Design and Performance Assessment of FixedComplexity Spectrally Efficient FDM Receivers

Safa Isam and Izzat DarwazehDepartment of Electronic and Electrical Engineering, University College London,

London WC1E 7JE, UKEmail: {sahmed, idarwazeh}@ee.ucl.ac.uk

Abstract—Spectrally Efficient FDM (SEFDM) signals employsnon-orthogonal and overlapping carriers to provide higher spec-trum utilization relative to Orthogonal FDM signals (OFDM).Complex detectors are employed to extract the signal from theintercarrier interference (ICI) created by the loss of orthogonal-ity. Sphere Decoder (SD) is proposed for SEFDM detection as analgorithm that achieves ML bit error rate (BER) performance.However, SD complexity is variable depending on the noise aswell as the conditioning of the system. In this paper, the useof Fixed complexity Sphere Decoder (FSD) for the detectionof SEFDM signal is proposed. The FSD is more suitable forhardware implementation as it eradicates the variable complexitycharacteristic of the Sphere Decoder algorithm whilst providingcompetitive bit error rate (BER) performance. The paper showshow the FSD can be applied to detect SEFDM signals andinvestigate the performance of the FSD in terms of the bit errorrate (BER). Simulations results show that the FSD results inminor error penalties that can be traded-off with complexity.

Index Terms—SEFDM, spectral efficiency, sphere decoder.

I. INTRODUCTION

Spectrally Efficient Frequency Division Multiplexing(SEFDM) systems are new systems that promise bandwidthsavings relative to the well known Orthogonal Frequency Divi-sion Multiplexing (OFDM). SEFDM employs non-orthogonaloverlapped carriers whereas the spectral efficiency (SE) isincreased by reducing the spacing between the sub-carriersand/or the per carrier transmission rate beyond the time-frequency orthogonality rule. Many variants of such sys-tems appeared independently under different names. FastOFDM (FOFDM) [1] and M-ary Amplitude Shift KeyingOFDM (MASK) [2] offered to halve spectrum utilizationbut are constrained only to one dimensional modulationssuch as BPSK and M-ary ASK while Spectrally EfficientFDM system (SEFDM) [3], High Compaction Multicarrier-Communications (HC-MCM) [4], Overlapped FDM system(Ov-OFDM) [5]; Multi-stream Faster than Nyquist Signaling(FTN) [6], [7] and Precoded SEFDM [8] promise variablespectral savings for two dimensional modulations. In theory,the SE enhancement approach is justified by the Mazo limitdefined in [9] and its extension to multicarrier systems recentlyin [6] showing that it is safe to increase the signaling rateby 20% without expecting any performance degradation. Inthis paper we adopt the signal notation of the SEFDM systemof [3], whereas any findings can be generalized to the othersystems with the appropriate change of notation.

Generation of the SEFDM signal can be efficiently per-formed with the standard Inverse Fast Fourier Transform(IFFT) [10], [11]. However, detection of SEFDM signals ischallenged by the need to extract the original signal from theintercarrier interference. The optimal detection of the SEFDMsignal requires brute force ML which can become extremelycomplex with the increase in the system size [3]. On the otherhand, using linear detection techniques such as Zero-Forcing(ZF) and Minimum Mean Square Error (MMSE) constrains thesize of the SEFDM system and the level of bandwidth savings[12]. Therefore, Sphere Decoder (SD) was proposed in [13]to provide ML performance at a much reduced complexity.Nevertheless, the SD complexity varies greatly depending onthe noise and the conditioning of the system. Attempts to tamethe complexity of the SD algorithm appeared in [14] in aquasi-optimal detector combining Semi-definite Programmingand SD, however, the complexity still remains variable. Froman implementation point of view, this variable complexitycan impede the efficient utilization of hardware resources.Therefore, in this paper, we propose the use of the Fixed com-plexity Sphere Decoder (FSD) algorithm for the detection ofSEFDM signal. The FSD algorithm is imported from MultipleInput Multiple Output (MIMO) into the SEFDM context. FSDrestricts the sphere wise search within a limited sub-space ofthe original SEFDM problem. Thereby, the FSD algorithm canno longer guarantee an optimum solution. However, its sub-optimality may be traded-off with complexity. In this work,we apply the FSD algorithm for SEFDM signal detection andshow that the FSD provides sub-optimal bit error rate (BER)performance with a significantly reduced and fixed complexity.

The rest of this paper is organized as follows: section IIprovides the signal model of the SEFDM system. Sections IIIand IV present the FSD based SEFDM detector and explain thedesign concepts of FSD SEFDM detector. Section V evaluates,through simulations, the performance of the proposed solutionsand section VI concludes the paper. Throughout this papervectors are denoted by uppercase characters and matrices byuppercase boldface characters.

II. SPECTRALLY EFFICIENT FDM SIGNAL

The SEFDM signal, x(t), of [3] is composed by modulatinga block of N QAM modulated input symbols on the non-

978-1-4244-8331-0/11/$26.00 ©2011 IEEE

ρN/α IFFT

S/P

X[0]

P/SX[ N-1]

Source

N/α FFT

ML S/Pr`

N-1

r `0

r(t)

r0

rN-1

s `0

s`N-1

((1-α)/α)N Zeros

Ignored

w(t)

s 0

s N-1

Ignored((1-α)/α)N Zeros

x(t)� �

���

Fig. 1. SEFDM block diagram.

orthogonal and overlapping sub-carriers as:

x (t) =1√T

∞∑l=−∞

N−1∑n=0

sl,nexp (j2πnα (t − lT )/T ) , (1)

where α is the bandwidth compression factor defined as α =ΔfT , Δf is the frequency separation between the sub-carriers,T is the SEFDM symbol duration, N is number of sub-carriersand sl,n denotes the symbol modulated on the nth sub-carrierin the lth SEFDM symbol. The fraction α characterizes thelevel of the bandwidth compression, with α = 1 correspondingto an OFDM signal.

By sampling the SEFDM symbol of index l = 0 from (1)at (T/Q) intervals where Q = ρN , and ρ is an oversam-pling factor, the discrete SEFDM signal is then expressed asX [k] = 1/

√Q∑Q−1

n=0 sn exp (j2παnk/Q) , where X [k] isthe kth time sample of the SEFDM symbol of index l = 0from x (t) in (1) and the factor 1/

√Q is a normalization

constant. The complete discrete model is then expressed inmatrix format as

X = ΦS, (2)

where X = [x0, · · · , xQ−1]′

is a vector of time samplesof x (t) in (1), S = [s0, · · · , sN−1]

′is a vector of input

symbols, [·]′ denoting a vector or matrix transpose operationand Φ is a Q × N matrix whose elements are φk,n =1/√

Q exp (j2παnk/Q) , for 0 ≤ n < N, 0 ≤ k < Q anddenotes the sampled SEFDM carriers.

Fig. 1 depicts a block diagram of an SEFDM system.The signal generation is realized by the Inverse Fast FourierTransform (IFFT) [11]. The received signal r (t) arrivescontaminated with Additive White Gaussian Noise (AWGN),w (t), therefore is related to the transmitted signal as

r (t) = x (t) + w (t) . (3)

The SEFDM receiver generates statistics of the incomingsignal by correlating r (t) with the conjugate carriers usingFast Fourier Transform (FFT) in analogy to the transmittedsignal, as shown in Fig. 1. The SEFDM signal statistics canbe represented as

R = CS + WΦ∗ , (4)

where R is a Q × 1 statistics vector, WΦ∗ is a Q × 1vector of AWGN noise samples correlated with the conjugate

carriers matrix Φ∗, [·]∗ denotes the conjugate transpose ofthe argument, and C = Φ∗Φ denotes the cross correlationcoefficient matrix where

cm,n = 1/Q(1 − ej2πα(m−n)/1 − e(j2πα(m−n)/Q)

). (5)

These statistics are fed to an ML detector to generate estimatesof the transmitted signal [3]. The ML estimates SML are givenas a least squares problem as:

SML = arg mins∈M

‖R − CS‖2 (6)

where ‖·‖ denotes the Euclidean norm and M is the constella-tion cardinality. However, the ML complexity grows dramat-ically with the increase in the number of carriers and/or thecardinality of the modulation alphabet. An improvement to theexhaustive enumeration techniques is given by sphere decodersthat achieve ML solution with a smaller computational effort.However, SD suffers from a random complexity that becomesoverly complex in the low SNR regimes [15] and [13]. SectionII-A below describes the SD algorithm.

A. The Sphere Decoder Algorithm

The Sphere Decoder (SD) algorithm, [13] and [15], relieson transforming the ML search space into a multi-dimensionalhypersphere. The SD examines candidate solutions that existwithin a pre-determined radius from the received signal statis-tics point. The SD estimate SSD is defined as

SSD = arg min ‖R − CS‖2, (7)

s∈M, ‖R−CS‖2≤g

where g is the initial radius of the search hypersphere. UsingCholesky decomposition [16], the problem in (7) can beexpressed as

SSD = arg min ‖L (P − S)‖2, (8)

s∈M, ‖R−CS‖2≤g

where L is an upper triangular matrix defined by the Choleskydecomposition as C∗C = L∗L and P is the unconstrained MLestimate of S defined as

P = (C∗C)−1CR. (9)

The SD algorithm then proceeds by examining all the nodesthat satisfy the radius constraint starting from the level numberN and moving downwards until reaching level number 1. Ateach level only the points that satisfy (8) are kept and theradius is updated accordingly. That is at level number y, allthe nodes that satisfy

|py − s|2 ≤ gy/l2yy, (10)

where py = py −∑Nj=y+1 (lyj/lyy) (sj − pj) and gy = g −∑N

j=y+1 ljj |py − s|2, are retained whilst any discarded nodewill eradicate all its predecessor nodes, and hence reduces theremaining search space compared to the ML search space.

In terms of the BER the SD algorithm manages to achieveML performance with a lower complexity (in worst cases it isequal to ML complexity). However, there is no guarantee as to

the level of complexity needed for a solution to be obtained.The complexity is highly dependent on the noise and in thisSEFDM context it is also dependent on the conditioning of thesystem. Therefore, it becomes a demanding task to design apractical system that can accommodate up to ML complexity.In addition, even if it is possible to dedicate the neededresources the design will be far from being optimium as thesystem will rarely use all the allocated hardware resources.To overcome this obstacle, a fixed complexity version ofthe SD algorithm was proposed in [17] and [18] for MIMOsystems. In the following section, we propose the use of afixed complexity sphere wise detector for SEFDM system.

III. FSD APPLICATION TO SEFDM SYSTEM

The Fixed complexity Sphere Decoder (FSD) fixes thevariable complexity of the SD and is therefore, more suitablefor hardware implementation [19]. The FSD algorithm fixesthe complexity of SD by restricting the search within a limitedsub-space of the problem. At every level, a fixed number ofnodes, termed here as the tree width, are examined. The FSDestimate is given by

SFSD = arg minS⊂H, s∈M

∥∥∥L(P − S)∥∥∥2

≤ g, (11)

where H is the search sub-space of the FSD algorithm ofwhich S is a candidate vector and g is the radius of the searchsphere. g corresponds to the distance from the ZF estimate

SZF and is given by g =∥∥∥R − CSZF

∥∥∥2

, where SZF is

obtained from SZF =∣∣C−1R

∣∣ , and |·| is a slicer function.The FSD relies on the assumption that as the search

continues downwards the levels, the probability of finding anode within the sphere decreases, that is

E [dN ] ≥ E [dN−1] · · · ≥ E [d1] , (12)

where di is number of candidate nodes for level i and E [·]is the expected value. This constraint was proved for MIMOsystems in [17] and [18] and here we extend the proof toinclude SEFDM system. Starting by emphasizing that anyretained node at search level y, should satisfy (10), we proceedto show that the RHS of (10) decreases as the search movesdownwards the spheres and this indicates the probability ofhaving constellation points satisfying the condition shoulddecrease. It is by definition that the radius gy satisfy

g ≥ gN−1 ≥ · · · ≥ g1. (13)

As for the lyy elements, recall that C = C∗C = L∗L, andthat lyy is given by

lyy =

√√√√cyy −y∑

k=1

ly,kl∗y,k , (14)

where

liy = (1/lyy)

(ciy −

y∑k=1

li,kl∗y,k

). (15)

�

�

�

�

RRetained NodeDiscarded Node

Fig. 2. The FSD tree search algorithm with tree width equal to 2.

However, for SEFDM system cyy is deterministic and can becalculated based on (5) as

cyy =1

Q2

N∑k=1

(1 − cos (2πα (k − y)))(1 − cos (2πα (k − y) /Q))

. (16)

(16) shows that cyy ≥ 0 therefore, it is straight forward that

lNN ≤ lN−1N−1 ≤ · · · ≤ l11. (17)

(13) and (17) lead to

g/lNN ≥ gN−1/lN−1N−1 ≥ · · · ≥ g1/l11 (18)

which in turn indicates that the probability of nodes satisfying(10) decreases as the search moves from level y to level y−1due to the shrinking property of the parameter gi/lii. Thisrelationship justify how the FSD can reach a solution evenwithin the limited search space.

IV. FSD DESIGN CONCEPTS

The choice of the subset H is crucial for the performanceof the FSD algorithm. In addition, the size of the subset willalso affect the probability of reaching the optimum solution.one main difference between SEFDM and MIMO systems isthat SEFDM aims to accommodate a large number of carrierswhereas the typical order of a MIMO system is 4×4, therefore,it is expected that the tree width for SEFDM will include ahigher number of nodes than MIMO, however, the ratio of thesearched nodes to the overall problem space can be smaller.

In this work, at each tree level we consider a fixed numberof nodes corresponding to the tree width. The decision on thenext level nodes is based on Schnorr-Euchner enumeration[20] of the constellation candidates. Schnorr-Euchner ordersthe children nodes based on their distances from the receivedsignal statistics vector or equivalently from the center pointdecided by P in (9). Fig. 2 illustrates the FSD algorithmfor a 4 carrier system with BPSK symbols or alternatively2 carrier system with 4QAM symbols. At each level, a fixednumber of nodes (=2) is retained, whereas the discarded nodesat that level result in discarding all their children nodes. Forthis particular example, the FSD tree width is fixed to 2.

Due to the limitation of the SD search in a sub-set of thepoints within the initial hypersphere, the quality of the initialestimate becomes crucial in determining the performance ofthe system. If the initial estimate is perturbed by contributions

rN-1

r `0

r`N-1

Ignored

N/αFFT

S/P FSDr(t)

r0

((1-α)/α)N Zeros

s`N-1

s `0

Tree width

� � �

Fig. 3. The FSD detector block diagram.

of the ill-conditioned system then the deviations from thereceived statistics point will propagate to the final solution. Atthe end of the search the retained points will be distinguishedby their distance from the statistics point and therefore, theeffects of the initial estimate will decide the probability ofhaving the optimal solution in the search subset. Fig. 3 depictsan FSD based SEFDM detector. The design of the FSD blockis simpler than SD and allows the trade-off of complexityversus error performance which can be implemented as a re-configurable architecture. Furthermore, the FSD is suitable forparallel processing as it allows for the independent checkingof candidate nodes within a tree level.

V. PERFORMANCE INVESTIGATIONS

Performance of the FSD algorithm for SEFDM detectionwas evaluated in AWGN channel by numerical simulations.BER performance is recorded for FSD and SD for differenttree widths and different system settings. Tests of the FSD inAWGN are carried out to verify that the proposed techniquesare fundamentally valid. The extension to fading channelsis straight forward using a joint channel equalization anddetection approach as in [21].

Fig. 4 shows the performance of the FSD detector fordifferent values of Eb/N0 for a 16 carrier system with α = 0.8carrying 4QAM input sumbols. The figure clearly shows thatthe FSD detector provides BER performance that approachesSD with the increase in the tree width which is effectively theincrease of the size of the search subset, whereas the BERenhancement with complexity is not linear. Noting that theML search space of the system is given by MN = 416(=4.3 × 109), and the simulations checked for tree widths of24 (= 16) , 27 (= 128) and 210 (= 1024), it is clear that theFSD requires a much reduced complexity at the cost of minorBER degradation.

Fig. 5 shows the FSD performance for different levels ofbandwidth compression. The figure shows that the FSD perfor-mance approaches SD with the increase in α. In addition, thefigure indicates that the deviation from the SD performance isinversely proportional to the complexity, which is an expectedresult of increasing the search subset size.

A. Complexity Analysis

The FSD enumerates a number of w nodes correspondingto the tree width. This means that from level k where k =log2 w, a number of w log2 M where M is the constellationcardinality will be examined. Where from the first level until

2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

BE

R

ZFOriginal SDFSD, tree width 16FSD, tree width 128FSD, tree width 1024Single Carrier

Fig. 4. BER performance of the FSD detector for α = 0.8 and 16 carriersystem carrying 4QAM symbols for maximum tree width of 16, 128 and1024.

0.7 0.8 0.9 110

−3

10−2

10−1

100

α

BE

R

ZFOriginal SDFSD, tree width 16FSD, tree width 128FSD, tree width 1024

Fig. 5. BER performance of different detectors for a 16 carrier systemcarrying 4QAM symbols for α = 0.7 to 1, α = 1 corresponds to OFDM fora maximum tree width of 16, 128 and 1024.

k, all the nodes will be retained, hence can be added to theinitialization of the decoder stage. Therefore, the search withineach level, with index < k, will correspond to the index of thatlevel. Furthermore, the decision on the distance of all the nodesin the the FSD subset is independent and can be performedsimultaneously. The overall number of effective nodes visitswill be equal to the N − k for a system employing w parallelprocessing capabilities.

Fig. 6 illustrates the complexity of detection by plottingthe average number of node visits for the original SD andFSD systems with three different tree widths. The figureshows how the SD complexity varies with Eb/N0 while theFSD complexity remains constant. In addition, the dynamicrange of the average number of node visits for SD is large,thus implementing the algorithm will require the allocation ofresources that can accommodate the highest possible numberof visits.

Fig. 7 shows the average decoding time for SD and FSDwith three different tree widths, although simulation times areusually affected by many other non relevant parameters to thecore of the algorithm; such as CPU power and memory, yetstill can act as indicators of the complexity. The figure againreflects the variability of the average decoding time of theSD algorithm. Therefore, the implementation of the SD needs

2 4 6 8 10 1210

1

102

103

104

105

Eb/N

0 [dB]

Ave

rage

num

ber

of n

ode

visi

ts

Original SDFSD, tree width 1024FSD, tree width 256FSD, tree width 16

Fig. 6. Average nodes visits for original SD and FSD for maximum treewidth of 16, 128 and 1024 for a 16 carriers SEFDM system carrying 4QAMsymbols for α = 0.8.

2 4 6 8 10 1210

−3

10−2

10−1

100

101

Eb/N

0 [dB]

Ave

rage

sim

ulat

ion

time

[sec

ond]

FSD, tree width 16FSD, tree width 128FSD, tree width 1024Original SD

Fig. 7. Average simulation time for original SD and FSD for maximum treewidth of 16, 128 and 1024 for a 16 carriers SEFDM system carrying 4QAMsymbols for α = 0.8.

to allocate resources that can account for maximum requireddecoding task whereas in FSD the complexity is fixed and islower than SD. The figure showed how the decoding time ofthe FSD increases with complexity, however, this time can beshortened using parallel processing to achieve the performancedepicted in Fig. 6. Overall, the numerical results confirms thatthe FSD provides good BER performance at a much reducedcomplexity; with the highest complexity tested being a verysmall fraction of the ML search space

(∼ 210/232).

VI. CONCLUSIONS

In this paper, the use of Fixed complexity Sphere Decod-ing (FSD) algorithm for Spectrally Efficient FDM (SEFDM)signals detection is presented. The FSD provides spherewise detection, therefore, is capable of ameliorating the ef-fects intercarrier interference inherent in the SEFDM signal.Furthermore, FSD offers fixed complexity and is therefore,more suitable for hardware implementation when comparedto the original Sphere Decoder (SD). Performance of theproposed detector was evaluated by numerical simulationsthat confirmed attractive BER performance. Results show thatFSD BER performance is sub-optimal; however, the slightdegradation in the BER performance of the FSD algorithm

is compensated for by the attractive low-fixed complexityand the potential for parallel processing, noting that the BERperformance can be improved by increasing the complexity.

REFERENCES

[1] M. R. D. Rodrigues and I. Darwazeh, “Fast OFDM: A Proposal forDoubling the Data Rate of OFDM Schemes,” in Proceedings of theInternational Conference on Telecommunications, vol. 3, pp. 484–487,June 2002.

[2] F. Xiong, “M-ary Amplitude Shift Keying OFDM System,” IEEETransactions on Communications, vol. 51, pp. 1638–1642, Oct. 2003.

[3] M. R. D. Rodrigues and I. Darwazeh, “A Spectrally Efficient FrequencyDivision Multiplexing Based Communications Cystem,” in Proceedingsof the 8th International OFDM Workshop, Hamburg, 2003.

[4] M. Hamamura and S. Tachikawa, “Bandwidth Efficiency Improvementfor Multi-Carrier Systems,” in Proc. 15th IEEE International Symposiumon Personal, Indoor and Mobile Radio Communications PIMRC 2004,vol. 1, pp. 48–52, Sept. 5–8, 2004.

[5] W. Jian, Y. Xun, Z. Xi-lin, and D. Li, “The Prefix Design and Perfor-mance Analysis of DFT-based Overlapped Frequency Division Multi-plexing (OvFDM-DFT) System,” in Proc. 3rd International Workshopon Signal Design and Its Applications in Communications IWSDA 2007,pp. 361–364, Sept. 23–27, 2007.

[6] F. Rusek and J. B. Anderson, “The Two Dimensional Mazo Limit,”in International Symposium of Information Theory, 2005. ISIT 2005.,vol. 57, pp. 970–974, 2005.

[7] F. Rusek and J. B. Anderson, “Multistream Faster than Nyquist Signal-ing,” IEEE Transactions on Communications, vol. 57, pp. 1329–1340,2009.

[8] S. Isam and I. Darwazeh, “Precoded Spectrally Efficient FDM System,”in 21th Personal, Indoor and Mobile Radio Communications Symposium2010, IEEE PIMRC’10,, 2010.

[9] J. E. Mazo, “Faster than Nyquist Signalling,” Bell Systems TechnicalJournal, vol. 54, pp. 429–458, Oct 1975.

[10] S. I. A. Ahmed and I. Darwazeh, “IDFT Based Transmitters for Spec-trally Efficient FDM System,” in London Communication Symposium,Sep 2009.

[11] S. Isam and I. Darwazeh, “Simple DSP-IDFT Techniques for Gener-ating Spectrally Efficient FDM Signals,” in IEEE, IET InternationalSymposium on Communication Systems, Networks and Digital SignalProcessing, pp. 20 – 24, Jul 2010.

[12] I. Kanaras, A. Chorti, M. Rodrigues, and I. Darwazeh, “An OptimumDetection for a Spectrally Efficient non Orthogonal FDM System,”in Proceedings of the 13th International OFDM Workshop, Hamburg,August 2008.

[13] I. Kanaras, A. Chorti, M. Rodrigues, and I. Darwazeh, “SpectrallyEfficient FDM Signals: Bandwidth Gain at the Expense of ReceiverComplexity,” in Proceedings of the International Conference On Com-munications, pp. 1–6, 2009.

[14] I. Kanaras, A. Chorti, M. Rodrigues, and I. Darwazeh, “A new quasi-optimal detection algorithm for a non orthogonal Spectrally EfficientFDM,” in Proc. 9th Int. Symp. Communications and Information Tech-nology ISCIT 2009, pp. 460–465, 2009.

[15] U. Fincke and M. Pohst, “Improved Methods for Calculating Vectorsof Short Length in a Lattice, Including a Complexity Analysis,” Math-ematics of Computation, vol. 44, no. 170, pp. 463–471, 1985.

[16] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge UniversityPress, 1985.

[17] L. G. Barbero and J. S. Thompson, “A Fixed-Complexity MIMODetector Based on the Complex Sphere Decoder,” in Proc. IEEE 7thWorkshop Signal Processing Advances in Wireless CommunicationsSPAWC ’06, pp. 1–5, 2006.

[18] L. G. Barbero and J. S. Thompson, “Fixing the Complexity of theSphere Decoder for MIMO Detection,” IEEE Transactions on WirelessCommunications, vol. 7, no. 6, pp. 2131–2142, 2008.

[19] M. S. Khairy, M. M. Abdallah, and S. E.-D. Habib, “Efficient FPGAImplementation of MIMO Decoder for Mobile WiMAX System,” inProc. IEEE Int. Conf. Communications ICC ’09, pp. 1–5, 2009.

[20] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, “Closest Point Searchin Lattices,” IEEE Transactions on Information Theory, vol. 48, no. 8,pp. 2201–2214, 2002.

[21] A. Chorti, I. Kanaras, M. Rodrigues, and I. Darwazeh, “Joint ChannelEqualization and Detection of Spectrally Efficient FDM Signals,” in 21thPersonal, Indoor and Mobile Radio Communications Symposium 2010,IEEE PIMRC’10,, Sep 2010.