A Lower Bound for the Coherence Block Length in Mobile Radio Channels

A lower bound for the coherence block (ChB) length in mobile radio channels is derived in this paper. The ChB length, associated with a certain mobile radio channel, is of great practical importance in future wireless systems, mainly those based on massive multiple input and multiple output (M-MIMO) technology. In fact, it is one of the factors that determines the achievable spectral efficiency. Firstly, theoretical aspects regarding the mobile radio channels are summarized, focusing on the rigorous definition of coherence bandwidth (BC) and coherence time (TC) parameters. Secondly, the uncertainty relations developed by B. H. Fleury, involving both BC and TC, are presented. Afterwards, a lower bound for the product BCTC is derived, i.e., the ChB length. The obtained bound is an explicit function of easily measurable parameters, such as the delay spread, mobile speed and carrier frequency. Furthermore, and especially important, this bound is also a function of the degree of coherence with which we define both BC and TC. Finally, an application example that illustrates the practical possibilities of the bound obtained is presented. As a further conclusion, the need to determine what degree of correlation is required to consider mobile channels as effectively flat-fading and stationary is highlighted.


Introduction
The concept of the coherence block (ChB) associated with a certain mobile radio channel is of great practical importance in future systems based on massive multiple input and multiple output (M-MIMO) technology. In fact, it is one of the factors that determines the achievable spectral efficiency [1][2][3][4]. The size of the ChB answers the fundamental question: for how long and in what frequency range can we consider the radio channel to be approximately constant? Answering this question is crucial since it determines the overhead incurred by periodically dedicating a certain number of carriers to transmit pilots that update the knowledge of the channel. Moreover, in massive MIMO systems, based on the use of a large number of antennas at the base station, the number of pilots dedicated to channel estimation is a limiting factor on the maximum number of users that the system can serve simultaneously. Therefore, it is of great importance to have a reasonable estimate of the duration of the ChB of different channels.
As a definition of the ChB, we textually adopt the proposal by Emil Björnson et al. in [5]: "A coherence block consists of a number of subcarriers and time samples over which the channel response can be approximated as constant and flat-fading. If the coherence bandwidth is B C and the coherence time is T C , then each coherence block contains N C = B C T C complex-valued samples". The coherence bandwidth (B C ) represents the frequency interval over which the channel responses are approximately constant, and that means that a group of subcarriers within the aforementioned frequency interval will see the channel in approximately the same way. Moreover, the coherence time of the channel (T C ) represents the time interval in which we can consider the channel approximately invariant with time, and thus during this interval the knowledge of the channel will be updated.
In order to estimate the length of the ChB, it is necessary to estimate the values of B C and T C corresponding to a specific mobile channel. According to the authors' knowledge, the literature tends to make a simple estimation of these parameters, considering them deterministic, although the stochastic nature of these channel characteristics is well known. Moreover, it is also obviated that in the mathematical definition of these parameters the "degree of coherence" that defines both B C and T C plays a central role [5] (remark 2.1). A rigorous definition of these parameters, B C and T C , leads us to consider the stochastic nature of the length of the ChB, allowing in the future to establish radio resource optimization strategies in M-MIMO systems beyond those already proposed, for example in [4]. Obtaining a lower bound for the length of the ChB is a first step in the direction of describing it in a more rigorous way than at present, and has in itself not only a theoretical value but also practical applications.
This paper presents a lower bound for the ChB length, N C , for wide-sense stationary uncorrelated scattering (WSSUS) channels. The contribution is based on a previous result of B. H. Fleury [6], who derived uncertainty relations involving B C and T C . The combination of both uncertainty relations makes it possible to obtain a lower bound of the number of samples that make up the ChB. This lower bound depends on the dispersion of the channel, the speed of the users, the carrier frequency, and, notably, on the level of correlation selected to define both B C and T C .

Rigorous Definition of B C and T C
Mobile radio channels, due to their complexity, need to be considered randomly time variant. Since Bello's seminal work [7], a statistical description of the radio channel based on four basic input-output functions that represent the channel in the domains of delay time (τ) and frequency (f ); and of the time (t) and the Doppler frequency (ν) has been widely accepted. Under the hypothesis that mobile radio channels can be considered in most cases WSSUS, the analysis is simplified. In this case, a complete description of the channel can be obtained using the autocorrelation functions of the basic functions of Bello [8]. If we start, as a basic Bello's function of the WSSUS channel, from the time-variant transfer function H(t,f ), we can define the impulsive response (IR) for a certain instant of time t 0 by means of the Fourier transform (FT) with respect to the variable f : Furthermore, the Doppler spread (DSpread) function is defined as the FT with respect to the variable t for a determined value of the frequency f 0 , as given in Equation (2).
From the IR and DSpread functions, the power delay profile (PDP) and the Doppler spectrum (DS) are defined as given in Equations (3) and (4), respectively.
Finally, we can obtain the autocorrelation function of the time-variant transfer function H(t,f ) with respect to the variables f and t, respectively, as: where ∆f and ∆t represent the lag in the frequency and time, respectively. Autocorrelation functions by definition have their maximum value at ∆f = 0 and ∆t = 0. From now on, we will consider the autocorrelation functions normalized to their maximum value, although we keep the notation in Equations (5) and (6). Once obtained and normalized to its maximum value at the origin, the values of B C at different correlation levels are defined as the maximum bandwidth over which the frequency correlation function is above a predetermined level (c f ), usually c f = 0.5, 0.7 or 0.9. Similarly, the coherence time T C is defined as the maximum time at which the time correlation function is above a predetermined level c t . Thus, the definition of both parameters depends on the level of correlation that we set. From a practical point of view, it is important that the correlation levels that are selected are sufficient to consider that H(t,f ) is approximately constant, with respect to the parameter f or t.

Time Duration and Bandwidth of a Pair of Signals
Two functions or signals related to each other through the FT are linked so that the width of one in a domain determines the width of the other one in the transformed domain. The property of the FT that determines that a compression of the abscissas in one domain means expansion of the same axis in the transform domain is well known. Consequently, the duration of a signal in the delay time domain is inverse to its bandwidth in frequency; and its time variation is inverse to the Doppler spectrum width.
In the case of mobile radio channels, it is common practice to measure the duration of the channel response in the delay time domain using the root mean square (RMS) delay spread (τ rms ), which is calculated as the square root of the second central moment of the PDP [8]. It is straightforward to demonstrate that the FT of the PDP is the autocorrelation of the time-variant transfer function H(t,f ) with respect to the variable f, as already shown in Equation (5). The duration of the autocorrelation function is usually measured by the B C at different correlation levels. Therefore, these two parameters, τ rms and B C , will maintain an inverse relationship between them according to the compression properties of the FT.
The relationship between B C and τ rms has received a lot of attention since the initial contribution of Gans [9], who proposed a simple relationship: where the parameter α depends on the shape of the PDP and the degree of correlation that defines B C [10]. The most significant contributions on how to determine α are summarized in [10]. In [11] a two-parameter model following the form B C = α τ −β rms was proposed. For the same reasons, there is an inverse relationship between the transformed pair given in Equation (6), i.e., between the width of the Doppler spectrum, S(ν), and the duration of the autocorrelation function, R t (∆t). Similar to τ rms , a measure of the width of S(ν) can be defined, the RMS Doppler spread, ν rms , calculated as the square root of the second central moment of S(ν); whereas the duration of the autocorrelation function in the time domain, R t (∆t), is usually measured using the T C parameter at different correlation levels. In mobile radio channels, the width of the Doppler spectrum is limited by the maximum Doppler frequency ν max = v/λ, where v represents the speed of the mobile user and λ is the wavelength associated with the frequency of operation. Therefore, this last parameter is widely used as a measure of the width of the DS. A widely accepted dual relation to Equation (7) is given in Equation (8), which is obtained from the classic Clarke's model [12]. The obtained T C value corresponds with a coherence time at a correlation level of 0.7.

Uncertainty Relations
The product delay time duration by bandwidth of pairs of functions related by the FT verifies uncertainty relations. These uncertainty relations have different formulations depending on the measure adopted for the time duration and frequency width of the signals [13]. One of the best known is the Heisenberg uncertainty, which can be demonstrated to be a property of the FT without further reference to physical interpretation [13]. The well-known relation ∆t ∆f ≥ 1⁄(4π) is verified for a pair of transformed functions when ∆t , ∆f are the square root of the second central moment of such functions. Fleury demonstrated in [6] two uncertainty relations that have great theoretical and practical interest. These inequalities are applicable to WSSUS channels and relate the durations of the basic functions of the channel in the four domains: where c f and c t represent the correlation levels on the frequency and time domains, respectively.

Lower Bound for the Coherence Block Length
In the uncertainty relationships Equations (9) and (10), the width of the PDP is measured by the τ rms and the width of the DS is measured by the parameter ν rms . From a practical point of view, the current values of τ rms and ν rms can be replaced by the maximum values expected in a given propagation environment. In the case of the RMS delay, the following inequality is considered: Moreover, taking into account that the Doppler spectrum is limited by the value of the maximum Doppler frequency it can be stated that: In Equation (12), c 0 represents the speed of light, v the relative speed between the transmitter and the receiver, and f 0 the carrier frequency. Considering the limits in Equations (11) and (12), the following inequalities can be written from Equations (9) and (10): Multiplying Equations (13) and (14) it yields: The inequality Equation (15) offers a lower bound to the duration of the ChB; this bound is conservative due to the relationships Equations (11) and (12) but, advantageously, relates it explicitly with easily obtainable parameters that are related to the dispersion of the channel, measured in terms of the maximum expected RMS delay spread, the speed of the mobiles and the carrier frequency. Furthermore, and especially important, this bound is related to the degree of coherence with which we define both B C and T C .

Results and Application Example
In this section we present, first, some representative experimental results showing compliance with the Fleury border Equation (9); and secondly, as an application example, the values obtained for the lower bound Equation (15) considering two bands of interest in 5G: those centered on 3.6 GHz and 26 GHz.

Experimental Verification of Fleury's Inequalities
Several authors have experimentally contrasted the inequality in Equation (9). Fleury presents in [6] the measured results of pairs (B C , τ rms ) along with the boundary defined in Equation (9). These results were measured in an indoor environment in the 900 MHz frequency band [14]; and clearly show the fulfilment of the inequality Equation (9). In [10], results that also comply with Equation (9) in the UHF band are presented. In [15], measured data are presented that confirm this boundary for a wide set of measurements carried out in the 1900 MHz frequency band in the Olympic Stadium of Athens. Regarding Equation (10), the authors are not aware of their experimental confirmation. Figure 1 shows the results obtained from 294 measured pairs (τ rms , B C ) in a meeting room in the 3 to 4 GHz frequency band. The pairs of values (τ rms , B C ) have been obtained from the corresponding 294 measurements of the IR of the channel in this band. The measurement system, as well as other details of interest, can be found in [16]. Fleury's confinement boundary Equation (9) is also represented using a solid line, showing that all measured pairs (τ rms , B C ) are above that boundary, as corresponds with WSSUS channels. It can be observed that the pairs (τ rms , B C ) are scattered on the plane, although they are always above the Fleury's boundary Equation (9). The dispersion decreases with the degree of correlation, and for c f = 0.9 the values are clearly concentrated around the boundary. This last fact was already observed and discussed in [16]. Moreover, in [16], it is shown how the dispersion of (τ rms , B C ) values is greater in line of sight (LOS) than in non LOS (NLOS).

Experimental Verification of Fleury's Inequalities
Several authors have experimentally contrasted the inequality in Equation (9). Fleury presents in [6] the measured results of pairs (BC, τrms) along with the boundary defined in Equation (9). These results were measured in an indoor environment in the 900 MHz frequency band [14]; and clearly show the fulfilment of the inequality Equation (9). In [10], results that also comply with Equation (9) in the UHF band are presented. In [15], measured data are presented that confirm this boundary for a wide set of measurements carried out in the 1900 MHz frequency band in the Olympic Stadium of Athens. Regarding Equation (10), the authors are not aware of their experimental confirmation. Figure 1 shows the results obtained from 294 measured pairs (τrms, BC) in a meeting room in the 3 to 4 GHz frequency band. The pairs of values (τrms, BC) have been obtained from the corresponding 294 measurements of the IR of the channel in this band. The measurement system, as well as other details of interest, can be found in [16]. Fleury's confinement boundary Equation (9) is also represented using a solid line, showing that all measured pairs (τrms, BC) are above that boundary, as corresponds with WSSUS channels. It can be observed that the pairs (τrms, BC) are scattered on the plane, although they are always above the Fleury´s boundary Equation (9). The dispersion decreases with the degree of correlation, and for cf = 0.9 the values are clearly concentrated around the boundary. This last fact was already observed and discussed in [16]. Moreover, in [16], it is shown how the dispersion of (τrms, BC) values is greater in line of sight (LOS) than in non LOS (NLOS).

Application Examples
As an initial application example, the values of the ChB length (NC) obtained for the lower bound Equation (15) are analyzed for two bands of interest in 5G, those centered on 3.6 GHz and 26 GHz. For each band, indoor and outdoor scenarios are considered, as well as two typical situations of low mobility (pedestrian), and high speed (vehicular). Table 1 shows the parameters defining the six propagation environments considered. In order to select representative values of τmax, measurements carried out by several authors in different environments have been considered. Thus, for indoor environments, values of 40 and 30 ns have been set to τmax for the 3.6 GHz [16] and 26 GHz frequency bands [17], respectively. Moreover, regarding the outdoor scenarios, values for τmax of 250 and 100 ns have been considered for the 3.6 GHz [18] and 26 GHz [19] bands, respectively. It should be noted that the proposed values are only representative and plausible values. Of course, there are many more experimental works than those referenced and site-specific values could be obtained for different propagation environments. However, the intention of this paper is to present the lower bound Equation (15), using this initial example to highlight its possibilities.  Figure 2 presents the values of the lower bound of the ChB length (NC) obtained according to Equation (15) for different levels of frequency correlation, for the six propagation environments considered. In this case, to simplify the analysis of the results, the temporal correlation level (ct) has been prefixed to 0.7. It can be seen how the length of the ChB decreases as the correlation level of BC increases according to the arccos(cf) function, and the value cf = 0.7 can be considered a breaking point from which the slope of the curves increases. The effect of the different dispersion of the channels is also observed, as there is a reduction in the length of the ChB due to the increase in the delay spread of the outdoor channel compared to the indoor one. Furthermore, it must be added that a greater

Application Examples
As an initial application example, the values of the ChB length (N C ) obtained for the lower bound Equation (15) are analyzed for two bands of interest in 5G, those centered on 3.6 GHz and 26 GHz. For each band, indoor and outdoor scenarios are considered, as well as two typical situations of low mobility (pedestrian), and high speed (vehicular). Table 1 shows the parameters defining the six propagation environments considered. In order to select representative values of τ max , measurements carried out by several authors in different environments have been considered. Thus, for indoor environments, values of 40 and 30 ns have been set to τ max for the 3.6 GHz [16] and 26 GHz frequency bands [17], respectively. Moreover, regarding the outdoor scenarios, values for τ max of 250 and 100 ns have been considered for the 3.6 GHz [18] and 26 GHz [19] bands, respectively. It should be noted that the proposed values are only representative and plausible values. Of course, there are many more experimental works than those referenced and site-specific values could be obtained for different propagation environments. However, the intention of this paper is to present the lower bound Equation (15), using this initial example to highlight its possibilities.  Figure 2 presents the values of the lower bound of the ChB length (N C ) obtained according to Equation (15) for different levels of frequency correlation, for the six propagation environments considered. In this case, to simplify the analysis of the results, the temporal correlation level (c t ) has been prefixed to 0.7. It can be seen how the length of the ChB decreases as the correlation level of B C increases according to the arccos(c f ) function, and the value c f = 0.7 can be considered a breaking point from which the slope of the curves increases. The effect of the different dispersion of the channels is also observed, as there is a reduction in the length of the ChB due to the increase in the delay spread of the outdoor channel compared to the indoor one. Furthermore, it must be added that a greater dispersion takes place in the 3.6 GHz band compared to the 26 GHz one. Finally, the effect of the speed of the users is also observed. The increase in speed between a user who moves on foot compared to another who travels in a vehicle at high speed produces, as is well known, a drastic decrease in the length of the ChB. Table 2 shows the N C values obtained for the representative frequency correlation levels, c f , of 0.5, 0.7 and 0.9. The N C values presented in the table, i.e., the minimum number of samples in the ChB, show variations of up to three orders of magnitude in different propagation environments and for the two frequency bands, also and very importantly, with the degree of correlation adopted for the definition of B C . dispersion takes place in the 3.6 GHz band compared to the 26 GHz one. Finally, the effect of the speed of the users is also observed. The increase in speed between a user who moves on foot compared to another who travels in a vehicle at high speed produces, as is well known, a drastic decrease in the length of the ChB. Table 2 shows the NC values obtained for the representative frequency correlation levels, cf, of 0.5, 0.7 and 0.9. The NC values presented in the table, i.e., the minimum number of samples in the ChB, show variations of up to three orders of magnitude in different propagation environments and for the two frequency bands, also and very importantly, with the degree of correlation adopted for the definition of BC.  In Figure 3, the variation of NC is represented against the correlation coefficients in frequency and time, but now varying these together according to Equation (15). Only the most critical case of a channel with a high mobility has been considered. The significant increase in NC that can be achieved if lower correlation values are set for both domains is shown. This example shows the interest in developing channel estimators that are capable of working with pilots spaced in time and frequency as much as possible. In addition, research on M-MIMO precoding and combining schemes that support the outdated channel state information (CSI) as best as possible also plays a key role.  In Figure 3, the variation of N C is represented against the correlation coefficients in frequency and time, but now varying these together according to Equation (15). Only the most critical case of a channel with a high mobility has been considered. The significant increase in N C that can be achieved if lower correlation values are set for both domains is shown. This example shows the interest in developing channel estimators that are capable of working with pilots spaced in time and frequency as much as possible. In addition, research on M-MIMO precoding and combining schemes that support the outdated channel state information (CSI) as best as possible also plays a key role.
In order to complete the analysis and show the validity of the lower bound proposed in Equation (15), that limit is applied to semi-empirical values of ChB lengths. To make a complete measurement of the length of the ChB in a specific environment, it is necessary to measure B C and T C for a set of sampling points in that environment. In our case, we have a wide set of B C measurements carried out in an indoor environment in the 3.2-4 GHz frequency band, already presented in [20]. However, we do not have measurements concerning the T C values in this environment, since the measurements in [20] were made for another purpose. To solve this problem, we propose a semi-empirical approach consisting of setting a theoretical value for T C , obtained from the Clarke's model, calculated according to Equation (8). The T C obtained takes a value of approximately 20 ms, corresponding with a coherence time at a correlation level of c t = 0.7 [12], for a typical indoor speed of 0.75 m/s, and at the central band frequency of 3.6 GHz. In order to complete the analysis and show the validity of the lower bound proposed in Equation (15), that limit is applied to semi-empirical values of ChB lengths. To make a complete measurement of the length of the ChB in a specific environment, it is necessary to measure BC and TC for a set of sampling points in that environment. In our case, we have a wide set of BC measurements carried out in an indoor environment in the 3.2-4 GHz frequency band, already presented in [20]. However, we do not have measurements concerning the TC values in this environment, since the measurements in [20] were made for another purpose. To solve this problem, we propose a semi-empirical approach consisting of setting a theoretical value for TC, obtained from the Clarke´s model, calculated according to Equation (8). The TC obtained takes a value of approximately 20 ms, corresponding with a coherence time at a correlation level of ct = 0.7 [12], for a typical indoor speed of 0.75 m/s, and at the central band frequency of 3.6 GHz. Figure 4 presents the cumulative distribution function (CDF) of the estimated NC for the frequency correlation levels (cf) of 0.5, 0.7 and 0.9. The results included show experimentally the validity of the lower bound for a real indoor channel, and they highlight the random nature of NC. The semi-empirical results are compared with the lower bound calculated according to Equation (15). It can be observed that the lower bound proposed by the authors in Equation (15) is conservative but can be used as an initial useful approach to establish the ChB length.
(a)  Figure 4 presents the cumulative distribution function (CDF) of the estimated N C for the frequency correlation levels (c f ) of 0.5, 0.7 and 0.9. The results included show experimentally the validity of the lower bound for a real indoor channel, and they highlight the random nature of N C . The semi-empirical results are compared with the lower bound calculated according to Equation (15). It can be observed that the lower bound proposed by the authors in Equation (15) is conservative but can be used as an initial useful approach to establish the ChB length. In order to complete the analysis and show the validity of the lower bound propo in Equation (15), that limit is applied to semi-empirical values of ChB lengths. To mak complete measurement of the length of the ChB in a specific environment, it is necess to measure BC and TC for a set of sampling points in that environment. In our case, have a wide set of BC measurements carried out in an indoor environment in the 3. GHz frequency band, already presented in [20]. However, we do not have measureme concerning the TC values in this environment, since the measurements in [20] were ma for another purpose. To solve this problem, we propose a semi-empirical approach c sisting of setting a theoretical value for TC, obtained from the Clarke´s model, calcula according to Equation (8). The TC obtained takes a value of approximately 20 ms, cor sponding with a coherence time at a correlation level of ct = 0.7 [12], for a typical ind speed of 0.75 m/s, and at the central band frequency of 3.6 GHz. Figure 4 presents the cumulative distribution function (CDF) of the estimated NC the frequency correlation levels (cf) of 0.5, 0.7 and 0.9. The results included show exp mentally the validity of the lower bound for a real indoor channel, and they highlight random nature of NC. The semi-empirical results are compared with the lower bound culated according to Equation (15). It can be observed that the lower bound proposed the authors in Equation (15) is conservative but can be used as an initial useful appro to establish the ChB length.
(a)  Furthermore, the value obtained following the rule-of-thumb proposed in [5] mark 2.1) is also represented in Figure 4. The value of TC is estimated as λ/4v, and BC 1/2Td, where Td is the delay spread, the delay difference between the shortest and long path in the PDP. According to the measurements [20], an average value for the Td para eter would be 150 ns in the indoor environment considered. Therefore, the ChB len (BCTC) takes a value of 92,592 samples, which is the limit represented with a dash line Figure 4. It is remarkable that this value is a deterministic estimate and thus, it does consider the level of coherence for which it is defined. In Figure 4, it can be seen that i a reasonable estimate when NC is defined for frequency correlation levels of 0.5; in fact least in this case, it behaves like the lower bound proposed; but it leads to an overestim tion of the length of ChB for correlation levels of 0.7 and 0.9. For cf = 0.7, it is observed t 19% of the values are below the one predicted by the rule-of-thumb; and for cf = 0.9, pr tically all the values (99%) are below the approach. Furthermore, the value obtained following the rule-of-thumb proposed in [5] (remark 2.1) is also represented in Figure 4. The value of T C is estimated as λ/4v, and B C as 1/2T d , where T d is the delay spread, the delay difference between the shortest and longest path in the PDP. According to the measurements [20], an average value for the T d parameter would be 150 ns in the indoor environment considered. Therefore, the ChB length (B C T C ) takes a value of 92,592 samples, which is the limit represented with a dash line in Figure 4. It is remarkable that this value is a deterministic estimate and thus, it does not consider the level of coherence for which it is defined. In Figure 4, it can be seen that it is a reasonable estimate when N C is defined for frequency correlation levels of 0.5; in fact, at least in this case, it behaves like the lower bound proposed; but it leads to an overestimation of the length of ChB for correlation levels of 0.7 and 0.9. For c f = 0.7, it is observed that 19% of the values are below the one predicted by the rule-of-thumb; and for c f = 0.9, practically all the values (99%) are below the approach.

Conclusions
Starting from the uncertainty relations obtained by Fleury, a lower bound has been derived for the length of the ChB. The bound explicitly relates to basic parameters of the channel, such as the maximum expected RMS delay spread, the carrier frequency as well as mobile speeds. Moreover, the dependence of the length of the ChB on the selected correlation levels in order to define both B C and T C , is quantified. Results for typical indoor as well as outdoor wireless situations (pedestrian and vehicular users) in the 3.6 and 26 GHz, along with those provided for experimental indoor data in the 3.2-4 GHz, show the validity of the lower bound presented. It must be pointed out that the impact of the correlation levels on the outdating of channel estimation is an aspect that must be analyzed in detail in future works.
The high dispersion of the values of channel parameters such as the B C , T C , τ rms and v rms along a propagation environment, underlines the need to consider the duration of the ChB as a random variable, and to analyze it from a statistical point of view, obtaining probability distributions that should allow us to obtain outage values, beyond a boundary that allows 100% compliance to be ensured but whose tightness is low. However, since the overhead in channel estimation in future mobile communication systems, especially in systems based on massive MIMO technology, depends on the length of the ChB, the proposed lower bound is a valuable engineering tool for assisting in the design and sizing of the physical layer.
Funding: This work has been supported by the Spanish Ministry of Economy, Industry and Competitiveness (TEC2017-86779-C2-1-R).