An extension of the RiMAX multipath estimation algorithm for ultra-wideband channel modeling

To describe the geometric properties of the electromagnetic waves of the MIMO propagation channel in terms of AoD, AoA, and ToA, multidimensional frequency domain channel sounding must be performed. This can be done with (virtual) MIMO array systems, consisting of M _T and M _R antennas at transmitter (Tx) and receiver (Rx), sampled at M _f frequency points. As such, the total amount of samples can be defined by M as follows:

An observation of the frequency response of a MIMO radio channel h can be modeled as the superposition of a deterministic part x(θ_smc) (determined by the SMC parameter set θ_smc) and a stochastic part d(θ_dan) (diffuse scattering and noise; determined by the DMC and noise (DAN) parameter set θ_dan). Both parameter sets will be described later in this section.

The deterministic part x(θ_smc) of the data model acts as the first order statistics of the radio channel, so that it can be interpreted as the mean of h, whilst the stochastic part d(θ_dan) describes the second-order statistics by means of its covariance matrix \(\boldsymbol {R}(\boldsymbol {\theta }_{\text {dan}}) \in \mathbb {C}^{M \times M}\), which will be discussed in Section 5.1.2. A realization of the radio channel h can be considered as a random variable distributed according to a complex multivariate Gaussian distribution \(\boldsymbol {h} \sim \mathcal {N}_{c} (\boldsymbol {x}(\boldsymbol {\theta }_{\text {smc}}), \boldsymbol {R}(\boldsymbol {\theta }_{\text {dan}}))\) [44] as follows:

An estimate of the most likely SMC and DMC and noise parameters can be found by maximizing the likelihood function of Eq. (3). Since this is not a trivial task, estimation frameworks such as the RiMAX algorithm will estimate \(\hat {\boldsymbol {\theta }}_{\text {smc}}\) and \(\hat {\boldsymbol {\theta }}_{\text {dan}}\) of the deterministic and stochastic arrays, such that they maximize the likelihood of observing the measured frequency response h of the radio channel. The objective is thus to find the parameters \(\hat {\boldsymbol {\theta }}_{\text {smc}}\) and \(\hat {\boldsymbol {\theta }}_{\text {dan}}\) that maximize the correlation with the measurement data. A maximum likelihood (ML) estimator for the parameters θ_smc and θ_dan has been proposed in [7, 22], exploiting the fact that the parameters of the two components of the channel model are asymptotically independent. Therefore, one can decouple the estimation problem into two separate estimation problems. The resulting RiMAX algorithm is iterative and alternates between the maximization of the likelihood function with respect to the parameters θ_smc and θ_dan. It has an approximately linear computational complexity in the number of propagation paths P and in the number of data samples M [7].

Based on the capability of the RiMAX algorithm to extract both parameter sets from the (virtual) array measurement data, the following structures for the deterministic and stochastic parameter sets (\(\boldsymbol {\theta }_{\text {smc}} \in \mathbb {C}^{P \times 4S}\) and \(\boldsymbol {\theta }_{\text {dan}} \in \mathbb {R}^{4 \times S}\), respectively) can be adopted:

In (4), φ _D , φ _A , τ _A , and γ are P×S matrices, where P is the number of SMCs extracted from the measurement data, and S is the number of sub-bands in which the total UWB bandwidth was partitioned (in analogy with the UWB-SAGE algorithm). As such, θ_smc is of size 4S×P. We will discuss the frequency-dependency of the geometrical parameters of the propagation paths in Section 4.2. Each row in the aforementioned matrices φ _D , φ _A , and τ _A contains the corresponding specular parameter for each of the \(p \in \mathcal {P}\) propagation paths (\(P = |\mathcal {P}|\) in total) and describes its frequency-dependency in each of the \(s \in \mathcal {S}\) sub-bands (\(S = |\mathcal {S}|\) in total). We note that the angular modeling was limited to that of the azimuthal plane, which is acceptable since most measurement campaigns are only performed with a planar array, with which the estimation of elevation parameters is not possible. The extension of the data model to the elevation domain is straightforward, where the AoD and AoA of the SMC will now have an extra elevation component together with an azimuthal component. We also consider only a single snapshot of the channel, such that the covariance matrix R(θ_dan) is only averaged over one observation of the channel. Whilst it would be more reliable to use several snapshots of the channel in a real measurement environment, we would then have to impose a parametric model to handle the time dependence of the SMC, or assume them to be independent and independent and identically distributed (i.i.d.) across snapshots in time. Since this is out of the scope for the purpose of this paper, we will leave this up for future work.

In (5), θ_dan contains the DMC parameters \(\left (\left [\boldsymbol {\alpha _{1}} \in \mathbb {R}^{1 \times S}, \boldsymbol {\tau _{d}} \in \mathbb {R}^{1 \times S}, \boldsymbol {\tau _{r}} \in \mathbb {R}^{1 \times S}\right ]\right)\) and the noise parameters (\(\boldsymbol {\alpha _{0}} \in \mathbb {R}^{1 \times S}\)) for each sub-band \(s \in \mathcal {S}\). A discussion of the model for the DMC can be found in [7]. The model is based on the observation that the PDP ψ(τ) [W] of the DMC and noise, which describes how the power of a signal is distributed over the time-delay domain, has a base time-delay τ _d related to the distance between the transmitter and receiver, together with an exponential decay over time-delay (see Eq. (6)), corrupted by complex additive white Gaussian noise with power α₀:

In Eq. (6), α₁, τ _d , τ _r , and α₀ are the four parameters which fully describe the DMC and noise characteristics of each sub-band and are gathered in the DMC and noise parameter vector θ_dan. In this work, we will assume that the DMC is spatially white at the transmit and receive side of the measurement system, meaning that they have constant angular power densities. It should be noted that recent works will assume the DMC to be spatially correlated with the SMC. For example, [25, 45] report a correlation between the location of SMC and DMC in the angular domain. In [27], the DMC is modeled as local clusters around the SMC. In [20, 46], it is proposed that the Power Angular Profile (PAP) is to be modeled by a uni-modal Von-Mises distribution.

The covariance matrix R(θ_dan) can be constructed by converting the model in Eq. (6) from the time-delay domain to the frequency domain. To do so, \(\boldsymbol {\kappa }(\boldsymbol {\theta }_{\text {dan}}) \in \mathbb {C}^{N_{F} \times 1}\) [W] is first introduced, which denotes a band-limited sampled version of the Fourier transform of Eq. (6), and can be defined for a bandwidth B and M _f frequency samples as follows:

in which \(\boldsymbol {e_{0}} \in \mathbb {N}^{M_{f} \times 1}\) is a unit vector defined as follows:

Furthermore, in Eq. (7), β _d is the normalized coherence bandwidth (dimensionless), calculated as follows:

in which B _d is the coherence bandwidth (Hz), normalized by the bandwidth per sub-band B (Hz). Furthermore, the frequency sampling interval Δ _f (Hz) can be written as a function of B and M _f as follows:

After calculating κ(θ_dan), the covariance matrix R(θ_dan) of the DMC and noise can be calculated by applying the Toeplitz-operator [7] as follows:

The original data model of the RiMAX estimation algorithm follows the narrowband assumption, hence stating that the SMC and DMC are Kronecker-separable in the spatial and frequency domains in order to keep the algorithm computationally viable [7]. In our measurements section, prior to the processing of the measurement data, we will check for the uncorrelated scattering assumption, which needs to hold in order to apply the Kronecker model. For a large MIMO configuration in a given communication system, the dimensions of the covariance matrix R(θ_dan) becomes too large to allow for a reliable estimation (remember that it has a size of [M×M]). This becomes a processing burden to accurately model the interaction between transmitter and receiver, so that both ends of a communication system need to be decoupled by applying the well-known Kronecker model. We refer to [47] for a discussion of its applicability and limitations.

In the Kronecker model, the covariance matrices at transmitter and receiver are assumed independent and separable, allowing the full covariance matrix of the channel to be expressed as the Kronecker-product of several smaller matrices. Thus, the covariance matrix R is assumed to have the following structure [7]:

in which \(\boldsymbol {R}_{T} \in \mathbb {C}^{M_{T} \times M_{T}}\) and \(\boldsymbol {R}_{R} \in \mathbb {C}^{M_{R} \times M_{R}}\) are the covariance matrices at the transmitter and receiver, respectively, describing the spatial distribution of the DMC at both ends, whereas \(\boldsymbol {R}_{f} \in \mathbb {C}^{M_{f} \times M_{f}}\) is the covariance matrix in the frequency domain. The term α₀I denotes the amount of complex additive white Gaussian measurement noise, which contributes to the stochastic part of the data model. The matrix I _M is an identity matrix of size [M×M].

In this extension of the RiMAX algorithm, we will assume that the transmitter and receiver are equipped with a uniform circular array (UCA), given that its angular resolution capability is uniform since the effective aperture does not change with azimuth angle. The extension to other array configurations is straightforward since only the steering matrices in the following subsection have to be adjusted. It should be noted that other configurations for the antenna array can be used in this algorithm as well, without much modification to the hereinafter explained matrices. The broadband MIMO radio channel model can be expressed in matrix notation to map the time-delay and angles of a propagation path to its complex notation in the frequency domain. We will therefore define the matrix \(\boldsymbol {B}_{f}(\boldsymbol {\tau }_{A}) \in \mathbb {C}^{M_{f} \times P}\) (dimensionless), which maps the time-delays τ _A of each propagation path \(p \in \mathcal {P}\) to its related complex exponential \(\phantom {\dot {i}\!}e^{-j2\pi \cdot m \Delta _{f} \cdot \tau _{A,p}}\) notation as follows:

Similarly, we can define the mapping of both the departing and the arriving angles φ _D and φ _A to the transmitting and receiving array responses \(\boldsymbol {B}^{s}_{T}(\boldsymbol {\varphi }_{D}) \in \mathbb {C}^{M_{T} \times P}\) (dimensionless) and \(\boldsymbol {B}^{s}_{R}(\boldsymbol {\varphi }_{A}) \in \mathbb {C}^{M_{R} \times P}\) (dimensionless) in each sub-band \(s \in \mathcal {S}\) as follows:

and

with r being the radius of the UCA, λ _s (m) the wavelength in each sub-band \(s \in \mathcal {S}\), and the vector function \(\boldsymbol {\rho }(M_{T/R}) \in \mathbb {R}^{M_{T/R} \times 1}\) (rad) mapping the angle between each antenna in the UCA and a chosen reference axis, defined as follows:

Both matrices B _T (φ _D ) and B _R (φ _A ) describe the complex frequency-dependent far-field beam pattern at each antenna array port at the transmitting and receiving array, respectively.

Now let us consider a measurement snapshot of the multidimensional UWB-MIMO channel h. This snapshot can be defined as the instantaneous frequency domain transfer function of the channel between each MIMO antenna pair and is modeled as a superposition of P discrete paths x plus a contribution of DMC and noise, written as follows:

The model for a single propagation path p in sub-band s is given by:

where the operator ♢ denotes Khatri-Rao (column-wise Kronecker) product and the variable \(\gamma ^{s}_{p}\) denotes the complex amplitude of path p in sub-band s. The superposition of P paths in sub-band s can then be written as follows:

in which B ^s (φ _D ,φ _A ,τ _A ) represents the steering vector in space and frequency for the entire array, and for all propagation paths, in sub-band s.

In a wideband scenario, the reflection coefficients of certain materials in the environment (e.g., furniture, cabinets) can no longer be considered constant with respect to frequency. In addition, the antenna array responses of transmitter and receiver will no longer be frequency-independent and can vary significantly over the total UWB bandwidth. To overcome this issue, the total UWB band was split up into S sub-bands to assure frequency stationarity in each sub-band. This is in agreement with the multiband radio UWB principle [48], where the total UWB band is split into multiple sub-bands that are separately processed by the receiver, to avoid problems with the restrictions on the analog RF circuit designs. Although the geometrical propagation characteristics (AoD, AoA, and ToA) are frequency-independent, the complex amplitude of each path will vary throughout the UWB frequency band. Hence, we can adopt the narrowband assumption in each sub-band, making it sufficient in terms of measurement accuracy to describe the directional characteristics of the antenna arrays at the center frequency of each sub-band.

In order to properly estimate propagation parameters over the UWB frequency band, an extension to the RiMAX algorithm was developed that can process UWB measurement data and will be referred to as UWB-RiMAX from now on. The global outline for this algorithm is broadly described in Fig. 1. In the next following subsections, we will go deeper into certain aspects of the algorithm.

In this subsection, we will discuss how the frequency-dependency of the SMC was modeled in the initialization procedure. As outlined in Fig. 1, the geometrical parameters φ _D , φ _A , and τ _A are kept constant over frequency, ensuring that these could not appear or disappear from one sub-band onto the next. In a latter part of the algorithm (estimation and optimization of the SMC (see Fig. 1), these geometrical parameters will be optimized in each sub-band separately, in order not to over-constrain the optimization procedure of the complex amplitudes γ. This also facilitates the optimization, as there is no strong dependency to be modeled between the sub-bands. Doing so, the geometrical parameters can thus vary slightly over frequency after the optimization procedure is carried out, but jointly estimating them in the initialization procedure ensures that they start their optimization from the same set of values.

In order to evaluate the UWB-RiMAX multipath estimation algorithm as described above, we have generated 200 synthetic channels with controlled parameters for the SMC and DMC parameters. In the following subsections, we first describe how these parameters were modeled in our evaluation procedure. Subsequently, we will describe how the estimated SMC and DMC are compared with their synthetically generated counterparts, in order to evaluate the performance of our algorithm.

In order to evaluate how well the synthetically generated SMC are estimated by the algorithm, we will make use of the multipath component distance (MCD) [54, 55]. The MCD (dimensionless) can be seen as a metric to define the “closeness” between two parameter sets in multipath parameter distance space. It was previously shown that it outperforms the Euclidean distance, and is a suitable metric for combining parameters that have different units (as is the case here with both angles and delays). The pairing of each UWB-RiMAX estimated SMC with its synthetically generated counterpart is done by searching for the smallest MCD between each (output) estimated SMC and the (input) synthetically generated SMC. The MCD \(\Omega _{\varphi _{i,j}}\) (dimensionless) between two distinct angles φ _i and φ _j , i.e., the angular distance between both, can be calculated as follows:

The MCD \(\Omega _{\tau _{i,j}}\) (dimensionless) between two distinct angles time-delay instances τ _i and τ _j can be written as follows:

with τ_std being the standard deviation of all delays τ, and Δτ_max calculated as follows:

The parameter ζ is a delay scaling factor to give the delay more weight in the MCD metric when necessary. This factor was chosen as 1, as originally proposed in [55]. Higher values can be chosen to give more weighting to the delay for the pairing of multipath parameters. The delay distance of the MCD was scaled with the normalized delay spread \(\frac {\tau _{\text {std}}}{\Delta \tau _{\text {max}}}\). The resulting MCD metric Ω_i,j between two multipath parameter sets i and j can then be calculated as follows:

which can be interpreted as the radius of a circle in the normalized multipath parameter distance space.

It should be noted that in our scenario, there is no one-to-one matching of the (output) estimated SMC to the (input) synthetically generated SMC. The fact that we allow multiple output SMCs to be matched to the same input SMC is to overcome the case when, e.g., the remainder of the signal after subtraction of previously detected paths at a certain time-delay instance, still contains a sufficient amount of power at this time-delay instance. Newly detected paths then have the same time-delay value and directional parameters, so that we allow them to be matched to the same input SMC.

All synthetically generated propagation paths in the APDP were superimposed with a DMC model, as described in Section 5.1.2. Figure 8 shows the resulting differences between all generated and estimated reverberation times as a function of UWB sub-band. This data is represented by means of box plots per frequency band, which indicate the first, second (median), and third quartiles of these differences, and the 1.5 interquartile range from both the lower and upper quartiles.

Figure 8 shows that a median error of less than 4 ns can be obtained for the lower frequencies in the UWB band. For the higher frequencies in the UWB band, a median error of less than 2 ns can be obtained. Overall, more than 75% of our simulation results show an absolute difference between true and estimated reverberation times of less than 4.6 ns in the worst-case scenario, which are more than acceptable results given the relatively low SNR of 20 dB in our evaluation.

Figure 9 depicts the box plots of the difference between the input and output powers for the total channel, as well as the contributions of both the SMC and DMC power.

From Fig. 9, we can see that the total power in the channel is very well estimated by the UWB-RiMAX algorithm, both at the lower and higher frequencies. We can also observe median differences between the true and estimated SMC powers ranging between –0.3 dB at the lower UWB frequencies, increasing to about 1.5 dB at the higher UWB frequencies. These are more than acceptable errors in the estimation of the SMC signal power, if we take into account that the error is still less than 1 dB up to 9 GHz. Next to that, we can observe median absolute differences between the true and estimated DMC powers of about –0.2 dB at the lower UWB frequencies, increasing to about –1 dB at the higher UWB frequencies. The fact that the power of both SMC and DMC is estimated slightly worse at the higher UWB frequencies might be resolved by softening the stop criterion at these frequencies, as it is currently influenced by the SNR of each path, which is thus lower at higher frequencies than at lower frequencies.

Overall, the values for the MCD metric show that our proposed algorithm is able to correctly estimate the most significant input propagation paths, and by analyzing the differences between the input and output SMC, DMC, and total powers, we can state that our proposed algorithm gives a fairly good agreement between the generated input values and the estimated output values.

Figure 15 shows the 10 strongest estimated geometrical propagation paths in the environment. Their length is an indication of their relative power.

From Fig. 15, we can see that our algorithm is able to estimate the correct geometrical propagation paths in the environment. For receiver positions 11 and 12, it looks like the algorithm estimated the wrong angles, but it should be noted that there was a metallic cabinet in the long side of the environment on which the paths apparently scattered from transmitter to receiver.

Figures 16, 17, and 18 show the mean values of the measured SMC, DMC, and SMC+DMC (total reconstructed) powers in the radio channel as a function of UWB frequencies. The power of the SMC part of the radio channel in each sub-band \(P^{s}_{SMC}\) can be calculated from Eq. (19), whilst the power of the DMC part of the radio channel in each sub-band \(P^{s}_{DMC}\) can be calculated from Eq. (39). The total reconstructed power in each sub-band \(P^{s}_{Tot}\) can then be calculated by summing over both \(P^{s}_{\text {SMC}}\) and \(P^{s}_{\text {DMC}}\). Figure 16 presents the results of the LoS scenarios, Fig. 17 presents the results of the OLoS scenarios, and Fig. 18 presents the results of the NLoS scenarios.

Figures 16, 17, and 18 show that the SMC+DMC (total reconstructed) powers match very well with the measured power in the channel across the UWB frequency band. For the LoS scenario, the difference between the measured and the reconstructed power resulted in a minimum (underestimated) value of –1.61 dB and a maximum (overestimated) value of 1.20 dB. For the OLoS scenario, the minimum difference was –1.38 dB, and the maximum difference was 0.55 dB. For the NLoS scenario, the minimum difference was –1.73 dB, and the maximum difference was 0.52 dB. Figure 16 shows that the SMC power dominates the DMC power in a LoS scenario, especially for the higher UWB frequencies. This can be explained by the fact for higher frequencies, the wavelength of the transmitted ray is small, meaning that an incident ray on a surface will encounter little effect from the roughness of the material and will reflect on it specularly. In an OLoS scenario, Fig. 17 shows that both the SMC and DMC power result in comparable power levels. This can be explained by the fact that the inherent necessity of a reflection from transmitter to receiver will automatically generate more DMC in the channel, originating from the roughness of the surface on which the reflection of the path occurs. The more reflections a propagation path undergoes, the higher the chance that the incident wave at the receiver will have encountered diffuse scattering along the way. In an NLoS scenario, Fig. 18 shows that the DMC power dominates the SMC power, especially for the lower UWB frequencies. This can be explained by the fact for lower frequencies, the wavelength of the transmitted ray is large, meaning that an incident ray on a surface with irregularities comparable in size to its wavelength will cause this ray to be scattered at many angles rather than just at one angle (as is the case with a specular reflection). This diffuse scattering typically occurs at lower frequencies, which explains why the contribution of the DMC is significantly higher than those of the SMC. The DMC power ratio will be discussed in the next section.

The DMC power ratio \(p^{s}_{r}\) can be quantified as the relative power attributable to the DMC part of the measured radio channel and can be written as follows in each sub-band s:

Figure 19 shows that the relative power attributable to the DMC part of the measured radio channel is higher for NLoS scenarios than for OLoS and LoS scenarios. In a LoS scenario, the presence of a strong direct path (LoS component) will dominate in power over the reflected paths, such that the relative DMC power ratio is lower than for OLoS or NLoS scenarios. The DMC power represents up to 50% of the total measured power for the lower UWB frequencies down to 30% for the higher UWB frequencies. As in the previous section, we know that this is due to the fact that an incident ray on an electrically small surface will encounter more effect from the roughness of its material, causing it to reflect on it diffusely. In contrast, in an OLoS scenario, the path between transmitter and receiver has to undergo one or more reflections, giving rise to more diffuse scattering along the way. This effect can be especially seen around 3 to 4 GHz. Finally, in an NLoS scenario, the DMC power represents up to 60% of the total measured power for the lower UWB frequencies (3.1 to 7 GHz), whilst it still represents up to 50% of the total measured power for the higher UWB frequencies (7 to 10.6 GHz). The necessity of a transmission from Tx to Rx in these NLoS scenarios between different media gives rise to more diffuse behavior of the waves, caused by the inherent roughness of the surfaces which the wave has to pass through. Overall, we can see that the DMC power ratio is lower for these higher frequencies, due to the fact that the encountered surfaces from Tx to Rx are electrically larger, resulting in more specular reflections.