low complexity adaptive channel estimation and qr

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Low Complexity Adaptive Channel Estimation and QR Decomposition for an LTE-ADownlink

Gangarajaiah, Rakesh; Nilsson, Peter; Edfors, Ove; Liu, Liang

Published in:2014 IEEE 25th Annual International Symposium on Personal, Indoor, and Mobile Radio Communication(PIMRC)

DOI:10.1109/PIMRC.2014.7136209

2015

Link to publication

Citation for published version (APA):Gangarajaiah, R., Nilsson, P., Edfors, O., & Liu, L. (2015). Low Complexity Adaptive Channel Estimation and QRDecomposition for an LTE-A Downlink. In 2014 IEEE 25th Annual International Symposium on Personal, Indoor,and Mobile Radio Communication (PIMRC) IEEE - Institute of Electrical and Electronics Engineers Inc..https://doi.org/10.1109/PIMRC.2014.7136209

Total number of authors:4

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https://doi.org/10.1109/PIMRC.2014.7136209

https://portal.research.lu.se/en/publications/ac870044-57f1-4f5a-bbc0-2f839f99ea61

https://doi.org/10.1109/PIMRC.2014.7136209

FINAL MANUSCRIPT: IEEE 25TH ANNUAL INTERNATIONAL SYMPOSIUM ON PERSONAL, INDOOR AND MOBILE RADIO COMMUNICATION (PIMRC). 2014 1

Low Complexity Adaptive Channel Estimation andQR Decomposition for an LTE-A Downlink

Rakesh Gangarajaiah, Peter Nilsson, Ove Edfors and Liang Liu Department of Electrical and InformationTechnology, Lund University, Sweden

Email: {rakesh.gangarajaiah,peter.nilsson,ove.edfors,liang.liu}@eit.lth.se

Abstract—This paper presents a link adaptive processor to per-form low-complexity channel estimation and QR decomposition(QRD) in Long Term Evolution-Advanced (LTE-A) receivers. Theprocessor utilizes frequency domain correlation of the propaga-tion channel to adaptively avoid unnecessary computationsinthe received signal processing, achieving significant complexityreduction with negligible performance loss. More specifically, awindowed Discrete Fourier transform (DFT) algorithm is used todetect channel conditions and to compute a minimum number ofsparse subcarrier channel estimates required for low complexitylinear QRD interpolation. Furthermore, the sparsity of subcarrierchannel estimates can be adaptively changed to handle differentchannel conditions. Simulation results demonstrate a reductionof 40%-80% in computational complexity for different channelmodels specified in the LTE-A standard.

Keywords—Wireless communication, MIMO, Channel estima-tion, OFDM, Adaptive signal processing.

I. I NTRODUCTION

The requirement for high speed wireless communication,limited frequency bands and fluctuating channel conditionshas made Multiple-Input Multiple-Output (MIMO) a widelyadopted technique in many radio standards, including the 3GPPLong Term Evolution-Advanced (LTE-A). To fully utilize thecapabilities of LTE-A systems, sophisticated signal processingoperations are required. Among others, accurate channel es-timation and the following channel matrix QR decomposition(QRD) are indispensable for advanced MIMO signal detectors,e.g., the K-Best detector [1], to recover transmitted informationfrom noisy received signals. Different algorithms for channelestimation are presented in [2] [3] and the most frequentlyused QRD algorithms are detailed in [4]. These essentialsignal processing operations of a MIMO system come with theprice of high computational complexity, preventing accurateimplementation of the corresponding algorithms in power andarea limited handheld devices.

MIMO is usually combined with Orthogonal FrequencyDivision Multiplexing (OFDM) to provide high spectral ef-ficiency. As a result, channel estimation and QRD have to beperformed on a tone-by-tone basis, making the complexity amore critical issue. To alleviate this complexity problem,au-thors in [5] present an interpolation technique, where channelestimation and QRD are performed only on pilot tones and theQRD for tones in between pilots is obtained by interpolation.However, the algorithm is implemented for a fixed interpo-lation distance and also requires a translation into anotherdomain for performing QRD interpolation. Moreover, such a

static interpolation strategy may degrade performance in highlyfrequency selective channels while resulting in unnecessarycomputations in channels with low frequency selectivity.

Mobile devices operate in fluctuating channel scenariosdepending on their surroundings and the LTE-A standardclassifies wireless channels into three main categories based onthe frequency selectivity, namely the EPA, EVA and ETU. TheEPA channel has a very low frequency selectivity and requiresfewer subcarrier channel estimates than the highly frequencyselective ETU channel to reach a target system bit errorrate (BER). Hence, a fixed solution such as the tone-by-toneapproach or the one presented in [5] is not efficient in terms ofthe number of computations performed. To alleviate the afore-mentioned problem, we propose an adaptive solution whichtakes into account the dynamic nature and frequency selectivityof the wireless channel. In detail, the proposed solution utilizesa windowed Discrete Fourier transform (DFT) based channelestimator to produce only a required number of subcarrierchannel estimates enabling very low complexity linear QRDinterpolation. The windowed DFT method also provides asimple way of detecting operating channel conditions, enablingthe adaptive processor to optimize the interpolation distance,measured in subcarriers, to reach a desired BER with thelowest computational efforts. To verify the proposed scheme,we simulated a simplified LTE-A downlink system with a4× 4 MIMO setup. Simulations performed with the EVA andETU channels show that the proposed method offers significantcomplexity saving over the traditional tone by tone method,with minor performance loss.

II. BACKGROUND

A MIMO system withM transmitter (Tx) and receiver (Rx)antennas can be modelled as

y = Hx+ n , (1)

where y = [y1, y2, ..., yM ]T is the received data vector,

H ∈ CM×M is the channel gain matrix between the antennas,x = [x1, x2, ..., xM ]

T is transmit data vector andn is theadditive white Gaussian noise (AWGN). To achieve a lowBER, the MIMO symbol detector has to minimize the error‖ y − Hx ‖2, wherex is the estimate of the transmit vectorand H is the estimated channel matrix obtained from a setof predefined pilot tones. Fig. 1 shows the structure of thesepredefined pilots in a4×4 LTE-A system and the redundancyin pilots enables good performance even under high frequencyselectivity.


0 0

P

P

P

P

An

ten

na

po

rt 3

An

ten

na

po

rt 2

P

P

P

Pilot tone Reserved pilot toneData tone

P

P

P

P

PAn

ten

na

po

rt 1

An

ten

na

po

rt 0

P

Time

Fre

qu

ency

Symb.

0

Symb.

4 Time slot

Fig. 1: Pilot structure of a4× 4 LTE-A system

Estimation of the channel gain matrixH is the first majorstep performed in order to recover data from (1) and accurateestimation enables data detection with a low BER. Further-more, since there areM2 paths between the Tx and Rx inM×M MIMO systems, estimators with lower complexity arepreferred. Once the estimateH is obtained, advanced decoderssuch as the K-best decoder solve (1) by decomposing thechannel gain matrixH into a product of a unitaryQ and anupper triangular matrixR. The unitary matrix is used to rotatethe received vector and the resulting linear system is solvedby using tree search techniques as

y = QRx+ n

Q∗y = Rx+Q∗n

y = Rx+ n

(2)

whereQ∗ is the Hermitian transpose of the unitaryQ matrix.The complexity of popular QRD algorithms, measured innumber of multiplications, isO(M3) [4]. Hence, in LTE-Areceivers, algorithms producing accurate channel estimationand QRD with low complexity are needed to utilize the fullpotential of MIMO systems.

The traditional approach for solving the signal detectionproblem has been by considering channel estimation and QRDas separate entities as shown in Fig. 2(a). Channel estimationis performed as an independent operation followed by QRDfor all the data tones. Several methods of channel estimationsuch as the Least Squares (LS), Robust Minimum Mean SquareEstimator (RMMSE), DFT and Matching Pursuit (MP) [2] [3]haven been proposed. DFT based estimators are a class of lowcomplexity estimators which utilize the DFT and inverse DFToperations to perform noise filtering, but suffer from a highnoise floor due to spectral leakage [6]. Authors in [7] suggesta method to utilize windows to weigh the data to minimize thespectral leakage enabling better performance. The advantageof DFT based estimators is their efficient hardware imple-mentation through Fast Fourier Transform (FFT) algorithms.The complexity of this implementation measured in terms oftotal multiplications isO

(

M2 (Nplog2 (Np) +Ndlog2 (Nd)))

whereNp is the number of pilot tones andNd is the numberof channel estimates produced for anM ×M MIMO system.

QRD which is performed on the channel estimates can beimplemented by several techniques such as the Given’s rotation(GR) [8], the Gram-Schmidt (GS) [9] or the Householder trans-form , but all have a complexityO(M3Nd). Even though both

(a)

Traditional Detection Link Adaptive Detection

(b)

Perform Pilot

LS estimation

Estimate channel

parameters

Partial channel

estimation

QRD

interpolation

Data detection

Perform Pilot

LS estimation

Full channel

estimationQRD

Interpolation

Data detection

Full channel

QRD

SNR

estimation

Fig. 2: Data detection flow: Traditional Vs Proposed

channel estimation and QRD are computationally intensive,ithas to be noted that in a typical LTE-A system using DFTbased channel estimators needs more computations for channelestimation than the QRD.

To reduce the high complexity of the above tone by toneapproach, interpolation techniques have been used as shownin Fig. 2(a). The channel estimates at the pilot positions areestimated, followed by QRD interpolation for the data tonesinbetween these pilot positions. A theoretical background whenusing the GS method to obtain lossless QRD interpolationis provided in [10] and a hardware implementation for anLTE-A frame structure utilizing only pilot positions to performQRD interpolation is presented in [5]. These techniques relyon mapping theQ and R matrices into a sub space wherepolynomial interpolation can be applied and the de-mappingthe interpolated matrices. Even though these methods aremore efficient, they do not utilize channel properties suchas frequency selectivity to further optimize the number ofsubcarrier channel estimates and QRDs performed. The fixedarchitecture of interpolating over pilot positions offerslittleflexibility and results in many redundant computations inchannels with low frequency selectivity and a performance lossin highly frequency selective channels.

III. L INK ADAPTIVE QR DECOMPOSITION

Mobile devices experience high frequency selectivity in richmultipath environments requiring channel estimation at datatones which are closely spaced whereas in low frequencyselectivity environments subcarrier channel estimates can beproduced at tones which are farther apart. Moreover, whenoperating in higher received Signal to Noise Ratios (SNRs),a more accurate channel estimation and QRD is requiredwhereas in lower SNRs the gain obtained by performing accu-rate channel estimation and QRD is lost due to the high noiselevels. Hence, by examining the current channel conditions, areduction in the total number of computations can be obtainedby tuning the channel estimator and QRD processor to executeonly the minimum number of required computations to reacha desired level of system performance.

Fig. 2(b) depicts such an adaptive approach where theoperating channel conditions such as the frequency selectivityare obtained by examining the LS estimates of the pilot tones


with the Signal to Noise Ratio (SNR) estimates obtainedfrom external processing units [11]. Based on these channelparameters, an adaptive channel estimator is utilized to produceestimates at tones much farther apart than the pilots whenoperating in an EPA channel or at tones much closer thanthe pilot positions when operating in the ETU channel. Thisestimated channel data is used by a low complexity QRDinterpolation method to approximate theQ and R matricesof the intermediate data tones. The proposed link adaptiveprocessor incorporates these ideas and the following sectionsintroduce the different components of this processor alongwith the methodology used to select the optimal distances forthe low complexity QRD interpolation. A DFT based channelestimator is used in the link adaptive processor due to itsreconfigurability and efficient hardware implementation and amethod of mitigating the spectral leakage is discussed next.

A. Windowed DFT channel estimator

DFT based estimators perform an inverse DFT operationon the pilot tones, weigh the taps according to a predefinedstrategy aimed at filtering noise and perform a DFT to producethe channel estimates at the data positions. Such an estimatorproducingNd estimates withNp pilots can be represented as

hd = FdWFHp hp, (3)

wherehd is the vector of channel estimates at data tones,Fd is a Nd × Nd DFT matrix, W is theNd × Np filteringmatrix,Fp is theNp×Np DFT matrix andhp is the vector ofchannel estimates at pilot tones. These estimators performwellat lower SNRs and are hardware efficient when implementedusing the FFT algorithm but suffer from a high noise floor dueto spectral leakage at higher SNRs [6].

The basic requirement of the proposed link adaptive proces-sor is the ability to recognize the operating channel conditions,which can be achieved by either analyzing the cyclic prefix orby an inverse DFT operation on the pilot tones. Fig. 3 showsthe average power in the different taps of LTE-A channelsobtained by using a 64 point inverse FFT operation on thepilot tones in a5MHz bandwidth downlink. We notice thatby analyzing the distribution of power in the first few taps ofthe inverse FFT output, the current channel conditions can bedetected. Furthermore, the inverse DFT operation is part ofthe DFT based estimator represented by (3) and enables thedetection of channel conditions at no additional cost.

Fig. 3 also shows the spectral leakage due to the non samplespaced channels which causes degraded performance at higherSNRs. Utilizing windows to reduce spectral leakage is a well-known method and [7] describes a method to improve theperformance of DFT based estimators at higher SNRs by usingHann windows. An inverse window operation is required toremove the effects of windowing and [7] uses a division bythe Hann window to achieve this. The overlap-add methodis another way of removing windowing effects and the linkadaptive processor uses this method as depicted in Fig. 4.The data from the first and last pilots in the5MHz spectrumis extended to produce128 points and the Hann windowingreduces spectral leakage. Finally, the windowing effects on the

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

IFFT Taps

No

rmal

ized

Po

wer

EPAEVAETU

Spectral Leakage

Filtering

Fig. 3: Average power in taps of a 5 MHz LTE-A downlink

0 20 40 60 80 100 120

−1

−0.5

0

0.5

1

Actual channel response

of 50 samples

Pilots

Extension for overlap add

Hann windowFFT width of 64 points

Extension for

overlap add

Fig. 4: Overlap add method to reduce spectral leakage

channel response is removed by using the overlap-add methodto produce interpolated channel estimates at the data tones.

B. QR interpolation

The windowed DFT based channel estimator enables thedetection of channel conditions such as frequency selectivity,which can be used to estimate the distance, measured insubcarriers, for QRD interpolation. The effects of interpolationerrors on the system BER can be minimized by adaptivelychanging the interpolation distance depending on channelconditions, for example, by choosing subcarriers which areclose when operating in highly frequency selective channels.

All unitary Q matrices of sizeM×M are part of the unitarygroupU(M) and any unitary matrixQ1 can be transformedinto another unitary matrixQ2 by using a rotation matrix ofthe formQ2Q

∗1. Authors in [12] present a method of obtaining

the intermediateQ matrices betweenQ1 andQ2 using

Q(s) = (Q2Q∗1)

sQ1 s ∈ R | 0 ≤ s ≤ 1 . (4)

If ‖ Q1 −Q2 ‖F= ǫ, where ǫ is a small constant, a linearinterpolation of the form

Q(s) = (1 − s)×Q1 + s×Q2, (5)

can be used to approximate (4). The correspondingR(s)matrix can also be approximated by

R(s) = (1 − s)×R1 + s×R2. (6)

These approximations lead to errors in QRD interpolationand a strategy to choose the correct interpolation distances,


50 60 70 80 900

5

10

15

0

5

30

35

Correlation

EPEET

100 100

a

50 60 0 80 900

1

3

4

5

6

x 10

Fig. 5: Interpolation error, Subcarrier distances and Correlation

to minimize the effect of these errors on BER for differentchannel scenarios is presented in the next section.

C. Coherence Bandwidth and Interpolation error

The coherence band width (Bcoh) of a wireless channel isdefined as the bandwidth over which the correlation of channelgains is higher than a specified limit [13]. QRD of channelgain matricesH1 andH2 of two correlated subcarriers resultsin Q1 and Q2 such that‖ Q1 − Q2 ‖F= ǫ, where ǫ is asmall constant. This enablesBcoh to be used as a parameterto evaluate the interpolation error due to approximations in (6).The LTE-A standard uses three main channel models and eachchannel exhibits a differentBcoh enabling the link adaptiveprocessor to choose between different bandwidths over whichQRD interpolation can be performed. Fig. 5(b) shows thedistances in sub carriers for different levels of correlation forthe three models. In the EPA model, gain matricesH1 andH2 which are 24 subcarriers apart show a correlation of 75%whereas the EVA and ETU models show 75% correlation forchannels around 15 subcarriers apart.

The uncoded BER of a wireless system operating in afrequency selective channel affected by AWGN is inverselyproportional to the SNR available at the receiver. The totalnoisen in a wireless receiver employing interpolation tech-niques can be expressed as

n = nsys + ninterp (7)

where nsys is the system noise andninterp is the noiseintroduced due to linear interpolation. A parameter

γ =ninterp

nsys

, (8)

can be used to decide the amount of interpolation error thatcan be introduced depending on the receiver SNR. The effectsof interpolation error on BER can be minimized by keepingγ small, which enables the link adaptive processor to increaseinterpolation distances at lower SNRs and to adaptively lowerthe distances at higher SNRs. Fig. 5(a) shows the dependenceof the errorninterp obtained by interpolatingQ matrices using

(5) between sub carriers spaced at different correlation levelsfor the three channel models used in LTE-A. The average errordue to interpolation in an EPA channel model is in the order of10−3 at correlation levels of 75% whereas the error in the fastfading ETU model is almost double that of the EPA model.

Using Fig. 5(a) and Fig. 5(b) the following strategy illus-trates how a link adaptive processor can be used to reduce totalcomputational complexity. An SNR of 20 dB at the receiverin a 4 × 4 MIMO system receiving 16 QAM data wouldcorrespond to ansys of 10−2. A link adaptive processorconfigured to operate with a value ofγ = 0.1 can operate withninterp = 10−3. If the current operating channel is detected tobe EPA, the processor will choose a correlation value of 75%in Fig. 5(a) corresponding to an interpolation distance of 24subcarriers in Fig. 5(b). Using a similar strategy, if operatingin an ETU channel, correlation of 85% is chosen leading tointerpolation distances of around 10 subcarriers. A lower valueof SNR will result in highernsys enabling the link adaptiveprocessor to choose subcarriers which have a lower correlationvalue leading to increased interpolation distances.

IV. RESULTS

A. Performance

The methodology described in the previous sections enablesus to choose interpolation distances adaptively to reduce com-plexity while maintaining the required level of BER. Table Ishows an example implementation with different interpolationdistances chosen for the three LTE-A channel models depend-ing on the SNR available at the receiver. The interpolationdistances are chosen so that minimal loss to BER is introducedwhen compared to the performance with perfect channel stateinformation (CSI).

TABLE I: Interpolation distances measured in subcarriers❳❳❳❳❳❳❳

ModelSNR (dB)

≤ 10 11 -15 16 -20 21 -25 26 -30 > 30

EPA 48 32 24 16 8 8

EVA 32 24 16 8 8 4

ETU 32 24 16 8 4 2

Fig. 6 shows two sets of uncoded BER curves for a4 × 4MIMO system with a K-Best decoder with K=10 operatingin the EVA channel. The first set of curves are obtainedby using perfect CSI and full channel QRD along with theproposed adaptive QRD (AQRD) with distances chosen fromTable I. These curves enable us to analyze the effects ofninterp on BER and it can be seen that the performanceloss is negligible when choosing the proposed interpolationdistances. The second set of curves are obtained by usingdifferent channel estimation techniques and QRD interpolationmethods. The DFT based estimator [2] with the proposedadaptive QRD shows significant degradation, with an errorfloor visible at higher SNRs due to spectral leakage. Use of theproposed windowed DFT based estimator and adaptive QRDimproves the performance at higher SNRs and is on par withperformance of a receiver employing the RMMSE estimator[14] with the QRD interpolator from [5].


12 14 16 18 20 22 24 26 28 3010�4

10�3

10�2

10�1

100

Perfect CSI + Full QRDPerfect CSI + AQRDWindowed DFT + AQRD

DFT[2] + AQRD

RMMSE[13] + QRD Interp[5]

Fig. 6: BER of an uncoded 16QAM system

B. Complexity analysis

A DFT based channel estimator used in the traditional flowof Fig.2(a) for anM × M MIMO system would require(Nplog2(Np) +Ndlog2(Nd))M

2 multiplications. Using theproposed windowed DFT instead and choosing the numberof channel estimationsX , changes the multiplication countof channel estimation to(3Nplog2(Np) + Xlog2(X))M2.Furthermore, the complexity of QRDs computed also reducesfrom NdM

3 to XM3.Fig. 7 shows the savings obtained by using the link adaptive

processor over the traditional tone by tone approach, withinterpolation distances chosen from Table I. The link adaptiveprocessor is designed to closely follow the BER obtained whenoperating with perfect CSI. Higher savings are obtained atlower SNRs as farther interpolation distances can be chosen,whereas this gain reduces at higher SNRs. Furthermore, EPAchannels need significantly lower number of computationsresulting in higher savings when compared to ETU channels.The shaded region in Fig. 7 indicates the range of possiblereductions when operating in different channel conditions.

C. Hardware requirements

The main advantage of DFT based channel estimators is theefficient hardware implementation. A typicalNd point radix-2 FFT is implemented using a pipelined structure [15]. Theorder of the output samples from the pipelined decimationin frequency (DIF) FFT algorithm is bit reversed with thefirst X = 2t where t ∈ Z output bins spaced at distancesof Nd

Xand the nextX outputs resulting in all bins at a

resolution ofNd

2X, enabling selective channel estimation at only

the desired frequency bins. For example, when operating inthe EPA channel with an SNR of 15 dB, from Table I, channelestimates which are spaced32 subcarriers apart are needed.The first 16 outputs from a pipelined512 point radix-2 DIFFFT are at the bins[0, 32, 64 . . .] corresponding to the desiredchannel estimates. This enables circuit level optimizations suchas clock gating which can be activated once the desired channelestimates are calculated leading to power savings.

QRD interpolation can be implemented using the methodsdescribed in [8] [9] and the proposed linear QRD interpolationcan be performed using only fixed multiplications and addi-tions leading to a negligible increase in complexity.

15 20 25 30 35

40

50

60

70

80

90

L�� A�� A ��

L�� A��

Reduction in

Multipli

Fig. 7: Range of possible reduction in multiplications

V. CONCLUSION

The proposed link adaptive processor is capable of decreas-ing the complexity of both channel estimation and QRD, whichare two important baseband signal processing operations. Theprocessor utilizes a windowed DFT based channel estimatorwhich not only suppresses the error floor due to spectralleakage but is also capable of producing channel estimatesat specified interpolation distances. Channel properties areused to identify these interpolation distances which add aminimal loss to BER while enabling the use of a simple QRDinterpolation technique. The BER performance of the proposedlink adaptive processor is on par with existing systems whileproviding a reduction in complexity of upto 40% in higherSNRs and 80% in lower SNRs. Hence, the link adaptiveprocessor is an attractive solution for adaptive processing invarying channel conditions for mobile LTE-A receivers.

ACKNOWLEDGMENT

This work is a part of the DARE project and the authorswould like to thank Lund University and the Swedish Foun-dation for Strategic Research.

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