System level simulation of LTE networksJosep Colom Ikuno, Martin Wrulich, Markus Rupp

Institute of Communications and Radio-Frequency EngineeringVienna University of Technology, Austria

Gusshausstrasse 25/389, A-1040 Vienna, AustriaEmail: {jcolom, mwrulich, mrupp}@nt.tuwien.ac.at

Web: http://www.nt.tuwien.ac.at/ltesimulator

Abstract—In order to evaluate the performance of new mobilenetwork technologies, system level simulations are crucial. Theyaim at determining whether, and at which level predicted linklevel gains impact network performance. In this paper we presenta MATLAB computationally efficient LTE system level simulator.The simulator is offered for free under an academic, non-commercial use license, a first to the authors’ knowledge. Thesimulator is capable of evaluating the performance of the Down-link Shared Channel of LTE SISO and MIMO networks usingOpen Loop Spatial Multiplexing and Transmission Diversitytransmit modes. The physical layer model is based on the post-equalization SINR and provides the simulation pre-calculated”fading parameters” representing each of the individual inter-ference terms. This structure allows the fading parameters to bepregenerated offline, vastly reducing computational complexityat run-time.

I. INTRODUCTION

The Long Term Evolution (LTE) standard, specified bythe 3rd Generation Partnership Project (3GPP) in Release 8,defines the next evolutionary step in 3G technology. LTE offerssignificant improvements over previous technologies such asUniversal Mobile Telecommunications System (UMTS) andHigh-Speed Packet Access (HSPA) by introducing a novelphysical layer and reforming the core network. The mainreasons for these changes in the Radio Access Network(RAN) system design are the need to provide higher spectralefficiency, lower delay, and more multi-user flexibility than thecurrently deployed networks [2].

In the development and standardization of LTE, as wellas the implementation process of equipment manufacturers,simulations are necessary to test and optimize algorithms andprocedures. This has to be performed on both, the physicallayer (link-level) and in the network (system-level) context.

While link-level simulations allow for the investigationof issues such as Multiple-Input Multiple-Output (MIMO)gains, Adaptive Modulation and Coding (AMC) feedback,modeling of channel encoding and decoding [3] or physicallayer modeling for system-level [4], system-level simulationsfocus more on network-related issues such as scheduling [5],mobility handling or interference management [6].

Along with the standardization process, commercially avail-able LTE simulators have been developed. Equipment vendors,to this effect, have also implemented their own, proprietarysolutions. Some universities and research centers have alsodeveloped such simulators, but to the authors’ knowledge nonewith publicly available source code.

The LTE system-level simulator [1] supplements an alreadyfreely-available LTE link-level simulator [7]. This combinationallows for detailed simulation of both the physical layerprocedures to analyze link-level related issues and system-levelsimulations where the physical layer is abstracted from linklevel results and network performance is investigated.

The license under which the simulators are published al-lows for academic research and a closer cooperation betweendifferent universities and research facilities. In addition, devel-oped algorithms can be shared under the same license again,facilitating the comparison and cross validation of algorithmsand results and making them more credible.

The LTE system-level simulator implementation offers ahigh degree of flexibility. For the implementation, extensiveuse of the Object-oriented programming (OOP) capabilities ofMATLAB, introduced with the 2008a Release have been made.

Having a modular code with a clear structure based inobjects results in a much more organized, understandable andmaintainable simulator structure in which new functionalitiesand algorithms can be easily added and tested.

This paper is organized as follows: in Section II we describethe overall structure of the LTE system-level simulator. InSection III we show how the physical layer has been abstractedin the link measurement model. Afterwards, we present thelink performance model in Section IV, and Section V presentsthe main uses of the simulator as well as some conclusions.

II. SIMULATOR OVERVIEW

While link-level simulations are suitable for developingreceiver structures [8], coding schemes or feedback strate-gies [9], it is not possible to reflect the effects of issuessuch as cell planning, scheduling, or interference using thistype of simulations. Simulating the totality of the radio linksbetween the User Equipments (UEs) and eNodeBs is animpractical way of performing system level simulations dueto the vast amount of computational power that would berequired [10]. Thus, in system-level simulations the physicallayer is abstracted by simplified models that capture its essen-tial characteristics with high accuracy and simultaneously lowcomplexity.

Figure 1 depicts a schematic block diagram of the LTEsystem-level simulator. Similarly to other system-level sim-ulators, the core part consists of: (i) a link measurementmodel [11] and (ii) a link performance model [12].

mobility management

link-performancemodel

link-measurementmodel

base-station deploymentantenna gain patterntilt/azimuth

micro-scale fading

interferencestructure

macro-scale fadingantenna gainshadow fading

throughput error rates error distribution

traffic model

link adaptation strategy

resource schedulingstrategy

precoding

network layout

power allocation strategy

Fig. 1. Schematic block diagram of the LTE system level simulator

The link measurement model abstracts the measured linkquality used for link adaptation and resource allocation. Onthe other hand the link performance model determines the linkBlock Error Ratio (BLER) at reduced complexity.

As figures of merit, the simulator outputs traces containingthroughput and error rates, from which their distributions canbe computed.

Implementation-wise, the simulator flow follows thepseudo-code below. The simulation is performed by defininga Region Of Interest (ROI) in which the eNodeBs and UEsare positioned and a simulation length in Transmission TimeIntervals (TTIs). It is only in this area where UE movementand transmission of the Downlink Shared Channel (DLSCH)are simulated.

for each simulated TTI domove UEsif UE outisde ROI then

reallocate UE randomly in ROIfor each eNodeB do

receive UE feedback after a given feedback delayschedule users

for each UE do1- channel state → link quality model → SINR2- SINR, MCS → link perf. model → BLER3- send UE feedback

Where, ”→” represents the data flow in and out of the simu-lator’s link abstraction model. In the MATLAB implementation,the separated structure in the pseudo-code is maintained,allowing for easy adding of new functionalities and algorithms.

III. LINK MEASUREMENT MODEL

In order to abstract the measured link quality, and as shownin the pseudocode, the Signal to Interference and Noise Ratio

x pos (m)

y p

os (

m)

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macroscopic pathloss [dB] shadow fading [dB]

Fig. 2. Left: Macrosopic pathloss LM,b11,uj , 70 dB MCL, θ3dB =65◦/15 dBi antenna, 128.1 + 37.6 log10(R [Km]) pathloss. Right: space-correlated shadow fading LS,b11,uj

(SINR) has been utilized as metric [13]. Specifically, a per-subcarrier post-equalization symbol SINR.

The link measurement model abstracts the measurements forlink adaptation and resource allocation and aims at reducingrun-time computational complexity by pregenerating as manyof the needed parameters as possible. This shifts most ofthe computational burden to an off-line task that pregeneratesand stores the results in trace files that can be (re-)used atsimulation time.

Special care has been taken as to account for the spatial andtime correlation of the channel present in a wireless cellularsystem. To this effect, the link quality model has been splitinto three parts, which are afterwards combined to obtainpost-equalitzation symbol SINR expressions: (i) macroscopicpathloss, (ii) shadow fading, and (iii) small-scale fading (SISOand MIMO).

A. Macroscopic pathloss

The macroscopic pathloss between an eNodeB sector andUE is used to jointly model both the propagation pathlossdue to the distance and the antenna gain. It is noted asLM,bi,uj , where bi denotes the i-th transmitter: 0 for theattached eNodeB (desired signal) and 1, . . . , Nint for theNint interfering eNodeBs and uj the j-th UE, which thendetermines the (x, y) position.

It is implemented as a pathloss map that can be computedonce and, as long as the network layout is kept the same, bereused. The map specifies for each point in the simulated ROIthe macroscopic pathloss between any point (x, y) and eachtransmitter.

Figure 2 (left) depicts a generated macroscopic pathloss mapusing a θ3dB = 65◦/15 dBi antenna [14] and and a distance-dependent pathloss of 128.1 + 37.6 log10(R [Km]) [15].

B. Shadow fading

Shadow fading, LS,bi,uj , is caused by obstacles in thepropagation path between the UE and the eNodeB and canbe interpreted as the irregularities of the geographical charac-teristics of the terrain introduced with respect to the averagepathloss obtained from the macroscopic pathloss model. It is

typically approximated by a log-normal distribution of mean0 dB and standard deviation 10 dB [14], [16].

While for simulating small scale fading a one-dimensionalrandom function of time may suffice [17], as the waveformchanges significantly even for small amounts of movement,this approach cannot adequately model the effects of shadowfading.

As shadowing effects occur over a large area, in order tobe able to capture the dynamics affecting macro-cell diversityin a realistic way a two-dimensional Gaussian process withappropriate spatial correlation is desirable [18]. For our mod-eling, a low-complexity method capable of introducing spacecorrelation into the Gaussian process while still preserving itsstatistical properties as well as inter-site correlation has beenused [19].

Figure 2 (right) depicts the resulting space-correlatedshadow fading map for a given eNodeB. A UE traversing theROI will experience a slowly changing pathloss (LS,b0,u0

) dueto shadow fading, LS,b0,u0

being correlated with LS,bi,u0, i =

1, . . . , Nint. Thus avoiding the unrealistic simulation scenariowhere spatially close UEs would have uncorrelated shadowfading losses.

C. Channel modeling

While the losses caused by the macroscopic pathloss andthe shadow fading are position-dependent and time-invariant,small-scale fading is modeled as a time-dependent process.

For each of the modeled MIMO transmission modes(Transmission Diversity (TxD) and Open Loop SpatialMultiplexing (OLSM)), a model based on a simple ZeroForcing (ZF) receiver has been developed. As of this version,systems with two transmit antennas have been modeled, butthe derived SINR expressions can be easily extended for theLTE transmit modes using four antenna ports. Based on thederived models, a trace of fading parameters modeling thetime-and-frequency variant behavior of the channel has beengenerated. These fading-parameters furthermore allow for ageneration prior to the system level simulation itself, whichreduces the run-time computational complexity significantly.

The channel modeling aims at computing a per-layer SINR.In LTE, a spatial layer is the term used for the differentstreams generated by spatial multiplexing. A layer thus can bedescribed as a mapping of symbols onto the transmit antennaports. Each layer is then identified by a (precoding) vector ofsize equal to the number of transmit antenna ports [20].

1) MIMO OLSM modeling: The LTE OLSM MIMO¸ trans-mission mode consists of a precoding for Spatial Multiplexing(SM) with large-delay Cyclic Delay Diversity (CDD) [21]. Inthis mode, the precoding is defined by: y(0)(i)

...y(Nt−1)(i)

=W (i)D(i)U

x(0)(i)...

x(ν−1)(i)

Where Nt specifies the number of transmit antennas and νthe number of layers, and D and U introduce the large-delayCDD.

Noting F the WDU matrix product, H0 the channel for thereceived signal, H1−Nint

the channel from the i-th interferingeNodeB, from a total of Nint, and an estimated channel H wehave

x = (H0F )+(H0F )x0 + (H0F )

+n+

Nint∑i=1

(H0F )+(HiF )xi,

where ’+’ denotes the pseudoinverse. Denoting A =(HF )+(HF ), B = (HF )+ and Ci = (H0F )

+(HiF ), anddenoting the matrix elements as aij , A[i, j] we can expressthe SINRi for the symbols received in layer i as

SINRi =|aii|2 Pi∑

j6=i

|aij|2 Pj + σ2ν∑

k=1

|bik|2 +Nint∑l=1

ν∑m=1

|cl,im|2 Pl,m

(1)Where Pi is the power received at layer i after macro

and shadow fading losses and σ2 the receiver noise, assumeduncorrelated. Assuming a homogeneous power distributionPl = Ptx/ν, we defined the following fading parameters:ζ = |aii|2 and ξ =

∑j 6=i |aij |2, which model channel

estimation errors (A = Iν×ν for perfect channel knowl-edge), ψ =

∑νk=1 |bik|2, which models the ZF receiver

noise enhancement, and θ =∑νm=1 |cl,im|2, modeling the

interference. We can then express SINRi,u for UE u as

SINRi,u =ζi LM,0,u LS,0,u Pl

ξiPl + ψiσ2 +

Nint∑l=1

θi,l LM,l,u LS,l,u Pl,m

(2)

Where LM,bi,u and LS,bi,u represent the macro and shadowfading pathlosses between the UE u and its attached eNodeB(for bi = 0) and its interferers (bi = 1, . . . Nint) respectively.

2) MIMO TxD modeling: For TxD, the precoding operationfor the two TX antenna case uses the Alamouti scheme [21],[22], which can be written as[

y0

y∗1

]︸︷︷︸y

=

[h(0) h(1)

h(1)∗ −h(0)∗

]︸ ︷︷ ︸

H

·[x0

x1

]︸︷︷ ︸x

+

[n0

n1

]︸ ︷︷ ︸n

Where h(0) and h(1) contain the channel coefficients fromthe first and second transmit antennas to all NR receiveantennas

Similarly to the OLSM case, for the H0 channel and Hi

(i = 1, ..., Nint) interfering channels, A = ˆH+0 H0, B = ˆH+

0

and Ci = ( ˆH0)+(Hi) have been defined. Then, similarly

to Equation (1), we obtain

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BLE

R

LTE BLER, CQIs 1-15, SISO AWGN, 5000 subframes

Fig. 3. BLER curves obtained from 1.4 MHz, single-user, 5000 subframeslong, SISO AWGN simulations for all 15 CQI values. From CQI 1 (leftmost)to CQI 15 (righmost)

SINRi =|aii|2 Pl

Pl

∑j 6=i

|aij |2 + σ2ν∑k=1

|bik|2 +Nint∑l=1

ν∑m=1

|cl,im|2 Pl,m

SINRi can then be expressed in an identical form asfor the OLSM case (Equation (2)) by having ζ = |aii|2,ξ =

∑j 6=i |aij |2, ψ =

∑νk=1 |hik|2 and θl =

∑νm=1 |cl,im|2.

3) SISO modeling: As for the Single-Input Single-Output(SISO) case, the SINR for a given subcarrier can be writtenas

SINR =Ptx

1

|h0|2σ2 +

Nint∑l=1

|hl|2

|h0|2Ptx,l

,

thus only a trace of the noise and the interference parameters1

|h0|2and|hl|2

|h0|2is needed.

IV. LINK PERFORMANCE MODEL

The link performance model determines the BLER at thereceiver given a certain resource allocation and Modulationand Coding Scheme (MCS). For LTE, 15 different MCSs aredefined, driven by 15 Channel Quality Indicator (CQI) values.The defined CQIs use coding rates between 1/13 and 1 com-bined with 4-QAM, 16-QAM and 64-QAM modulations [23].

To asses the BLER of the received Transport Blocks (TBs),a set of Additive White Gaussian Noise (AWGN) link-levelperformance curves are employed. The SINR-to-BLER map-ping then requires of an effective SINR value γeff , obtainedfrom mapping the set of sub-carrier-SINRs assigned to the UETB to an AWGN-equivalent SINR. Figure 3 shows the SISOAWGN¸ BLER curves the link performance model utilizes [7].

The Exponential Effective Signal to Interference and NoiseRatio Mapping (EESM) [24], [25], [26] is the method currentlyused to obtain a TB effective SINR γeff which can be used

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I

Fig. 4. CQI mapping. BLER=10% points from the BLER curves (left) andSINR-to-CQI mapping function (right)

to map to the BLER obtained from AWGN link-level simula-tions. The effective SINR γeff is obtained by performing thefollowing non-linear averaging of the several Resource Block(RB) SINRs:

γeff = EESM(γi, β) = −β · ln

(1

N·N∑i=1

e−SINRiβ

).

Where N is the total number of sub-carriers to be averagedand β is calibrated by means of link level simulations to fitthe compression function to the AWGN BLER results [13].

It is possible to consider not all of the TB sub-carriersbut only a subset, as long as the frequency spacing betweentwo SINR values does not exceed half of the coherencebandwidth [13]. Thus, in our channel traces, we reduced theamount of memory needed for a simulation by using only twosub-carrier SINRs per RB to obtain γeff .

Using AWGN BLER curves, γeff is mapped to BLER. Itis then decided via a coin toss whether the given receivedTB was received correctly and ACK reporting is subsequentlygenerated.

Related to the link performance model, the CQI feedbackreporting provides the eNodeB with a figure of merit of thestate of the channel of the UE. For the CQI feedback strategy,the SINR-to-CQI mapping is realized by taking the 10% pointsof the BLER curves, obtaining the mapping shown in Figure 4.The obtained CQIs are afterwarfds floored to obtain the integerCQI values that are reported back to the eNodeB.

V. MAIN USES, SIMULATION RESULTS AND CONCLUSIONS

In this paper we present a LTE system level simulatorcapable of simulating LTE SISO and MIMO networks usingTxD or OLSM transmit modes and offered for free under anacademic, non-commercial use license.

The main purpose of this tool is to assess the networkperformance increase of new scheduling algorithms (Figure 5).

Testing Fractional Frequency Reuse (FFR) strategies im-plemented at the scheduler level, as well as the networkimpact of different receiver types and channel quality feedbackstrategies, provided accurate modeling of those, can also betested. SINR optimization (Figure 6) via electrical [27] andmechanical tilting and with pathloss maps imported from

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Fig. 5. UE and cell throughput CDFs: TxD and OLSM, max C/I and roundrobin schedulers

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Fig. 6. Sector SINR, calculated with distance dependent macroscale pathlossonly (left) [15] and additional lognormal-distributed space-correlated shadowfading [19] (right)

network planning tools can also be easily added, thus enablingvalidation simulations against network planning tools, whichusually use an even more abstract modeling of the physicallayer

The availability of its source code allows its results tobe cross-checked and validated and researchers to comparealgorithms in a standardized system. Together with the LTElink-level simulator [7], it forms, to the best of the authors’knowledge, the only link-and-system LTE simulation suiteopenly available for research purposes.

ACKNOWLEDGMENT

The authors would like to thank the whole LTE researchgroup for continuous support and lively discussions. This workhas been funded by the Christian Doppler Laboratory forWireless Technologies for Sustainable Mobility, as well asmobilkom austria AG, KATHREIN-Werke KG, and the In-stitute of Communications and Radio-frequency Engineering.The views expressed in this paper are those of the authors anddo not necessarily reflect the views within the partners.

REFERENCES

[1] [Online]. Available: http://www.nt.tuwien.ac.at/ltesimulator/[2] E. Dahlman, S. Parkvall, J. Skold, and P. Beming, 3G Evolution: HSDPA

and LTE for Mobile Broadband. Academic Press, Jul. 2007.

[3] J. Colom Ikuno, M. Wrulich, and M. Rupp, “Performance and modelingof LTE H-ARQ,” in Proc. ITG International Workshop on SmartAntennas (WSA), Berlin, Germany, Feb. 2009.

[4] M. Wrulich and M. Rupp, “Computationally efficient MIMO HSDPAsystem-level evaluation,” EURASIP Journal on Wireless Communica-tions and Networking, vol. Volume 2009, 2009.

[5] C. Shuping, L. Huibinu, Z. Dong, and K. Asimakis, “Generalizedscheduler providing multimedia services over HSDPA,” in Proc. IEEEInternational Conference on Multimedia and Expo, 2007, pp. 927–930.

[6] M. Castaneda, M. Ivrlac, J. Nossek, I. Viering, and A. Klein, “Ondownlink intercell interference in a cellular system,” in Proc. IEEE18th International Symposium on Personal, Indoor and Mobile RadioCommunications (PIMRC), 2007, pp. 1–5.

[7] C. Mehlfuhrer, M. Wrulich, J. C. Ikuno, D. Bosanska, and M. Rupp,“Simulating the long term evolution physical layer,” in Proc. of the 17thEuropean Signal Processing Conference (EUSIPCO 2009), Glasgow,Scotland, Aug. 2009.

[8] M. Simko, C. Mehlfuhrer, M. Wrulich, and M. Rupp, “Doubly Dis-persive Channel Estimation with Scalable Complexity,” in Proc. ITGInternational Workshop on Smart Antennas (WSA), Bremen, Germany,Feb. 2010.

[9] S. Schwarz, M. Wrulich, and M. Rupp, “Mutual information basedcalculation of the precoding matrix indicator for 3GPP UMTS/LTE,” inProc. ITG International Workshop on Smart Antennas (WSA), Bremen,Germany, Feb. 2010.

[10] M. Wrulich, W. Weiler, and M.Rupp, “HSDPA performance in a mixedtraffic network,” in Proc. IEEE Vehicular Technology Conference (VTC)Spring 2008, May 2008, pp. 2056–2060.

[11] M. Wrulich, S. Eder, I. Viering, and M. Rupp, “Efficient link-to-systemlevel model for MIMO HSDPA,” in Proc. of the 4th IEEE BroadbandWireless Access Workshop, 2008.

[12] K. Brueninghaus, D. Astely, T. Salzer, S. Visuri, A. Alexiou, S. Karger,and G.-A. Seraji, “Link performance models for system level simulationsof broadband radio access systems,” Sept. 2005.

[13] M. of WINNER, “Assessment of advanced beamforming and MIMOtechnologies,” WINNER, Tech. Rep. IST-2003-507581, 2005.

[14] Technical Specification Group RAN, “E-UTRA; LTE RF system scenar-ios,” 3rd Generation Partnership Project (3GPP), Tech. Rep. TS 36.942,2008-2009.

[15] ——, “Physical layer aspects for E-UTRA,” 3rd Generation PartnershipProject (3GPP), Tech. Rep. TS 25.814, 2006.

[16] W. C. Lee, Mobile Communications Engineering. McGraw-Hill Pro-fessional, 1982.

[17] W. C. Jakes and D. C. Cox, Microwave Mobile Communications. Wiley-IEEE Press, 1994.

[18] X. Cai and G. Giannakis, “A two-dimensional channel simulation modelfor shadowing processes,” Vehicular Technology, IEEE Transactions on,Nov. 2003.

[19] H. Claussen, “Efficient modelling of channel maps with correlatedshadow fading in mobile radio systems,” Sept. 2005.

[20] S. Sesia, I. Toufik, and M. Baker, LTE, The UMTS Long Term Evolution:From Theory to Practice. John Wiley & Sons, 2009.

[21] Technical Specification Group RAN, “E-UTRA; physical channels andmodulation,” 3rd Generation Partnership Project (3GPP), Tech. Rep. TS36.211 Version 8.7.0, May 2009.

[22] S. Alamouti, “A simple transmit diversity technique for wireless com-munications,” IEEE Journal on selected areas in communications, 1998.

[23] Technical Specification Group RAN, “E-UTRA; physical layer pro-cedures,” 3rd Generation Partnership Project (3GPP), Tech. Rep. TS36.213, Mar. 2009.

[24] E. Tuomaala and H. Wang, “Effective SINR approach of link to systemmapping in OFDM/multi-carrier mobile network,” in Mobile Technology,Applications and Systems, 2005 2nd International Conference on, Nov.2005.

[25] X. He, K. Niu, Z. He, and J. Lin, “Link layer abstraction in MIMO-OFDM system,” in Cross Layer Design, 2007. IWCLD ’07. InternationalWorkshop on, Sept. 2007.

[26] R. Sandanalakshmi, T. Palanivelu, and K. Manivannan, “Effective SNRmapping for link error prediction in OFDM based systems,” in Infor-mation and Communication Technology in Electrical Sciences (ICTES2007), 2007. ICTES. IET-UK International Conference on, Dec. 2007.

[27] L. Thiele, T. Wirth, K. Brner, M. Olbrich, V. Jungnickel, J. Rumold, andS. Fritze, “Modeling of 3D Field Patterns of Downtilted Antennas andTheir Impact on Cellular Systems,” in Proc. ITG International Workshopon Smart Antennas (WSA), Berlin, Germany, Feb. 2009.