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On the Clustering of Radio Channel Impulse Responses Using Sparsity-Based Methods Ruisi He, Member, IEEE, Wei Chen, Member, IEEE, Bo Ai, Senior Member, IEEE, Andreas F. Molisch, Fellow, IEEE, Wei Wang, Member, IEEE, Zhangdui Zhong, Jian Yu, and Seun Sangodoyin, Student Member, IEEE
Abstract—Radio channel modeling has been an important research topic, as the analysis and evaluation of any wireless communication system requires a reliable model of the channel impulse response (CIR). The classical work by Saleh and Valenzuela and many recent measurements show that multipath component (MPC) arrivals in CIRs appear at the receiver in clusters. To parameterize the CIR model, the first step is to identify clusters in CIRs, and a clustering algorithm is thus needed. However, the main weakness of the existing clustering algorithms is that the specific model for the cluster shape is not fully taken into account in the clustering algorithm, which leads to erroneous clustering and reduced performance. In this paper, we propose a novel CIR clustering algorithm using a sparsity-based method, which exploits the feature of the Saleh–Valenzuela (SV) model that the power of the MPCs is exponentially decreasing with increasing delay. We first use a sparsity-based optimization to recover CIRs, which can be well solved using reweighted 1 minimization. Then, a heuristic approach is provided to identify clusters in the recovered CIRs, which leads to improved clustering accuracy in comparison to identifying clusters directly in the raw CIRs. Finally, a clustering enhancement approach, which employs the goodness-of-fit (GoS) test to evaluate clustering accuracy, is used to further improve the performance. The proposed algorithm incorporates the anticipated behaviors of clusters into the clustering framework and enables applications with no prior knowledge Manuscript received November 09, 2015; revised February 10, 2016; accepted March 19, 2016. Date of publication March 25, 2016; date of current version May 30, 2016. Part of this work is presented at the 2016 IEEE Vehicular Technology Conference, Nanjing, China, May 15-18. This work was supported in part by the National Natural Science Foundation of China under Grant 61501020 and Grant 61401018, in part by the China Postdoctoral Science Foundation under Grant 2015M570030, in part by the State Key Laboratory of Rail Traffic Control and Safety under Grant RCS2016ZJ005, in part by the Natural Science Base Research Plan in Shanxi Province of China under Grant 2015JM6320, and in part by the National 863 Project under Grant 2014AA01A706. (Corresponding author: Bo Ai.) R. He is with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China, and also with the Beijing Key Laboratory of Traffic Data Analysis and Mining, Beijing Jiaotong University, Beijing 100044, China (e-mail:
[email protected]). W. Chen and B. Ai are with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China, (e-mail:
[email protected];
[email protected]). A. F. Molisch and S. Sangodoyin are with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089 USA (e-mail:
[email protected];
[email protected]). W. Wang is with the Institute of Communications and Navigation, German Aerospace Center (DLR), 82234 Wessling, Germany (e-mail:
[email protected]). Z. Zhong and J. Yu are with the School of Computer and Information Technology, Beijing Jiaotong University, Beijing 100044, China (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2016.2546953
of the clusters, such as number and initial locations of clusters. Measurements validate the proposed algorithm, and comparisons with other algorithms show that the proposed algorithm has the best performance and a fairly low computational complexity. Index Terms—Channel impulse response (CIR), clustering, convex optimization, sparse representation.
I. I NTRODUCTION
A
CCURATE channel models are a prerequisite for the design and performance analysis of any wireless communication system [1]. For wideband systems, which encompass almost all cellular and Wireless LAN systems in typical environments, the channel impulse response (CIR) is essential, as it determines intersymbol interference as well as achievable frequency diversity and other important characteristics. Stochastic CIR models describe the stochastic properties of the arrival delays and amplitudes of resolvable multipath components (MPCs). Among the CIR models, the Saleh–Valenzuela (SV) model [2] is the most popular one. It employs the concept of clustered MPCs, i.e., describing groups (clusters) of MPCs in the delay domain, where both the first MPC in each cluster and the MPCs within each cluster are assumed to have exponentially decaying amplitudes, despite with different decay time constants. While originally based on empirical measurements in medium-sized offices, the SV model has been found to fit the measurements well in a variety of wideband channels [3]–[6], ultrawideband (UWB) channels [7]–[9], and millimeter-wave channels [10]– [12]. Parameterization of SV models using measurements is thus useful to develop a reliable CIR model for communication system design. To parameterize SV model, the first step is to identify clusters from CIRs, which has been done manually by visual inspection [13]–[15], as the human eye is good at the detection of patterns and structures even in noisy data. However, the procedure of visual inspection is cumbersome and tiring for a large amount of measurement data, and is thus not feasible for many practical clustering implementation. In addition, this approach is subjective, and different persons may provide different clustering results. Automatic clustering of CIRs overcomes some of the drawbacks of visual inspection and has been an active area of research in the past decade. The main challenges in automatic clustering of CIRs are as follows: 1) the notion of clusters tends
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to be intuitive rather than well defined; 2) the number of clusters is usually unknown; 3) the similarity of MPCs is difficult to quantify; and 4) the cluster shapes assumed by a specific model are difficult to incorporate into the clustering algorithm. The KPowerMeans algorithm [16] is the most popular algorithm used in the clustering of radio channels in the past years [17]–[19]. It is based on the KMeans algorithm [20], which is a hard partitional approach and directly divides data objects into some prespecified number of clusters. KMeans is typically used with a Euclidean metric for computing the distance between points and cluster centers; therefore, it can easily find spherical or ball-shaped clusters in data. The KPowerMeans algorithm introduces the power of the MPCs to augment the standard KMeans concept. In the KPowerMeans algorithm, upper and lower bounds on the number of clusters have to be known a priori. The appropriate clustering result is finally determined based on some indices, which emphasize the compactness of each cluster and isolation between the clusters. In [21], the MPC distance (MCD) is proposed to quantify the similarity between MPCs. A small value of MCD means that two MPCs are close to each other and can be grouped into the same cluster. It is found that MCD can improve the performance of the KPowerMeans algorithm [22]. Determination of the number of clusters is an important part of KPowerMeans. In [23], the performances of several cluster validity indices are evaluated and compared, to select the best estimation of the number of clusters. It is found that the Xie-Beni index [24] generally has the best performance, though none of the indices is able to always predict correctly the desired number of clusters. We finally note that KPowerMeans is most suitable for the evaluation of spatiotemporal measurements, i.e., when both the delay and angle of arrival (and/or angle of departure) are available. Under those circumstances, compact clusters can be identified well. Other clustering algorithms for CIRs have also been reported in the literature. In [25], the Fuzzy-c-means algorithm is used as an alternative to the KPowerMeans. It is found that with random initialization, the Fuzzy-c-means algorithm outperforms the KPowerMeans. However, the cluster power threshold and delay scaling factor need to be manually adjusted to have reasonable results. In [26], an automated identification of clusters in CIRs is proposed, where the objective is to fit a series of exponential curves to the measured CIRs by minimizing the root-mean-squared error (RMSE). A similar idea is used in [27] to cluster UWB CIRs. However, the clustering performance of this approach is sensitive to the threshold of RMSE, and the algorithm is computationally intractable. In [28], a statistical technique is proposed to cluster CIRs by dividing the data into multidimensional analysis regions. In [29], the hidden Markov model framework is used to learn the parameters of the MPCs’ distribution and classify the MPCs in the CIRs. In [30], CIRs are clustered using region competition [31] and the amplitude distribution of MPCs is incorporated into the clustering algorithm. None of these algorithms consider the anticipated behaviors of MPCs in CIRs. As reported later in this paper, the feature that the power of MPCs is exponentially decreasing with increasing delay significantly affects clustering accuracy, and so that taking into account this modeled property of CIRs will enable us
to improve performance. Moreover, the above clustering algorithms in the literature generally require many user-specified parameters, such as number of clusters, cluster initialization, and measure of similarity, and the outcomes of these algorithms are sensitive to the settings. In our previous work of [32], we briefly introduce the idea of using convex optimization to cluster CIRs. In this paper, we extend our previous work and propose a novel CIR clustering framework using a sparsity-based method. Instead of clustering the raw CIRs, we first recover the ideal CIRs by solving a sparsity-based optimization problem that incorporates the characteristics of the CIRs, and then identify the clusters using the recovered CIRs. The key strength of the proposed algorithm lies in its capability of incorporating the characteristics of CIRs and requiring no prior knowledge of the clusters. Measurements are used to validate the proposed algorithm, and it is found that the proposed algorithm has a lower computational complexity and better performance in comparison to the state of the art. The remainder of this paper is organized as follows. Section II describes the problem of CIR clustering and the channel model behind it. A measurement campaign, which provides data for the analysis in this paper, is also introduced. Section III shows the main idea of the proposed sparsity-based clustering algorithm, and the detailed framework of the algorithm is presented in Section IV. Section V shows the results of CIR clustering and validates the proposed algorithm. Finally, Section VI concludes the paper. II. R ADIO C HANNEL M ODEL A. Data Collection We use UWB data measured at the University of Southern California (USC), Los Angeles, CA, USA, to state our problem and validate our proposed algorithm. For space reasons, we only give a brief summary; details of the measurement campaign can be found in [33]. The measurements were performed in a warehouse facility at USC, i.e., an indoor scenario. Fig. 1 shows the photo of the warehouse environment. The warehouse comprises large open halls for storing items such as books and computers. The ceiling, floor, and walls surrounding the large open hall are made of reinforced bricks and concrete while concrete pillars serving as structural supports for the ceiling in the warehouse. The storage halls were demarcated into several aisles, with each aisle comprising rows of two layered metallic storage racks. In our measurements, the separation distances between the transmitter (TX) and the receiver (RX) were 5, 10, 15, 20, and 25 m, respectively. We selected five positions for the line-of-sight (LOS) scenario at each TX–RX distance separation, and these positions experience different surrounding objects and shadowing statistics. At each position, 64 realizations of the small-scale fading components were measured by moving TX and RX over 8 positions (with a 50-mm step interval), respectively. During the measurements, there were no moving objects or personnel in the warehouse, so that the channel was static during each measurement run. We used a HP8720ET vector network analyzer (VNA) to measure the complex transfer function of channel. The VNA
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the CIR of the SV model can be expressed as h(τ ) =
Kl L
αl,k exp(jφl,k )δ(τ − Tl − τl,k )
(2)
l=1 k=1
Fig. 1. Photo of USC Warehouse Facility in [33].
was calibrated including a 20-m-long coaxial cable (which is used to connect the TX and RX ends) and a 23-dB low-noise amplifier (LNA). A stepped frequency sweep was conducted for 1601 points in the 2–8-GHz frequency range. The delay resolution was 0.167 ns, and the frequency resolution was 3.74 MHz. Two omnidirectional antennas were used with a height of 1.78 m in the measurements. The channel transfer function H of each measured location was extracted from the VNA data. The transfer function was further transformed to the delay domain by using an inverse Fourier transform with a Hann window to suppress sidelobes. The resulting CIR h can be used to state our problem and validate our proposed algorithm in the following. Note that the influence of small-scale fading is removed by averaging the instantaneous CIRs over the 8 × 8 TX/RX local positions.
B. Channel Impulse Response For wireless communications, the transmission medium is the wireless propagation channel that links TX and RX. The signal can get from the TX to the RX via a number of different propagation paths, which is called the multipath effect. Each of the paths is usually characterized by a complex gain and a propagation delay. In this case, the CIR h has the form
h(τ ) =
N
αn exp(jφn )δ(τ − τn )
(1)
n=1
where αn and φn are the amplitude gain and phase of the nth path, respectively. N is the total number of propagation paths. τn is the arrival time of the nth path. δ(·) is the Dirac delta function. However, measurements have shown that MPCs tend to arrive in groups, i.e., “clusters” [2], [34]–[36]. The most popular model to reflect this fact is the SV model [2], which is based on a doubly stochastic Poisson process. An important feature of the SV model is that the powers of the MPCs within a cluster decrease exponentially with delay, and the power of the clusters follows a (different) exponential distribution. Mathematically,
where αl,k and φl,k are the amplitude gain and phase of the kth path within the lth cluster, respectively. L is the total number of clusters and Kl is the total number of paths for the lth cluster. Tl is the arrival delay of the lth cluster. τl,k is the excess arrival delay of the kth path within the lth cluster. In the SV model, the phase φl,k is assumed as an independent random variable which is uniformly distributed between 0 and 2π. The amplitude gain αl,k , for given values of Tl and τl,k , is fading with a conditional mean square value that is a monotonically decreasing function of Tl and τl,k . The average power of αl,k is expressed as Tl τl,k 2 2 |αl,k | = |α0,0 | · exp − · exp − (3) Γ Λl A1
A2
2
where |α0,0 | is the average power of the first MPC in the first cluster. A1 and A2 represent for inter- and intracluster decay of MPC power with delay, respectively. Γ and Λl are the cluster and MPC power decay constants, respectively. For the convenience of the following analysis, the power delay profile (PDP) P is used instead of the CIR h(τ ). The PDP P specifies the power of a signal received through a multipath channel as a function of delay, and it can be mathematically 2 described by P (τ ) = E{ |h(τ )| } for a band-limited transfer function [1]. Since analyzing the PDP on a linear scale will often neglect the weak MPCs [37], we convert to a logarithmic scale instead. In such case, the two terms of exponential decays of the power with delay in (3) are converted to linear decays. Note that in the following, we assume ergodicity which holds if the bandwidth of the analyzed system is not too large [38]. C. Problem Description A large body of channel measurements validates the SV model [39]–[41]. An example plot of an SV-shaped PDP is shown in Fig. 2(a), based on the LOS UWB measurements in [33]. A visual inspection clearly shows that there are 5 clusters in the PDP of Fig. 2(a), and the clusters start at 50, 120, 175, 214, and 248 ns, respectively. The power of the first MPC in each cluster roughly follows the linear decay function with delay, which is indicated by the red line, whereas the remaining MPCs in each cluster roughly follow the linear decay functions with different slopes. The two terms of A1 and A2 in (3) are well reflected by the data. However, even though the visual inspection can be easily used to cluster PDPs (with a consideration of the features of A1 and A2 ), most automatic clustering algorithms fail to offer results in agreement with the visual inspection. An example plot of the clustering using KMeans algorithm [20] is shown in Fig. 2(b), where KMeans fails to partition PDPs into the SV-shaped clusters. The standard KMeans algorithm considers PDPs in Fig. 2(b) as two-dimensional (2-D) (delay and power) points. It finds a partition such that the squared error between
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a high slope at the first MPC within each cluster, and the slope can thus be used for the cluster identifications. The above idea can be formulated as the following optimization problem [42]: 2 ˆ ˆ (4) min P − P + λ Ω2 · Ω1 · P ˆ P
Fig. 2. Example plots of the measured PDP using UWB LOS measurements in [33] when TX–RX separation distance is 15 m. The magenta lines represent the least-squared regression of PDPs within clusters. (a) Clustering based on visual inspection. The first peak in each cluster is marked with black circles for clarity, whose least-squared regression fit is plotted with red line. (b) Clustering based on KMeans algorithm. Different clusters are plotted with different colors. The black parts of the curves represent noise data, which were not used in the clustering.
the empirical mean of a cluster and the points in the cluster is minimized.1 Since KMeans uses the Euclidean distance to measure the similarity of MPCs, it fails to incorporate the anticipated behaviors of CIRs, such as the relationship between power and delay in (3). Therefore, the tail of one PDP cluster is usually grouped into the next cluster, which apparently does not agree with the assumption of the SV model. In the following, we propose a new automatic clustering algorithm to incorporate the above behaviors of CIRs. III. M AIN I DEA The proposed clustering algorithm involves solving a sparsity-base optimization problem. The main idea can be summarized as follows. 1) We assume that CIRs statistically follow the trend of (3), i.e., powers of MPCs generally decrease with delays in terms of A1 and A2 . 2) Then, we consider the measured PDP vector P as the given signal and try to recover an original unknown sigˆ which is close to P and has the formulation nal vector P ˆ are the of (3) using convex optimization, where P and P ˆ vectors of P (τ ) and P (τ ), respectively. We use a method to enhance the sparsity of the solution. ˆ to identify the clusters. As 3) Finally, we use the curve of P ˆ generally has shown in (3), the curve of the dB-scaled P 1 The KMeans algorithm requires two user-specified parameters: number of clusters and cluster initialization. They were manually determined based on the visual inspection before the implementation of Fig. 2(b), which enables KMeans to have its best performance.
⎡
2
0
where ·x represents x norm operation and 0 norm operation, which returns the number of nonzero coefficients. P and ˆ have dimension N , and λ is a regularization parameter. Ω1 P is the finite-difference operator in the form of (5), shown at the bottom of the page, where Δτ represents the minimum resolvable delay difference of data. Equation (5) is used to calculate ˆ and is able to deal with both sparse and nonthe slope of P sparse data of MPCs. Ω2 is used to obtain the turning point at which the slope changes significantly and can be expressed as ⎡ ⎤ 1 −1 0 . . . . . . 0 ⎢ 0 1 −1 . . . . . . 0 ⎥ ⎢ ⎥ ⎢ .. . . . . . . . . .. ⎥ ⎢ . . . . . ⎥ Ω2 = ⎢ . . (6) ⎥ ⎢ ⎥ . . ⎣0 0 . 1 −1 0 ⎦ 0 0 . . . . . . 1 −1 (N −2)×(N −1) ˆ ˆ folThe term λΩ2 · Ω1 · P ensures that the recovered P 0 lows the anticipated behavior of term A2 in (3). It also implies that the proposed algorithm favors a small number of clusters to avoid overparameterization. We introduce the term Ω2 · Ω1 in ˆ as it is more (4) to better identify clusters from the recovered P, ˆ sensitive to the slope change of P. The anticipated behavior of ˆ using a clustering term A1 in (3) can be incorporated into P enhancement approach which will be given shortly. The optimization problem in (4) is nondeterministic polynomial (NP) hard, and a prevailing approach for solving optimization problems with a sparse penalty is to relax the 0 norm by the convex 1 norm [43]–[45]. However, the 1 norm minimization could have structure error, i.e., its global minimum is not the most sparse solution, which leads to the increase of the number of clusters. In order to obtain a more accurate clustering result, we use the reweighted 1 norm minimization approach [46], which finds a sparse solution and has a low computational complexity in comparison to the 0 norm minimization [42]. IV. S PARSITY-BASED C LUSTERING A LGORITHM In this section, we summarize the detailed framework of the proposed sparsity-based clustering algorithm.
Δτ Δτ − 0 ··· ··· 0 ⎢ |τ1 − τ2 | |τ1 − τ2 | ⎢ Δτ Δτ ⎢ − ··· ··· 0 0 ⎢ |τ2 − τ3 | |τ2 − τ3 | ⎢ ⎢ .. .. .. .. .. .. Ω1 = ⎢ . . . . . . ⎢ ⎢ . Δτ Δτ ⎢ .. − 0 0 0 ⎢ ⎢ |τN −2 − τN −1 | |τN −2 − τN −1 | ⎣ Δτ Δτ − 0 0 ··· 0 |τN −1 − τN | |τN −1 − τN |
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(5)
(N −1)×N
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A. Reweighted 1 Minimization For the optimization problem of (4), we use the reweighted 1 minimization [46], which employed the weighted norm and iterations to enhance the sparsity of the solution. This algorithm consists of the following steps. 1) Set the iteration count m to zero and set the initial weight (0) wn = 1, n = 1, . . . , N . The weight parameter is used later to guarantee a sparse solution in the reweighted 1 minimization. 2) Solve the following weighted 1 minimization problem: ˆ ˆ (m) = arg min P P − P 2 (7) (m) ˆ ≤ Lmax s.t.W · Ω2 · Ω1 · P 1
where Lmax is the maximum number of clusters to be identified in each PDP. Superscript (m) represents the mth iteration. W(m) is a diagonal matrix, expressed as ⎤ ⎡ (m) w1 0 ... 0 ⎥ ⎢ .. ⎥ ⎢ 0 w(m) . . . . ⎥ ⎢ 2 (m) (8) =⎢ . W ⎥. .. .. ⎥ ⎢ . . . 0 ⎦ ⎣ . (m) 0 . . . 0 w(N −2) 3) Update the weights as 1 wn(m+1) = , n = 1, . . . , N. (m) Pˆn + ε
(9)
Parameter ε is used to provide stability and to ensure ˆ (m) does not strictly that a zero-valued component in P prohibit a nonzero estimate at the next step. Specifically, the reweighted 1 minimization favors a sparse solution, ˆ (m) to be zeros, and the updated which forces many P (m+1) weight wn tends to be infinite. The value of ε affects how well the reweighted 1 minimization approximates the NP hard optimization problem (4), and the convergence rate of the algorithm, which is mainly of concern in the numerical optimization (which is out the scope of this paper) rather than physical explanation. In practice, ε can be chosen as the smallest value allowed in the computing environment, or some value that is much smaller to the expected nonzero magnitudes of P. 4) Terminate on convergence or when m attains a specified maximum number of iterations M . Otherwise, increment m and go to step 2. It is found that convergence mostly occurs when m = 3 in our PDP clustering, and we thus set M = 10 to have reasonable results. The above iterative algorithm is found to have high quality of reconstructions with low computational cost and is thus used ˆ that is used for identifying clusters. to recover P B. Cluster Identification ˆ using the reweighted 1 An example plot of the recovered P, minimization, is shown in Fig. 3(a). Compared with the raw P
Fig. 3. Cluster identification using the same data of Fig. 2. (a) PDP recovery ˆ which can be used for the cluster (dB). (b) Plots of Φ and Ω(N −1)×N · P, identification (derivatives of signal, dB).
ˆ reasonably tracks the variation of P with noise and fading, P and has a piecewise linear curve, which successfully partitions at the delay that a cluster partition naturally occurs. To better ˆ we define the following vector as identify the partitions of P, ˆ (10) Φ = Ω2 · Ω1 · P (N −2)×1
ˆ at which which can be used to indicate the turning point of P the slope changes significantly. Fig. 3(b) shows an example plot of Φ, where we can see that a positive peak of Φ generally corresponds to a cluster partition. The delay index nc (where 1 ≤ nc ≤ N − 2) of the cth cluster can thus be identified as follows: S := {nc |Φnc ≥ Cth }
(11)
where nc is the element of the set S. 1 ≤ c ≤ Nc and Nc is the total number of clusters. Cth is the threshold to identify clusters. ˆ (which can be referred to a slope of P) ˆ The term of Ω1 · P is also plotted in Fig. 3(b) for comparison, where we can see that Φ has a better clustering accuracy if the peak searching approach is used for clustering. It is further found that the absolute value of Φ is generally small for the region with large delays. This is mainly caused by the decreasing power of PDPs with delay. Therefore, a Cth that decreases with delay should be used to better identify clusters with large delays. Here, we provide a heuristic implementation approach of (11) as follows: {nc |Φnc ≥ Cth , if 0 ≤ nc ≤ 0.3 · N } S := {nc |Φnc ≥ 0.5 · Cth , if 0.3 · N < nc ≤ N } . (12) In our clustering, a Cth = 1 is found to have a reasonable performance. As shown in Fig. 3(b), the clusters are correctly identified using (12). C. Clustering Enhancement In Sections IV-A and IV-B, the clusters are identified from the measured PDPs, under the consideration of the anticipated behavior of term A2 in (3). However, the anticipated behavior of term A1 has not been considered so far, which could be used to improve the clustering performance. Fig. 4(a) shows an example of the clustering result only using the anticipated behavior of term A2 (i.e., the above steps in Sections IV-A and IV-B). It
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Fig. 4. Cluster enhancement using UWB measurements in [33] when TX-RX separation distance is 5 m. (a) Without clustering enhancement. (b) With clustering enhancement. (c) Without clustering enhancement. (d) With clustering enhancement. In (a) and (b), different clusters are plotted with different colors. The black curves represent noise data and the magenta lines represents the least-squared regression of PDPs within clusters. The first peaks in each cluster are marked with black circle. In (c) and (d), the magenta square represents the cluster power.
can be seen that the second and third clusters should be a part of the first one according to the SV model, so that the dB-scaled cluster power decreases linearly with delay. In the following, we present the steps of clustering enhancement to incorporate the anticipated behavior of term A1 in (3) and to improve the clustering results. 1) Set the initial Lmax to an arbitrarily large integer such as Lmax = 30. 2) Do all the steps in Sections IV-A and IV-B to have the initial clustering results. 3) Find the delay index of the first MPC peak n ˜ c within the cth cluster,2 which can be easily done by a peak search within each cluster and selecting the peak with the maximum power. 4) Denote Qc as the power of the cth cluster, i.e., the power summation of all the MPCs in the cth cluster. Using all nc , Qc ), we develop a least-squared linear Nc points of (˜ regression curve g(n). Then, we examine the goodnessof-fit (GoS) of g(n) using the coefficient of determination R, which is defined as [47] 2 (Qc − g(˜ nc )) (13) R = 1 − c 2 Qc − Qc c
where (·) denotes the sample mean value of the set (·). R measures how successful the fit of g(n) is in explaining the variation of the data. It ranges from −∞ to 1, with a value closer to 1 indicating that the regression model fits the data better.3 2 Note that for the measurements with limited bandwidth, generally n
˜c. c
