New Insights in Optimal Pilot Symbol Patterns forOFDM Systems

Michal Šimko1, Paulo S. R. Diniz2, Qi Wang1 and Markus Rupp1

1 Institute of Telecommunications, Vienna University of Technology, Vienna, Austria2 DEL POLI/UFRJ, PEE COPPE/UFRJ, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

Email: msimko@nt.tuwien.ac.atWeb: http://www.nt.tuwien.ac.at/ltesimulator

Abstract—Nowadays, most wireless communication systemsutilize coherent detection which implies the necessity of multi-plexing reference symbols between data-symbols for the purposeof channel estimation. In Long Term Evolution (LTE), these ref-erence symbols can consume up to 14.2% of the total bandwidth.In this work, we search for an optimal pilot-symbol pattern designusing a post-equalization Signal to Interference and Noise Ratio(SINR) framework. We show how close to optimally choose thedistance between adjacent pilot-symbols and how to distributethe available power between pilot and data symbol at the sametime. We confirm the performance of our analytical solution bymeans of simulation. Compared to a system with a fixed distancebetween pilot-symbol and unit power distribution, the proposedsystem configuration yields a gain of around 30% in terms ofcapacity.

Index Terms—LTE, Power distribution, OFDM, MIMO.

I. INTRODUCTIONCoherent detection is widely utilized in current systems for

mobile wireless communications. One of the most importantaspects of such a system is its pilot-symbol pattern. In thispaper, we analyze state-of-the-art knowledge about pilot-symbol patterns and provide a novel design that maximizesan approximation of the capacity.

A. Related WorkIn the past, many researchers tried to find an optimal pilot-

symbol pattern design using various criterions. In [1, 2], theauthors chose the Mean Square Error (MSE) of a channelestimator as the cost function for the design of the pilot-symbolpatterns. They showed that equipowered, equispaced pilot-symbols led to the lowest MSE. Authors of [3, 4] proposed adesign pattern that maximized the system’s capacity. There aremany different approaches for designing pilot-symbol patternsbased on the minimization of Bit Error Ratio (BER) [5] orSymbol Error Ratio (SER) [6].

Authors of [7–10] showed how to distribute available poweramong data and pilot-symbols given a certain pilot-symbolpattern. The authors utilized the post-equalization Signal toInterference and Noise Ratio (SINR) under imperfect chan-nel knowledge as the cost function to solve the formulatedproblem. This relatively simple framework allows to treat theproblem analytically and to find a solution independent of theactual channel realization.

B. Contribution

The main contributions of this paper are:• By maximizing an approximation of the channel capacity,

we derive the optimal distance between adjacent pilot-symbols in block fading scenarios and at the same timethe optimal power distribution between pilot and datasymbols.

• As with our previous work, all data, tools, as wellimplementations needed to reproduce the results of thispaper can be downloaded from our homepage [11].

The remainder of the paper is organized as follows. InSection II, we describe the mathematical system model fortransmitting pilots and data over a Multiple Input MultipleOutput (MIMO) channel. In Section III, we briefly describethe post-equalization SINR expression for Zero Forcing (ZF)equalizers with imperfect channel knowledge. In Section IV,we summarize relevant knowledge about the performanceof linear channel estimators. We formulate the optimizationproblem for optimal pilot-symbol design in Section V. Finally,we present simulation results in Section VI and conclude ourpaper in Section VII.

II. SYSTEM MODEL

In this section, we briefly introduce a transmission modelsuitable for our further derivation. A received symbol inthe frequency domain of an Orthogonal Frequency DivisionMultiplexing (OFDM) system at the nr-th receive antenna canbe written as

ỹnr =

Nt∑nt=1

H̃nt,nrxnt + ñnr , (1)

where H̃nt,nr ∈ CNsub×Nsub represents the channel matrix inthe frequency domain between the nt-th transmit and the nr-th receive antennas. The transmitted signal vector is referredto as xnt , the received signal vector as ỹnr . The vectorñnr ∈ CNsub×1 denotes additive white zero mean Gaussiannoise with variance σ2n on an antenna nr. In the case of atime-invariant channel, the channel matrix H̃nt,nr appears asa diagonal matrix. In this paper, we limit our discussion to thetime-invariant case.

msimkoText BoxCopyright 2013 IEEE. Published in the proceedings of WCNC 2013, Shanghai, China, 2013.

Specifically, the vector xnt ∈ CNsub×1 in Equation (1)comprises the precoded data symbols xd,nt ∈ CNd×1 and thepilot-symbols xp,nt ∈ PNp×1 from the set of all possible pilot-symbols P defined in Long Term Evolution (LTE), at the nt-thtransmit antenna placed by a suitable permutation matrix P

xnt = P[xTp,nt x

Td,nt

]T. (2)

Figure 1 shows an example with the pilot-symbols multi-plexed between the data symbols, where every D-th symbolis a pilot-symbol.

D D DD

pilot symbol data symbol

Fig. 1. Pilot pattern with equidistant reference symbols represented by blackdots multiplexed within available resources with distance D.

The precoding process on a subcarrier k can be describedas

[xd,1,k · · ·xd,Nt,k]T= Wk [s1,ks2,k · · · sNl,k]

T, (3)

where xd,nt,k is a precoded data symbol at the nt-th transmitantenna port and the k-th subcarrier, Wk ∈ CNt×Nl is a uni-tary precoding matrix at the k-th subcarrier and snl,k ∈ D1×1is the data symbol of the nl-th layer at the k-th subcarrier.Here, D is the set of available modulation alphabets. The dataelements from Equation (3) are stacked into a vector to forma data symbol vector xd,nt .

For the derivation of the post-equalization SINR, we use aMIMO input-output relation at the subcarrier level, given as:

yk = HkWksk + nk. (4)

The matrix Hk ∈ CNr×Nt denotes the MIMO channel matrixat the k-th subcarrier. This matrix is obtained by means ofthe Fourier transformation of the physical channel in thetime domain. The vector sk consists of the data symbols ofall layers at the k-th subcarrier. The vector nk representsadditive white zero-mean Gaussian noise with variance σ2n ata subcarrier k. We denote the effective channel matrix by

Gk = HkWk. (5)

Furthermore, the average power transmitted on each of the Nllayers is denoted by σ2s . The power transmitted on each dataposition is σ2d, while it is σ

2p on each pilot position.

III. POST-EQUALIZATION SINR

In this section, we consider a time-invariant scenarioand briefly describe an analytical expression for the post-equalization SINR of a MIMO system using a ZF equalizerbased on imperfect channel knowledge. More details can befound in our previous work [9].

If perfect channel knowledge is available at the equalizer,the ZF estimate of the data symbol sk is given as

ŝk =(GHkGk

)−1GHk yk. (6)

The data estimate ŝk defined in Equation (6) results in a post-equalization SINR of the m-th layer given as [12, 13]

γm,k =σ2s

σ2neHm

(GHkGk

)−1em

, (7)

where the vector em is an Nl × 1 zero vector with a oneon the m-th element. This vector extracts the signal on thecorresponding layer m after the equalizer.

Let us proceed to the case of imperfect channel knowledge.We define the perfect channel as the channel estimate plus theerror matrix due to the imperfect channel estimation

Hk = Ĥk +Ek, (8)

where the elements of the matrix Ek are random variables,statistically independent of each other with zero mean andvariance σ2e . Inserting Equation (8) in Equation (4), the input-output relation changes to

yk =(Ĥk +Ek

)Wksk + nk. (9)

Since the channel estimation error matrix Ek is unknown atthe receiver, the ZF solution is given again by Equation (6),but the channel matrix Hk is replaced by its estimate Ĥk,which is known at the receiver

ŝk =(ĜHk Ĝk

)−1ĜHk yk, (10)

with the matrix Ĝk being equal to ĤkWk.Applying a ZF equalizer, Equation (10) leads to the SINR

on the m-th layer [9]

γm,k =σ2s

(σ2n + σ2eσ

2d) e

Hm

(ĜHk Ĝk

)−1em

. (11)

IV. CHANNEL ESTIMATION

In this section, we summarize relevant knowledge aboutchannel estimation under time-invariant channels. Althoughwe limit our discussion only to a Least Squares (LS) channelestimator, note that based on the results shown in [9], allconcepts can be easily applied also to other linear channelestimators.

It has been shown that equidistant pilot-symbols are optimalin terms of capacity and MSE [1, 2]. Therefore, we considerequidistant pilot patterns with distance D between adjacentpilots. An example of such a pattern is shown in Figure 1.

MSE of a linear channel estimator can be expressed as [9]

σ̃2e = ce (D)σ2nσ2p

+ d (D) , (12)

where ce (D) and d (D) are real constants depending on thedistance D between adjacent pilot-symbols that determine theperformance of the channel estimators.

The coefficient ce (D) from Equation (12) is purely deter-mined by the pilot-symbol pattern and the coefficient d (D)is additionally dependent on the channel autocorrelation func-tion [9]. Figure 2 shows the behavior of these coefficients overdistance between two adjacent pilot-symbols. Let us first con-sider ce (D). At low distances it grows rapidly with increasingdistance between two pilot-symbols. At rather large distances,it remains almost constant. The value of the coefficient d (D)is first close to zero and once after the distance between twopilots is greater than the coherence bandwidth of the channel,it grows with increasing distance between two pilot-symbols.

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

pilot distance D

d (s

olid

lin

e)

dVehA

dPedB

dPedA

ce

c e(d

ashe

d lin

e)

Fig. 2. Coefficients ce (D) and d (D) over distance between two adjacentpilot-symbols for an LS channel estimator for various ITU channel models.

Inserting Equation (12) into Equation (11), assuming thatavailable transmit power is evenly distributed between alllayers σ2s =

σ2dNl

and simplifying the expression, we obtainthe SINR expression:

γm,k =1

NleHm(GHkGk

)−1em

f(σ2p, σ

2d, D

), (13)

for which the power allocation function f(σ2p, σ

2d, D

)is given

as

f(σ2p, σ

2d, D

)=

1

σ2n

(1σ2d

+ ce(D)σ2p

)+ d (D)

. (14)

Note that the power allocation function is independent ofthe channel realization. In this work, we consider the case,when the available transmit power is utilized. Therefore, weintroduce a variable poff is defined as the power offset betweenthe power of the pilot-symbols and the data symbols, denotedby

σ2d = poffσ2p. (15)

When transmitting the available transmit power, the powerstransmitted at the pilot and the data symbols are interconnectedas

σ2dNd + σ2pNp = Nd +Np, (16)

where Nd and Np are numbers of the data and pilot-symbols,respectively. With Equation (15) and Equation (16), the powerallocation function depends only on the power offset poff andthe distance between adjacent pilot-symbols D, therefore fromnow on, it is denoted by f (poff , D).

V. PILOT DESIGN

In this section, we analyze pilot-symbol patterns using apost-equalization SINR framework. In general, certain amountof resources like bandwidth and power are available for trans-mission of data and pilot-symbols. If more resources are usedfor data transmission, there is less left for the transmissionof the pilot-symbols, leading to a degradation of the channelestimation performance. Clearly, there is an equilibrium forthe distribution of the available resources.

The optimal distance between two adjacent pilot-symbolsand power distribution between pilot and data symbols cannotbe found exclusively by maximizing the post-equalizationSINR. This would lead to a solution with small distancebetween adjacent pilot-symbols, which would decrease theavailable bandwidth for data transmission. Therefore, anothertype of cost function is required that allows to include apenalty due to the bandwidth occupied by the pilot-symbols.Capacity is one of the possible choices for the new costfunction

C = B log2 (1 + γm) , (17)

where B is the bandwidth utilized for the data transmission.Inserting the post-equalization SINR into the capacity expres-sion allows to consider the effect of the channel estimationerror and power allocation in a very simple way [14]. Thepost-equalization SINR Equation (13) can be multiplicativelydecomposed into two parts, the channel independent powerallocation function Equation (14) and a function that dependson the actual channel fh (G) = 1

NleHm(GHk Gk)−1

em. In order

to maximize the system’s capacity without taking into accountan actual channel realization, we approximate Equation (17)at high SINR by

C ≈ B log2 (f (poff , D) fh (G)) (18)≈ B log2 (f (poff , D)) +B log2 (fh (G))

and at low Signal to Noise Ratio (SNR) by

C ≈ log2 (e)Bf (poff , D) fh (G) . (19)

The only part influenced by changing the distance be-tween pilots and by adjusting the power allocation isB log2 (f (poff , D)) for high SNR and Bf (poff , D) for lowSNR values, respectively. Therefore, we define the cost func-tion as

fcost (poff , D) =

{Bf (poff , D) ; SNR < SNRtrB log2 (f (poff , D)) ; SNR > SNRtr

(20)

where SNRtr is an SNR threshold value, at which the capacityapproximation for low SNR values is replaced by that at high

SNR values. With the above defined cost function, we canformulate our optimization problem as

maximizepoff ,D

fcost (poff , D)

subject to Ndσ2d +Npσ2p = Nd +Np

B = const(21)

For a fixed distance D, the optimal value of power distri-bution poff can be efficiently found employing techniques ofconvex optimization. For solving the optimization problem inEquation (21) we utilized CVX, a package for specifying andsolving convex programs [15, 16]. After finding optimal valuesof the variable poff for all possible distances D, we maximizethe cost function in order to find the optimal combination ofthe variables poff and D.

Figure 3 depicts example of the cost function at an SNR of20 dB with the VehA channel model.

-10-5

05

10

020

4060

800

1

2

3

4

5

6

poffD

cost

fun

ctio

n

Fig. 3. Cost function plotted over pilot distance D between adjacent pilot-symbols and different power offset poff between pilot and data symbols forSNR= 20dB and VehA channel model.

Figure 4 shows the optimal choice of distance betweenadjacent pilot-symbols and the corresponding power offsetbetween data and pilot-symbols over SNR for various ITUchannel models. With increasing SNR, a better channel es-timate is required, therefore the optimal distance betweenneighboring pilot-symbols is decreasing. At the same time,the power offset between pilot and data symbols is decreasingwith increasing SNR, which corresponds to less and less powerassigned to the pilot-symbols, since there is less noise at thepilot-symbols.

VI. SIMULATION RESULTS

In this section, we present simulation results and com-pare the transmission system’s capacity setting an optimaldistance between adjacent pilot-symbols and with optimalpower distribution between pilot and data symbols, againsta system using fixed distance between pilot-symbols and unitdistribution of power between data and pilot-symbols. All data,tools and scripts are available online [11] in order to allowother researchers to reproduce the results shown in this paper.Table I shows the most important simulator settings.

0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

SNR [dB]

D (

solid

lin

e)

0

2

4

6

8

10

p off

[dB

] (d

ashe

d lin

e)

VehAPedBPedA

Fig. 4. Optimal choice of the distance between adjacent pilot-symbols Dand the optimal power offset between data and pilot-symbols poff over SNRfor various ITU channel models.

TABLE ISIMULATOR SETTINGS FOR FAST FADING SIMULATIONS

Parameter ValueBandwidth 1.4 MHz

Number of transmit antennas 1Number of receive antennas 1

Receiver type ZFTransmission mode Open-loop spatial multiplexing

Channel type ITU VehA, PedA, PedB [17]

Figure 5 shows capacity over distance between adjacentpilot-symbols D and over variable poff at SNR = 20 dB.We observe that the capacity maximum matches with themaximum of the cost function in Figure 3.

-20-10

010

20

020

4060

800

1

2

3

4

5

poffdistance between pilots

capa

sity

[bi

t/s/

Hz]

Fig. 5. System’s capacity plotted over pilot distance D between adjacentpilot-symbols and different power offset poff between pilot and data symbolsfor SNR= 20dB and VehA channel model.

Capacities of two different systems are compared in Fig-ure 6. The solid lines represent a system using the optimaldistance between adjacent pilot-symbols and optimal powerdistribution between pilot and data symbols. Optimal valuesfor both of the variables are shown in Figure 4. The dashed linein Figure 6 represents a system using fixed distance betweenadjacent pilot-symbols D = 3 and no power distribution

between pilot and data symbols. We chose D = 3, becausethis is the distance used in an LTE system [18]. Note, that thegain achieved for one OFDM symbol will not be achieved bythe subframe structure utilized by LTE systems, consisting ofmore OFDM symbols, in which some of the OFDM symbolsdo not carry any reference symbols.

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

SNR [dB]

capa

city

[bi

t/s/

Hz]

optimal D, poffD=3, poff=0 dB

PedAPedBVehA

Fig. 6. Capacity versus SNR for system with optimal pilot distance Dbetween pilot-symbols and optimal power offset poff (solid line) and forsystem with fixed distance between pilot-symbols (D = 3) and no powerdistribution (dashed line).

VII. CONCLUSIONIn this paper, we tackled the problem of optimal pilot-

symbol design for time-invariant scenarios. We delivered an-alytical expressions for the optimal distance between adjacentpilot-symbols and the optimal power distribution betweenpilot-symbols and data symbols. The optimal values wereconfirmed by means of simulations and compared to a systemusing a fixed distance between adjacent pilot-symbols andunit power distribution. Such system is outperformed by anoptimal system by approximately 30% in terms of capacity.Note that we considered only single OFDM symbol, thereforethe gain is smaller, when using subframe structure, where someOFDM symbols do not carry any pilot-symbols. In this paper,we showed that at low SNR values the number of necessarypilot-symbols can be decreased, but their power have to besignificantly increased compared to the power of the datasymbols. On other hand, in order to obtain optimal systemperformance at high SNR values, pilot-symbols pattern withhigher density have to be utilized, though with smaller poweroffset.

ACKNOWLEDGMENTSThe authors would like to thank the LTE research group

for continuous support and lively discussions. This workhas been funded by the Christian Doppler Laboratory forWireless Technologies for Sustainable Mobility, KATHREIN-Werke KG, and A1 Telekom Austria AG. The financial support

by the Federal Ministry of Economy, Family and Youth andthe National Foundation for Research, Technology and Devel-opment is gratefully acknowledged. They are also thankful toCNPq and FAPERJ for their financial support of the secondauthor.

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