally efficient PS associated to a CE technique for SU and MU-mMIMO systems.

2) Deriving analytical expressions for the normalized mean square error (NMSE) and the Cramer-Rao lower bound (CRLB) [19] of the estimation technique, when considering the proposed SE.

3) Performance comparison of the proposed method against state-of-the-art techniques in terms of NMSE, PAPR, SE and computational complexity.

4) Performance comparison of the complete MU-mMIMO system considering the OS-based FBMC receivers with the considered methods in terms of bit-error rate (BER) over the 5G QuaDRiga (QUAsi Deterministic RadIo channel Generator) channel under the 3D urban macro-cell non-line-of-sight (3D-UMa-NLOS) scenario.

The rest of the paper is organized as follows. In Section II, we present the principles of FBMC/OQAM and go through relevant existing FBMC frequency domain and time-domain CE techniques. The proposed PSs along with the MU and SU CE techniques are detailed in Section III. In Section IV, we provide the analytical derivation of the NMSE and the CRLB

of the estimation technique when considering the proposed PS along with the corresponding computational complexity. Section V assesses the efficiency of the proposed CE approach compared to state-of-the-art solutions and verifies the accuracy of our claim regarding computational complexity through simulations. Section VI provides a performance comparison over the realistic 5G Quadriga channel model. Section VII concludes the paper.

## II. FBMC/OQAM: PRINCIPLES AND CHANNEL ESTIMATION

To lay the groundwork for the proposed CE technique, in this section we will discuss relevant time domain and frequency domain CE techniques for FBMC/OQAM systems available in the literature.

### A. FBMC/OQAM principles

The FBMC modulation can be applied using either the frequency spread (FS)-based [20] or the polyphase network (PPN)-based [21] implementation. The latter is usually employed on the transmitter side due to its lower complexity [22]. Similar to OFDM, the PPN implementation consists of a size-$M$ IFFT, followed by a PPN for the filtering stage. The PPN complexity is directly related to the overlapping factor $K$. Hence, lower computation complexity and latency are achieved by using a short prototype filter (PF) (i.e. $K = 1$). To preserve orthogonality in the real field, the OQAM modulation scheme is employed. As a result, the PPN output signal is delayed by $M/2$ samples before being overlapped with the signal generated at the previous iteration. Therefore, if $M$ is the total number of subcarriers and $a_{m,n}$ is the pulse-amplitude modulated (PAM) symbols at time slot $n$, subcarrier index $m$, then the baseband signal $s(k)$ is expressed as

$$s(k) = \sum_{n=-\infty}^{+\infty} \sum_{m=0}^{M-1} g\left(k - n\frac{M}{2}\right) (-1)^{nm} a_{m,n} \phi_n(m) e^{j2\pi \frac{km}{M}}, \quad (1)$$

where $\phi_n(m) = i^{n+m}$ is a phase shift introduced between adjacent PAM symbols to preserve the orthogonality in the real field. Considering a time-invariant channel, the received signal can be expressed as follows

$$r(k) = \sum_{\ell=0}^{L-1} h(\ell)s(k - \ell) + \eta(k) \quad (2)$$

where $h$ is the impulse response of the channel, $L$ is the number of taps, and $\eta$ is the additive white Gaussian noise. The demodulated symbol at the frequency bin $m$ and time symbol $n$ can be written as

$$\hat{a}_{m,n} = \sum_{k=-\infty}^{\infty} r(k)g\left(k - n\frac{M}{2}\right) e^{-j2\pi mk/M} \phi_n^*(m). \quad (3)$$

where * denotes the complex conjugate. In order to retrieve the real transmitted symbols, it is necessary to apply equalization on $\hat{a}_{m,n}$ and extract the real part. However, the main focus of this work is to address the CE problem of FBMC/OQAM systems. Therefore, state-of-the-art equalization techniques based on the overlap-save algorithm are employed for the rest of the study [23], [24].
## B. Frequency domain channel estimation

In FBMC/OQAM systems, CE is a more complex task due to the absence of complex orthogonality. To avoid interference with the transmitted data, the transmitted pilots must be real valued and a sufficient number of GIs must be inserted between the transmitted pilots and data symbols. However, the received signal is still corrupted by intrinsic interference resulting from the adjacent transmitted data.

In [25], the authors reviewed several preamble-based CE methods including the pairs of pilots (POP) method [5]. This latter technique involves transmitting two pilots at adjacent time and frequency locations. The pilots are transmitted on two neighboring subcarrier indices. Assuming the channel response remains constant over these two pilot positions, four equations with four unknowns can be derived, which can then be solved. Although the POP method offers simplicity, its estimation accuracy is poor, leading to important BER degradation [25]. Alternatively, [5] introduces a more robust method known as the IAM. The key innovation of this technique is its ability to convert the typically problematic intrinsic interference into a beneficial feature. Specifically, if the interference terms can be managed, additional power can be recovered, thus improving the accuracy of the estimation. This is achieved by repeating the sequence vector [1, 1, -1, -1] across all allocated subcarriers used as pilots. To prevent uncontrollable intrinsic interference from neighboring FBMC symbols, all other subcarrier values are set to zero. These subcarriers are not transmitted through the channel, instead functioning solely as a GI. Indeed, the IAM technique suffers from a high PAPR making it subject to the non-linearity distortions of the power amplifier.

In [26], preamble-based least squares (LS) CE for FBMC/OQAM was analyzed, showing that FBMC/OQAM with sparse preambles, where only specific number of subcarriers carry pilots and the other subcarriers are nulled, performs similarly to CP-OFDM. Later, in [7], an iterative CE technique was proposed to lower inter-symbol interference (ISI) from neighboring symbols using a shorter preamble featuring only one-column frequency domain pilots. All previously mentioned CE approaches either suffer from high PAPR or are limited to short delay spreads. If this prerequisite is not fulfilled, these methods will experience a significant penalty in performance.

## C. Time domain channel estimation

As an alternative research direction to Frequency domain models, time domain CE models were proposed. According to [14], by substituting (1) and (2) in (3), the received pilot vector $P_r$ can be expressed as follows

$$ P_r = Ah + \eta \quad (4) $$

where $h = [h(0), ..., h(L-1)]$ is a vector containing the channel taps, $\eta = [\eta_{0,n}, ..., \eta_{M-1,n}]$ is the noise vector, and $A$ is an $M \times L$ matrix with its entry on the $m^{th}$ row and $l^{th}$ column is defined as

$$ A(m,l) = \sum_{k=-\infty}^{\infty} \sum_{m'=0}^{M-1} \sum_{n'=-\infty}^{+\infty} a_{m',n'} g \left( k - l - n' \frac{M}{2} \right), \quad (5) \\ \times g \left( k - n \frac{M}{2} \right) e^{j2\pi(m'-m)k/M}, \\ \times e^{j\pi(m'+n'-m-n)/2} e^{-j2\pi m'l/M}. $$

It is important to note that the orthogonality restriction imposed by FBMC/OQAM systems in the real domain, added to the filtering operation, leads to a correlation of the noise vector elements in (4).

Therefore, the conventional LS CE method used for OFDM systems [27] cannot be readily applied [14]. To solve this issue, the authors in [14] included the channel noise correlation matrix C in the LS method as follows

$$ \hat{h} = (A^H C^{-1} A)^{-1} A^H C^{-1} P_r. \quad (6) $$

The authors of [14] have shown that applying (6) outperforms the frequency domain IAM method for SU scenario. However, in a MU scenario, this method requires extra GIs to separate the pilot signals of different users, hence penalizing PS.

To avoid the insertion of GIs between user pilots, the authors in [28] proposed an alternative PS that relies on interleaving user pilots in time and frequency. Furthermore, each user employs $N_p$ instead of $M$ pilots, where $N_p$ corresponds to the channel length of a user and three FBMC GIs were inserted between the pilots and data symbols to shield the pilots from interference. While spectrally efficient thanks to the absence of GIs, this method leads to intrinsic interference due to the experienced correlation between the estimates of different user channels. To avoid a performance penalty, it is necessary to explicitly consider intrinsic interference when applying (6) during the estimation process. However, this leads to a higher computational complexity. Moreover, the authors of [28] simply considered an exponentially decaying channel in their simulations. Therefore, the performance may differ for more realistic channels.

Other alternative methods, which operate on a different basis, aim to restore complex orthogonality in FBMC/OQAM systems, enabling conventional OFDM CE techniques. This is achieved by spreading symbols in time or frequency [29], [30], leading to coded FBMC/OQAM waveform. While this strategy effectively addresses the issue of intrinsic interference and enables conventional OFDM CE techniques, it comes with significant drawbacks, including increased sensitivity to doubly-selective channels [31] and increased computational complexity.

Next, we propose a new PS that enables the use of the conventional LS CE method while improving its accuracy and avoiding GI insertion between users pilots. Moreover, the proposal requires only one FBMC symbol as a GI between pilots and data symbols.

# III. PROPOSED PILOT STRUCTURE

## A. Single-user scenario

Zadoff-Chu (ZC) sequences [32] represent one of the pilot sequence families used in the 5G new radio (NR) wireless
<table><caption>TABLE I
EVALUATION OF Ψ<sub>g</sub>(0, m) OF THE NPR1 PF [1].</caption><thead><tr><th>m</th><th>Ψ<sub>g</sub>(0, m)</th></tr></thead><tbody><tr><td>1</td><td>0.564</td></tr><tr><td>2</td><td>0</td></tr><tr><td>3</td><td>0.0667</td></tr><tr><td>4</td><td>0</td></tr><tr><td>5</td><td>0.0023</td></tr><tr><td>6</td><td>0</td></tr></tbody></table>

system. Each sequence has a constant amplitude zero auto-correlation (CAZAC) property that is preserved even when processed by an IFFT/FFT since the output samples remain a ZC sequence. This leads to a low PAPR, making such sequences an attractive choice for CE. However, due to the fact that FBMC systems only exhibit orthogonality within the real domain, it is not possible to load adjacent subcarriers with complex-valued pilots, since the latter must be shielded from interference. To circumvent this limitation, exploiting only subcarriers with even or odd indices ensures non-overlapping subcarriers, allowing complex-valued pilots to be loaded without intrinsic interference. This behavior can be explained by the ambiguity function of the PF defined by

$$ \Psi_g(n, m) = \sum_{k=-\infty}^{\infty} g\left(k + \frac{nM}{2}\right) g^*\left(k - \frac{nM}{2}\right) e^{j \frac{2\pi km}{M}}, \quad (7) $$

In fact, we have

$$ \Psi_g(n, 2m) = 0 \quad \forall (n, m) \in \Omega, $$

along with $\Omega = [-\infty, +\infty] \times [-\frac{M}{2}, \frac{M}{2}]$.

We opt to set $n = 0$ since the pilot symbols are sent on the first FBMC symbol. Table I presents the ambiguity function for the NPR1 [1] PF as calculated using equation (7) with a total of $M = 512$ subcarriers. As shown in Table I, the ambiguity function of the NPR1 PF satisfies $\Psi_g(0, 2m) = 0$ for even subcarrier indices.

In light of this observation, we propose assigning complex pilots to even frequency indices and keeping the neighboring FBMC subcarrier positions vacant or inactive. Therefore, if $\mathbf{P} = [p_0(0), p_2(1), \dots, p_M(M/2)]^T \in \mathbb{C}^{M/2 \times 1}$ is a vector containing complex-valued elements generated from a ZC sequence, the transmitted pilots $\mathbf{P}_T$ are defined as

$$ \mathbf{P}_T = \begin{cases} p_m(k) & \text{if } m = 2k, \text{ where } 0 \le k \le M/2 - 1 \\ 0, & \text{otherwise} \end{cases}, \quad (8) $$

along with

$$ p_m(k) = e^{-\frac{j\pi k^2}{M/2}} \quad \text{where } 0 \le k \le M/2 - 1. \quad (9) $$

Following the proposed PS, the complex-valued ZC sequences can still be employed for FBMC/OQAM, along with the CE techniques considered for OFDM. However, a sufficient number of GIs must be introduced among the training and data symbols to ensure their total isolation, resulting in a data rate loss. In fact, the required number of FBMC GIs depends on the overlapping factor of the considered PF. Therefore, we propose

<figure><img src="image_3.png" /><figcaption>Fig. 1. Proposed FBMC/OQAM PS for uplink SU CE.</figcaption></figure>

to employ the NPR1 short PF which requires only two FBMC symbols for channel estimation: one symbol is designated for carrying the pilot signals, while the other serves as a GI to shield the pilots from interference. While the aforementioned advantage is particularly relevant for short PFs, the structure is flexible and can be employed with long PFs as well, though at the cost of increased GIs to maintain isolation. Figure 1 depicts the proposed PS in the uplink.

Applying an FFT to the received pilots in equation (2), and by adopting the filtering stage in the time domain, the received pilot symbols at index $n = 0$ can be expressed as

$$ \mathbf{P}_r = \mathbf{A} \odot \mathbf{H} + \eta \quad (10) $$

where $\odot$ is the Hadamard product, $\mathbf{H} \in \mathbb{C}^{M \times 1}$ is the CFR with its $m^{th}$ element being $H(m) = \sum_{\ell=0}^{L-1} h(\ell)e^{-j2\pi m\ell/M}$ and $\mathbf{A} \in \mathbb{C}^{M \times 1}$ with its $m^{th}$ entry defined as

$$ A(m) = \sum_{k=-\infty}^{+\infty} \sum_{m'=0}^{M-1} p_{m'} g^2(k) e^{j2\pi(m'-m)k/M}. \quad (11) $$

As mentioned earlier, the noise correlation must be included in the LS channel due to the lack of complex orthogonality. Therefore, we compute the noise correlation denoted by $\rho_{\eta_{m_1}, \eta_{m_2}}$ between subcarriers $m_1$ and $m_2$. Hence, $\rho(\eta_{m_1}, \eta_{m_2})$ can be computed as follows

$$ \begin{aligned} \rho(\eta_{m_1}, \eta_{m_2}) &= \frac{\mathbb{E}[\eta_{m_1} \eta_{m_2}^*] - \mathbb{E}[\eta_{m_1}] \mathbb{E}[\eta_{m_2}^*]}{\sigma^2}, \\ &= \sum_{k=-\infty}^{\infty} g^2(k) e^{j2\pi \frac{k(m_2-m_1)}{M}} \end{aligned} \quad (12) $$

where $\sigma^2$ is the noise variance and $\eta_m$ is defined as follows

$$ \eta_m = \sum_{k=-\infty}^{\infty} \eta(k) g(k) e^{j2\pi \frac{km}{M}}, \quad (13) $$

We opt now to set $m_1 = 0$ for the sake of clarity and simplicity in our analysis. However, it is important to note that the correlation remains consistent for other values of $m_1$. In fact, using equation (7), we have $\rho_{\eta_{m_1}, \eta_{m_2}} = \Psi_g(0, m_2)$, and from Table I, we have $\rho(\eta_0, \eta_{2m_2}) = 0$. Therefore, the assumption of uncorrelated noise at even subcarrier indices is valid, and the conventional frequency domain LS channel estimation method used in OFDM systems can be readily applied
without considering the noise correlation matrix, reducing the complexity as shown below

$$ \hat{\mathbf{H}}_{\text{LS}} = \mathbf{P}_{\text{T}}^* \times \mathbf{P}_{\text{r}}. \quad (14) $$

In fact, the estimated channel has most of its power concentrated on a few first samples $L'$. Therefore, it is advantageous to denoise the above channel estimate. Denoising is the process of eliminating noise from a noisy signal. The estimated channel in (14) can be seen as corrupted by noise. Hence, the resulting CIR can be considered a noisy signal, and denoising it can lead to an improved CIR estimation. Denoising is performed by pruning the CIR coefficients. This can be done using the Discrete Cosine Transform (DCT) [33] or the Discrete Fourier Transform (DFT)-based approach [34]. However, when the path delays are not integer multiples of the sampling period, causing leakage in the CIR, the DFT-based method becomes unsuitable for CE. Hence, the DCT-based method will be employed instead. It starts by applying a DCT of size $M/2$ on the estimated CFR $\hat{\mathbf{H}}_{\text{LS}}$ as follows

$$ \hat{\mathbf{h}}'_{\text{DCT}} = \mathbf{DCT}(\hat{\mathbf{H}}_{\text{LS}}), \quad (15) $$

In fact, $\hat{h}'_{\text{DCT}}[n] \approx \eta[n]$ when $n$ is larger than the channel delay spread $L'$. To accurately select the significant channel taps, we first estimate the noise power as follows

$$ \hat{\sigma}_{\eta}^{2} = \frac{1}{\frac{M}{2} - L'} \sum_{n=L'}^{\frac{M}{2}-1} |\hat{h}'_{\text{DCT}}[n]|^{2}. \quad (16) $$

Next, we select the significant channel taps according to the following rule

$$ \hat{h}''_{\text{DCT}}[n] = \begin{cases} \hat{h}'_{\text{DCT}}[n], & \text{if } |\hat{h}'_{\text{DCT}}[n]|^2 > \hat{\sigma}_{\eta}^2, \\ 0, & \text{otherwise} \end{cases}, \quad (17) $$

Eventually, the channel estimate is then obtained by taking the IDCT of equation (17) as follows

$$ \hat{\mathbf{H}}_{\text{IDCT}} = \mathbf{IDCT}(\hat{\mathbf{h}}''_{\text{DCT}}), \quad (18) $$

Note that the above equation gives the channel estimate only for the even subcarrier indices since the odd subcarriers were vacant. Therefore, the channel on the odd subcarriers can be interpolated using Makima's method in [35].

## B. Multi-user scenario

In the MU scenario, $U$ users share the same subband. Ensuring the orthogonality for the user pilots of the uplink typically calls for the insertion of GIs. However, most of the available solutions in the literature are not efficient in terms of spectral usage, as discussed previously. The use of ZC sequences in this context is particularly advantageous due to their low cross-correlation properties. This characteristic minimizes pilot contamination, allowing for clearer channel estimation even when multiple users transmit simultaneously. Furthermore, the constant amplitude of ZC sequences, which results in a low PAPR, offers significant benefits. Indeed, CE relies on pilot signals, often allocated a higher power to enhance accuracy. However, a high PAPR in the preamble

<figure><img src="image_4.png" /><figcaption>Fig. 2. Proposed FBMC/OQAM PS for uplink MU CE.</figcaption></figure>

signal can cause significant distortion if the peak power exceeds the linear operating range of the power amplifier, thus degrading the quality of the received pilots and adversely impacting CE performance. The constant amplitude property of ZC sequences addresses this challenge, ensuring reliable CE. Thus, ZC sequences provide a reliable framework for efficient channel estimation in MU-mMIMO systems. To address the issue of poor spectral usage while leveraging the advantages of ZC sequences, we extend the SU PS along with the time domain CE technique that was proposed in Section III-A to a MU uplink scenario. We show how the extended PS reduces MU CE to SU. In this regard, we propose to circularly shift the pilots of different users within the same band as shown in Figure 2. Note that the frequency resources used for pilot signals of different users are mutually inclusive. This means that each frequency slot is used by all users, however, the pilot signal of user $u$ is a circular shift of the user $u+1$ pilot signal. This translates mathematically as follows

$$ \mathbf{P}_{\text{Tu}} = \begin{cases} p_m^u(k) & \text{if } m = 2k, \text{ where } 0 \le k \le M/2 \\ 0, & \text{otherwise} \end{cases}, \quad (19) $$

along with

$$ \begin{gathered} p_m^{u+1}(k) = p_i^u(k), \\ i = \text{mod}_{L_{\text{ZC}}} \left( m + \frac{L_{\text{ZC}}}{U} \right), \end{gathered} \quad (20) $$

such that

$$ \frac{L_{\text{ZC}}}{DS} \ge U, \text{ and } i > L' \quad (21) $$

where $\text{mod}_x$ is the modulo $x$ operator, $L_{\text{ZC}} = \frac{M}{2}$ is the ZC sequence length, and $DS$ is the average delay spread of all users.

The absence of GIs between user pilots leads to intrinsic interference. The presence of this latter translates into an inter-user dependency of the information used for computing the CE. As a result, intrinsic interference from a user $u_2$ to another user $u_1$ denoted by $\zeta_{u_2}^{u_1}$ must be considered for CE. Therefore, with $U$ being the total number of users, the demod-
<figure><img src="image_5.png" /><figcaption>Fig. 3. CIR for 3 users before the denoising stage over the multi-path 5G QuaDRiGa channel under the '3GPP-3D-UMa-NLOS' scenario.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Channel estimation for uplink MU and SU.</figcaption></figure>

ulated pilot signal on even subcarrier indices $m$, time index 0 can be expressed as

$$p_m = \sum_{u=1}^{U} \sum_{k=-\infty}^{\infty} \sum_{m'=0}^{M-1} p_{m'}^u g^2(k) \times e^{j2\pi(m'-m)k/M} \times H_u(m) + \eta_m, \quad (22)$$

where $H_u(m)$ is the $u^{th}$ user channel frequency response at subcarrier $m$. To ease notations, the above equation can be expressed in a vector form as follows

$$\bar{\mathbf{P}} = \bar{\mathbf{P}}' \odot \bar{\mathbf{H}} + \bar{\eta}, \quad (23)$$

where

$$\begin{align*}
\bar{\mathbf{H}} &= [\mathbf{H}_1, \dots, \mathbf{H}_U]^T, \\
\mathbf{H}_u &= [H_u(0), H_u(2), \dots, H_u(M/2-2)]^T, \\
H_u(m) &= \sum_{\ell=0}^{L_u-1} h_u(\ell) e^{-j2\pi m \ell / M}, \\
\bar{\mathbf{P}} &= [\mathbf{P}_{r_1}, \dots, \mathbf{P}_{r_U}]^T, \\
\mathbf{P}_{r_u} &= [p_0^u, p_2^u, \dots, p_{M/2-2}^u]^T, \\
\bar{\eta} &= [\eta_0, \dots, \eta_u]^T,
\end{align*}$$

and

$$\bar{\mathbf{P}}' = \begin{bmatrix} \mathbf{P}_1 & \zeta_1^2 & \cdots & \zeta_1^U \\ \zeta_2^1 & \mathbf{P}_2 & \cdots & \zeta_2^U \\ \vdots & \vdots & \ddots & \vdots \\ \zeta_U^1 & \zeta_U^2 & \cdots & \mathbf{P}_U \end{bmatrix}, \quad (24)$$

where $\mathbf{P}_u$ and $\zeta_q^u \in \mathbb{C}^{M/2 \times 1}$ are vectors with their $m^{th}$ entry defined as

$$P_u(m) = \sum_{k=-\infty}^{\infty} \sum_{m'=0}^{M-1} p_{m'}^u g^2(k) \times e^{j2\pi(m'-m)k/M}, \quad (25)$$

$$\begin{align}
\zeta_{q'}^q &= \sum_{k=-\infty}^{\infty} \sum_{m'=0}^{M-1} p_{m'}^q g^2(k) \times e^{j2\pi(m'-m)k/M} \tag{26} \\
&= P_{q'} \left( \text{mod}_{L_{ZC}} \left(m + \frac{L_{ZC}}{U}\right), l \right). \nonumber
\end{align}$$

Using equation (14) and neglecting noise correlation as previously discussed in equation (12), the LS CE can be obtained as

$$\hat{\mathbf{H}}_{LS} = \overline{\mathbf{P}}_{T_u}^* \times \overline{\mathbf{P}}, \quad (27)$$

where $\hat{\mathbf{H}}_{LS} = [\hat{\mathbf{H}}_1, \dots, \hat{\mathbf{H}}_U]$ and $\overline{\mathbf{P}}_{T_u} = [\mathbf{P}_{T_1}, \dots, \mathbf{P}_{T_U}]$. In fact, the proposed PS for a MU scenario shifts the intrinsic interference caused by the user pilots to the noise part. In other words, the estimated channel can be divided into two parts: the CIR part and the noise plus user interference part. In light of this definition, equation (15) is modified as follows

$$\hat{\mathbf{h}}_{\text{DCT}}^{\mathbf{q}} = \begin{cases} \text{DCT}_{\frac{M}{2}}(\hat{\mathbf{H}}_{\mathbf{q}}), & \text{if } 0 \le n \le L_q - 1 \\ \sum_q \zeta_q^{\mathbf{q}} \mathbf{H}_{q'} + \eta_{\mathbf{q}}, & \text{otherwise} \end{cases}. \quad (28)$$

where $L_q$ is the delay spread of the $q^{th}$ user channel.

To clarify equation (28), we plot in Figure 3 the CIR estimate for the first user before the denoising stage, under the multi-path 5G QuaDRiGa channel with the '3GPP-3D-UMa-NLOS' scenario and 3 users. The figure clearly illustrates how the proposed PS, based on circular shifts in the frequency domain, effectively separates the users' channels in the time domain. Specifically, the interference from the second and third users appears outside the effective time region of the first user's CIR. This temporal separation ensures that the CIRs of different users do not overlap, significantly reducing multi-user interference that would otherwise degrade the estimation accuracy. As a result, the DCT-based denoising stage can suppress the shifted interference and retain only the relevant CIR taps. Additionally, Figure 3 highlights the benefit of using the DCT instead of the DFT; the DCT provides smoother edges in the CIR, which helps reduce leakage and enhances the overall estimation quality.

Similarly as explained before in Section III-A, all the second part in equation (28) can be neglected which modifies equation (16) to

$$\hat{\sigma}_{\eta_q}^2 = \frac{1}{i - L'} \sum_{n=L'}^{i-1} |\hat{\mathbf{h}}_{\text{DCT}}^{\mathbf{q}}[n]|^2, \quad (29)$$
where i is the circular shift applied for the qth user defined in (20).

Hence, the intrinsic interference in MU CE can be ignored and therefore, equation (24) reduces to a block diagonal matrix with its entries defined as diag(P₁, P₂, ..., P_U). As a result, the CE for MU reduces to a single-user one, where the channel estimate of the uth user can be computed by adopting equations (17) and (18). To complement the mathematical formulation of the proposed CE technique, Figure 4 illustrates the sequential steps outlined previously considering two users.

## IV. PROPOSED ANALYTICAL STUDY

To assess the efficiency of the CE technique, we propose to analytically derive both the CRLB and the NMSE for the estimators when considering the proposed PS.

### A. NMSE derivation

As part of this analysis, we reconsider Equation (18), which can be reformulated as follows

$$ \hat{\mathbf{H}}_{\text{IDCT}} = \mathbf{D}^{\dagger} \mathbf{W} \mathbf{D} \hat{\mathbf{H}}_{\text{LS}} = \mathbf{\Phi}_{\mathbf{W}} \hat{\mathbf{H}}_{\text{LS}} \quad (30) $$

where $\mathbf{\Phi}_W = \mathbf{D}^{\dagger}\mathbf{W}\mathbf{D}$, $\mathbf{D}$ is the DCT transform matrix of size $(\frac{M}{2} \times \frac{M}{2})$, $\dagger$ denotes the Hermitian transpose operation, and $\mathbf{W}$ is a weighting matrix of size $(\frac{M}{2} \times \frac{M}{2})$ defined as follows

$$ \mathbf{W} = \text{diag} \left( \left[ \mathbf{1}_{L'}, \mathbf{0}_{\frac{M}{2}-L'} \right] \right) \quad (31) $$

In this diagonal matrix, the first $L'$ diagonal elements are set to 1, while the remaining diagonal elements are set to 0. This configuration ensures that only the most significant $L'$ components are retained, thereby reducing noise and improving estimation accuracy. The auto-covariance matrix $\mathbf{R}_{ee}^{\dagger}$ of the estimation error is defined as

$$ \mathbf{R}_{ee}^{\dagger} = \mathbb{E}[(\mathbf{\Phi}_{\mathbf{W}}\hat{\mathbf{H}}_{\text{LS}} - \mathbf{H})(\mathbf{\Phi}_{\mathbf{W}}\hat{\mathbf{H}}_{\text{LS}} - \mathbf{H})^{\dagger}] \quad (32) $$

Moreover, considering $\mathbf{e}_{LS}$ as the estimation error of the LS estimate, we have

$$ \hat{\mathbf{H}}_{\text{LS}} \approx \mathbf{H} + \mathbf{e}_{\text{LS}} \quad (33) $$

Therefore, by substituting equation (33) in (32), we have

$$ \mathbf{R}_{ee}^{\dagger} = \mathbb{E} \left[ \left( (\mathbf{\Phi}_{\mathbf{W}} - \mathbf{I})\mathbf{H} + \mathbf{\Phi}_{\mathbf{W}}\mathbf{e}_{\text{LS}} \right) \left( (\mathbf{\Phi}_{\mathbf{W}} - \mathbf{I})\mathbf{H} + \mathbf{\Phi}_{\mathbf{W}}\mathbf{e}_{\text{LS}} \right)^{\dagger} \right] $$

where $\mathbf{I}$ is the identity matrix.

knowing that $\mathbf{H}$ and $\mathbf{e}_{LS}$ are independent<fnref n="1" />, we can simplify the expression to

$$ \mathbf{R}_{ee}^{\dagger} = (\mathbf{\Phi}_{\mathbf{W}} - \mathbf{I})\mathbf{R}_{HH}^{\dagger}(\mathbf{\Phi}_{\mathbf{W}} - \mathbf{I})^{\dagger} + \mathbf{\Phi}_{\mathbf{W}}\mathbf{R}_{eLS}^{\dagger}\mathbf{e}_{LS}^{\dagger} \mathbf{\Phi}_{\mathbf{W}}^{\dagger} $$

Where $\mathbf{R}_{eLS}^{\dagger}$ is the auto-covariance matrix of the LS estimation error which can be derived as follows

$$ \mathbf{R}_{eLS}^{\dagger} = \mathbb{E}[\mathbf{e}_{LS} \mathbf{e}_{LS}^{\dagger}] = \sigma^2 \quad (34) $$

and $\mathbf{R}_{HH}^{\dagger}$ is the autocovariance matrix of $\mathbf{H}$.

Assuming a random channel model with $L$ sample-spaced

independent Rayleigh fading paths, each with a power delay profile $\alpha(\ell)$, such that the $\ell^{th}$ channel tap $h(\ell)$ follows a $\mathcal{N}(0, \alpha(\ell))$ distribution, the autocovariance matrix of $\mathbf{H}$ can be expressed mathematically as

$$ \mathbf{R}_{HH}^{\dagger} = \mathbb{E}[\mathbf{H}\mathbf{H}^{\dagger}] = \mathbf{F}\mathbb{E}[\mathbf{h}\mathbf{h}^{\dagger}]\mathbf{F}^{\dagger} = \mathbf{F}\mathbf{R}_{hh}\mathbf{F}^{\dagger} \quad (35) $$

where $\mathbf{F}$ is the $M \times L$ DFT matrix, and $\mathbf{R}_{hh}$ is the autocovariance matrix of $\mathbf{h}$, mathematically expressed as

$$ \mathbf{R}_{hh} = \text{diag} \left( [\alpha(0), \alpha(1), \dots, \alpha(L-1)] \right) \quad (36) $$

Since the estimator is unbiased, then the NMSE of this estimator can be calculated as the trace of the autocovariance matrix $\mathbf{R}_{eLS}^{\dagger}$ as follows

$$ \text{NMSE} = \text{tr}(\mathbf{R}_{ee}^{\dagger}) \quad (37) $$

### B. CRLB derivation

For an unbiased estimator, the CRLB is defined as the trace of the inverse of the Fisher information matrix $\mathbf{I}_{fisher}$ as follows [36]

$$ \text{CRLB} = \text{tr}(\mathbf{I}_{\text{fisher}})^{-1} \quad (38) $$

along with

$$ \mathbf{I}_{\text{fisher}} = -\mathbb{E} \left[ \frac{\partial^2}{\partial(\bar{\mathbf{H}}) \partial(\bar{\mathbf{H}}^{\dagger})} \log f(\bar{\mathbf{P}}; \bar{\mathbf{H}}) \right] \quad (39) $$

where $f$ is the multivariate probability density function (PDF) of the observation vector $\bar{\mathbf{P}}$ given $\bar{\mathbf{H}}$ is known, which can be expressed mathematically as follows

$$ f(\bar{\mathbf{P}}; \bar{\mathbf{H}}) = \frac{\exp\left(-\frac{1}{2}(\bar{\mathbf{P}} - \bar{\mathbf{P}}' \times \bar{\mathbf{H}})^{\dagger}\Sigma^{-1}(\bar{\mathbf{P}} - \bar{\mathbf{P}}' \times \bar{\mathbf{H}})\right)}{\sqrt{(2\pi)^N |\Sigma|}} \quad (40) $$

with $\Sigma$ being the covariance matrix with its entries defined in equation (12), $|\Sigma|$ is its determinant, and $N$ is the number of observed variables.

To derive the analytical expression for the CRLB using Equation (38), we start by computing the log-likelihood function of $f$ as follows

$$ \log f(\bar{\mathbf{P}}; \bar{\mathbf{H}}) = -\mathcal{N}\log(2\pi\Sigma) - |\bar{\mathbf{P}}^{\dagger}\Sigma^{-1}\bar{\mathbf{P}} - \bar{\mathbf{P}}^{\dagger}\Sigma^{-1}\bar{\mathbf{P}}'\mathbf{H} - \mathbf{H}^{\dagger}\bar{\mathbf{P}}'\Sigma\bar{\mathbf{P}} + \mathbf{H}^{\dagger}\bar{\mathbf{P}}'\Sigma^{-1}\mathbf{H}\bar{\mathbf{P}}'| \quad (41) $$

Taking the expectation of the second derivative of the log-likelihood, we obtain

$$ \mathbb{E} \left[ \frac{\partial^2}{\partial(\bar{\mathbf{H}}) \partial(\bar{\mathbf{H}}^{\dagger})} \log f(\bar{\mathbf{P}}; \bar{\mathbf{H}}) \right] = \bar{\mathbf{P}}^{\dagger} \Sigma^{-1} \bar{\mathbf{P}}' \quad (42) $$

Therefore, the analytical expression for the CRLB is given by

$$ \text{CRLB} = \text{tr} \left( \bar{\mathbf{P}}^{\dagger} \Sigma^{-1} \bar{\mathbf{P}}' \right)^{-1} \quad (43) $$

<footnote n="1">¹This simplification is valid under the assumption that $\mathbf{H}_{LS}$ is taken on even subcarrier indexes</footnote>
### C. Complexity Analysis

In this section, we analyze the computational complexity of the proposed technique in the context of a MU scenario, comparing it with conventional CE methods in terms of the required number of real-valued multiplications (RMs). We denote the number of RMs required for a specific operation $x$ as $C_{RM}(x)$. From Figure 4, the proposed technique involves the following 4 computational steps:

#### Hadamard Product

The first step involves $U \times \frac{M}{2}$ complex multiplications. Since multiplying two complex numbers requires three real-valued multiplications, the number of RMs for the Hadamard product is:

$$C_{RM}(\odot) = \frac{3}{2}MU.$$

#### DCT and IDCT

The DCT/IDCT has approximately the same computational complexity as the FFT. According to [37], the number of RMs, $C_{RM}(FFT_N)$, for a Fourier transform of size $N$ is $C_{RM}(FFT_N) = N \log_2(N) - 3N + 4$. Since the DCT/IDCT is performed on size $\frac{M}{2}$ for each user, the required number of RMs of each is therefore:

$$C_{RM} \left( \frac{\text{IDCT}}{\text{DCT}} \frac{M}{2} \right) = U \times \left[ \frac{M}{2} \log_2 \left( \frac{M}{2} \right) - \frac{3}{2} M + 4 \right].$$

#### Denoising Stage

The denoising stage involves calculating the noise power $\sigma^2$. The number of RMs required for this step is:

$$C_{RM}(\sigma^2) = \left( 3 \sum_{1}^{U} \left( \frac{M}{2} - L_u \right) + U \right). \quad (44)$$

#### Makima's Interpolation

For each interval between two known points, Makima interpolation consists of two steps: Slope Calculation and Cubic Polynomial interpolation. To interpolate between $\frac{M}{2}$ points, we need to calculate $(\frac{M}{2} - 1)$ slopes, where each slope requires one RM. For the second step, each polynomial segment between the $(\frac{M}{2} - 1)$ intervals requires five RMs. Therefore, the total number of RMs for Makima interpolation is:

$$C_{RM}(\text{Makima}) = U \times 6 \left( \frac{M}{2} - 1 \right).$$

Consequently, the total number of RMs required for the proposed technique is the sum of the RMs from the above steps, expressed as:

$$C_{RM} = U \times \left( 3\left(\frac{M}{2} - 1\right) + M \log_2\left(\frac{M}{2}\right) \right) + 3\sum_{1}^{U}\left(\frac{M}{2} - L_u\right). \quad (45)$$

<table><caption>TABLE II
PAPR COMPARISON OF DIFFERENT METHODS WITH A SUBBAND SIZE OF 300 SUBCARRIERS AND A TOTAL NUMBER OF SUBCARRIERS OF 512</caption><thead><tr><td>Methods</td><td>PAPR value (dB)</td></tr></thead><tbody><tr><td>IAM-C</td><td>26.53</td></tr><tr><td>ICM</td><td>12.12</td></tr><tr><td>Method in [14]</td><td>12.12</td></tr><tr><td>Method in [28]</td><td>11.06</td></tr><tr><td>Proposed method</td><td>5.36</td></tr><tr><td>OFDM</td><td>2.59</td></tr></tbody></table>

### V. SIMULATION RESULTS

In assessing the performance of the proposed PS, it is imperative to consider the PAPR as a fundamental metric. PAPR is critically important in communication systems, especially in MU scenarios, as it significantly impacts the efficiency of power amplifiers and the integrity of the transmitted signal [38]. Indeed, the non-linearity of the power amplifier generates distortion on the transmitted signal which degrades the BER performance. Therefore, we prioritize the evaluation of PAPR for the competing methods such as [10], [13], [14], [28] prior to any other performance analyses. Following this initial assessment, we will evaluate the NMSE, the SE, and the computational complexity of the proposed technique in both SU and MU scenarios, benchmarking our results against existing CE methods documented in the literature.

#### A. PAPR Comparison

A well-known drawback of multicarrier transmitters is that they suffer from high PAPR. In a multicarrier system, the PAPR is defined as the ratio of peak power $P_{peak}$ to average power $P_{avg}$ and can be expressed in decibels (dB) as follows:

$$PAPR_{\text{dB}} = 10 \log_{10} \left( \frac{P_{\text{peak}}}{P_{\text{avg}}} \right) \quad (46)$$

Table II shows the PAPR performance of the proposed method compared to various existing CE techniques, including IAM, ICM, and the methods presented in [14], [28]. The ICM method uses a frequency-domain training sequence structured as $[C_0, C_1, \dots, C_{M-1}]^T$, where $C_0, C_1, \dots, C_{M-1}$ are randomly generated binary values from $\{-1, 1\}$ [10]. On the other hand, the IAM-C method which employs a frequency-domain training sequence of $[1, -j, -1, j, \dots, 1, -j, -1, j]^T$, will be used for comparison as it offers the best performance among the three IAM variants [13]. The NPR1 short PF is used with a total of $M = 512$ subcarriers in the simulations.

As shown in the table, the IAM-C method suffers from a significantly high PAPR (26.53 dB) making it susceptible to non-linear distortions in the power amplifier. This high PAPR results from the deterministic/periodic nature of the pilot sequence in the frequency domain, which translates to high PAPR in the time domain. While the ICM scheme greatly reduces the PAPR (12.12 dB), it does so at the expense of system performance. Indeed, it has been demonstrated that frequency-domain CE models like IAM and ICM generally perform worse than time-domain CE models [39].

Since our goal is to benchmark against high-performing CE approaches, we have omitted frequency-domain models in subsequent comparisons since that would only serve to
<figure><img src="image_7.png" /><figcaption>Fig. 5. NMSE performance comparison of the proposed technique compared with the one in [28] for SU. The CRLB was added for reference.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. NMSE performance comparison of the proposed technique compared with the ones in [14] and [28] for MU scenario with 4 users. The CRLB was added for reference.</figcaption></figure>

confirm already well-established results showing limitations in estimation accuracy compared to time-domain models [14], [39]–[41]. Therefore, to maintain both the focus and the clarity of our analysis, frequency-domain models are intentionally excluded from further performance evaluations. We instead benchmark the proposed PS against competitive time-domain CE methods that better reflect state-of-the-art performance.

In contrast, the proposed PS significantly reduces the PAPR (5.36 dB) compared to both the ICM and IAM schemes and achieves a lower PAPR compared to the methods in [14], [28]. This outcome is expected, as the proposed method employs the ZC CAZAC sequence, which is well-known for its low PAPR properties. Additionally, the use of only half the number of subcarriers further contributes to the reduction in PAPR. Similar to the ICM method, the methods in [14], [28] exhibit lower PAPR than the IAM method due to the non-deterministic nature of their pilot sequences.

The OFDM waveform exhibits the lowest PAPR, primarily because it does not employ a prototype filter. In contrast, the ramp-up and ramp-down of the prototype filter used in other methods introduce a slight increase in PAPR.

In conclusion, since the methods in [14], [28] exhibit favorable PAPR characteristics, they will be considered for further comparisons in the subsequent sections of this work.

## B. Estimation performance

We evaluate in this section the performance of the proposed CE technique compared to that in [14] and [28] for SU and MU scenarios, using the NMSE metric. We consider 1 user which employs the PS in Figure 1, with $M = 128$ subcarriers, 4-QAM modulation and the MMB4 PF [42]. A random channel model with exponential power delay profile of $\alpha(l) = e^{-\beta l}$ is considered as in [28], where $l \in [0, ..., L-1]$.

1) **SU:** Figure 5 compares the NMSE of the CE technique when employing the proposed PS for SU to that in [28] with $L = 32$. The authors in [28] had already shown that the number of pilots has a negligible effect on the accuracy of the CE, a finding which our results also confirm. Applying

the proposed PS and CE technique described in Section III-A which mitigates intrinsic and inter-user interference, improves the NMSE when compared to [28] as depicted in the same figure. In the SU case, this gain stems from the DCT-based denoising stage, which leverages the sparsity of the CIR in the time domain. Specifically, the DCT-denoising stage enables energy compaction within the first few taps of the estimated CIR, while attenuating residual noise in the remaining taps, as described in Equation (17). This improvement is further substantiated by analytical validation using Equation (37). Furthermore, the NMSE closely approaches the CRLB bound as depicted in the same figure which validates the efficiency of the proposed technique.

2) **MU:** Figure 6 compares the NMSE of the proposed CE technique to that in [14] and [28] for a MU scenario with 4 users and $L = 32$. While not applying any specific processing to target intrinsic interference as described in Section III, the proposed PS inherently suppress interference from adjacent users by utilizing a phase rotation on their pilots. This design shift ensures that interference does not affect the CE of the target user. Specifically, the circular shifts applied to user pilots ensure that any interference falls outside the effective time-support of the intended user channel, and is thus excluded during the DCT-based denoising stage. This latter, effectively cleans the estimated channel response, further enhancing performance. As a result, the proposed methods outperform the competing techniques from prior art while considerably reducing the required complexity. This enhancement is also supported by analytical validation using Equation (37). Furthermore, the NMSE closely approaches the CRLB bound, validating the efficiency of the proposed technique, as depicted in the same figure.

## C. Evaluation of computational complexity

Figure 7 illustrates the number of RMs needed to estimate the channel for $U$ users using the proposed method, in comparison with the approaches from [14] and [28]. This
<figure><img src="image_9.png" /><figcaption>Fig. 7. Comparison of the number of RMs required for estimating the channel for *U* users.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Dual pilot structure.</figcaption></figure>

comparison is made for $M = 128$ subcarriers and a channel length of $L_u = 16$ for each user. The methods in [14] and [28] rely on equation (6) for CE. While [14] estimates the channel using all subcarriers, the method in [28] utilizes only $L_u \times U$ subcarriers, thus reducing the computation needed for CE. Furthermore, in both [14] and [28], the matrices **A** and **C** in equation (6) are pre-known at the transmitter and receiver. Consequently, the number of real multiplications required for [14] is $U \times (4ML_u)$, and for [28] it is $4 \times (UL_u)^2$. The complexity of the proposed method is calculated using equation (45). As the number of users increases, the overall computational complexity rises for all methods. However, the proposed method achieves the lowest computational complexity (6760 RMs for 8 users) since it eliminates the need for matrix multiplications entirely. In contrast, the approach in [28] is more efficient than that of [14] because it estimates the channel using fewer subcarriers. Yet, when the number of users reaches 8, all subcarriers must be allocated for CE, causing the complexity of [28] to match that of [14] (65536 RMs). Thus, the advantage of the proposed method lies in its streamlined approach, significantly reducing the computational load, especially as the number of users increases, without compromising accuracy.

## VI. PERFORMANCE OVER THE REALISTIC QUADRIGA CHANNEL MODEL

In this section, we study the performance of the proposed CE technique in terms of BER considering the FBMC-OS receiver over the static and non-static 5G QuaDRiga channel in two cases: first by employing the proposed CE technique and second by assuming that the channel is perfectly estimated at the receiver side. The FBMC-OS receiver was proposed initially in [23] and then extended to MU-mMIMO systems in [24]. The NPR1 short PF is employed where the number of non-truncated filter coefficients is set to $N_G = 7$ for an SIR of 55 dB [1]. For the OS-FBMC based receiver, the up-sampling factor used for the OS technique is set to $N_{UF} = 4$ which is a good compromise between system performance and complexity. This is reflected by the $OS_4$-FBMC receiver notation. Specific simulation parameters which will be used in this section are presented in Table III considering the 16-QAM constellation.

<table><caption>TABLE III SIMULATION PARAMETERS FOR THE QUADRIGA CHANNEL MODEL</caption><thead><tr><td>Parameters</td><td>Value</td></tr></thead><tbody><tr><td>Carrier frequency</td><td>3.5 GHz</td></tr><tr><td>Channel model</td><td>'3GPP_3D_UMa'</td></tr><tr><td>Channel type</td><td>NLOS</td></tr><tr><td>Total number of subcarriers</td><td>512 subcarriers</td></tr><tr><td>Sub-band size</td><td>300 subcarriers</td></tr><tr><td>BS height</td><td>25 m</td></tr><tr><td>Number of BS antennas</td><td>64</td></tr><tr><td>Number of vertical/horizontal panels</td><td>2/1</td></tr><tr><td>Vertical and horizontal antennas spacing</td><td>42.85 mm</td></tr></tbody></table>

### A. QuaDRiGa Simulation tool

QuaDRiGa (QUAsi Deterministic RadIo channel Generator) [43] was designed to simulate MIMO radio channels tailored to specific network setups. It integrates features from the Spatial Channel Model (SCM) [44] and WINNER channel models [45], along with methodologies that support quasi-deterministic multi-link tracking of user mobility in dynamic environments. In our simulations, QuaDRiGa generates realistic radio channels, with parameters aligned with the 5G standard [46] and a base station configuration of 64 cross-polarized antennas.

### B. BER performance over static channel

Figure 9 illustrates the BER performance of the FBMC-$OS_4$ receiver for the two cases discussed earlier and highlights the effect of interpolation when users are stationary. The proposed CE technique leverages Makima interpolation to estimate the channel on the odd subcarriers. This choice is based on the observation that channel coefficients exhibit limited variation across adjacent subcarriers [4]. To evaluate the impact of this interpolation and quantify any performance loss, we introduce a dual PS. In this structure, each user transmits a second preamble symbol, where odd subcarriers are directly loaded with pilots instead of the even ones, as shown in Figure 8. This modification eliminates the need for interpolation, as the channel on odd subcarriers is directly
<figure><img src="image_11.png" /><figcaption>Fig. 9. BER Performance of the FBMC-OS₄ receiver using the proposed PSs (Figures 2 and 8) over the 5G QuaDRiGa Channel in the '3D-UMa-NLOS' Scenario with 2 Users and 64 Receiving Antennas.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. BER performance of the FBMC-OS₄ receiver by employing the proposed PS scheme compared to [28] over the non-static 5G QuaDRiGa channel under the '3D-UMa-NLOS' scenario with speeds of 5 and 250 km/h, 2 users and 64 receiving antennas.</figcaption></figure>

estimated from the second transmitted preamble symbol. We compare the BER performance of this dual-preamble scheme with the original interpolated scheme in the same figure. As demonstrated, the performance difference is marginal, with a degradation less than 0.2 dB, which remains within acceptable limits. This minimal loss confirms that the channel response varies only slightly between adjacent subcarriers, validating the effectiveness of the interpolation approach.

### C. BER performance under non-static channel

We now evaluate the BER performance of the proposed scheme under non-static channel conditions, utilizing the 5G QuaDRiGa model with the 3D-UMa-NLOS scenario. The evaluation considers two users moving at speeds of 5 km/h and 250 km/h, comparing the proposed method against the benchmark technique presented in [28]. The results, illustrated in Fig. 10, demonstrate that the proposed interpolation-based scheme maintains robust performance despite the dynamic nature of the channel. In comparison to a dual-preamble baseline, the proposed approach exhibits only a slight performance loss, approximately 0.2–0.3 dB at a target BER of $10^{-4}$. This minor degradation suggests that, despite the Doppler effect, the channel variation across adjacent subcarriers remains limited.

More significantly, the proposed method consistently outperforms the joint estimation technique in [28] by approximately 1.4 dB across both low and high mobility conditions. This performance gain stems from two key structural advantages: First, the pilot design uses circular frequency shifts to ensure temporal separation between the users' channel responses, allowing for clean decoupling in the time domain. Second, a DCT-based denoising step smooths the channel estimates and suppresses residual inter-user interference at the edges. Together, these features enhance estimation accuracy, particularly under mobility. In contrast, the method in [28] adopts a joint estimation framework in which each user's channel is entangled with all others through a coupled interference matrix. This structure lacks modularity and robustness, because estimation errors or mismatches for one user due to

Doppler effects can propagate and degrade the performance of all users. Our method, by enabling per-user estimation, avoids such dependencies and offers better scalability, resilience, and practicality for real-world multi-user deployments.

A notable degradation in CE performance is observed as the speed increases from 5 km/h to 250 km/h. This effect is expected in preamble-based PSs, as they may eventually fail to track rapid channel variations accurately. Under such conditions, scattered pilot configurations are generally more effective. However, this approach complicates PAPR reduction techniques, is less favorable than preamble-based PSs in the context of MU-mMIMO systems, and lies outside the scope of this work.

### D. Breakdown of the proposed PS and the corresponding Solution

As shown in equation (21), the number of users that can be supported for CE using the proposed PS is constrained by the channel DS. Exceeding this user limit will result in degraded CE performance.

To analyze this, we plot the BER performance of the FBMC-OS receiver versus the number of users in Figure 11, employing the PS in Figure 2 over both the 5G QuaDRiGa channel and the Rayleigh fading channel with an exponential decay and DS of 16 samples. From Figure 11, we observe that the BER performance begins to degrade at 4 users for the 5G QuaDRiGa channel. According to equation (20), the shift between two users is 37 samples, which means that the DS of the channel must not exceed this value. However, in the '3D-UMa-NLOS' scenario of the 5G QuaDRiGa channel, the DS is a random variable that may exceed 37 samples, leading to interference between CIRs of users and thus causing performance degradation in CE.

To address this issue, an additional preamble symbol can be assigned to support additional users at a different time instant, mitigating interference between CIRs and allowing the system to accommodate a higher DS. This improves BER
<figure><img src="image_13.png" /><figcaption>Fig. 11. BER performance vs. number of Users at $Eb/N_0 = 17$ dB: Performance breakdown of the proposed PS over exponential decaying Rayleigh channel (DS = 16 samples) and the 5G QuaDRiGa channel considering the FBMC/OQAM waveform.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. SE loss comparison between the proposed method and the methods in [14], [28] when using long and short PFs.</figcaption></figure>

performance, as shown in the same figure, although it comes at the cost of reduced SE, since 2 preambles are required instead of 1 for CE. In contrast, for the Rayleigh fading channel with a delay spread of 16 samples, and following equation (21), the proposed PS can support 6 users with negligible interference between the CIR of different users, as it is well within the system capacity to handle this DS. This explains why no improvement in BER performance was observed when allocating an additional preamble to serve the additional users when considering the Rayleigh channel.

## E. Evaluation of the SE loss

In this section, we assess the SE loss resulting from the training overhead needed for CE over the '3D-UMa-NLOS' scenario of the 5G Quadriga channel. The simulation parameters are listed in Table III, using both the NPR1 short PF and the long MMB4 PF. For the short PF, 1 GI is used to shield the pilots from interference, resulting in the allocation of 2 FBMC symbols for CE, as shown in Figure 1. In contrast, the long PF requires 4 GIs, leading to the allocation of 5 FBMC symbols for CE. The SE loss for the method in [14] can be expressed as $SE_{loss} = \frac{(N_{GI}+1) \times U}{L_{frame}}$, since every supported user requires a distinct training symbol. However, for [28] and the proposed method, it is the same and given by $SE_{loss} = \frac{k \times (N_{GI}+1)}{L_{frame}}$, where $N_{GI}$ denotes the number of GIs, $L_{frame}$ is the frame length, and k is the number of pilot symbols needed to support U users expressed as $k = [U \times \frac{DS}{L_{ZC}}]$.

Figure 12 evaluates the SE loss, assuming $L_{frame} = 36$ symbols using the two equations above. The method in [14] suffers from a high SE loss, as each supported user requires a distinct training symbol, since all subcarriers are used for estimating the CIR. On the other hand, the proposed method and the method in [28] shows a significantly reduced SE loss compared to [14], as it does not require a dedicated training symbol for each user. However, the number of users served per pilot symbol is limited by the DS of the channel. For example, when using the 5G Quadriga channel, 3 users can be supported

per pilot symbol. Therefore, for every additional set of 3 users, an extra preamble symbol is required, leading to increased SE loss, as depicted in the figure. Moreover, when using the long MMB4 PF, the SE loss is higher, as a larger number of GIs are required. Indeed, as the number of users increases, the method in [14] exhibits around 97% SE loss when supporting 7 users, requiring 35 out of 36 symbols for CE, which makes it impractical for real-world implementations.

## VII. CONCLUSION

In this paper, we proposed a novel PS design for FBMC/OQAM systems considering a SU scenario. Differently to available solutions, the proposed model does not impose the symbol interval to be much longer than the maximum channel delay spread. We then extended these proposals for a joint MU CE approach in a MU scenario and mMIMO systems, in which the pilots for different users are interleaved in frequency by a simple circular shift without the need for any GIs between them. This results in a high reduction in training overhead which becomes significant when the number of users increases. Accordingly, the proposed method significantly improves PS compared to the available solutions in the literature that require a large number of GIs for separating different user pilots, while offering the best performance. In addition to achieving the lowest PAPR, and improved SE, we conducted an analytical study of the NMSE and the CRLB bound for the estimation technique when augmented with the proposed PSs for both SU and MU scenarios. The performed validation shows that the results obtained through the proposed analytical study are consistent with those derived from simulations, confirming the efficiency and accuracy of our approach. Furthermore, the proposed PS associated with the conventional OFDM CE technique notably reduces computational complexity when the number of users increases.

## ACKNOWLEDGEMENT

This work was supported in part by the Beyond 5G project funded by the French government as part of the economic recovery plan, namely "France Relance", and the investments
for the future program and in part by the PEPR 5G and Future Networks project funded by the Agence Nationale de la Recherche (ANR) grant NF-PERSEUS : "ANR-22-PEFT-0004".

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