CN108989249B - Large-scale MIMO beam domain channel tracking method in high-speed rail scene - Google Patents

Large-scale MIMO beam domain channel tracking method in high-speed rail scene Download PDF

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CN108989249B
CN108989249B CN201810670443.XA CN201810670443A CN108989249B CN 108989249 B CN108989249 B CN 108989249B CN 201810670443 A CN201810670443 A CN 201810670443A CN 108989249 B CN108989249 B CN 108989249B
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beam domain
speed rail
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channel tracking
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CN108989249A (en
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徐友云
周乔
李大鹏
陈建平
王云峰
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Nanjing Ticom Tech Co ltd
Nanjing University of Posts and Telecommunications
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    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/0202—Channel estimation
    • H04L25/0204—Channel estimation of multiple channels
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    • H04B7/0413—MIMO systems
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
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    • H04L25/0228—Channel estimation using sounding signals with direct estimation from sounding signals
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Abstract

The invention provides a large-scale MIMO beam domain channel tracking method under a high-speed rail scene, which is characterized in that beam domain channels are respectively established according to the standard of LTE-R aiming at the large-scale MIMO high-speed rail scene
Figure DDA0001708031020000011
A transfer equation of the beam domain channel autocorrelation R and a state equation between the received signal y and the transmitted signal s; then, according to the obtained state equation and transfer equation, performing wave beam domain channel tracking by using Kalman filtering; using the tracking result, the error nrmse is tracked by the normalized channelnAnd performing performance evaluation. The method greatly reduces the pilot frequency overhead and simultaneously reduces the complexity of a channel tracking algorithm, thereby improving the channel tracking performance of the whole high-speed rail large-scale MIMO system.

Description

Large-scale MIMO beam domain channel tracking method in high-speed rail scene
Technical Field
The invention belongs to the technical field of wireless communication, and particularly relates to a large-scale MIMO beam domain channel tracking method in a high-speed rail scene.
Background
In recent years, with the great popularization of high-speed railways, wireless communication technology in high-mobility scenes is gradually becoming a research hotspot. With the advent of the 5G (fine Generation communication) era, when the speed of a train reaches or exceeds 350km/h, a 5G communication system is required to be capable of providing good service for users therein, mainly in terms of high transmission rate and low delay. At present, the most popular high-speed rail Mobile communication System is GSM-R (Global System for Mobile Communications-Railway), the supported maximum transmission rate is lower than 200Kbps, and the 5G requirement is that in a high-speed Mobile scene, the data transmission rate can reach 150Mbps or even higher, and obviously, the traditional GSM-R cannot meet the requirement of future high-speed rail communication. Large-scale MIMO (Multiple-input Multiple-output) is one of the key technologies of 5G, and can fully utilize the dimensional resources of the wireless space, thereby greatly improving the throughput of the future wireless communication system. Therefore, the combination of massive MIMO and high-speed rail communication is one of effective means for improving the data transmission rate and the frequency spectrum utilization rate of the high-speed rail wireless mobile communication system. For a TDD large-scale MIMO high-speed rail communication system, the pilot frequency overhead in the training process can be increased linearly along with the increase of the total number of user antennas; for an FDD large-scale MIMO high-speed rail communication system, the feedback overhead in the training process is in direct proportion to the number of antennas at the base station side. Therefore, ensuring a certain channel tracking accuracy while reducing the pilot overhead is a difficult problem in a massive MIMO high-speed rail communication system.
Through the search of the prior art documents, Zhang Cheng et al published a sentence entitled "Location-aid Channel Tracking and kalman Transmission for HST Massive MIMO Systems (for Location-based Channel Tracking and Downlink Transmission of high-speed rail Massive MIMO Systems)" in IET Communications, oct.2017,11(13), pp.2082-2088 (journal of the british institute of engineering and technology, 10 months 2017, volume 11, 13, page 2082), which introduces a time correlation model into a Massive MIMO spatial Channel model and performs time-domain Channel Tracking by using filtering. The method has the disadvantages that when the time domain channel tracking is carried out, the pilot frequency overhead is large, and the complexity of a channel tracking algorithm is large.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a large-scale MIMO beam domain channel tracking method in a high-speed rail scene.
The purpose of the invention is realized by the following technical scheme, and the communication system is provided based on the communication system, the communication system belongs to two widely adopted high-speed rail communication systems, a base station and the top of a train are assumed to be deployed in a uniform linear array, M (M > >1) antennas are configured on the side of the base station, N (M > N, N > >1) terminals are configured on the top of the train, and each terminal has 1 antenna.
The specific communication process is as follows: the train top antenna array is responsible for receiving signals from the base station, and users communicate with the train top antenna array through WiFi AP deployed in the train. The invention only considers the layer of base station and train top antenna array communication.
Considering the downlink, since there are M antennas on the base station side, M beams can be formed, 1 beam is allocated to each terminal, and N beams are allocated in total, then for the ith (1 ≦ i ≦ N) terminal, the massive MIMO channel frequency response matrix is:
Figure BDA0001708030000000021
wherein a isp,iRepresents the attenuation coefficient of the p-th propagation path, and ap,i～CN(0,1)，dp,iDenotes the distance between the transmitting antenna m and the receiving antenna n along the path p, λ is the carrier wavelength, dp,iU (0, 1)/λ, l is the index of the subcarrier, τp,iThe time delay of the p-th transmission path is shown, and j is a complex number field.
When the number of base station antennas is enough and the antenna spacing is half wavelength, V is equivalent to a discrete Fourier transform matrix, i.e.
Figure BDA0001708030000000022
And ViIs the ith column of V, indicates the beam basis vector assigned to the ith terminal, and each terminal is assigned ViAre different from each other.
Figure BDA0001708030000000023
Representing beam domain informationThe lane matrix, in particular form as follows,
Figure BDA0001708030000000024
wherein St,iIndicating the number of paths in the coverage of the ith beam.
The large-scale MIMO wave beam domain channel tracking method based on the high-speed rail scene comprises the following steps:
s1, respectively establishing beam domain channels according to the standard of LTE-R for large-scale MIMO high-speed rail scenes
Figure BDA0001708030000000031
A transfer equation of the beam domain channel autocorrelation R and a state equation between the received signal y and the transmitted signal s;
specifically, a transfer equation and a state equation required by kalman filtering can be obtained by a method that a time-dependent model is considered to be introduced into channel modeling after a large-scale MIMO high-speed rail space channel model is constructed in a beam domain. According to the standard of LTE-R, the angle of the beam is hardly changed in the adjacent ct (channel Coherence time), i.e. the path aod (angle of department) in the beam coverage is basically unchanged. Thus, for the ith beam, there is the following relationship:
Figure BDA0001708030000000032
ψp,i,n+1＝ψp,i,n
wherein psip,i,nIs AOD, n denotes the subscript of CT, ρ is the time-dependent coefficient, and ρ ═ J0(2πfDT), wherein J0(. represents a zeroth order Bessel function of the first kind, fDIs the doppler shift and T is the channel coherence time. bp,i,n+1Represents a random variation independent of the attenuation coefficient, and bp,i,n+1～CN(0,1)。
Assuming that the ith beam covers p paths (containing one main path and a few scatter paths), in the nth CT, the beam domain channel matrix is rewritten:
Figure BDA0001708030000000033
let beta bep,iDoes not change with time, wherein
Figure BDA0001708030000000034
Then in the (n + 1) th CT, the beam domain channel matrix is:
Figure BDA0001708030000000035
further, the method can be obtained as follows:
Figure BDA0001708030000000041
wherein
Figure BDA0001708030000000042
(
Figure BDA0001708030000000043
Indicating that the left and right sides obey the same probability distribution), then
Figure BDA0001708030000000044
S2, tracking the wave beam domain channel by Kalman filtering according to the state equation and the transfer equation obtained in the step S1;
s3, using the result of S2, tracking error nrmse through normalized channelnAnd performing performance evaluation.
Preferably, the transfer equation in S1 is:
Figure BDA0001708030000000045
Rp,i,n+1＝ρ2Rp,i,n+(1-ρ2)Rp,i,n+1
the state equation is:
Figure BDA0001708030000000046
wherein the content of the first and second substances,
Figure BDA0001708030000000047
a channel representing the p path of the ith beam within the nth CT; ρ is a time correlation coefficient, and ρ is J0(2πfDT), wherein J0(. represents a zeroth order Bessel function of the first kind, fDIs the doppler shift, T is the channel coherence time; gamma rayp,i,n+1Is a transfer equation parameter; rp,i,nDenotes the channel autocorrelation, s, of the p-th path of the ith beam in the n-th CTp,i,nTime-domain pilot signals representing the nth CT, i.e.
Figure BDA0001708030000000048
wi,nCN (0,1) represents the noise of the nth CT,
Figure BDA0001708030000000049
the beam-domain pilot signal representing the nth CT, i.e.
Figure BDA00017080300000000410
Figure BDA00017080300000000411
Wherein ViRepresenting the beam basis vector, y, assigned to the ith terminal antennap,i,nIndicating the received signal of the ith terminal at the nth CT.
Preferably, the beam domain channel tracking is divided into a prediction stage and an adjustment stage, the prediction stage does not require a pilot, and the adjustment stage requires a pilot. If the CSI is predicted only through Kalman filtering, the channel tracking performance is worse and worse along with the continuous accumulation of channel tracking errors; if the channel tracking amount is continuously adjusted, since the high-speed rail channel is time-varying rapidly, frequent adjustment will bring large pilot overhead. Therefore, a reasonable solution is to compromise the prediction time and the adjustment time, so that good channel tracking performance can be achieved while reducing the pilot overhead.
If the prediction phase is from tpA CT unit consisting of a regulation phase tdOne time of Kalman filtering is t ═ tp+tdAnd (3) CT.
Preferably, in the S2, when n is 0,1, L t-1 for the subscript n indicating CT in one kalman filter<＝tpWhen the reaction temperature is 1, adding a catalyst,
Figure BDA0001708030000000051
ERp,n+1|n+1＝ERp,n+1|n＝ρ2ERp,n|n+(1-ρ2)Rp,n+1
when n is>tpAt-1, then:
Figure BDA0001708030000000052
ERp,n+1|n＝ρ2ERp,n|n+(1-ρ2)Rp,n+1
Figure BDA0001708030000000053
Figure BDA0001708030000000054
Figure BDA0001708030000000055
where superscript B denotes the beam domain channel, ERp,nDenotes the channel tracking error of the p-th path of the i-th beam in the n-th CT, Kp,nRepresenting a kalman filter factor.
Preferably, the S3 specifically includes the following steps:
defining normalized channel tracking error
Figure BDA0001708030000000061
Wherein the beam basis vector ViIs such that
Figure BDA0001708030000000062
And minimum.
Preferably, the state equation and the transfer equation are established based on a communication system of a beam domain channel matrix, and the beam domain channel matrix is expressed as:
Figure BDA0001708030000000063
wherein a isp,iRepresents the attenuation coefficient of the p-th propagation path, and ap,i～CN(0,1)，dp,iDenotes the distance between the transmitting antenna m and the receiving antenna n along the path p, λ is the carrier wavelength, dp,iU (0, 1)/λ, l is the index of the subcarrier, τp,iRepresenting the time delay of the p-th transmission path, St,iIndicating the number of paths in the coverage of the ith beam.
The invention has the beneficial effects that: large-scale MIMO high-speed rail channel tracking is carried out from a beam domain, compared with a time domain pilot signal, the time domain pilot signal is an MX 1 column vector, the beam domain pilot signal is one-dimensional, and the invention reduces the pilot overhead from M to 1; in addition, compared with the time domain, the invention changes the complexity of the channel tracking algorithm from O (M)3) Reduced to o (m), the resulting channel tracking error is reduced during the adjustment phase compared to the time-domain channel tracking. The pilot frequency overhead is greatly reduced, and the complexity of a channel tracking algorithm is reduced, so that the channel tracking performance of the whole high-speed rail large-scale MIMO system is improved.
Drawings
FIG. 1: the system model established in the invention is a schematic diagram.
FIG. 2: the algorithm in the invention implements a flow chart.
FIG. 3: and comparing simulation results of channel tracking errors under multiple terminals.
Detailed Description
The invention discloses a large-scale MIMO wave beam domain channel tracking method in a high-speed rail scene. Referring to fig. 1, the communication system and the channel model thereof according to this embodiment are:
a widely adopted two-hop high-speed rail communication system is established, and a uniform linear array is assumed to be arranged at a base station and the top of a train, 64 antennas are arranged at the side of the base station, 41 terminals are arranged at the top of the train, and each terminal is provided with one antenna. The specific communication process is as follows: the train top antenna array is responsible for receiving signals from the base station, and users communicate with the train top antenna array through WiFi AP deployed in the train. The invention only considers the layer of base station and train top antenna array communication.
Considering the downlink, since there are 64 antennas on the base station side, 64 beams can be formed, each terminal is allocated with one beam, and 41 beams are allocated in total, so that for the ith (1 ≦ i ≦ 41) terminal, the massive MIMO channel frequency response matrix is:
Figure BDA0001708030000000071
wherein a isp,iRepresents the attenuation coefficient of the p-th propagation path, and ap,i～CN(0,1)，dp,iDenotes the distance between the transmitting antenna m and the receiving antenna n along the path p, λ is the carrier wavelength, dp,iU (0, 1)/λ, l is the index of the subcarrier, τp,iIndicating the time delay of the p-th transmission path.
When the number of base station antennas is enough and the antenna spacing is half wavelength, V is equivalent to a discrete Fourier transform matrix, i.e.
Figure BDA0001708030000000072
And ViIs the ith column of V, indicates the beam basis vector assigned to the ith terminal, and each terminal is assigned ViAre different from each other.
Figure BDA0001708030000000073
Which represents the beam domain channel matrix, is embodied in the following form,
Figure BDA0001708030000000074
wherein St,iIndicating the number of paths in the coverage of the ith beam.
For the above communication system, the present invention provides a beam domain channel tracking method, which is shown in fig. 2 and includes the following steps:
the method comprises the steps of firstly, aiming at a large-scale MIMO high-speed rail scene, establishing a transfer equation and a state equation under a beam domain channel according to the standard of LTE-R.
Specifically, after a massive MIMO high-speed rail space channel model is constructed in a beam domain, a time correlation model is considered to be introduced into channel modeling. According to the standard of LTE-R, the angle of the beam obtained by calculation hardly changes in the adjacent ct (channel Coherence time), i.e. the path aod (angle of future) in the beam coverage is substantially unchanged. Thus, for the ith beam, there is the following relationship:
Figure BDA0001708030000000075
ψp,i,n+1＝ψp,i,n
wherein psip,i,nIs AOD, n denotes the subscript of CT, ρ is the time-dependent coefficient, and ρ ═ J0(2πfDT), wherein J0(. represents a zeroth order Bessel function of the first kind, fDIs the doppler shift and T is the channel coherence time. bp,i,n+1Represents a random variation independent of the attenuation coefficient, and bp,i,n+1～CN(0,1)。
Assuming that the ith beam covers P paths (including one main path and a few scatter paths, P is 3 in this embodiment), in the nth CT, the beam domain channel matrix is rewritten:
Figure BDA0001708030000000081
let beta bep,iDoes not change with time, wherein
Figure BDA0001708030000000082
Then in the (n + 1) th CT, the beam domain channel matrix is:
Figure BDA0001708030000000083
further, the method can be obtained as follows:
Figure BDA0001708030000000084
wherein
Figure BDA0001708030000000085
(
Figure BDA0001708030000000086
Indicating that the left and right sides obey the same probability distribution), then
Figure BDA0001708030000000087
In conclusion, the kalman filtering condition can be obtained:
transfer equation:
Figure BDA0001708030000000088
Rp,i,n+1＝ρ2Rp,i,n+(1-ρ2)Rp,i,n+1
the state equation is as follows:
Figure BDA0001708030000000091
Figure BDA0001708030000000092
representing the time-domain pilot signal, wi,nCN (0,1) represents noise;
Figure BDA0001708030000000093
representing beam-domain pilot signals, i.e.
Figure BDA0001708030000000094
Figure BDA0001708030000000095
And secondly, tracking the beam domain channel by using Kalman filtering.
And obtaining a transfer equation and a state equation required by Kalman filtering from the second step, and then tracking the beam domain channel by utilizing the Kalman filtering. The Kalman filtering comprises two stages of prediction and adjustment, pilot frequency is not needed in the prediction stage, and the pilot frequency is needed in the adjustment stage. If the CSI is predicted only through Kalman filtering, the channel tracking performance is worse and worse along with the continuous accumulation of channel tracking errors; if the channel tracking amount is continuously adjusted, since the high-speed rail channel is time-varying rapidly, frequent adjustment will bring large pilot overhead. Therefore, a reasonable scheme is to compromise the prediction time and the adjustment time, so that good channel tracking performance can be obtained while pilot overhead is reduced.
Assuming that the prediction phase consists of 5 CTs and the adjustment phase consists of 5 CTs, one kalman filter consists of 10 CTs. The whole beam domain channel tracking process is as follows:
for n ═ 0,1, L9, when n < ═ 4, then there are:
Figure BDA0001708030000000096
ERp,n+1|n+1＝ERp,n+1|n＝ρ2ERp,n|n+(1-ρ2)Rp,n+1
when n >4, then there are:
Figure BDA0001708030000000097
ERp,n+1|n＝ρ2ERp,n|n+(1-ρ2)Rp,n+1
Figure BDA0001708030000000098
Figure BDA0001708030000000101
Figure BDA0001708030000000102
where the superscript B denotes the beam domain channel, Rp,nRepresenting the beam-domain channel autocorrelation matrix, ERp,nIndicating beam domain channel tracking error, Kp,nRepresenting a kalman filter factor.
And thirdly, solving the normalized channel tracking error for performance evaluation.
Defining normalized channel tracking error
Figure BDA0001708030000000103
Additional beam basis vector ViIs such that
Figure BDA0001708030000000104
And minimum.
Compared with the prior art, the method performs large-scale MIMO high-speed rail channel tracking from the beam domain, compared with the time domain pilot signal which is a column vector of 64 multiplied by 1, the beam domain pilot signal is one-dimensional, and the method reduces the pilot overhead from 64 to 1; in addition, compared with the time domain, the invention changes the complexity of the channel tracking algorithm from O (M)3) The channel tracking error is reduced to o (m), and the channel tracking error is reduced in comparison with the time domain channel tracking during the adjustment stage.
In order to verify the effectiveness of the beam-space Channel Tracking algorithm proposed in this embodiment, a Channel Tracking algorithm entitled "Location-aid Channel Tracking and Downlink Transmission for HST Massive MIMO Systems" (Location-based Channel Tracking and Downlink Transmission for high-speed rail Massive MIMO Systems) "published by Zhang Cheng et al in IET Communications, oct.2017,11(13), pp.2082-2088 (journal of the british engineering and technical society of engineering, 2017, 10 th volume 11, 13 th volume, 2082 nd page 2088) is used as a comparison algorithm.
The simulation experiment platform of this embodiment is Matlab, and monte carlo simulation is used to independently perform 1000 experiments to obtain nrmse. The initial AOD distribution of 41 LOS (line of sight) paths of the terminals is as follows: 60 DEG: 3 DEG: 60 DEG, the AOD of the other two scattering paths in the beam coverage are farther from the original AOD1 deg. The antenna pitch wavelength ratio d/λ is 0.5, and T is defined as 0.07/fDThe time correlation coefficient ρ is 0.9522, which ensures that the CSI is approximately constant within a CT, and the rest of the simulation parameters are the same as the reference.
As can be seen from fig. 3, except for the prediction-only algorithm, the channel tracking errors of the other two channel tracking algorithms are accumulated in the first 5 CTs and eliminated in the last 5 CTs. The channel tracking error of the algorithm herein is slightly less than the reference algorithm during the adjustment phase and slightly higher than the reference algorithm during the prediction phase, and both are superior to the prediction-only algorithm. In addition, the algorithm has the biggest characteristic that the pilot frequency cost is far smaller than that of a reference algorithm, and because the adjusting stage is more complicated than the predicting stage, the complexity of the two algorithms in the adjusting stage is compared, as shown in the following table, as can be seen in the following table: the complexity of the algorithm herein is much less than the reference algorithm.
Figure BDA0001708030000000111
Finally, it should be noted that: the above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.

Claims (5)

1. A large-scale MIMO wave beam domain channel tracking method under a high-speed rail scene is characterized in that: the method comprises the following steps:
s1, respectively establishing beam domain channels according to the standard of LTE-R for large-scale MIMO high-speed rail scenes
Figure FDA0002893728170000015
A transfer equation of the beam domain channel autocorrelation R and a state equation between the received signal y and the transmitted signal s;
s2, tracking the wave beam domain channel by Kalman filtering according to the state equation and the transfer equation obtained in the step S1;
s3, using the result of S2, tracking error nrmse through normalized channelnPerforming performance evaluation;
the transfer equation in S1 is:
Figure FDA0002893728170000011
Rp,i,n+1＝ρ2Rp,i,n+(1-ρ2)Rp,i,n+1
the state equation is:
Figure FDA0002893728170000012
wherein the content of the first and second substances,
Figure FDA0002893728170000013
a channel representing the p path of the ith beam within the nth CT; ρ is a time correlation coefficient, and ρ is J0(2πfDT), wherein J0(. represents a zeroth order Bessel function of the first kind, fDIs the doppler shift, T is the channel coherence time; gamma rayp,i,n+1Is a transfer equation parameter; rp,i,nDenotes the channel autocorrelation, s, of the p-th path of the ith beam in the n-th CTp,i,nTime-domain pilot signal, w, representing the nth CTi,nCN (0,1) represents the noise of the nth CT,
Figure FDA0002893728170000014
beam domain pilot signal representing the nth CT, where ViRepresenting the beam basis vector, y, assigned to the ith terminal antennap,i,nIndicating the received signal of the ith terminal at the nth CT.
2. The method of claim 1, wherein the large-scale MIMO beam domain channel tracking method in a high-speed rail scene comprises: the beam domain channel tracking is divided into a prediction stage and an adjustment stage, if the prediction stage is from tpA CT unit consisting of a regulation phase tdOne time of Kalman filtering is t ═ tp+tdAnd (3) CT.
3. The method of claim 1, wherein the large-scale MIMO beam domain channel tracking method in a high-speed rail scene comprises:
the subscript n, n is 0,1, L t-1 for CT in the one-time kalman filtering in S2, when n is<＝tpAt-1, then:
Figure FDA0002893728170000021
ERp,n+1|n+1＝ERp,n+1|n＝ρ2ERp,n|n+(1-ρ2)Rp,n+1
when n is>tpAt-1, then:
Figure FDA0002893728170000022
ERp,n+1|n＝ρ2ERp,n|n+(1-ρ2)Rp,n+1
Figure FDA0002893728170000023
Figure FDA0002893728170000024
Figure FDA0002893728170000025
where superscript B denotes the beam domain channel, ERp,nDenotes the channel tracking error of the p-th path of the i-th beam in the n-th CT, Kp,nRepresenting a kalman filter factor.
4. The method of claim 1, wherein the large-scale MIMO beam domain channel tracking method in a high-speed rail scene comprises: the S3 specifically includes the following steps:
1. defining normalized channel tracking error
Figure FDA0002893728170000026
Wherein the beam basis vector ViIs such that
Figure FDA0002893728170000027
Minimum, where P is the total propagation path number, ER is the estimation error covariance, and R is the channel covariance.
5. The method of claim 1, wherein the large-scale MIMO beam domain channel tracking method in a high-speed rail scene comprises: the state equation and transfer equation are established based on a beam domain channel matrix, the waveThe beam domain channel matrix is represented as:
Figure FDA0002893728170000028
wherein a isp,iRepresents the attenuation coefficient of the p-th propagation path, and ap,i～CN(0,1)，dp,iDenotes the distance between the transmitting antenna m and the receiving antenna n along the path p, λ is the carrier wavelength, dp,iU (0, 1)/λ, l is the index of the subcarrier, τp,iRepresenting the time delay of the p-th transmission path, St,iIndicating the number of paths in the coverage of the ith beam.
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