Designing far too much time overhead. FDD

Designing of an
Optimized Pilot Beam Pattern For MU-MIMO Systems


Abstract—In   this   project,  channel 
for  massive multiple-input multiple-output (MIMO) systems with a large number of transmit antennas at
 the  base  station
 is considered, and
 a new algorithm for pilot  beam  pattern design
 for optimal channel  estimation  under the assumption of Gauss-Markov channel  processes is proposed. The proposed algorithm
designs the optimal pilot beam pattern sequentially by exploiting the statistics of the channel, antenna correlation, and temporal correlation. The algorithm provides a sequentially
optimal sequence of pilot beam patterns for a given set of system
parameters. Numerical
results show the effectiveness
of the proposed algorithm.

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MIMO systems which
are using very large number of transmitting antennas and receiving antennas at
the base station are called massive MIMO systems. This is an active research
area in achieving high spectral efficiency 2-5. By using simple signal
processing with the number of transmit antennas Massive MIMO systems can provide
performance scaling 2. In practice, such benefits can be limited by channel
estimation accuracy 6.
Perfect channel
estimation can be infeasible
for massive MIMO systems due
to orthogonal training sequences for channel estimation are limited either by
the channel coherence time or by the interference from multiple users in
neighboring cells.

To solve the problems engaged in  channel estimation for massive
systems, most of
the recent works considered the time-division duplexing (TDD) technique in which reciprocity benefits are used
for exploiting channel reciprocity 6, 1.  In
 this  case,
 the  pilot overhead related problems for  channel estimation can  be reduced by using uplink channel sounding because the required
orthogonal training sequences become
dependent on  the  number
of serviced users and independent
the number of transmit
antenna  at  the  base  station. In the frequency-division duplexing (FDD)
technique, channel estimation for Massive MIMO systems becomes more challenging.
This is due to traditional  small  array
 (e.g.,  two,  four,  or
 eight  antenna) MIMO
channel sounding approaches require far  too  much time overhead. FDD has limited worn on massive MIMO
channel estimation techniques 4, 17.  By
exploiting spatial correlation or closed-loop training
these techniques improves the channel estimation performance. FDD systems require potentially
substantial feedback overhead when ever transmit channel adoption
is needed 10–9


This project we
are considering the problem
of downlink channel
estimation in FDD massive MIMO systems. We developed a new pilot beam pattern design for orthogonal pilot sequences
which are bounded by the channel coherence time. To minimize the channel estimation mean square error (MSE) we proposed an efficient algorithm which provides the sequentially optimal pilot beam pattern. By
using  second-order statistics of the channel, the temporal correlation, and the signal-to-noise ratio (SNR) jointly
the pilot beam pattern at each training instance
can be derived.




         Vectors and matrices are written in boldface with
matrices in capitals. All vectors are column vectors. AT, AH, and A? indicate the transpose, Hermitian transpose, and the complex conjugate of A, respectively. Ai,j   denotes the element of A   at  the  i-th
 and  j-th  column.  diag(d1 , ··· , dn )  is the
 diagonal matrix composed of  elements d1 ,
··· , dn  and



diag(A)  gives a vector containing the diagonal elements of matrix  A.  For  a  vector
 a,  we  use  ||a|| for  2-norm. For a matrix A,
tr(A)  and var(A)
denote the trace of A, 
and variance operator, respectively.
Kronecker product
is ?, and vec(A) operator creates a column vector by stacking the elements of A
column wise. E{x} represents the expectation of x, and In stands for the identity matrix of size n.



A.System Set-Up


       Here we
are considering a massive MIMO system with Nt  transmit
antennas and a single receive antenna (Nt >>1), as shown in
1. The signal received at nth Symbol time is
given by           



       Where sn is the Nt  × 1 transmitted symbol vector at time n, hn is the Nt  × 1 MISO channel vector at time n, and wn is a zero-mean
independent and identically distributed (i.i.d.) complex Gaussian noise at time n
with covariance ?2 .
We assume that under a state-space model the channel is Rayleigh-faded and time-varying, i.e., the ?rst-order stationary Gauss-Markov process gives the channel dynamic.




By Satisfying Lyapunov equation



Where  bk is a zero-mean and temporally independent plant Gaussian vector, a is the temporal fading correlation coefficient.





MMSE filter













Massive MIMO system model, n=lM+m


For stationary,

 for all k.


          Assume that the transmission takes place by  continuously slotting  with M
 consecutive symbols as one slot and each slot is composed of a
data transmission period of Md symbols
and a training period of Mp   symbols (M= Mp+Md). During training  periods,
the channel is estimated by transmitting a
 sequence  of  properly 
designed  known
pilot transmit vectors



.(Note that

 is the pilot beam pattern at time n at training
symbol time n). Unknown
data is transmitted during data transmission periods. During training period, based on the
estimated channel transmit beam forming can be applied.


B. Channel Estimation



           Based on the current and all previous observations we are  considering the  minimum mean square error (MMSE) channel
estimation during training periods, i.e.,


 is all observations made
during the pilot transmission
up to symbol time n, and is given by



The system equation (1) can be rewritten as



Then,  (2) and (4) form a state-space model and the optimal channel estimation is given by Kalman ?ltering and prediction applied
to this state-space model 11. During the
training period, a measurement update step at each and every symbol time is
available due 
to  the
 known pilot  pattern, and
 the  Kalman
channel estimate and the error covariance matrix are given by 11











are the  prediction error and estimation covariance
matrices, respectively, and is defined as





 During the data transmission period,
based on the last channel estimate of previous training period the channel
is predicted
without the measurement Update step as 11



Where i  = 1,…, Md.
 During the data transmission period, based on the current channel estimate the transmit beam
forming can be applied

forming 5, 16 can be applied for maximum rate of transmission. Based on the current channel estimate, the beam forming weight vector for maximal ratio transmit beam forming is given by



where dk is the k-th data symbol with signal power



this section,
for channel estimation we proposed the best pilot beam pattern design method by considering
the estimation of Mean Square Error (MSE) criterion in the previous section.
Both signal-to-noise ratio (SNR) and the training based channel capacity 14
are directly related to channel estimation MSE.


Note equation
(9) shows that during the l-th data transmission period the  channel
 estimation  error
 depends  only
 on  Rh,  a  and the  estimation
 error  covariance  matrix

 at  the last pilot symbol time. By properly designing the pilot
 beam  pattern  sequence

, we need to minimize the estimation MSE,

 at the last pilot symbol time, Since a  and Rh are given.

        Note that

 is a function of

. S should be jointly optimized to minimize the MSE  at time

. Since the impact
of S on  

 is intertwined,
such joint optimization is too complicated. Since the MSE at

 for each and every

should be optimized for the

-th data transmission period, furthermore optimal channel
estimation at

 for some l is not the only optimization goal. Hence, to design the pilot beam pattern sequence
we adopted a greedy sequential optimization approach. That is, at
time n we optimized pilot

 to minimize


 at all pilot
symbol time

 starting from n==1


Problem 1:

 is given for  each pilot symbol time starting from 1 to n,  for
all pilot symbol time

, design

 such that




       In MIMO systems we need to perform
channel estimation at each receiver antenna separately. So in this project, we are considering
MISO case only. The MISO result obtained here can
be directly applied
to MIMO systems. Joint processing across the multiple receive antennas of MIMO systems for channel estimation
is beyond the
scope of the current project.


The proposed Algorithm


The following
proposition gives the solution to the problem 1 in MISO case.


Proposition 1: A scaled dominant eigenvector of the error covariance matrix of
Kalman prediction for time n gives all the 
previous pilot sn’ (n’< n), pilot beam pattern sn at time n by minimizing . Poof: case1) : From (7)  can be written as     Since  (13) can be rewritten as            Note that equation (14) is the generalized Rayleigh quotient with respect to the pencil . Thus, equation(14) satisfies the following bound15 for any Non-zero vector ,.           Where and  are the smallest and largest eigen values of  , respectively, and optimal  is given by eigen vector of the kalman prediction error covariance matrix  corresponding to scaled by .     Case 2:  In this case, before the ?rst pilot symbol time n in the l-th slot, we have Md   prediction steps without measurement update steps. The measurement update of the form (7)  at time n is valid when  replaced by the kalman prediction for time based on all the previous pilot beam patterns and is given by         Thus, just by replacing  with  in the case(1) the proof in Case 1) is applicable to this case.        Thus, the kalman prediction error covariance matrix gives  the optimal sn for given pilot , . Interestingly, it can be shown that the pilot beam pattern sn obtained from (13) is equivalent to the ?rst principal component direction of  given by            Note  that by performing eigen decomposition (ED) of the kalman prediction error covariance matrix with size Nt × Nt we can obtain the optimal sn at each pilot  symbol  time  n. Since Nt  is large for massive MIMO systems, this  is  computationally expensive. By exploiting the eigen-space of the Kalman prediction  error  covariance matrix  property provided by the following proposition we can construct an efficient beam pattern design algorithm.   Proposition 2:  For any and , The Kalman prediction error covariance matrix   and the Kalman ?ltering error covariance matrix    generated with sequentially optimal    obtained from Proposition 1 are simultaneously diagonalizable with Rh,  under the assumption of   Proof : proof is by introduction. Let  be the ED of . First note that . For any pilot symbol time suppose that is the ED of  , where and . By proposition 1,  is given by a scaled eigen vector of  gives sn corresponding to the largest eigen value . Then  is obtained from the measurement update (7), and is given by     Thus, and  are simultaneously diagonalizable. After prediction step (6),  and  are simultaneously diagonalizable. Since  and Rh has the same set of eigen vectors as  and   in the first training period.  Now during the ?rst data transmission period consider a symbol time n. For this case, the prediction error covariance matrix is given by                Where i = 1,... , Md.  Thus, during the ?rst data period any prediction error covariance matrix is simultaneously diagonalizable with  for . since this kalman recursion repeats, we have proved the claim   Algorithm 1 Designing of sequentially optimal pilot beam pattern   Require: perform the ED of . Store and .  where While l=0,1,…do                                 for m=1 to M do                                  if  then end if end for end while   Thus, for the sequentially optimal pilot beam pattern design  all Kalman prediction error covariance matrix have the same set of eigenvectors as Rh. Note that (19) shows how the eigen values of the channel prediction error covariance matrix change during the pure prediction and step (17) shows how a sequentially optimal pilot beam pattern at time n reduces the channel estimation error by changing the eigen value distribution.  Hence for obtaining the sequence of optimal pilot beam patterns we can design an efficient algorithm  that could minimize the channel estimation Mean Square Error (MSE) at each symbol time. The algorithm is summarized in algorithm1. In  the  proposed algorithm, during the  training period the  dominant eigen value ?i is  tracked at  each  symbol time. That is, the maximum eigen value index is searched and the related eigen vector ui is employed because the pilot beam pattern for the corresponding symbol time. After incorporating the reduction of the prominent eigen value by the measurement update based on the pilot beam pattern and the eigen value change by the conjecture step, the dominant eigen value index for the next time step is searched again and this operation iterates. Typically the proposed algorithm can be run only if the channel statistics a, SNR, Rh and the slot  information (Mp , Md) known, and the optimal sequence of pilot beam patterns depends on  these  parameters. We have a nontrivial optimal sequence unless until the antenna elements are uncorrelated. At the beginning of the transmission session the necessary parameters can be shared between the transmitter and the receiver, and then the receiver can run the algorithm to know the currently used pilot pattern by itself. Note that the proposed algorithm requires the  ED  of  Rh  only once and all  other computation is simple arithmetic.                          IV.NUMERICAL RESULTS          In this section, we have provided the numerical results for evaluating the  performance of  the  proposed algorithm. Since  massive MIMO  systems  assume that the transmit antennas must be much more greater than 1 (i.e, ),  here we are considering  Nt = 32 transmit antennas and a single receiving antenna, and set M  = 8 and Mp = 4 due to the insufficient pilot training. We adopt  a symbol duration of 100?s and carrier frequency of 2.5GHz. We compare two different cases: v = 3km/h, 30km/h in which a = 0.9999, 0.9995, respectively. The considered     channel spatial correlation model is the quadratic exponential correlation model 16, 17, given by                                              (20) where  |r| <  1  and  normalized so  that  tr(Rh )  = 1.  The channel estimation performance was measured by the trace of the Kalman estimation error covariance, averaged under w 1, 000 Monte Carlo runs. The noise variance  is determined according to the  SNR, i.e.,                          And the received Signal-to-noise ratio (SNR) is defined as                           imposed by imperfect channel estimation. We first compared the performance of the proposed approach to several existing methods 18 in Fig. 2. For the randomly and orthogonal pilot methods based on the Kalman filter, we considered a round-robin selection for the initialized pilot beam habits. Fig. 2 shows the channel state fast during tracking periods, the suggested algorithm tracks and also guarantees the received high SNR gain. Because of our tracking of unreal distribution of the channel Minimum Mean Square Error MMSE, the proposed method converges faster as the antenna spatial correlation boosts by comparing Fig. 2(a) with (b). As compared to the perfect channel estimation case in Fig. 2(b) the proposed method produces an SNR loss of 2dB. Since the channel MMSE increased during data transmission periods, as expected (6) the fast fading process with v  = 30km/h in  Fig. 3, the channel MMSE and the received SNR shows a  repetitive trajectory curve. Due to the increased temporal fading correlation affecting the spectral distribution of the channel MMSE there is some loss exist in the received SNR and channel estimation, however, the proposed method still shows good performance.                                           V. CONCLUSIONS   By using ?rst-order stationary Gauss-Markov channel process we proposed a new algorithm for the designing of optimal pilot beam pattern for Multi User Massive MIMO systems. 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