Optimal Pilot Symbol Power Allocation in LTE

1. Copyright 2011 IEEE. Published in the proceedings of VTC-Fall 2011, San Francisco, CA, USA, 2011.
Optimal Pilot Symbol Power Allocation in LTE
ˇ
Michal Simko, Stefan Pendl, Stefan Schwarz, Qi Wang, Josep Colom Ikuno and Markus Rupp
Institute of Telecommunications, Vienna University of Technology
Gusshausstrasse 25/389, A-1040 Vienna, Austria
Email: msimko@nt.tuwien.ac.at
Web: http://www.nt.tuwien.ac.at/ltesimulator
Abstract—The UMTS Long Term Evolution (LTE) allows the B. Contribution
pilot symbol power to be adjusted with respect to that of the
In this paper, we derive analytical expressions for optimal
data symbols. Such power increase at the pilot symbols results
in a more accurate channel estimate, but in turn reduces the power allocation for a MIMO system with Zero Forcing (ZF)
amount of power available for the data transmission. In this equalizer under imperfect channel state information. We uti-
paper, we derive optimal pilot symbol power allocation based lize the post-equalization SINR, as the optimization function,
on maximization of the post-equalization Signal to Interference which is analogous to the throughput maximization in a real
and Noise Ratio (SINR) under imperfect channel knowledge.
system.
Simulation validates our analytical mode for optimal pilot symbol
power allocation. The main contributions of the paper are:
Index Terms—LTE, Channel Estimation, OFDM, MIMO. • By maximizing the post-equalization SINR, we deliver
optimal values for the pilot symbol power adjustment
in MIMO Orthogonal Frequency Division Multiplexing
I. I NTRODUCTION (OFDM) systems.
• The post-equalization SINR expression is derived for a
Current systems for cellular wireless communication are ZF receiver under imperfect channel knowledge.
designed for coherent detection. Therefore, channel estimator • We analytically derive the Mean Square Error (MSE)
is a crucial part of a receiver. UMTS Long Term Evolution expression of the Least Squares (LS) channel estimator
(LTE) provides a possibility to change the power radiated at utilizing linear interpolation.
the pilot subcarriers relative to the that at data subcarriers. • Simulation results with an LTE compliant simulator [6, 7]
Clearly, this additional degree of freedom in the system design confirm optimal values for pilot symbol power.
provides potential for optimization. • As with our previous work, all data, tools, as well
implementations needed to reproduce the results of this
A. Related Work paper can be downloaded from our homepage [8].
The remainder of the paper is organized as follows. In
In order to optimize the pilot symbol power allocation Section II we describe the mathematical system model. In
a model that takes into account the pilot power adjusting, Section III, we derive the post-equalization SINR expression
receiver structure and channel estimation error at the same for ZF with imperfect channel knowledge. The channel estima-
time, is needed. It has been shown by simulation that pilot tors of this work are briefly discussed in Section IV and their
symbol power allocation has a strong impact on capacity [1]. MSEs are derived. We formulate the optimization problem for
Authors of [2] show by simulation the impact of different optimal pilot symbol power allocation in Section V. Finally,
power allocations on the system’s Bit Error Rate (BER). we present LTE simulation results in Section VI and conclude
However, their analysis is based on Signal to Noise Ratio our paper in Section VII.
(SNR) so that they only approximate the impact of imperfect
channel knowledge on BER for Binary Phase-shift Keying II. S YSTEM M ODEL
(BPSK) modulation. In [3], optimal pilot symbol allocation In this section, we briefly point out the key aspects of LTE
is derived analytically for Phase-shift Keying (PSK) modula- relevant for this paper, as well as an system model.
tion of order two and four, using BER as the optimization In the time domain the LTE signal consists of frames with
criterion. In [4] optimal pilot symbol power in Multiple Input a duration of 10 ms. Each frame is split into ten equally long
Multiple Output (MIMO) system is derived based on lower subframes and each subframe into two equally long slots with
bound for capacity. Authors of [5] investigate power allocation a duration of 0.5 ms. Depending on the cyclic prefix length,
between pilot and data symbols for MIMO systems using post- being either extended or normal, each slot consists of Ns = 6
equalization Signal to Interference and Noise Ratio (SINR) as or Ns = 7 OFDM symbols, respectively. In LTE, the subcarrier
the optimization function. However, they only approximate the spacing is fixed to 15 kHz. Twelve adjacent subcarriers of one
SINR expression and their model is tightly connected with slot are grouped into a so-called resource block. The number
a Linear Minimum Mean Square Error (LMMSE) channel of resource blocks in an LTE slot ranges from 6 up to 100,
estimator. corresponding to a bandwidth from 1.4 MHz up to 20 MHz.

2. A received OFDM symbol in frequency domain can be III. P OST- EQUALIZATION SINR
written as In this section, we derive an analytical expression for the
Nt post-equalization SINR of a MIMO system under imperfect
ynr = Xnt hnt ,nr + nnr , (1) channel knowledge and ZF equalizer given by the system
nt =1 model in Equation (4). In the following section, we omit the
where the vector hnt ,nr contains the channel coefficients in subcarrier index k, since the concept we present is independent
the frequency domain between the nt -th transmiter and nr -th of it.
receiver antennas and nnr is additive white zero mean Gaus- If the equalizer has perfect channel knowledge available,
sian noise with variance σn at the nr -th receiver antenna. The
2 the ZF estimate of the data symbol s is given as
diagonal matrix Xnt = diag (xnt ) comprises the precoded −1
ˆ = GH G
s GH y. (9)
data symbols xd,nt and the pilot symbols xp,nt at the nt -th
transmit antenna placed by a suitable permutation matrix P s
The data estimate ˆ given by Equation (9) results in the post-
on the main diagonal equalization SINR of the l-th layer given as [10]
T
xnt = P xT t xT t . (2) σs
2
p,n d,n γl = −1 , (10)
σn eH (GH G)
2
l el
The length of the vector xnt is Nsub corresponding to the
number of subcarriers. Let us denote by Np and Nd , the where the vector el is an Nl × 1 zero vector with the l-th ele-
number of pilot symbols and the number of precoded data ment being 1. This vector serves to extract the corresponding
symbols, respectively. The precoded data symbols xd,nt are layer power after the equalizer.
obtained from the data symbols via precoding Let us proceed to the case of imperfect channel knowledge.
T T
We define the perfect channel as the channel estimate plus the
[xd,1,k · · · xd,Nt ,k ] = Wk [s1,k s2,k · · · sNl ,k ] , (3) error matrix due to the imperfect channel estimation
where xd,nt ,k is a precoded data symbol at the nt -th transmit ˆ
H = H + E, (11)
antenna port and the k-th subcarrier, Wk is the precoding
matrix at the k-th subcarrier and snl ,k is the data symbol of where the elements of the matrix E are independent of each
the nl -th layer at the k-th subcarrier. other with variance σe . Inserting Equation (11) in Equa-
2
Note that according to Equation (2), also the vectors ynr , tion (4), the input output relation changes to
hnt ,nr and wnr can be divided into two parts corresponding ˆ
to the pilot symbol positions and to the data symbol positions. y = H + E Ws + n. (12)
For the derivation of the post-equalization SINR, we will Since the channel estimation error matrix E is unknown at
use the input-output relation at the subcarrier level, given as: the receiver, the ZF solution is given again by Equation (9),
ˆ
but channel matrix H is replaced by its estimate H, which is
yk = Hk Wk sk + nk . (4)
known at the receiver
Matrix Hk is the MIMO channel matrix of size Nr × Nt . −1
s ˆ ˆ
ˆ = GH G ˆ
GH y, (13)
Furthermore, matrix Wk is a unitary precoding matrix of size
Nt × Nl . In LTE the precoding matrix can be chosen from a
ˆ ˆ
with matrix G being equal to HW. The error after the
finite set of precoding matrices [9]. The vector sk comprises
the data symbols of all layers at the k-th subcarrier. We denote equalization process using ZF is given as
the effective channel matrix that includes the effect of the −1
s ˆ ˆ
ˆ − s = GH G ˆ
GH (EWs + n) . (14)
channel and precoding by Gk
Gk = Hk Wk . (5) From Equation (14) we can compute the MSE matrix
H
Furthermore, let us denote the average data power transmitted MSE = E (ˆ − s) (ˆ − s)
s s (15)
at one layer by σs , the total data power by σx , and the average
2 2
−1
pilot symbol power by σp 2 = σn + σx σe
2 2 2 ˆ ˆ
GH G ,
1 with σe being the MSE of the channel estimator. Equation (15)
2
σs = E
2
sl,k 2
= , (6)
2
Nl directly leads to the SINR of the individual layers
Nt
σx =
2
E xd,nt 2 σs
2
2 = 1, (7) γl = −1 . (16)
nt =1 2 2 2 ˆ ˆ
(σn + σe σx ) eH GH G el
l
Nt
σp =
2
E xp,nt 2
2 = 1. (8) Note, that in practice variables σs , σx and σn are replaced by
2 2 2
nt =1 their estimates.

3. Interpolation By plugging (18) into (20), expanding the square and using
the identities
npi
Channel gain
E Δh∗i hpi = E
p hp = 0, (22)
xpi i
c2 = 1 − c1 , (23)
e p1 p2 pm i pn we obtain for the LS MSE at an interpolated position:
Subcarrier index (i,i) (p ,pn ) (p ,pm )
Fig. 1. Linear inter- and extrapolation of the channel on pilot subcarriers to σei = Rh,h + c2 (Rh,h
2
1
n
+ σn ) + c2 (Rh,h
2
2
m
+ σn )
2
intermediate subcarriers. (p ,pn ) (i,p ) (p ,i)
+2c1 c2 Rh,h
m
− 2c1 Rh,hn − 2c2 Rh,h
m
IV. C HANNEL E STIMATION (24)
In this section, we present state-of-the-art channel estimators (i,j)
and derive analytical expressions for their MSE. Due to the Here, Rh,h denotes the element in the i-th row and j-th
orthogonal pilot symbol pattern utilized in LTE, the MIMO column of matrix Rh,h = E hhH . For an extrapolated
channel can be estimated as Nt Nr individual Single Input position, we utilize the two consecutive neighboring pilot
Single Output (SISO) channels. Therefore, in the following positions, p1 and p2 , to obtain the estimated channel element
section, we omit the antenna indices. according to:
ˆ ˆ e − p1 ˆ ˆ
A. LS Channel Estimation he = hp1 + (hp2 − hp1 ) (25)
p2 − p1
The LS channel estimator [11, 12] for the pilot symbol
c1
positions is given as the solution to the minimization problem:
2 Then, the same Formula (24), with appropriately modified
ˆp ˆ
hLS = arg min yp − Xp hp = X−1 yp
p (17) indices, applies for the extrapolated MSE σee as well. The
2
ˆ
hp 2
MSE at an extrapolated position is generally larger than at an
The remaining channel coefficients at the data subcarriers have interpolated position, because the cross correlation between the
to be obtained by means of interpolation. In this work, we channels at position e and p2 is smaller than those between
utilize linear interpolation of the pilot subcarriers on each the channels at position i and pn .
antenna to obtain the channel vector h containing the channel The total MSE of LS channel estimator is given as mean
elements of all subcarriers in one OFDM symbol. over all subcarriers. Due to the dense pilot symbol pattern and
In order to derive the theoretical MSE, we distinguish three thus strong correlation over frequency, the total MSE can be
cases, as shown in Figure 1: expressed as
• MSE at a pilot subcarrier pi : σep
2
σe = ce σn ,
2 2
(26)
• MSE at an interpolated subcarrier i: σei
2
• MSE at an extrapolated subcarrier e: σee
2
where ce is a constant. Equation (26) follows from Equa-
At a pilot position pi the estimated channel element equals: tion (23) and Equation (24).
ˆ yp np
hpi = i = hpi + i (18) B. LMMSE Channel Estimation
xpi xpi
The LMMSE channel estimator requires the second order
Δhpi
statistics of the channel and the noise. It can be shown that
The MSE of the channel estimator is given by the noise the LMMSE channel estimate is obtained by multiplying the
variance σep = σn . To obtain the variance at an interpolated
2 2
LS estimate with a filtering matrix ALMMSE [13–15]:
position consider the following equation to compute the in-
ˆ ˆ ˆp
hLMMSE = ALMMSE hLS (27)
terpolated channel element hi from the two neighboring pilot
ˆ p and hp :
positions h m ˆ
n
In order to find the LMMSE filtering matrix, the MSE
ˆ ˆ i − pm ˆ ˆ
hi = hpm + ·(hpn − hpm ) (19) 2
pn − pm σe = E
2 ˆp
h − ALMMSE hLS , (28)
2
c1
has to be minimized, leading to
Then, the MSE can be computed as:
−1
2 ALMMSE = Rh,hp Rhp ,hp + σn I
2
, (29)
σi = E
2 ˆ
hi − hi (20)
2
where the matrix Rhp ,hp = E hp hH is the channel autocor-
p
2
=E ˆ ˆ ˆ
hi − hpm + c1 · (hpn − hpm ) (21) relation matrix at the pilot symbol positions, and the matrix
2 Rh,hp = E hhH is the channel crosscorrelation matrix.
p

4. x 106
To derive the theoretical MSE, we plug Equation (29) into 1.8
Equation (28): 1.7
4x4
LS channel estimator
σe =E
2 2 ˆp
h − (Rh,hp (Rhp ,hp − σn I)−1 hLS ) · (30)
1.6
1.5
H
h − (Rh,hp (Rhp ,hp − 2 ˆp
σn I)−1 hLS ) 1.4
f(o)
2x2
1.3
After a straightforward manipulation, the average MSE at each 1.2 LMMSE channel estimator
subcarrier is expressed as 1.1
1x1
1 −1 1
σe =
2
tr Rh,h − Rh,hp Rhp ,hp + σn I
2
Rhp ,h . -10 -5 0 5 10
Nsub poff[dB]
Fig. 2. Power allocation function f (poff ) for different numbers of transmit
V. P OWER A LLOCATION antennas and LS (solid line) and LMMSE (dashed line) channel estimators
In this section, we describe the problem of optimal pilot TABLE I
VALUES OF THE PARAMETERS OF f (poff ) FOR DIFFERENT NUMBER OF
power allocation in LTE based on the maximization of the TRANSMIT ANTENNAS FOR 1.4 MH Z BANDWIDTH , ITU P EDA [17]
post-equalization SINR under imperfect channel knowledge. CHANNEL MODEL , LS AND LMMSE CHANNEL ESTIMATORS
Although the shown results are from the application to LTE, Parameter Tx = 1 Tx = 2 Tx = 4
the presented concept can be applied to any MIMO OFDM Nd 960 912 864
Np 48 96 144
system.
If we increase the power at the pilot symbol by a factor c2 ,
p LS
the MSE of the channel estimator improves by the factor c2 ce 0.3704 0.3704 0.5556
p poff,opt [dB] ≈5.8 ≈3.6 ≈3.5
σe
2
σe =
˜2 . (31) LMMSE
c2
p ce 0.0394 0.0394 0.0544
poff,opt [dB] ≈-0.7 ≈-2.8 ≈-3.2
However, the power of the data symbols has to be decreased by
a factor c2 in order to keep the total transmit power constant. The target is to find an optimal value of poff,opt that max-
d
Obviously, the two factors c2 and c2 are connected. For this imizes the post-equalization SINR while keeping the overall
p d
purpose, we define a variable poff , expressing the power offset transmit power constant
between the mean energy of the pilot symbols and the data maximize γl (37)
symbols, and refer to it as pilot offset. The variables c2 , c2
p d
poff
and poff are interconnected as follows: subject to Nd σx + Np σp = const
2 2
Np + N d In order to maximize the post-equalization SINR, the power
cp = (32)
poff Nd + Np allocation function f (poff ) in the denominator of Equation (35)
Np + Nd has to be minimized. The minimum of the power allocation
cd = Np
= poff cp (33)
Nd + poff function f (poff ) can be found by simply differentiating it and
solving for 0. By these means, the optimal value of the variable
Plugging in the variables c2 and c2 in Equation (16), we obtain
d p poff is given as solution to the following expression:
the SINR expression with adjusted power of the pilot symbols
−2 (poff Nd + Np ) p−3 + 2 (poff Nd + Np ) Nd p−2 + ce = 0.
2
off off
σs c2
2
(38)
γl = d
. (34)
σe 2 2
2 −1
σn +
2
c2 σx cd eH (GH G)
l el An analytical solution for Equation (38) can be obtained by
p
If we insert Equation (32) and Equation (33) into Equation (34) means of Ferrari’s solution [16]. Figure 2 depicts f (poff ) for
and simplify the expression, we obtain: different numbers of transmit antennas for LTE. Typical values
of parameters Nd , Np and ce are given in Table I. Note,
1 that although Nd and Np depend on the utilized bandwidth,
γl = , (35)
Nl σn eH (GH G)−1 el
2
the minimum of f (poff ) is independent of it, since Nd and
l
(Nd +Np )2
f (poff )
Np scale with the same constant with increasing bandwidth.
for which the function f (poff ) is given as The value of ce is different for four transmit antennas due
to the lower number of pilot symbols at the third and fourth
2 1
f (poff ) = (poff Nd + Np ) + ce . (36) antenna. The last row of Table I gives the optimal values of
p2
off poff,opt for different numbers of transmit antennas and an ITU
We refer to it as power allocation function. Note, that it is PedA [17] type channel model. While utilizing the LMMSE
independent of channel realization and noise power. Therefore, channel estimator, one obtains negative values for poff,opt .
it is possible to find an optimum value for pilot symbol power This means, that optimally, the pilot symbol power has to
allocation independent of the SNR and channel realization. be reduced in order to maximize the post-equalization SINR.

5. TABLE II
S IMULATOR SETTINGS FOR FAST FADING SIMULATIONS linear interpolation. Throughput simulation results validate the
Parameter Value accuracy of our analytical model for the optimum pilot power
Bandwidth 1.4 MHz adjustment. All data, tools and scripts are available online in
Number of transmit antennas 1, 2, 4
Number of receive antennas 1, 2, 4 order to allow other researchers to reproduce the results shown
Receiver type ZF in the paper [8].
Transmission mode Open-loop spatial multiplexing
Channel type ITU PedA [17] ACKNOWLEDGMENTS
MCS 9
coding rate 616/1 024 ≈ 0.602 The authors would like to thank the LTE research group
symbol alphabet 16 QAM for continuous support and lively discussions. This work
has been funded by the Christian Doppler Laboratory for
3.5 Wireless Technologies for Sustainable Mobility, KATHREIN-
Werke KG, and A1 Telekom Austria AG. The financial support
3 by the Federal Ministry of Economy, Family and Youth
and the National Foundation for Research, Technology and
2.5
2x2
Development is gratefully acknowledged.
throughput [Mbit/s]
2
4x4
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