Designing of an

Optimized Pilot Beam Pattern For MU-MIMO Systems

Abstract—In this project, channel

estimation

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.

LITERATUR

SURVEY

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

MIMO

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

of

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.

A.Notation

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

row

and j-th column. diag(d1 , ··· , dn ) is the

diagonal matrix composed of elements d1 ,

··· , dn and

×2

}

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.

The

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.

II. SYSTEM MODE L

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

Fig.

1. The signal received at nth Symbol time is

given by

w

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

Tracking

Tracking

Fig.1.

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

p

Based on the current and all previous observations we are considering the minimum mean square error (MMSE) channel

estimation during training periods, i.e.,

where

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

With

and

,where

and

and

are the prediction error and estimation covariance

matrices, respectively, and is defined as

Where,

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

Eigen-beam

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

III. THE PROPOSE D PIL OT BEAM

PATTERN DESIGN

In

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

given

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

the

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.

A.

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. For a given set
of
system
parameters with low computational complexity, the proposed algorithm jointly exploits
the
antenna correlation, temporal
correlation and statistics of channel and to provide
a sequentially optimal pilot
beam pattern.
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