Joint Channel Equalization and Detection ofSpectrally Efficient FDM Signals

Arsenia Chorti1, Ioannis Kanaras2, Miguel R.D. Rodrigues3 and Izzat Darwazeh2

1CCM Dep., EIS, Middlesex University, Hendon, London, NW4 4BT, UK, a.chorti@mdx.ac.uk, a.chorti@ee.ucl.ac.uk2EEE Dep., University College London, London WC1E 7JE, UK, {i.kanaras, i.darwazeh}@ee.ucl.ac.uk

3Instituto de Telecomunicacoes, University of Porto, 4169 - 007 Porto, Portugal, mrodrigues@dcc.fc.up.pt

Abstract—This paper investigates the transmission in timedispersive channels of Spectrally Efficient Frequency DivisionMultiplexed (SEFDM) signals, where carrier orthogonality isintentionally violated in order to increase bandwidth efficiency.Sufficient statistics of the transmitted SEFDM signal can beobtained by projecting the received signal onto an orthonormalbase generated at the receiver using an Iterative Modified GramSchmidt (IMGS) procedure. In order to reduce the compu-tational complexity resulting from Inter-Carrier Interference(ICI), detection has been implemented based on a RegularizedSphere Decoding (RSD) algorithm. The proposed scheme waspreviously tested in Additive White Gaussian Noise (AWGN) forvarious SEFDM signal parameters. In the present work, theseresults are extended to account for the effect of time dispersivechannels. Randomly generated SEFDM symbols are used aspilots to provide estimates of the channel impulse response insystems with or without cyclic prefixes. A joint equalization-detection is subsequently performed in a RSD stage. We showthat it is possible to detect optimally SEFDM signals of smalldimensionality (e.g. N = 32), with up to 20% bandwidthgain with respect to OFDM systems of the same symbol-rate.This indicates that the wireless transmission of non orthogonalSEFDM signals is tangible.

I. INTRODUCTION

A fundamental limit in communication theory, the Nyquistrate, stems from the fact that in a bandwidth of B Hertz (base-band) and in a time interval of T secs there exist approximately2BT base functions [1]. This result underlines the fact thatgiven the above configuration in the time-frequency domain,2BT data symbols can be independently Maximum Likelihood(ML) demodulated. In a N carrier OFDM system bearing NM -QAM complex information symbols, the base functions areconveniently chosen as rectangular time windowed orthogonalcomplex exponentials, efficiently generated by an IFFT.

Conversely, a less known result is that assuming enhancedsignal processing power is available at the receiver, faster thanNyquist signaling is possible [2], relying on vectorial ML de-tection [3], [4]. In [5] it has been demonstrated that there existsa lower limit in the down-scaling of the Bandwidth-SymbolPeriod Product (BSPP), below which signal detectabilityabruptly deteriorates. This threshold is known as the Mazolimit and represents the shrinking of the detector decisionregions. For rectangular time windowed BPSK and 4-QAMFDM signals, this limit has been evaluated at 0.802 × 2BT ,signifying that OFDM performance is attainable even with a

reduction of about 20% of the transmission spectrum. Specif-ically for 1D modulations, Fast-OFDM schemes doublingthe data rate have been proposed [6], [7]. Conversely, for2D modulations, decreasing the BSPP below the Mazo limitresults in a severe deterioration of the theoretically achievableBER performance.

In [8] the feasibility of faster than Nyquist SEFDM systemswith small dimensionality (N < 40) in AWGN has beendemonstrated. In the receiver, the signal sufficient statisticsare produced by performing signal correlation with an or-thonormal base that spans the SEFDM signal space. Thesymbol detection can then be realized on a ML principle,implemented efficiently in terms of algorithmic complexityusing a Regularized Sphere Decoder (RSD) [9]. Furthermore,in [10] the limiting factor of the large RSD algorithmiccomplexity in low SNRs was addressed. A novel combinedSemi-Definite Programming (SDP) and Pruned Sphere De-coder (PSD) detector was described and demonstrated to attainquasi-optimal performance at substantially reduced complexityeven in low SNR cases.

In the present work we further examine the performance ofnon-orthogonal SEFDM signals in static multipath channels.Both systems with and without a cyclic prefix are examinedand channel estimation based on randomly generated SEFDMpilot symbols is discussed. Signal detection and channel equal-ization is then combined in a single RSD.

This paper is organized as follows. In section II we de-scribe the SEFDM system model, while in section III weexamine the effect of time dispersive channels on the pro-posed SEFDM signals. Channel estimation, based on randomlygenerated SEFDM pilot symbols, is outlined in section IVwhile simulation results concerning the error performance andcomplexity of the joint equalization-detection RSD schemeare presented in section V. Finally, section VI summarizes themain contributions.

II. FASTER THAN NYQUIST SEFDM

A. The SEFDM Transceiver

The original SEFDM transceiver is described in [11], whilein [8] a RSD detection approach is discussed. A high data rateinput stream is split into N parallel low data rate streams. Thelatter modulate, according to a specific modulation scheme oflevel M , N SEFDM subcarriers fα,n(t), n = 0, . . . , N −

2010 IEEE 21st International Symposium on Personal Indoor and Mobile Radio Communications

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1, whose frequency separation Δf is only a fraction of theinverse of the SEFDM symbol period T , i.e.,

Δf =α

T, with α < 1. (1)

Thus, the required bandwidth is reduced by approximately(1−α)N

T , at the expense of the loss of orthogonality betweenthe carriers. The transmitted signal, time windowed within theSEFDM symbol period, is given by:

s (t) =1√T

N−1∑n=0

dnfα,n(t) =1√T

N−1∑n=0

dnej2πnΔft, 0 ≤ t < T

(2)where dn represents the n-th complex modulation symbol.

Denoting h(t) the channel impulse response and n(t)AWGN with Power Spectral Density (PSD) N0

2 , the receivedsignal r(t) is expressed as:

r(t) = h(t) ⊗ s(t) + n(t) (3)

where ⊗ denotes convolution.The proposed receiver consists conceptually of two stages.

The first stage uses a bank of N correlators to extract N suffi-cient statistics from the received signal. The second stage usesa combined detector-equalizer implemented as a RSD. Thecorrelator orthonormal base spans the SEFDM signal spaceand is generated from the SEFDM carriers using an IMGSorthonormalization method [12]. As a result, we generate anorthonormal base {b0(t), b1(t), . . . , bN−1(t)}.

In the case of an ideal channel, i.e. h(t) = δ(t), the vectorof sufficient statistics at the output of the bank of correlatorscan be described in matrix representation as follows

z = Md + w, (4)

where z = [zi] is a vector of N sufficient statistics andM = [Mij ] is the N × N projections matrix of the SEFDMcarriers on the conjugates of orthonormal base. Also, d = [di],while w = [wi] is a vector containing N independent Gaussianrandom variables of zero mean and variance σ2 = N0

2 .

B. Discrete Time Model

The analysis of the effect of time dispersive channels onSEFDM signals is facilitated by employing a discrete timedomain equivalent model. The transmitted SEFDM symbol isequivalently expressed in column vector form as

s = [si], i = 0, . . . , N − 1 (5)

with si =1√N

N−1∑k=0

dkej2π αikN . (6)

Equations (5) and (6) suggest that in analogy to a classicOFDM transmitter that uses an Inverse Fast Fourier Trans-form (IFFT) for signal generation, an Inverse Fractional FastFourier Transform (IFrFFT) algorithm can be employed for thegeneration of the SEFDM signal with complexity O(N log N)[13]. IMGS has algorithmic complexity O(N3), however theprojection vectors need only be pre-evaluated once off-line.

S

PInner

Product

IMGS

0z

1z

1Nz

Symbol Sink

Joint EqualizerDetector

0d

1d

1ˆNd

P

S

0d

1d

1Nd

FDM signal

Symbol Source IFrFFT

0s

1s

1Ns

P

S

CHANNEL

0b 1b 1bN...

S

P

Fig. 1. Conceptual diagram of an SEFDM modem employing an IFrFFTstage.

The equivalent discrete time signal model can be re-writtenas:

s = F−αd (7)

where F−α denotes the Inverse Fractional Discrete Fourier(IFrDF) matrix;

F−α = [F−αkn ], with F−α

kn = ω−αkn and ω1√N

e−2πj 1N . (8)

The discrete Fourier matrix is simply F1. Denoting B = [bi]the matrix of the IMGS base vectors at the bank of correlators,the cross-correlation matrix M is equivalently expressed as:

M = BF−α. (9)

The conceptual diagram of the SEFDM modem employing anIFrFFT is depicted in Fig. 1.

III. JOINT DETECTION-EQUALIZATION USING THE RSD

A straightforward approach to circumvent the effect ofmultipath channels is employed in OFDM; part of the symbolto be transmitted is repeated in the cyclic prefix. The signalthus appears periodic which offers the substantial advantage ofchannel matrix diagonalization in OFDM . This very appealingfeature does not however hold in the case of the proposedSEFDM scheme. Conversely, in the case where estimates ofthe channel impulse response can be obtained, we will outlinehow a joint equalization-detection can be employed in SEFDMsystems based on the RSD algorithm. We will investigate theapproach in the case where (i) no cyclic prefix is present and(ii) a cyclic extension of the SEFDM signal is employed. Toperform channel estimation we rely on randomly generatedpilot SEFDM symbols, noting that the focus of this work is noton the construction of optimal channel estimation strategies butrather on the feasibility and reliability of the proposed systemsfor wireless applications.

178

A. SEFDM without CP in Multipath Channels

We consider a static multipath channel that can be modeledas a finite impulse response (FIR) filter with an L-tap channelimpulse response (CIR) h = [h0, h1, . . . , hL−1]T . Further-more, time and frequency synchronization are consideredperfect to facilitate calculations. The received signal r duringthe i−th SEFDM symbol can be expressed as [14]:

ri = Csi + CTsi−1 + n (10)

where the indices i and i − 1 indicate the symbol index, ndenotes a vector of samples of the AWGN, C is a N × NToeplitz matrix with first column [h0, h1, . . . , hL−1, 0, . . . , 0]T

and first row [h0, 0, . . . , 0]. CT is a N × N Toeplitz matrixthat indicates the tail of the channel that is responsible for theInter-Symbol Interference (ISI), i.e. the first column of CT is[0, 0, . . . , 0]T and the first row [0, 0, . . . , hL−1, . . . , h1].

The received signal is correlated with the orthonormal baseB so that the statistics vector at the output of the correlatorsis expressed as:

zi = BCF−a︸ ︷︷ ︸M′

di + BCTF−a︸ ︷︷ ︸M′′

di−1 + w. (11)

1) ISI Removal: To perform the removal of the ISI weassume that an estimate of the CIR h is available, so that wecan construct estimates of the matrices C, CT, M′ and M′′.The removal of the ISI can be performed by straightforwardsubstraction of the ISI term, i.e.

M′′ = BCTF−a, (12)

zi = zi − M′′di−1. (13)

The initialisation of (13) is based on the regular transmissionof null symbols as in Digital Audio Broadcasting systems.Based on this approach, as long as the BER is low, most ofthe time the perviously detected symbol di−1 will be errorless,i.e di−1 = di−1 most of the time.

2) Symbol Detection in the Presence of SEFDM ICI: :Similar to the case where only an AWGN channel is present,we rely on a RSD to perform joint equalization and symboldetection. After the removal of the ISI, we feed the RSD withthe ISI free statistics vector zi and the estimated Gram matrixM′ that includes the effect of the multipath channel:

M′ = BCF−a. (14)

In this approach, the channel equalizer is embedded in thesymbol detector.

B. SEFDM with CP in Multipath Channels

In analogy to the OFDM case, an SEFDM system withcyclic prefix offers the advantage of rendering the ISI cancela-tion stage obsolete. The insertion/deletion of the CP eliminatesthe ISI. Furthermore, the linear convolution of the transmittedsignal with the channel is now converted into a cyclic convo-lution. The received signal is expressed as:

ri = Ccsi + n (15)

where Cc is the N × N circulant matrix with first column[h0, h1, . . . , hL−1, 0, . . . , 0]T . The received signal is then cor-related with the IMGS orthonormal base B, providing thestatistics vector:

zi = BCcF−a︸ ︷︷ ︸Q

d + w. (16)

Assuming as previously that an estimate h of the CIR isavailable, we get an estimate of the system Gram matrix Q:

Q = BCcF−a. (17)

The RSD detector is then fed with zi and Q.

IV. CHANNEL ESTIMATION

In OFDM systems either pilot symbols or pilot tones arecommonly employed to provide estimates of the CIR andnumerous publications have investigated in the past optimalstrategies towards this end, e.g. [15]. In this work, we focuson the effect of static time dispersive channels and thereforewe rely on pilot symbols for channel estimation. Withoutloss of generality, we investigate channel estimation basedon randomly generated pilot symbols. In the following, pilotsymbols

p = [p0, p1, . . . , pN−1]T (18)

are assumed to be transmitted over suitable regular timeintervals. Thus, during the transmission of a pilot symbol p,the transmitted signal is expressed as:

a = [ai], i = 0, . . . , N − 1 = F−ap. (19)

Next, we investigate the effect of the CIR estimation error inRSD SEFDM systems both with and without a cyclic prefix.

A. Channel Estimation in SEFDM Systems without CP

When the SEFDM signal is transmitted without the insertionof a CP, the received signal (carrying the pilot symbol) can beexpressed as:

r = Ph + n (20)

where P = [Pk,n] is N × L. P can be expressed asthe rightmost part of the N × N Toeplitz matrix A =[Ak,n] with first row aT = [ai]T and first column[a0, 0, . . . , 0, aN−1, aN−2, . . . , a1]T , i.e.

Pk,n = Ak,n, k = 0, . . . , N − 1, n = 0, . . . , L − 1. (21)

In the above, we assume that during the transmission of thepilot symbol no ISI is present (the pilot symbol is transmittedafter a null SEFDM symbol as in real OFDM systems, e.g.Digital Audio Broadcasting). It is straightforward to show [14]that the ML estimate of h is simply the Zero-Force (ZF)estimate:

h = (PHP)−1PHr. (22)

179

B. Channel Estimation in SEFDM Systems with CP

When a CP is employed then the received signal (carryingthe pilot symbol) can be expressed as:

r = Pch + n (23)

where Pc = [Pck,n] is the N ×L rightmost part of the N ×Ncirculant matrix Ac = [Ack,n] with first row aT = [ai]T , i.e.

Pck,n = Ack,n, k = 0, . . . , N − 1, n = 0, . . . , L − 1. (24)

The ML estimate of h is the ZF estimate of (23):

h = (PcHPc)−1Pc

Hr. (25)

C. MSE in Channel Estimation

The Mean Square Error (MSE) in the channel estimates inthe case of OFDM systems depends solely on the PSD N0

2of the AWGN channel [14]. However, in SEFDM systems theMSE depends additionally on on the amount of ICI betweendistinct subcarriers as inferred by (22) and (25). In particular,the MSE in the absence or presence respectively of CP can beexpressed as:

MSEno CP = E{Tr{[h− h]H[h − h]}

}, (26)

MSECP = E{Tr{[h− h]H[h − h]}

}. (27)

In Figs. 2 and 3 we have estimated the MSE based on (26)and (27) for SEFDM signals with and without CP for differingvalues of α (α = 1 is the OFDM case) and Eb/N0. The resultsare averaged over 1000 randomly generated pilot symbols ina 4-QAM N = 32 carriers SEFDM system with T = 4 μsec,in two different static multipath channels with CIRs:

Channel1 :h1 = [0.8765, 0, 0,−0.2279]T, (28)

Channel2 :h2 = [0.8765,−0.2279, 0, 0, 0.1315, 0, 0,−0.4032eiπ

2 ]T .(29)

Channel2 has a much worse effect on the quality of thereceived signal as it has double the number of paths ofChannel1. In particular, the effective impulse response ofChannel2 is 1/4th of the N = 32 carrier SEFDM signal.In both cases, the MSE deteriorates with decreasing Eb/N0

and decreasing fractional carrier spacing α = ΔfT , rangingfrom α = 1 (OFDM) to α = 0.8 (the Mazo limit) and is,as expected, higher in Channel2. Furthermore, the MSE ishigher for SEFDM systems without CP. In order to deriveperformance lower bounds, the simulation results provided inthe rest of the paper are based on SEFDM systems withoutCP, assuming ISI cancelation as described in III-A1.

V. BER AND ALGORITHMIC COMPLEXITY

MEASUREMENTS

The efficacy of the techniques described above is illustratedthrough three sets of simulation results in Channel1 andChannel2. In all simulations the modulation used is 4-QAMand the SEFDM symbol period is T = 4 μsec, while no cyclic

8 10 12 14 16 18 20

10−3

10−2

10−1

MS

E in

Cha

nnel

Est

imat

ion

Eb/N

0 in dB

α=1 Channel 1α=1, Channel 2α=0.95, Channel 1α=0.95, Channel 2α=0.9, Channel 1α=0.9, Channel 2α=0.85, Channel 1α=0.85, Channel 2α=0.8, Channel 1α=0.8, Channel 2

Fig. 2. MSE in channel estimation of a 4-QAM N = 32 carriers SEFDMsignal without CP in Channel1 and Channel2. α = 1 corresponds toOFDM.

8 10 12 14 16 18 20

10−3

10−2

10−1

MS

E in

Cha

nnel

Est

imat

ion

Eb/N

0 in dB

α=1 Channel 1α=1, Channel 2α=0.95, Channel 1α=0.95, Channel 2α=0.9, Channel 1α=0.9, Channel 2α=0.85, Channel 1α=0.85, Channel 2α=0.8, Channel 1α=0.8, Channel 2

Fig. 3. MSE in channel estimation of a 4-QAM N = 32 carriers SEFDMsignal with CP in in Channel1 and Channel2. α = 1 corresponds toOFDM.

extension is employed. All results are averaged over channelestimates obtained from randomly generated pilot SEFDMsymbols. It is worth noting that in the systems examined theoverall gain in spectral efficiency is due to (i) the decrease upto 20% in the carrier spacing (α = 0.8; the Mazo limit) and(ii) the gain (typically up to 20%−25%) by not using a cyclicprefix. In all cases, the performance and detection complexityis compared to classic OFDM which corresponds to a SEFDMsystem with α = 1.

In the first set of results, we use Channel2 and simulatea 4-QAM SEFDM system with N = 24 carriers. We setEb/N0 = 15 dB and demonstrate BER measurements, ave-raged over 1000 random pilot symbols, of the proposed jointequalization-detection approach including the ISI cancelation

180

0.8 0.85 0.9 0.95 110

−6

10−5

10−4

10−3

10−2

10−1

Carriers frequency separation α

BE

R fo

r E

b/N0 o

f 15

dB

Equalization ISI cancellationPerfect equalization and ISI cancelleationEqualization without ISI cancellationNo equalization and ISI cancellation

Fig. 4. BER of N = 24 carriers SEFDM signal without CP in Channel2for Eb/N0 = 15 dB. α = 1 corresponds to OFDM.

step. The results are compared to BER measurements for thesame SEFDM system when (i) no equalization is employed,(ii) with equalization but without ISI cancelation and (iii) withequalization and ISI cancelation based on errorless estimatesof the channel CIR (assuming perfect a-priori knowledge). Theresults are depicted in Fig. 4 and demonstrate the effectivenessof the proposed approach, as the non-equalized SEFDM signalis practically undetectable (BER in the region of [0.3, 0.5]).

The BER curve for perfect equalization corresponds theupper bound in signal detectability in the examined channel.Some of the SEFDM signals (α < 1) have slightly improvederror performance compared to the respective OFDM signal(α = 1) due to the fact that the examined SEFDM signalshave different spectra and are therefore affected by differentportions of the channel frequency response. Although thiseffect in fading channels will be averaged out, it underlinesthe fact that the proposed SEFDM scheme does not lead toloss in signal detectability in multipath channels. Finally, theresults demonstrate a substantial performance gain when usingthe simple ISI cancelation scheme.

In the second set of results, we examine 4-QAM SEFDMsignals with N = {24, 32} carriers in Channel1 andChannel2 for varying values of the fractional carrier spacingα = {0.8, 0.85, 0.9, 0.95, 1} and Eb/N0 = 12 dB. Thepresented results are averaged over 100 randomly generatedpilot symbols, each used for the equalization of 500 randomdata SEFDM symbols (105 × {24, 32} bits per simulationpoint). The simulated SEFDM systems lack a CP and ISIcancelation is performed in all cases.

The BER measurements in Fig. 5 demonstrate the efficiencyof the proposed approach for joint equalization-detection forSEFDM signals. The BER of the SEFDM signal with N = 32carriers is slightly lower compared to the BER of the respectiveSEFDM signal with N = 24 carriers. This is the consequenceof the fact that the channel impulse response affects a larger

0.8 0.85 0.9 0.95 1

10−4

10−3

10−2

Carrier frequency separation α

BE

R fo

r E

b/N0=

12 d

B

N=24 Channel 2N=32, Channel 2N=24 Channel 1N=32 Channel 1

Fig. 5. BER of N = {24, 32} SEFDM signals in Channel1 andChannel2 for Eb/N0 = 12 dB. α = 1 corresponds to OFDM.

0.8 0.85 0.9 0.95 110

2

103

104

105

Carriers frequency separation α

Num

ber

of R

SD

Tre

e N

ode

Vis

its

N=24 Channel 2N=32, Channel 2N=24 Channel 1N=32 Channel 1

Fig. 6. Visits per Node of the RSD tree of N = {24, 32} SEFDM signalsin Channel1 and Channel2 for Eb/N0 = 12 dB. α = 1 corresponds toOFDM.

proportion of the SEFDM symbol in the latter system. In thecase of Channel1, as α decreases the BER slightly increases,indicating that in this case the MSE in channel estimation hassome small effect in the system performance. On the otherhand, in the case of Channel2, the effect of MSE in channelestimation is masked under the effect of the channel itselfand the BER of SEFDM signals with fractional spacing α ={0.8, 0.85, 0.9} is slightly better than the BER of the respectiveOFDM signal.

Conversely, implementation constraints with respect to thesignal dimension are imposed by the algorithmic complexity,which increases steeply with increasing N and decreasingfractional carrier separation α. The algorithmic complexity isdepicted in Fig. 6, measured through the average number ofvisits in the nodes of the RSD tree. The algorithmic complexity

181

8 9 10 11 12 13 14 15 16 17

10−5

10−4

10−3

10−2

Eb/N

0 in dB

BE

R fo

r va

ryin

g E

b/N0 fo

r α=

0.8

N=24 Channel 2N=32, Channel 2N=24 Channel 1N=32 Channel 1

Fig. 7. BER of α = 0.8 N = {24, 32} SEFDM signals in Channel1and Channel2 vs Eb/N0.

8 10 12 14 16 18 2010

2

103

104

105

106

Eb/N

0 in dB

Num

ber

of S

D T

ree

Nod

es V

isits

for α

=0.

8

N=24 Channel 2N=32, Channel 2N=24 Channel 1N=32 Channel 1

Fig. 8. Visits per Node of the RSD tree of α = 0.8 N = {24, 32} SEFDMsignals in Channel1 and Channel2 vs Eb/N0.

further depends on the channel noise levels, both in terms ofEb/N0 [8] as well as on the number of wireless paths. SEFDMsystems with α > 0.9 have similar complexity to the respectiveOFDM systems.

Finally, in the third set of results we examine the BERperformance and algorithmic complexity of 4-QAM SEFDMsignals with 20% bandwidth gain, α = 0.8. SEFDM sig-nals with N = {24, 32} carriers are transmitted throughChannel1 and Channel2 for different noise levels. In Figs.7 and 8 we plot the BER and algorithmic complexity versusthe Eb/N0 for 100 randomly generated pilot symbols over500 random SEFDM symbols each. It is noteworthy that thealgorithmic complexity substantially decreases with increasingEb/N0, suggesting that the proposed SEFDM scheme is fea-sible for high SNR applications. For sufficiently high Eb/N0,higher dimensionality SEFDM signals can be introduced, thus

offering a viable approach towards the enhancement of thespectral efficiency of future communication systems.

VI. CONCLUSIONS

Investigation of the feasibility of non orthogonal SEFDMsystems, with increased spectral efficiency, in time dispersivechannels is reported. We have provided a coherent analysisof the effect of static multipath channels on SEFDM signals,examining systems with or without a cyclic signal extension.In relation to [8] we propose a joint equalization-detectionapproach based on employing a RSD stage. In particular, it isshown that for practical Eb/N0 values the proposed receivercould afford the computational cost of the detection of a 4-QAM SEFDM signal of N = 32 carriers with up to 20%bandwidth saving relative to a standard OFDM signal whileadditionally not using a CP. The present work serves as a proofof concept for the employment of non-orthogonal SEFDMsystems in wireless applications. In the future, we will examinethe effect of multipath fading on the proposed SEFDM schemeas well as investigate alternative detection techniques aimingat further decreasing the algorithmic complexity.

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