Robust Tomlinson-Harashima Precoding for Two-Way Relaying

Robust T omli nson-Hara shima Preco ding for T w o-W a y Relaying Saeideh Mohammadkhani ∗ , Amir Hosein Jafa r i Ele ctric al Engine ering Dep artment, Ir an Universit y of Scienc e and T e chnolo gy, T ehr an, I r an George K. Ka r agiannidis, F ellow, IEEE Dep artment of Ele ctric al and Com puter Engine ering, A ristotle University of Thessaloniki, Thessaloniki 541 24, Gr e e c e Abstract Most of the non-linea r tr ansceivers, which are based on T omlinson Ha rashima (TH) preco ding and hav e b e en prop osed in the literature for t wo-w ay r elay net works, assume pe rfect c hannel state informatio n (CSI). In this paper, we prop ose a novel and ro bus t TH prec o ding s cheme for tw o-wa y relay netw orks with multiple antennas at the tr a nsceiver and the relay no des. W e assume im- per fect CSI and that the channel uncertain t y is bo unded by a s pherical region. F urthermore, w e consider the sum o f the mean square error as the o b jective function, under a limited power constraint for tr ansceiver and relay no des. Sim- ulations are provided to ev aluate the performa nce and to v alida te the efficiency of the prop osed scheme. Keywor ds: Imper fect channel state information, T omlinson Haras hima Preco ding, worst-case optimization. 1. In tro duction By using netw or k co ding, tw o-wa y relay netw o r ks hav e attracted a sig nif- icant attention, due to its adv antage in terms of spectra l efficiency [1]. On ∗ Corresp onding author Email addr esses: s_moha mmadkhani@ elec.iust.ac.ir (Saeideh M ohammadkhani), amirjafa ri@iust.ac .ir (Amir Hosein Jafar i), geokarag @auth.gr (G eorge K. Karagiannidis, F ellow, IEEE) the other hand, multiple − input − m ultiple − o utput (MIMO) technique enhances spatial diversit y , thr o ughput and reliability . The combination o f MIMO and t wo-w ay r elaying with precoding at b oth source and relay no des shows the ben- efits of them. In addition, non-linea r preco ding at the tr ansmitter, in the form o f TH with linea r r elay preco der and linea r minim um mean square er ror (MMSE ) receiver provides a b etter bit error- rate (BER) p erfor mance in compariso n to linear source preco der [2]. The per formance of a MIMO rela ying system dep ends on the av ailable c han- nel state infor mation (CSI). How e ver, in most practical cases, CSI is imperfect, due to quantization err or o r ina ccurate channel estimation, whic h is a result of insufficient training seq uences o r low signa l-to-noise ra tio (SNR), feedback error s, etc.. Therefor e, this impe rfectness must be explicitly considered in the estimated channel that is led to some robust designs, which a re less sensitive to the optimizatio n error s. In genera l, there ar e t wo t yp es of robust desig ns : sto chastic [3 ] and worst c ase [4]. In the worst-cas e, the channel error is co n- sidered to belo ng to a pre de fined uncertaint y region and the fina l goal is the optimization of the w orst sys tem perfor mance for each error in this reg ion. In the sto chastic approa ch, a sto chastic viewp oint is chosen to lo ok to the problem and the req uir ed r obustness is ac q uired f rom a pr o babilistic feature. Regarding the sto chastic appro ach adv antages, the worst-case design is necessa r y to take absolute robustness , i.e., guar anteed p erfor mance with probability o ne. TH preco ding is mor e sensitive on the channel estimation er r ors, compared to linea r pr eco ding tec hniques, due to its nonlinear nature. Specifically , in the presence of c hannel imp erfectnes s , the per formance o f TH prec o ding would be deteriorated critically [2]. In [5]-[7], robust linear preco ding w ere considered for one way netw ork . TH pr eco ding design in o ne w ay relay netw ork with p erfect CSI was prop osed in [8] and ro bust co nsideration were done in [9]-[11]. How ever, to our best kno wledge, the research on r obust TH pr eco ding desig n for tw o wa y MIMO net works is missing. In this pap er, we prop ose a TH preco ding scheme for tw o-way MIMO relay systems, where both source and r elay nodes a re equipp ed with m ultiple anten- 2 1 s 1 d 1 B 1 x 1 F r F 1 H 2 G 2 Γ 2 s 2 d 1 v ˆ 2 v 2 v 2 x 2 B 2 F 2 H 1 G 1 Γ 1 ˆ v Source 1 Source 2 Relay Figure 1: MIMO Two -W ay Relay with TH Pr ecoding. nas. F urthermo re, p erfect CSI of the source-r elay links and imp erfect CSI o f the relay-destination links are av ailable a t the r elay node. The aim is to minimize the sum o f mean squar e e r ror (MSE) a t each receiver no de, keeping the tr a nsmit power o f relay a nd source no des less than a thresho ld. Notations - The low er case and upper c ase b oldface letters indicate the v ec- tors and matrices, resp ectively . ( . ) H , ( . ) T , ( . ) − 1 , || . || , | . | , tr ( . ), E ( . ) and I N represent Hermitian, T ransp ose, in version, F rob enius norm, determinant, trace of a matrix, statistica l exp ecta tio n, and an identit y matrix of size N , re sp e c - tively . 2. System Mo del And Probl em Descriptio n W e consider a MIMO tw o w ay r elay sys tem including t wo m ultiple an tenna no des w ith N t antennas, which exchange their information with the help of o ne relay no de, equipp ed with N r antennas as shown in Fig. 1. The informa tio n exchange b etw een no des 1 a nd 2 is p erformed in tw o time s lots. In the fir st, no des 1 a nd 2 co nc ur rently fed their information, s i = [ s i, 1 , ...s i,N t ], int o the TH preco der. The re sulted v ector of signals at each transmitter no de is x i = C − 1 i v i , (1) where C i = B i + I N t is a low er left triangular matrix with unit diag onal elements and v i = s i + d i contains mo dified data symbols, wher e d i is such that the 3 real and imaginar y co mpo nent s of x i are constra ined to be within the reg ion ( − √ M , √ M ] which M is the num b er of co ns tellation po ints in the M-a ry QAM mo dulation scheme. In addition, the en tries of x i is co nsidered as E ( x i x H i ) = σ 2 x i I . After the nonlinear o per ation, the v ecto r x i is multiplied with an N t × N t preco der ma trix F i , i = 1 , 2 a nd for ward to relay . The received signals at the relay a nt ennas are Y r = H 1 F 1 x 1 + H 2 F 2 x 2 + n r , (2) where H i , i = 1 , 2 is the N r × N t channel ma trix b etw een the no de i and the relay no de and n r is the additiv e white complex Gaussian noise vector at relay with σ 2 n r . In the second time slot, the received signal in the relay is m ultiplied by a n N r × N r linear preco ding matrix F r and forward t o the receivers. x r = F r Y r = F r H 1 F 1 x 1 + F r H 2 F 2 x 2 + F r n r . (3) The received signal at the receivers ca n be wr itten as y i = G i F r H 1 F 1 x 1 + G i F r H 2 F 2 x 2 + G i F r n r + n i , (4) where G i , i = 1 , 2 is N t × N r channel m atrix b etw een r elay and i th receiver. The following assumptions are made abo ut the CSI: 1. The r eceiver no des hav e av ailable p erfect CSI o f the eq uiv a lent channels betw een transmitter-re lay-rece iver, G i F r H i F i , G i F r H j F j that i, j = 1 , 2 , i 6 = j . The equiv alent channel can b e es tima ted b y using the sent training sequence from transmitter and received in receiver after passing from relay . 2. The sour ce-relay channel, F i , is perfected estimated at the relay by using a training sequence. 3. The relay-receiver CSI is not perfect at the relay , due to limitation in the rate of feedback link from receiver to r elay o r due to feedback error. Based on ab ov e as sumptions, the self-interference can b e co mpletely remov ed. Therefore, ¯ Y i = G i F r H j F j x j + G i F r n r + n i , i, j = 1 , 2 , i 6 = j (5) 4 Due to its simplicity , a linear receiv e r is us e d at eac h r eceiver to retrieve the transmitted signa ls. Denoting Γ i as the N t × N t matrix at the i re c e iver, the estimation of the trans mitted signal v ector can b e expres sed as ˆ v i = Γ i G 1 F r H j F j x j + Γ i G i F r n r + Γ i n i , (6) where i, j = 1 , 2 , i 6 = j . Note that if v i can be estima ted at the destination, s i can be recov er ed by modulo ope r ation. In this pa pe r, we consider the minimization of the sum MSE of tw o receiv er no des in order to estimate v i sub ject to transmit p ow er constra int at the relay and tr ansmitter no de s . Optimiza tion is jointly done ov er TH prec o ding ma trices C i , F i , linear relay preco der F r and linear equalizer at the receiver Γ i . Thus the optimization problem can b e formulated as min Γ i , F i , F r , C i ,i =1 , 2 mse 1 + mse 2 s.t. P T ≤ P r,t , P 1 ≤ P 1 ,t , P 2 ≤ P 2 ,t , (7) where mse i is the MSE a t the i th receiver, P T , P i , i = 1 , 2 a re the tra nsmit power of re lay a nd i th tr ansmitter and P r,t , P i,t are the maximum power which can be used by the relay and i th tr a nsmitter. The MSE at the i th rece iver node can be wr itten as mse i = E ( k ˆ v i − C j x j k 2 ) = E ( k ( Γ i G i F r H j F j − C j ) x j k 2 )+ σ 2 n r ( k Γ i G i F r k 2 )+ σ 2 n i k Γ i k 2 , (8) where j = 2 if i = 1 and j = 1 if i = 2 . The transmit pow er of the relay no de is P T = σ 2 x 1 tr ( F r H 1 F 1 F H 1 H H 1 F H r ) + σ 2 x 2 tr ( F r H 2 F 2 F H 2 H H 2 F H r ) + σ 2 n r tr ( F r F H r ) (9) The transmit p ow er of i th transmitter no de can be deno ted as P i = σ 2 x i tr ( F i F H i ) , i = 1 , 2 . (10) W e ass ume tha t the infor mation for the channels be tw een r elay-receivers are not p erfect. Therefore , by considering the p o pula r metho ds for channel 5 estimation, we have G i = ˆ G i + ∆G i , (11) where ˆ G i is the e stimated channels and ∆ G i is the channel err or matrice that is bo unded by spherical, i.e. S g = { a ∈ C : || a || 2 ≤ σ 2 g i } , ∆G ∈ S g . (12) It should b e no ted that the actual error is unknown and only the upper b ound, ε 2 g is known. When the channel error e x ists, there are infinite go als and constraints for th e problem and it is unsolv a ble. In the res t of pap er, w e attempt to obtain a solution for (7) with CSI er rors. 3. Robust THP Design T o solve the optimization pro blem in (7), using a worst-case desig n, we co uld transform it to a simpler pro ble m as min Γ i , F i , F r , C i ,i =1 , 2 max ∆G i mse 1 + mse 2 s.t. P T ≤ P r,t , P 1 ≤ P 1 ,t , P 2 ≤ P 2 ,t . (13) It can see from (13 ), the optimization is done ov er the error of the channels. T o this end, the channel error is considered in sum MSE expressio n. max ∆G i mse i ≤ σ 2 x j k ( Γ i ˆ G i F r H j F j − C j k 2 + σ 2 x j σ 2 g i k Γ i k 2 k F r H j F j k 2 + σ 2 n r k Γ i ˆ G i F r k 2 + σ 2 n r σ 2 g i k Γ i k 2 k F r k 2 + σ 2 n i k Γ i k 2 . (14) By consider ing this fact that the p ow er cons traints ar e not rela ted to Γ i , we minimize the obtained sum mse over Γ i ∂ ∂ Γ ∗ i = 0 ⇒ σ 2 x j Γ i ˆ G i F r H j F j F H j H H j F H r ˆ G H i − σ 2 x j C j F H j H H j F H r ˆ G H i + σ 2 x j σ 2 g i Γ i k F r H j F j k 2 + σ 2 n r Γ i ˆ G i F r F H r ˆ G H i + σ 2 n r σ 2 g i Γ i k F r k 2 + σ 2 n i Γ i = 0 , (15) 6 and Γ i = σ 2 x j C j F H j H H j F H r ˆ G H i × ( σ 2 x j ˆ G i F r H j F j F H j H H j F H r ˆ G H i + σ 2 x j σ 2 g i k F r H j F j k 2 I + σ 2 n r ˆ G i F r F H r ˆ G H i + σ 2 n r σ 2 g i k F r k 2 I + σ 2 n i I ) − 1 . (16) By r eplacing the obtained Γ i in the (14), we o btain the following expression f or mse mse i = tr ( σ 2 x j C j ( I − σ 2 x j D H j E H i ( A i I + B i + E i D j D H j E H i ) − 1 × E i D j ) C H j ) , (17) where A i = σ 2 x j σ 2 g i || F r D j || 2 + σ 2 n r σ 2 g i || F r || 2 + σ 2 n i , E i = ˆ G i F r , D i = H i F i , B i = σ 2 n r E i E H i . (18) By applying the matrix inv ersion lemma ( A + B C D ) − 1 = A − 1 − A − 1 B ( D A − 1 B + C − 1 ) B A − 1 , we obtain mse i = tr ( σ 2 x j C j ( I + σ 2 x j D H j E H i ( A i I + B i ) − 1 E i D j ) − 1 C H j ) . (19) In the second step, the o ptimization must b e done ov e r F i , F r , C i . Since the problem (10) b y co nsidering (19) is nonconv ex, a glo bally optimal so lutio n of F i , F r , C i is difficult to o bta in with reasonable computational complexit y . W e develop an iterative algorithm. Before do ing optimization, using the relation betw een trace a nd determinant , the MSE e xpression is changed. Indeed, a low er b ound for MSE is co ns idered as mse i = | I + σ 2 x j D H j E H i ( A i I + B i ) − 1 E i D j | − 1 N . (20) In this r elation, we use tr ( X ) ≥ N | X | 1 N with a N × N p ositive semidefinite matrix, X . If the X is diagonal and ha ve equal diag onal elemen ts, this relation is establis hed for equa lit y . Here, we use this fact | C i C H i | = 1 and | XY | = | YX | . In a ddition, since minimizing | X | ( − 1) is equiv alent to ma ximizing | X | , ther efore, 7 the equiv alent equation can be expr essed as max F 1 , F 2 , F r | I + σ 2 x 2 D H 2 E H 1 ( A 1 I + B 1 ) − 1 E 1 D 2 | + | I + σ 2 x 1 D H 1 E H 2 ( A 2 I + B 2 ) − 1 E 2 D 1 | s.t. σ 2 x 1 tr ( F r D 1 D H 1 F H r ) + σ 2 x 2 tr ( F r D 2 D H 2 F H r ) + σ 2 n r tr ( F r F H r ) ≤ P r,t σ 2 x i tr ( F i F H i ) ≤ P i,t , i = 1 , 2 . (21) In or der to solve the eq uiv alent master problem, w e consider the following sin- gular v alue decomp osition (SVD) H = [ H 1 , H 2 ] = U h Λ (1 / 2) h V H h , H 1 = U h Λ (1 / 2) h V H h, 1 , H 2 = U h Λ (1 / 2) h V H h, 2 , V H h = [ V H h, 1 , V H h, 2 ] , (22) and ˆ G = [ ˆ G T 1 , ˆ G T 2 ] T = U g Λ (1 / 2) g V H g , ˆ G 1 = U g, 1 Λ (1 / 2) g V H g , ˆ G 2 = U g, 2 Λ (1 / 2) g V H g , U g = [ U T g, 1 , U T g, 2 ] T , (23) where the dimension of U h , Λ h , V h are N r × N r , N r × N r ,2 N t × N r , r esp ec- tively a nd the dimensio n o f U g , Λ g , V g are 2 N t × N r , N r × N r , N r × N r , re- sp ectively . In a ddition, SVD of F i and F r are given b y F i = X i Λ (1 / 2) i Z i and F r = X r Λ (1 / 2) r Z r . Pr op osition 1: By using equiv alent decompo sition, the mse relation in (23) can be maximized such that X r = V g , Z r = U h , X 1 = V h, 1 , X 2 = V h, 2 . (24) Pr o of : Deno te T and R a re Hermitian and p ositive definite. Then, function | I N + T − 1 R | , is max imized when T and R commute and ha ve eigenv alues in 8 opp osite order. Two matr ices T and R are c ommut e whe n TR = R T . By using this Lemma, the mse is maximized when pr e c o der and relay matr ices hav e diagonal structure and follow prop osed design. By re pla cing so urces preco de r and re lay preco der structur e in mse i , P T and P i , w e o btain relations in (25). Even after the ab ov e transfo r mation, the optimization problem is nonconv ex over optimization co efficients. W e apply a conv ex optimization metho d to optimize the functions with r esp ect to each v ariable and intro duce an alterna ting optimization algorithm to s o lve them. Therefore, we div ide obtained problem in (25) into three sub- problem and apply the prop osed algor ithm for eac h subproblem. max Λ r , Λ 1 , Λ 2 2 X i,j =1 ,i 6 = j | I + σ 2 x i Λ i Λ h Λ g Λ r (( σ 2 x i σ 2 g j k Λ (1 / 2) r Λ (1 / 2) h Λ (1 / 2) i k 2 + σ 2 n r σ 2 g j k Λ (1 / 2) r k 2 + σ 2 n j ) I + σ 2 n j Λ r Λ g ) − 1 | s.t. σ 2 x 1 tr ( Λ r Λ 1 Λ h ) + σ 2 x 2 tr ( Λ r Λ 2 Λ h ) + σ 2 n r tr ( Λ r ) ≤ P r,t σ 2 x i tr ( Λ i ) ≤ P i,t , i = 1 , 2 (25) Firstly , by introducing a ux iliary v ar iables t k and t ′ k , (25) is transformed to relation in (26). F or each subproblem, we in tr o duce slack v ariables β k , β ′ k as a upper b ound for the denominator of relation in (26) and define f ( t k , β k ) = β k ( t k − 1) and f ( t ′ k , β ′ k ) = β ′ k ( t ′ k − 1). T o deal with nonconv ex co nstraints f ( t k , β k ) and f ( t ′ k , β ′ k ), we replace them by its conv ex upp er b ound and iter- atively solve the resulting pr oblem by judiciously updating the v ariables un til conv er g ence. T o this end, for a g iven φ k for all k , we define G ( t k , β k , φ k ) , φ k 2 β 2 k + 1 2 φ k ( t k − 1) 2 which obta in b y co nsidering the inequality of a rithmetic and geometric means of φ k β 2 k and φ − 1 k ( t k − 1) 2 and φ k = t k − 1 β k . This pro cedure 9 is also applied for f ( t ′ k , β ′ k ). max x,t k ,t ′ k N r Y k =1 t k + N r Y k =1 t ′ k s.t. σ 2 x 1 λ 1 ,k λ h,k λ g,k λ r,k ( σ 2 x 1 σ 2 g 2 k Λ (1 / 2) r Λ (1 / 2) h Λ (1 / 2) 1 k 2 + σ 2 n r σ 2 g 2 k Λ (1 / 2) r k 2 + σ 2 n 2 ) + σ 2 n 2 λ r,k λ g,k ≥ t k − 1 σ 2 x 2 λ 2 ,k λ h,k λ g,k λ r,k ( σ 2 x 2 σ 2 g 1 k Λ (1 / 2) r Λ (1 / 2) h Λ (1 / 2) 2 k 2 + σ 2 n r σ 2 g 1 k Λ (1 / 2) r k 2 + σ 2 n 1 ) + σ 2 n 1 λ r,k λ g,k ≥ t ′ k − 1 σ 2 x 1 tr ( Λ r Λ 1 Λ h ) + σ 2 x 2 tr ( Λ r Λ 2 Λ h ) + σ 2 n r tr ( Λ r ) ≤ P r,t σ 2 x i tr ( Λ i ) ≤ P i,t , i = 1 , 2 (26) By a pplying the mentioned pro cedur e, it is seen that (26 ) ca n b e transformed to second order cone pro gramming (SOCP ) over ea ch v ar ia ble. The SOCP representation of (26) is shown in (27). T he main ingredient in arriving a t the SOCP repr e sentation is the fact that hyperb olic c onstraint u v ≥ z 2 is equiv alent 10 to || [2 z ( u − v )] T || ≤ ( u + v ). max x,t k ,t ′ k τ + τ ′ s.t. k [2 v 1 ,j 1 t 2 j 1 − 1 − t 2 j 1 ] T k ≤ t 2 j 1 − 1 + t 2 j 1 , j 1 = 1 , 2 , ..., 2 q − 1 k [2 v m,j 1 v m − 1 , 2 j m − 1 − v m − 1 , 2 j m ] T k ≤ v m − 1 , 2 j m − 1 + v m − 1 , 2 j m , m = 2 , ..., q , j m = 1 , ..., 2 q − m k [2 τ v q − 1 , 1 − v q − 2 , 2 ] T k ≤ v q − 1 , 1 + v q − 2 , 2 k [2 v ′ 1 ,j 1 t ′ 2 j 1 − 1 − t ′ 2 j 1 ] T k ≤ t ′ 2 j 1 − 1 + t ′ 2 j 1 , j 1 = 1 , 2 , ..., 2 q − 1 k [2 v ′ m,j 1 v ′ m − 1 , 2 j m − 1 − v ′ m − 1 , 2 j m ] T k ≤ v ′ m − 1 , 2 j m − 1 + v ′ m − 1 , 2 j m , m = 2 , ..., q , j m = 1 , ..., 2 q − m k [2 τ ′ v ′ q − 1 , 1 − v ′ q − 2 , 2 ] T k ≤ v ′ q − 1 , 1 + v ′ q − 2 , 2 σ 2 x 1 λ 1 ,k λ h,k λ g,k λ r,k ≥ φ k 2 β 2 k + 1 2 φ k ( t k − 1) 2 ( σ 2 x 1 σ 2 g 2 k Λ (1 / 2) r Λ (1 / 2) h Λ (1 / 2) 1 k 2 + σ 2 n r σ 2 g 2 k Λ (1 / 2) r k 2 + σ 2 n 2 ) + σ 2 n 2 λ r,k λ g,k ≤ β k σ 2 x 2 λ 2 ,k λ h,k λ g,k λ r,k ≥ φ k 2 β ′ 2 k + 1 2 φ k ( t ′ k − 1) 2 ( σ 2 x 2 σ 2 g 1 k Λ (1 / 2) r Λ (1 / 2) h Λ (1 / 2) 2 k 2 + σ 2 n r σ 2 g 1 k Λ (1 / 2) r k 2 + σ 2 n 1 ) + σ 2 n 1 λ r,k λ g,k ≤ β ′ k σ 2 x 1 tr ( Λ r Λ 1 Λ h ) + σ 2 x 2 tr ( Λ r Λ 2 Λ h ) + σ 2 n r tr ( Λ r ) ≤ P r,t σ 2 x i tr ( Λ i ) ≤ P i,t , i = 1 , 2 (27) After convergence iterations and r eplacing obtained matrices F i and F r in optimization problem, we minimize MSE function ov er C i . The optimum C i can be obtained b y using prop os ed approach in [12]. 4. Simulations and Discussion In this section, w e present the computer simulation results of our prop os ed robust non-linear THP transceiver des ig n. W e sim ulate a MIMO tw o-way relay system with N r = N t = 4. The channel matrices ar e mo deled by copmlex Gaussina ra ndom v ariables zer o mean and unit v ariance. Noise v a riances at the relay and a t the r eceivers are a lso assumed similar a nd equal to σ 2 n r = σ 2 n 1 = σ 2 n 2 = 0 . 1. All simulation results were av era g ed over 1 0 00 indep endent realizations of the fading channels. 11 1 2 3 4 5 6 7 8 9 10 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 Iterations Sum of MSE P r,t =5dB P r,t =10dB P r,t =15dB Figure 2: Sum of MSE v ersus the n umber of iterations for differen t v alues of P r,t , where σ 2 g 1 = σ 2 g 2 = σ 2 g = 0 . 01. Fig. 2 depicts the convergence behavior of the pro po sed optimization algo- rithm and its required num b er of iteratio ns for different p ow er co nstraint on the transmitters. This figure co nfirms tha t algorithm converge after a few it- erations. Fig. 3 displays the effect o f c hannels uncertaint y . Two err or b ounds σ 2 g = 0 . 01 , 0 . 05 are considered. The idea l ca se with p erfect CSI, i.e. σ 2 g = 0, is also co nsidered. When σ 2 g is incr eased, the uncer taint y in channel co efficients grows. There fore, the MSE is increased with increasing channel uncertaint y . 5. Conclusion This pap er studied the robust TH preco ding for tw o relay netw o rk. It is assumed that the CSI is imperfect. W e aim to minimize the maximum of the sum of MSE s ub ject to transmit power of relay and transmitters is low er than a pr edefined threshold. The spherical mo del is used to characterize uncer taint y of the channels. 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