Performance and Complexity of Adaptive LatticeReduction in Fading Channels

Alan T. Murray and Steven R. WellerSchool of Electrical Engineering and Computer Science

University of NewcastleCallaghan, NSW 2308, Australia

Email: {alan.murray,steven.weller}@newcastle.edu.au

Abstract—Lattice-reduction-aided detection (LRAD) has beenshown to considerably increase the performance of linearMultiple-Input Multiple-Output (MIMO) detection systems. Inprevious work, we introduced Adaptive Lattice Reduction asa way to exploit channel correlation to yield lattice reduction-based detectors exhibiting considerably reduced complexity. Inthis paper, we review Adaptive Lattice Reduction and examinethe performance and complexity of this detection scheme in bothtemporally and frequency fading channels. For typical mobileenvironments, for example, we demonstrate a complexity reduc-tion approaching 33% on a 4× 4 system, without compromisingperformance.

Index Terms—multiple-input multiple-output (MIMO), symboldetection, lattice reduction (LR).

I. INTRODUCTION

With increasing sized constellations being used in conjunc-tion with spatial multiplexing Multiple-Input Multiple-Output(MIMO) systems, the complexity of even polynomial-timedetection methods can be computationally burdensome [1]. Incomparison, linear detection methods increase in complexitylinearly proportional to the constellation size and numberof transmit antennas used. However, basic linear detectionstrategies suffer significantly from noise enhancement due tonon-orthogonality of the channel matrix.

Lattice-reduction-aided detection (LRAD) has been identi-fied as a way of improving the performance of linear detectionstrategies. LRAD performs lattice reduction (LR) on thechannel matrix to find a more orthogonal set of reduced basisvectors. The Lenstra-Lenstra-Lovasz (LLL) algorithm [2] isone such lattice reduction algorithm used to find this set. Per-formance studies suggest that lattice-reduction-aided detectionand derivatives are well suited low complexity solutions whenlarge constellations are used [1], [3].

In this paper we describe the system setup, give a briefoverview of LRAD and provide a summary of the LLLalgorithm in Sections II, III and IV respectively. We showin Section V that channel correlation can be exploited toyield lattice reduction-based detectors exhibiting considerablyreduced complexity. This is achieved via the introductionof a previously introduced technique [4] we call AdaptiveLattice Reduction (ALR). We present results in Section VIwhich illustrate the extent of this complexity reduction in bothtemporally and frequency selective fading channels and finallyconclude in Section VII.

II. SYSTEM MODEL AND SETUP

We use the complex-valued Nt transmit and Nr receiveantenna MIMO model:

y = Hx + n. (1)

The nth transmit antenna is mapped to the constellation setAn with elements drawn from

X ={a+ ib+

1 + i

2, a, b ∈ Z

}, (2)

the complex shifted integer grid. The complex vector xof length Nt represents the transmitted symbols where theelement xn ∈ An is the symbol transmitted from the nth

antenna. Some (but not all) constellations can be considereda subset of X. For example, rectangular constellations such asQPSK or m-QAM can be considered a subset of X but notcircular constellations such as 8-PSK.

The channel is modelled by H, an Nr by Nt complex-valuedmatrix representing the overall channel and has elementshm,n complex-valued non-dispersive fading coefficients of thechannel between the nth transmit and the mth receive antenna.Gaussian noise is represented by the length Nr vector n withE {|n|} = 0 and E

{∣∣nnH∣∣} = σ2. Finally, y represents the

vector of received symbols of length Nr.Many works use the real decomposition of the complex-

valued MIMO transmission model [5], [6]. Lattice reductionmethods can operate on both real and complex integer latticesand in particular the LLL algorithm has been extended forcomplex lattice reduction [7].

Maximum-likelihood (ML) detection finds the vector ofsymbol estimates x which minimises the euclidean distanceproblem as follows:

x = arg minx∈ANt

‖y −Hx‖2. (3)

This problem is NP-hard, and the complexity is exponentiallyproportional to the constellation set size and number oftransmit antennas. Various heuristic approaches exist whichapproximately solve this problem in less than exponentialtime. Some detectors, such as sphere detectors, typically havecomplexity subexponentially proportional to the constellationset size and number of antennas and whilst their complexity is

suitably low for small constellations and low numbers of trans-mit antennas, their complexity can become considerable aseither increase. Linear detection approaches have complexityproportional to the number of transmit antennas only. Lineardetection approaches use the inverse of the channel to generatean estimate of the transmitted symbols.

Zero Forcing (ZF) is a linear detection method which esti-mates x by finding the least squares solution to (1). However,correlation between the columns of H results in significantnoise enhancement during quantisation of the least squaressolution.

III. LATTICE REDUCTION AIDED DETECTION (LRAD)

The channel matrix H can be considered a generator matrixfor the lattice HXNt of which the received symbols areelements. As such, MIMO detection can be considered equiv-alent to the closest lattice point problem. From lattice theory,methods exist which attempt to reduce noise enhancement.Lattice basis reduction [8, §2.6.1] does this by finding amore orthogonal set of basis vectors using the following stepsadapted from [6] and detailed in [9]:

1) Find reduced lattice basis2) Use pseudoinverse of reduced basis to form estimates3) Quantise estimates to X4) Transform result to find transmitted constellation points

The reduced lattice basis is found by optimising the generatingmatrix, in this case the channel matrix. Complex integer linearcombinations of the column vectors of H are taken to formthe closer to orthogonal reduced matrix HL which spans thesame set of points HXNt ≡ HLXNt and so

HL = HT or H = HLT−1, (4)

where T is a unimodular matrix with complex integer entriesand det(T) = ±1, therefore T−1 also contains only complexinteger entries.

As in [6], by finding an equivalent and closer to orthogonalset of the basis vectors, HL, noise enhancement is reducedwhen quantisation is performed. Importantly, as T−1 and xboth contain only integer spaced entries, so does T−1x andso symbol detection or quantisation is merely rounding to thegrid X.

We can quantitatively measure the orthogonality of a matrixusing metrics such as condition number or orthogonalitydefect. As in [10, §4.6.2], we measure the orthogonality defectof matrices using:

δ(H) =∏Nt

k=1 ‖hk‖|det(H)|

, (5)

where hk is the kth column of H, δ(H) ≥ 1 for all H andδ(H) = 1 if and only if the columns of H are orthogonal.

To assess the impact of lattice redution on orthogonalitydefect, we generated 106 randomly chosen H ∈ C4×4, andcomputed the lattice reduced HL. The method used to performlattice reduction is detailed in Section IV. The orthogonalitydefect was found using (5) before and after lattice reduction.

This gives a quantifiable method to examine the performanceof lattice reduction. The result, as shown in Fig. 1, is a consid-erable reduction in orthogonality defect. It is this improvementthat reduces noise enhancement in linear detection methodsand reduces the error rate of LRAD based systems.

IV. LENSTRA-LENSTRA-LOVASZ (LLL) ALGORITHM

In this section we review the description of a complex LLLreduction algorithm [7]. We extend this description with thepurpose of bringing context to enhancement that will followin Section V. We define:• Hi as the squared euclidean norm of the orthogonal

vectors produced by the Gram-Schmidt Orthogonalisation(GSO) of H.

• µij as the ratio of the length of the orthogonal projectionof the ith basis onto the jth orthogonal vector and thelength of the jth orthogonal vector.

• HiL and Ti represent the values of the reduced basis and

transform after the ith step of the LLL algorithm• Initially, H0

L = H and T0 = INt

• k is the index of the current column of H being processedsuch that 2 ≤ k ≤ Nt

The LLL algorithm then consists of three steps:1) H and µ are computed using a modified GSO procedure

[11]2) Size reduction aims to make basis vectors shorter

and more orthogonal by asserting the condition that|Re(µk,j)| ≤ 0.5 and |Im(µk,j)| ≤ 0.5 for all j < k

3) Basis vectors hk−1 and hk are swapped if some condi-tion is satisfied such that size reduction can be repeatedto make basis vectors shorter

Size reduction and basis vector swapping iterates until theswapping condition is not satisfied by any pair of hk−1 and hk.The resultant basis is then said to be reduced. The swappingcondition for LLL reduction, also called the Lovasz condition,is:

Hk < (δ − |µk,k−1|2)Hk−1, (6)

where δ with 14 < δ < 1 is a factor selected to achieve a good

quality-complexity trade off [2].After each swapping step, Hk−1, Hk and some of the µi,j

values needed to be updated. Techniques can be employed tominimise the number and frequency of recalculations of Hand µ elements [11].

V. COMPLEXITY REDUCTION

Considerable complexity reduction can be achieved in fad-ing channels through the technique we call Adaptive LatticeReduction (ALR). In a slowly fading channel, the elements ofH change slowly and are therefore correlated in time. It hasbeen demonstrated that for these channels the transform matrixT is constant for several renditions of the channel [4]. Assuch, a complexity reduction can be achieved by not directlyrecalculating the reduced basis HL every time the channelvaries.

0 1 2 3 4 5 6 7 8 9 100

0.10.20.30.40.50.60.70.80.91

Orthogonal ity De f ec t

Pro

bab

ilit

y

HHL

Fig. 1. Cumulative density of the orthogonality defect for non-reduced andreduced basis channel matrices

From (4) we can calculate the new reduced basis by simplymultiplying a new H by a previously calculated T. Even if Tis not the optimal basis transform, the reduced basis still spansthe same space as the new H, it is simply not as orthogonal.

Furthermore, as (4) holds true after every step of the LLLalgorithm, further complexity reduction can be achieved byusing a previously calculated T to “pre-reduce” a new Hin order to reduce the number of steps required to computethe new reduced basis. We can simply pre-reduce a newH by calculating H′ = HTA, where TA is a previouslycalculated basis transform. The aim is that this pre-reducedbasis is already closer to orthogonal and many steps of theLLL algorithm can be skipped.

To generate the new reduced basis and transform, weperform lattice reduction on the pre-reduced matrix H′ tocalculate H′L. This procedure calculates a new basis transformTB such that H′L = (HTA)TB. The two transforms can becombined to calculate the transform T = TATB such that

H′L = HT (7)

T can then be used to pre-reduce the next H. Alternatively,if the initial value T0 is initialised to the previous transformvalue TA, the transform produced by the LLL algorithm willalready be T and the explicit multiplication T = TATB canbe skipped.

Finally, again as (4) holds true after every step of theLLL algorithm, the algorithm can be terminated after a pre-determined maximum number of steps. This is important as itcan be used to constrain the complexity of the algorithm. Thiswould be particularly relevant for a hardware implementation.By combining pre-reduction and early termination, the algo-rithm will progressively move toward the optimum reducedbasis whilst capping the number of iterations performed oneach rendition of H.

VI. RESULTS

Simulations have shown that progressively updating thetransform matrix by using this pre-reduction technique yields

a significant reduction of the number of steps needed in theLLL algorithm.

A. Temporally Fading Channels

1) Setup: We reuse the setup from Fig. 1, except this timemake use of a time fading channel. We selected to base oursimulations around specifications similar to that used in mobile3GPP environments. Our simulations were developed using afreely available library of mathematical, signal processing andcommunication routines which includes a substantial multipathfading channel simulator and common COST and ITU channelmodels [12]. These parameters are outlined in Table I.

TABLE ITEMPORAL FADING SIMULATION PARAMETERS

Property ValueTx / Rx Antennas 4× 4

Fading Type Jakes Doppler SpectrumFading Method Rice Method of Exact Doppler Spread

Carrier Frequency 2100 MHzSampling Frequency 1.92 MHz

Fading Period 960 received symbolssuccessive blocks are uncorrelated.

The sampling frequency selected represents the slowestsampling frequency available in 3GPP systems. This causesthe highest normalised Doppler frequency and as such can beconsidered a worst case.

2) Quality: We varied the maximum Doppler velocity from1km/h up to the maximum 3GPP Long Term Evolution targetof 350km/h and examined the orthogonality defect of eachchannel basis before reduction, after lattice reduction and afteradaptive lattice reduction. As expected, the cumulative densityof the orthogonality defect did not vary as the velocity wasincreased. Significantly, neither did that of the reduced andadaptively reduced channel basis.

Fig. 2 shows that the cumulative density of the basisorthogonality defect is indistinguishable between the adaptiveand unmodified reduction implementations. This confirms thatthe quality of bases are independent of the maximum Dopplervelocity, an intuitively appealing result as the expectationof each individual channel basis is not perturbed by theintroduced correlation. Therefore, it can be expected that fora given channel model, adaptive lattice reduction will offerequivalent performance at lower complexity.

3) Complexity: In order to perform a complexity com-parison, we counted the number of floating point operations(FLOPS) performed by the LLL algorithm with complexadditions counting as 2 and complex multiplications as 6operations. To provide a fair comparison, we also included thecost of the pre-reduction matrix-matrix multiplication used byadaptive lattice reduction. It should be noted that the complex-ity of this multiplication accounts for 480 FLOPS or more thanhalf of the complexity of adaptive lattice reduction. The resultsproved interesting as the complexity of the adaptive schemedid not increase considerably even as the maximum Dopplervelocity was increased. This is shown in Table II where the

0 1 2 3 4 5 6 7 8 9 100

0.10.20.30.40.50.60.70.80.91

Orthogonal ity De f ec t

Pro

bab

ilit

y

HHL

H!L

Fig. 2. Cumulative density of the orthogonality defect for a correlated fadingchannel

percentage of saved FLOPS does not vary considerably, evenat the high velocity of 350km/h.

TABLE IICOMPLEXITY COMPARISON BETWEEN ADAPTIVE AND NON-ADAPTIVE

LATTICE REDUCTION AS THE MAXIMUM DOPPLER VELOCITY IS VARIED.

Velocity FLOPS % saved(km/h) LR ALR

10 1209 821 32.1%60 1210 821 32.1%

120 1207 821 32.0%350 1207 821 31.9%

In order to find the limiting case of the adaptive scheme,we examined the complexity as the channel update frequencywas reduced. This was performed by increasing the numberof received symbols between successive channel updates, orin other words by holding the measured value of H static forseveral received symbols. This effectively increases the nor-malised Doppler frequency as seen by the channel reductionalgorithm. This simulation was run at the maximum 3GPPtarget of 350km/h. As can be shown in Fig. 3, the complexityof the adaptive scheme slowly approaches that of the non-adaptive scheme.

4) Performance: In order to compare the error rate perfor-mance of adaptive and non-adaptive lattice reduction we choseto implement Subspace Lattice Reduction Aided Detection(SLRAD) [13]. This particular detector was chosen due to itsability to produce a candidate list suitable for generating softinformation and close to ML performance. Adaptive LatticeReduction only varies the complexity of the lattice reductioncomponent of the detector so whilst SLRAD indeed includescomponents with complexity not examined, this complexity isequal between adaptive and non-adaptive detection. We usedthe simulation parameters from Table II and measured the biterror rate performance when QPSK was used for modulation.

The performance of both adaptive and non-adaptive SLRADwas shown to be very close to the performance of MLdetection, as Fig. 4 illustrates. This confirms our statement

0 160 320 480 640 800 9600

200

400

600

800

1000

1200

1400

Symbols per Channel Update

FLO

PS

LRALR

Fig. 3. Complexity comparison between adaptive and non-adaptive latticereduction as the number of symbols between channel updates is varied.

0 2 4 6 8 10 1210!4

10!3

10!2

10!1

SNR Eb

N 0

BE

R

MLSLRADA!SLRAD

Fig. 4. Performance comparison between adaptive and non-adaptive latticereduction at 350 km/h.

from Section VI-A2 that as adaptive and non-adaptive latticereduction generates bases with equivalent distributions of or-thogonality defect, the performance of detectors implementingeither method will also be equivalent.

We also examined the performance of SLRAD and MLdetection as the channel update frequency was reduced. Aswith Fig. 3 we increased the number of received symbolsbetween successive channel updates. Fig. 5 shows how theperformance of ML and both SLRAD based implementationsdeteriorated as the update frequency was reduced. Notably, theadaptive scheme matched the performance of the non-adaptivescheme, again as predicted in Section VI-A2. Furthermore, wecan see that the performance of all three detection methodshas deteriorated significantly when the channel is updatedonly every 200 received symbols. At this point, the adaptivescheme still offers a significant complexity reduction as shownpreviously in Section VI-A3.

B. Frequency Selective Fading Channels

0 2 4 6 8 10 1210!4

10!3

10!2

10!1

SNR Eb

N 0

BE

R

MLSLRADA!SLRAD

50 S y m/Up d

100 S y m/Up d

200 S y m/Up d

Fig. 5. Performance comparison between adaptive and non-adaptive latticereduction at 350km/h as the number of symbols between channel updates isvaried.

1) Setup: A Frequency selective fading channel can beconsidered as a set of correlated narrow-band frequency-flatfading channels. In systems employing Orthogonal FrequencyDivision Multiplexing (OFDM), multipath causes correlatedfading across the narrow-band subcarriers of which modu-lated symbols are transmitted and received. We can thereforeexamine the performance of Adaptive Lattice Reduction insuch channels. Rather than simulating a full OFDM system,we instead take the Fourier Transform of the channel impulseresponse. This gives us a correlated fading envelope and allowsthe examination of the frequency domain response of differingmultipath channels. As such we implicitly presume that anappropriately long cyclic prefix has been chosen to avoid bothInter-Symbol Interference (ISI) and Inter-Carrier Interference(ICI). These parameters are outlined in Table III.

TABLE IIIFREQUENCY SELECTIVE FADING SIMULATION PARAMETERS

Property ValueTx / Rx Antennas 4× 4

Fading Type Tapped Delay LineFading Method Frequency ResponseChannel Profile Various ITU and COST multipath profiles

Carrier Frequency 2100 MHzSampling Frequency 30.88 MHzSubcarriers / Tones 2048

I.I.D. Fading per multipath

2) Complexity: Using the same approach from SectionVI-A3, we measured the complexity of adaptive and non-adaptive lattice reduction for various multipath channels. Thereis a reasonable correlation between the RMS Delay Spreadof the channel impulse response and the complexity of theadaptive scheme, as shown in Table IV.

Significant complexity reduction can be achieved in chan-nels with a relatively minimal RMS Delay Spread, or in“Rural Area” (RA) COST profiles. As the RMS Delay Spreadincreases with the “Typical Urban” (TU) profiles, the com-

0 20 40 60 80 100 120 140 1600

500

1000

1500

2000

RMS De lay Spread (Sample s)

FL

OP

S

LRALR

Fig. 6. Complexity comparison between adaptive and non-adaptive latticereduction for different multipath channels.

plexity of adaptive scheme aproaches and then begins toexceed that of the non-adaptive scheme. In channels withmoderate spread, or “Bad Urban” (BU) profiles, the adaptivescheme’s complexity further exceeds that of the non-adaptivescheme. Finally, channels with a delay spread over 80 samples,or “Hilly Terrain” (HT) profiles, sees the adaptive scheme’scomplexity reduction significantly exceed that of the non-adaptive scheme.

We can see the correlation between RMS Delay Spreadand Complexity in Fig. 6. Each of the channel profiles isrepresented by a marker for each scheme. As expected thecomplexity of non-adaptive lattice reduction is relatively flat,whereas the complexity of adaptive lattice reduction is roughlydependant on RMS Delay Spread.

TABLE IVCOMPLEXITY COMPARISON BETWEEN ADAPTIVE AND NON-ADAPTIVE

LATTICE REDUCTION FOR DIFFERENT MULTIPATH CHANNELS.

Channel RMS Delay FLOPS % savedSpread (Samples) LR ALR

Delta 0 1150 820 28.7%ITU Pedestrian A 1 1212 883 27.1%COST207 RA6 3 1192 889 25.4%COST259 RAx 3 1203 890 26.0%COST207 RA 4 1227 906 26.2%ITU Vehicular A 11 1215 1093 10.1%COST259 TUx 15 1207 1046 13.4%ITU Pedestrian B 20 1206 1190 1.3%COST207 TU12 31 1203 1219 -1.4%COST207 TU12alt 31 1198 1217 -1.6%COST207 TU6alt 33 1209 1279 -5.8%COST207 TU 33 1200 1267 -5.6%COST207 BU 74 1207 1418 -17.5%COST207 BU6alt 74 1217 1423 -17.0%COST207 BU12 76 1206 1529 -26.9%COST207 BU12alt 78 1202 1534 -27.6%COST259 HTx 93 1214 1965 -61.9%COST207 HT6alt 121 1227 1901 -54.9%ITU Vehicular B 123 1206 1943 -61.2%COST207 HT12 153 1208 1925 -59.4%COST207 HT 155 1207 1886 -56.3%COST207 HT12alt 157 1201 1933 -60.9%

VII. CONCLUSION

In this paper we have examined the complexity and per-formance of Adaptive Lattice Reduction in various temporallyand frequency selective fading channels. We have shown thatsubstantial complexity reduction can generally be achieved byexploiting the static nature of the LRAD transform matrix invarious fading channels. It has been confirmed through real-istic simulations that Adaptive Lattice Reduction can be usedwithout compromising performance. Adaptive Lattice Reduc-tion has considerable potential for improving the performanceand reducing complexity in future practical implementationsof LRAD based detection schemes for MIMO communicationsystems.

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