# Exploring MIMO Correlation Matrix Distance for Cell-free V2I Channel Characterization

Eric Pierre Simon, Pierre Laly, Valentin Picquet, Davy P Gaillot

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## To cite this version:

Eric Pierre Simon, Pierre Laly, Valentin Picquet, Davy P Gaillot. Exploring MIMO Correlation Matrix Distance for Cell-free V2I Channel Characterization. IEEE Antennas and Wireless Propagation Letters, In press, 10.1109/LAWP.2025.3622260. [hal-05314213](https://hal.science/hal-05314213v1)

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Submitted on 14 Oct 2025

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# Exploring MIMO Correlation Matrix Distance for Cell-free V2I Channel Characterization

Eric P. Simon, Pierre Laly, Valentin Picquet, Davy P. Gaillot

***Abstract***—This letter investigates the stationarity characteristics of vehicle-to-infrastructure (V2I) channels in a cell-free context, based on 5.89 GHz field measurements. The Correlation Matrix Distance (CMD) metric is used to quantify the stationarity time of both individual access point (AP)-to-user channels and the collective channel formed by stacking multiple APs. Results show that with two APs, the collective channel's stationarity time is similar to that of the most non-stationary AP. However, when the number of APs increases to four, the collective channel's stationarity time decreases significantly, falling below that of any individual AP. This demonstrates that increasing spatial distribution in cell-free architectures amplifies the impact of local non-stationarities, especially near APs. These findings highlight the importance of thoroughly characterizing stationarity in cell-free systems, particularly as the network scales.

***Index Terms***—Cell-free massive MIMO, stationarity time, vehicular networks.

## I. INTRODUCTION

Cell-free massive MIMO is a promising wireless technology where a large number of distributed access points (APs) jointly serve user equipments (UEs), eliminating the notion of traditional cell boundaries [1]. In such architectures, the collective channel formed by stacking the channels from all APs to a UE and the respective spatial correlation matrices play a pivotal role, especially in centralized processing schemes that exploit the joint spatial structure of the network.

Future vehicle-to-infrastructure (V2I) systems are expected to leverage cell-free architectures to deliver seamless, high-reliability, and high-capacity connectivity. However, the rapid mobility inherent to V2I scenarios challenges the traditional assumption of wide-sense stationarity (WSS) in wireless channel models [2]. In cell-free systems, this challenge is amplified: the stationarity of the collective channel, rather than that of individual AP-UE links, becomes a critical factor influencing the design of beamforming [3], pilot assignment [4] and resource allocation algorithms [5], [6]. Yet, to the knowledge of the authors, the stationarity of the collective channel obtained from the time evolution of the spatial correlation matrices have never been experimentally investigated in the literature.

The main contributions of this work are as follows. (i) A measurement campaign was conducted in a realistic vehicular cell-free scenario using the MaMIMOSA channel sounder [7]

for various driving tests. (ii) Two cell-free configurations with the same number of emitting antennas and a single-antenna user were considered to compare deployment strategies for system-level perspective: one with two APs each equipped with four antennas, and another with four APs each with two antennas. (iii) For these two configurations, the stationarity of both individual AP-UE channels and the collective channel were analyzed using the Correlation Matrix Distance (CMD) metric [8]–[11], which quantifies the temporal evolution of spatial correlation matrices. The use of this metric enables a detailed analysis of the spatial correlation matrices dynamics, in contrast to other metrics such as spatial divergence or shadow fading correlation [12].

Our results can be summarized as follows. (i) In the 4-AP configuration, the collective channel exhibits significantly lower stationarity times compared to the 2-AP case with the same total number of antennas. (ii) For the 2-AP (4-AP) configuration, the stationarity time of the collective channel is close to (lower than) the lowest stationarity time among the individual AP-UE channels. This behavior is due to the aggregation of the fastest-varying components from all APs, which produces a more dynamic and less stationary global channel. (iii) These trends were consistently observed across all driving tests.

In conclusion, as the number of APs increases, the non-stationarity of the wireless channel is expected to become more pronounced. Moreover, the validity of radio channel models relying on the WSS assumption may be fundamentally limited when the AP density is very large. These findings emphasize the need for system designers to account for non-stationarity effects in both algorithm design and hardware implementation to ensure robust deployment of cell-free vehicular networks.

## II. EXPERIMENTAL SET UP

The propagation channels were measured using the MaMIMOSA radio channel sounder [7], [13], [14] at a central frequency of 5.89 GHz and a bandwidth of 80 MHz. The transmitted signals were modulated using orthogonal frequency-division multiplexing (OFDM) with 8192 subcarriers, each spaced 12.21 kHz apart. To accommodate the vehicular environment, the streaming mode of MaMIMOSA was employed, corresponding to the frame structure illustrated in Fig. 1. Each frame was transmitted every 1 s and began with a 51.2 µs preamble, followed by 8192 OFDM symbols, resulting in a total frame duration of 998.77 ms. Each Tx antenna transmitted a pilot signal every eight subcarriers, corresponding to $\Delta f = 97.7$ kHz, with a per-antenna subcarrier shift

E.P. Simon, Pierre Laly, Valentin Picquet, Davy P. Gaillot are with Univ. Lille, CNRS, UMR 8520-IEMN laboratory, F-59000 Lille, France. e-mail: eric.simon@univ-lille.fr. This work was supported by the French National Research Agency as part of France 2030 5G NF-PERSEUS and NF-YAKARI projects. It was also supported by the CPER RITMEA project co-financed by the European Union with the European Regional Development Fund, the French state, and the Hauts-de-France Region Council.
applied to interleave the Tx antenna signals. For each antenna, 818 subcarriers were recorded. It is noted that one OFDM symbol out of every 8 was recorded during this measurement campaign, yielding a snapshot time $\Delta t = 975.36$ µs.

<figure><img src="image_3.png" /><figcaption>Fig. 1. MaMIMOSA streamshot frame structure.</figcaption></figure>

Eight 500-meter-long radio-over-fiber (RoF) links, each with an attenuation loss of 1 dB/km, were connected to the MaMIMOSA system (see Fig. 2(a)) and the Tx antennas to emulate a cell-free network comprising a total of eight antennas. The RF-optical converters incorporated power amplifiers to compensate for insertion losses, introducing a noise figure of 27 dB. The output power of each RF chain was set to 0 dBm, and a 15 dB gain power amplifier was inserted between the optical/RF module and the antenna, resulting in a 15 dBm transmit power and a received signal-to-noise ratio (SNR) of 40 dB. All coaxial cables in the setup had a calibrated length of 2.5 m. To ensure accurate channel measurements, a hybrid calibration procedure was conducted using one of the RoF links to account for differences between the RoF and coaxial connections.

<figure><img src="image_4.png" /><figcaption>Fig. 2. (a) MaMIMOSA transmission system with 500m long RoF links - (b) and (c) Tx linear array configurations with 4 and 2 patch antennas per AP - (d) Omnidirectional Rx antennas.</figcaption></figure>

### III. MEASUREMENT CAMPAIGN

The channel measurement campaign was conducted on the Cité Scientifique Campus of the University of Lille, France. The MaMIMOSA transmission system works as the central processing unit (CPU) of the cell-free network. At the transmitting access points (APs), 6.5 dBi patch antennas were utilized. The Tx antennas were mounted at a height of 2.90 m.

Two cell-free network configurations were evaluated. In the first configuration (Config. 1), two APs, each equipped with four antennas, were deployed at positions A and B. In the second configuration (Config. 2), four APs, each with two antennas, were deployed at positions A, B, C, and D (see Fig. 2(b), (c) for the Tx antenna configuration and Fig. 3 for the AP positions). The distance between positions A and C was 24 m, whereas the distance between B and D was

25 m. The receiver was a van equipped with four EM6116 omnidirectional antennas (see Fig. 2(d)), each with a gain of 2 dBi. These antennas were mounted on the rooftop at a height of 2.90 m and spaced 92 cm apart. The MaMIMOSA receiving (Rx) system was housed inside the van. Both the Tx and Rx units were powered by Li-ion batteries, enabling up to eight hours of continuous operation. Each measured MIMO matrix was $N_{\text{Tx}} \times N_{\text{Rx}} \times N_{\text{frame}} \times N_{\text{OFDM}} \times M_f$, where $N_{\text{Tx}} = 8$ is the total number of Tx antennas, $N_{\text{Rx}} = 4$ the number of Rx antennas, $N_{\text{frame}}$ the number of measured frames, $N_{\text{OFDM}} = 1024$ the number of OFDM symbols recorded by frame and $M_f = 818$ the number of recorded frequencies. We also define $N_{\text{snap}} = N_{\text{frame}}N_{\text{OFDM}}$ the total number of recorded snapshots.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Setup of the investigated linear array configurations with 4 (config. 1) and 2 (config. 2) patch antennas per AP. The white track is from A (C) to B (D) and the red track is from A (C) to A (C), respectively.</figcaption></figure>

The measurement scenario included different driving tests for config. 1: one from point A to point B (AB) over a distance of 100 meters, passing through a roundabout in between, another from B to A (BA), and a last one where the vehicle drove from point A, around the roundabout, and returned to point A (AA). The average speed was 30 km/h. For config. 2, the driving tests C to C (CC), C to D (CD), D to C (DC) and D to D (DD) were considered.

### IV. CMD METRIC AND STATIONARITY TIME

Let $\mathbf{h}(t, f) = [\mathbf{h}_1(t, f)^T, \dots, \mathbf{h}_L(t, f)^T]^T$ be the $LN \times 1$ collective channel vector at time $t$ and frequency $f$ between $L$ APs equipped with $N$ antennas and the single-antenna user, where $\mathbf{h}_l(t, f)$ is the $N \times 1$ channel vector between AP $l$ and the user. The corresponding $LN \times LN$ channel correlation matrix is $\mathbf{R}(t, f) = E\{\mathbf{h}(t, f)\mathbf{h}^\mathrm{T}(t, f)\}$. In this study, only results for Rx antenna 2 (rear-left of the van) are presented because similar trends are obtained with the other Rx antennas.

The CMD is the distance between two correlation matrices evaluated at times $t$ and $t'$ for a given frequency $f$:

$$d_{\text{corr}}(t, t', f) = 1 - \frac{\operatorname{tr}\{\mathbf{R}(t, f)\mathbf{R}(t', f)\}}{\|\mathbf{R}(t, f)\|_F \|\mathbf{R}(t', f)\|_F}, \quad (1)$$

where $\operatorname{tr}\{\cdot\}$ represents the trace operator, and $\|\cdot\|_F$ denotes the Frobenius norm. The metric ranges from 0 (identical matrices) to 1 (completely uncorrelated), providing a clear quantification of the structural changes in the antenna correlation over time or space. When applied to a single AP $l$, the CMD is obtained by setting $\mathbf{h}(t, f) = \mathbf{h}_l(t, f)$.
In this paper, the channel correlation matrices are estimated by averaging the channel matrix realizations on both time and frequency dimensions as follows:

$$ \hat{\mathbf{R}}(i\Delta t, k\Delta f) = \frac{1}{L_t} \frac{1}{L_f} \sum_{l=i}^{i+L_t-1} \sum_{n=k}^{k+L_f-1} \mathbf{h}(l\Delta t, n\Delta f)\mathbf{h}(l\Delta t, n\Delta f)^H \quad (2) $$

where $i = 0, \dots, N_{\text{snap}} - L_t$ and $k = 0, \dots, M_f - L_f$.

The selection of $L_t$ represents a trade-off: it must be large enough to ensure accurate estimation of correlation matrices while remaining sufficiently small to guarantee that the included snapshots have consistent statistical properties. In [15], it is suggested that the measurement window for obtaining the average power in a mobile radio signal should range between 20 and 40 wavelengths. Based on this guideline, in this study, correlation matrices were estimated over a window of 40 wavelengths. This corresponds to a time interval of approximately 244 ms, encompassing 250 consecutive MIMO channel snapshots, assuming an average vehicle speed of $v = 30$ km/h at the measurement central frequency.

In the frequency domain, $L_f = 5$ subbands centered around the central frequency are selected, which remains below the coherence bandwidth of the channel usually measured in these suburban scenarios.

The CMD can be computed at each individual AP using the $N \times N$ spatial correlation matrix per AP where $N = 4$ for config. 1 and 2 for config. 2, respectively. Another option is to compute from all APs by building a collective channel that is constructed for both configurations by stacking the channel vectors of each AP (2 for config. 1 and 4 for config. 2), resulting in an overall spatial correlation matrix of size $8 \times 8$.

Using the CMD described above, time-varying stationarity intervals can be determined for a frequency $f$, either for the individual APs or the collective AP. For simplicity, the frequency index is dropped from now on. Following the approach in [9], the time-varying stationarity duration, $T_s(i\Delta t)$, is defined as the maximum time interval during which the CMD stays below a specified threshold $\alpha$ as follows:

$$ T_s(i\Delta t) = \Delta \operatorname{idx}^{(i)} \times \Delta t \quad (3) $$

where $\Delta \operatorname{idx}^{(i)} = \operatorname{idx}_{\max}^{(i)} - \operatorname{idx}_{\min}^{(i)}$ with:

$$ \operatorname{idx}_{\min}^{(i)} = \max j \text{ s.t. } d_{\text{corr}}(i\Delta t, j\Delta t) \ge \alpha \quad (4) $$
with $0 \le j \le i - 1$

$$ \operatorname{idx}_{\max}^{(i)} = \min j \text{ s.t. } d_{\text{corr}}(i\Delta t, j\Delta t) \ge \alpha \quad (5) $$
with $i \le j \le N_{\text{snap}} - L_t$

In the literature, $\alpha$ is usually set at 0.2 [9], [11], [12].

## V. MEASUREMENT RESULTS

First, the AA driving test for Config. 1 is analyzed. Figure 4 shows a top view of the CMD for the AP located in A, while Figure 5 shows the corresponding stationarity time as defined in (3). The Ao (Ar) label corresponds to the Rx crossing this AP on the outbound (return) direction.

For ease of interpretation, Fig. 6 shows the Doppler profile for one antenna of AP A (antenna 2). At $t = 2$ s, as the

<figure><img src="image_6.png" /><figcaption>Fig. 4. CMD at AP A for Config. 1 and AA driving test.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Stationarity time $T_s$ at AP A for Config. 1 and AA driving test.</figcaption></figure>

van passes the AP, the radial speed undergoes a sudden sign change, then gradually returns to zero as the vehicle passes through the roundabout at $t = 14$ s. Upon passing the AP again on the return path (at 25 s), the radial speed exhibits another abrupt sign change. These instances of AP passage correspond to high levels of non-stationarity, as the rapid change in the angle between the receiver antenna and the transmitter array strongly modifies the channel correlation matrix. This effect is reflected in Fig. 5, where the stationarity time drops to about 100 ms at these critical moments.

After passing AP A, the stationarity time increases significantly and stabilizes at a value around 5.5 s until the roundabout. During the maneuver through the roundabout, the stationarity time decreases to about 2.5 s. A similar trend is observed on the return trip, with additional peaks reaching up to 9.5 s. The regions of departure ($t = 2$ s) and arrival ($t = 25$ s) on AP A are easily observed on the CMD in Fig. 4. The CMD lacks symmetry with respect to the anti-diagonal. This might be attributed to the fact that the vehicle travels on the right side of the road before reaching the roundabout (outbound direction) and on the left after passing it (return direction). Moreover, since Rx antenna 2 is positioned in the rear-left corner of the van, these factors create significant asymmetries in the Line-of-Sight (LoS) characteristics, leading to strong
<figure><img src="image_8.png" /><figcaption>Fig. 6. Time-varying Doppler spectrum for Config. 1 and AA driving test using antenna 2. The colorbar presents the normalized relative gain in dB.</figcaption></figure>

different propagation conditions between the outbound and return journeys. For instance, a 10 dB difference in the main path is observed in the Doppler spectrum (Fig. 6) at $t = 9$ s (5 seconds before the roundabout) and $t = 19$ s (5 seconds after the roundabout).

Figure 7 shows the CMD for the second AP, located at position B. A higher degree of non-stationarity is observed compared to AP A. This is primarily due to the limited presence of a LoS component, which occurs only when the vehicle crosses the roundabout.

<figure><img src="image_9.png" /><figcaption>Fig. 7. CMD at AP B for Config. 1 and AA driving test.</figcaption></figure>

Figure 8(a) shows the empirical cumulative density functions (CDFs) of the aggregate time-varying stationarity times estimated individually for both APs and for the collective channel. The collective channel is constructed by stacking the channel vectors of each AP, resulting in an overall spatial correlation matrix of size $(8 \times 8)$. The CMD is then computed from this matrix to determine the corresponding stationarity times, similar to the APs. Consistent with the CMD results in Figs. 4 and 7, AP A exhibits longer stationarity times compared to AP B. The stationarity times for the collective channel is very close to that of AP B.

Next, we examine the second configuration (four APs in A, B, C, and D, each equipped with two antennas). Figure 8(b)

<figure><img src="image_10.png" /><figcaption>Fig. 8. CDFs of the aggregate stationarity times, (a) config. 1 and AA driving test; (b) config. 2 and CC driving test.</figcaption></figure>

<table><caption>TABLE I
MEDIAN VALUE OF $T_s$ [s].</caption><thead><tr><th></th><th>AP A</th><th>AP B</th><th>AP C</th><th>AP D</th><th>Collective</th></tr></thead><tbody><tr><td>Config. 1 AA</td><td>2.61</td><td>1.95</td><td>N/A</td><td>N/A</td><td>2.08</td></tr><tr><td>Config. 1 AB</td><td>1.97</td><td>0.83</td><td>N/A</td><td>N/A</td><td>0.82</td></tr><tr><td>Config. 1 BA</td><td>1.95</td><td>1.62</td><td>N/A</td><td>N/A</td><td>1.22</td></tr><tr><td>Config. 2 CC</td><td>6.14</td><td>4.25</td><td>9.28</td><td>4.2</td><td>0.64</td></tr><tr><td>Config. 2 CD</td><td>4.25</td><td>2.78</td><td>8.27</td><td>4.81</td><td>0.81</td></tr><tr><td>Config. 2 DC</td><td>7.60</td><td>3.66</td><td>12.25</td><td>5.81</td><td>0.72</td></tr><tr><td>Config. 2 DD</td><td>6.42</td><td>4.62</td><td>7.60</td><td>5.52</td><td>0.70</td></tr></tbody></table>

shows the empirical CDFs of the stationarity times for each AP and the collective channel. In contrast with Config. 1, it is now observed that the collective channel has the lowest stationarity times in all cases. This is expected because additional non-stationary effects arise as the number of AP is increased. Hence, this results in a more rapidly changing global spatial correlation structure. This finding is confirmed by the median value of $T_s$ presented in Table I for the two configurations and additional driving tests.

## VI. CONCLUSION

Cell-free architectures are based on the principle that each user is served by a large set of cooperating APs. However, our work shows that as the number of APs per user increases, the spatial correlation matrix of the collective channel incorporates additional localized non-stationarities. This is particularly evident near APs where the receiver trajectory causes abrupt propagation changes, leading to reduced stationarity times. This conclusion was observed in all of the driving tests.

These results motivate future investigations with a larger number of distributed APs to characterize network-wide stationarity more thoroughly as the system scale increases under various driving scenarios. Such studies will be crucial for understanding the trade-offs between spatial diversity and temporal coherence in next-generation cell-free networks operating in vehicular and other high-mobility environments.
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