Two-Stage Detector for SC-FDMA Transmissionover MIMO ISI Channels

Marcel Jar, Emil Matus, Esther Perez-Adeva, Eckhard Ohlmer, and Gerhard FettweisVodafone Chair Mobile Communication Systems

Technische Universitat DresdenDresden, Germany

marcel.jar,matus,esther.perez,eckhard.ohlmer,fettweis@ifn.et.tu-dresden.de

Abstract—In this paper, a two-stage detector is proposed forsignals transmitted using multiple antenna configurations overfrequency-selective channels. For the first stage, a frequencydomain equalizer applying the minimum mean-squared errorcriterion is used to mitigate the intersymbol interference intro-duced by the channel’s non-flat frequency response. Following,the output of this frequency domain equalizer, assumed to beintersymbol interference free, is used as an input for a spheredetector. The sphere detector then processes its input, returninglog-likelihood ratios for all transmitted bits from all transmittingantennas. These log-likelihood ratios can be used, for instance,as inputs for decoders following the sphere detector in thereceiving chain. The proposed method is shown to be well suitedfor the uplink scenario of the Long Term Evolution (LTE)Advanced standard, where high-modulation order signals can betransmitted by multiple antennas using a single-carrier frequencydivision multiplexing access (SC-FDMA) scheme that is prone tosuffer from the effects of intersymbol interference.

I. INTRODUCTION

In order to accommodate a growing demand for higher datarates in wireless communications, new standards have beenproposed and deployed over the past few years. Currently,wireless providers worldwide are increasing their capacityby switching from Third Generation (3G) technologies tothe Long Term Evolution (LTE) standard [1]. Some of themost important characteristics of this standard are the useof multiple-input multiple-output (MIMO) antenna systems1,frequency-division modulation schemes using large band-widths, and high-order modulations.

Multiple antenna technologies can theoretically increasethe throughput by a factor of min (NTx, NRx), where NTx

and NRx denote the number of transmitting and receivingantennas, respectively [3]. However, each receiving antennaperceives a combination of the signal transmitted by alltransmitting antennas, resulting in multi-antenna interference(MAI). Moreover, although the capacity scales linearly withthe bandwidth, large bandwidths frequently result in experi-encing channels with a non-flat frequency response, resultingin the appearance of intersymbolic interference (ISI). ISI intro-duces correlation between symbols transmitted over differenttime instants, making the task of estimating the transmittedsignal more computationally intensive.

1MIMO systems were first proposed only for the downlink scenario.However, they were later extended to the uplink scenario for the LTEAdvanced standard [2].

ISI is circumvented in the LTE downlink, i.e., base station(eNodeB) to mobile user equipment (UE), by making use oforthogonal frequency-division multiplexing access (OFDMA).OFDMA divides the frequency spectrum in sub-bands andassigns a different subcarrier to each band, effectively trans-forming ISI MIMO channels into a set of parallel frequency-flat, hence ISI-free, MIMO sub-channels [4]. Since eachsub-channel is flat, powerful detection methods, like spheredetection [5], can be used for each channel individually.

Despite its great potential, OFDMA presents two majordrawbacks. First, it is very sensitive to carrier frequencysynchronization. Second, and more importantly, it results ina high peak-to-average power ratio (PAPR), making it illsuited for the uplink scenario of cellular networks, where thetransmitter is normally a battery powered device [6].

In order to maintain the PAPR at appropriate levels formobile devices, the LTE standard employs a single-carrierfrequency division multiplexing access (SC-FDMA), insteadof OFDMA, for the uplink scenario. SC-FDMA shares manycharacteristics with OFDMA, being able to dynamically allo-cate frequency bands to its users, while resulting in a muchlower PAPR [6]. However, as opposed to OFDMA, SC-FDMAtransmissions are prone to suffer from ISI.

For SC-FDMA receivers, to keep the computational com-plexity low, a linear frequency domain equalizer (FDE) is nor-mally employed [2]. A linear FDE normally uses the minimummean-squared error (MMSE) criterion to mitigate both ISI andMAI at the same time. However, the noise introduced by thechannel, which is assumed here to be additive white Gaussiannoise (AWGN), is also filtered resulting in an undesirablenoise-enhancement effect and correlation, which shifts thechannel memory from the impulse response into the noise.

In this work, a novel two-stage detector scheme is proposed.It employs a linear FDE to suppress the ISI, while leaving thetask of mitigating the MAI to a more robust detector (in thiswork, a sphere detector). It will be shown in what followsthat, as long as the interference is largely dominated by theMAI, this two-stage detector can result in a significant gain inperformance.

This paper is organized as follows. In Section II, a de-scription of the basic receiver for the uplink scenario of theLTE Advanced standard is presented. Following, a two-stagereceiver is proposed and analyzed in Section III. Monte-Carlo

978-1-4673-0762-8/12/$31.00 ©2012 IEEE 391

Fig. 1. Schematics diagram for a SC-FDMA LTE uplink scenario transmitter.

simulation results are presented in Section IV, indicating thesignificant gains in performance that the proposed two-stagereceiver has over FDE detection. Concluding remarks are laiddown in Section V.

II. BASIC SYSTEM MODEL

For the sake of simplicity, the main processing steps forSC-FDMA transmission for the LTE uplink scenario arepresented here for a 2× 2 MIMO system, as shown in Fig. 1.Nevertheless, all results can be straightforwardly extended toother MIMO configurations.

The process starts for the ith layer with the encoder re-ceiving a vector of bits and outputting an encoded vector ~ci.Following, the encoded vector passes through a rate matchingdevice that will puncture or repeat some bits, in order toaccommodate the transmitted signal to the available resources,resulting in ~c′i. After puncturing/repeating, the signal is inter-leaved and modulated, resulting in a vector of symbols ~si.Denoting by ~sNi the output of the serial to parallel converterfor the ith layer in Fig. 1, and by FN a Fourier matrix of sizeN , the output of the discrete Fourier transforms (DFT) of sizeN is given by

~SNi = FN~sNi.

The vector ~SNi is mapped to N , among M > N possiblesubcarriers, via a resource mapper. This mapping can be repre-sented by an M×N matrix DMN with N ones correspondingto the used frequencies and zeros elsewhere. Hence,

~SMi = DMN~SNi = DMNFN~sNi.

Following, the signal is sent back to the time domain, usingan inverse DFT of size M , F†M , resulting in

~sMi = F†M~SMi = F†MDMNFN~sNi,

where the operator † denotes the Hermitian of a matrix.Prior to transmission, the signal ~sMi is converted back from

parallel to serial and a cyclic prefix is added, resulting in ~scpMi.Stacking up the output of both cyclic prefix adders, ~scpM0

and ~scpM1, and doing the same for the received signal at eachreceiving antenna, ~rcpM0 and ~rcpM1, the relationship betweentransmitted and received signal is given by[

~rcpM0

~rcpM1

]=

[H00 H01

H10 H11

] [~scpM0

~scpM1

]+

[~n0~n1

],

where ~ni is a vector of additive white Gaussian noise com-ponents of size M and variance σ2, and each sub-channelmatrix Hij denotes the M ×M channel matrix between theith transmitting and the jth receiving antenna.

Assuming that there are at most L echoes, these matriceshave the format

Hij =

h0ij 0 · · · 0... h0ij

. . ....

hL−1ij

.... . .

...

0 hL−1ij

. . ....

......

. . ....

0 0 · · · h0ij

.

At the receiver, as shown in Fig. 2 the cyclic-prefix is removedand the signal is converted from serial to parallel, resulting,for each layer, in ~rMi. The overall effect of adding a cyclicprefix at the transmitter, and removing it at the receiver, is thatit makes the channel to appear circular to the receiver [6]. Thiscircular property, as it will be shown later, greatly simplifiesthe task of the FDE. To simplify notation, the sub-channelswill be assumed to be circular, from now on, and have thecyclic prefix attached to it, resulting in[

~rM0

~rM1

]=

[Hcp

00 Hcp01

Hcp10 Hcp

11

] [~sM0

~sM1

]+

[~n0~n1

].

After the size M DFT and resource remapper, the receiversignal, in the frequency domain, is given by[

~RN0

~RN1

]=

[H00 H01

H10 H11

] [~SN0

~SN1

]+

[~N0

~N1

],

where each sub-channel matrix Hij , obtained according to (1),is a diagonal matrix due to the DFT eigenstructure propertyof circulant matrices [4]. For this scenario, there is still spatialcoupling in the frequency domain between signals transmittedby different antennas, however there is no more couplingbetween signals transmitted at different subcarriers. For the ithsubcarrier, the relationship between transmitted and receivedsignal is[

RiN0

RiN1

]=

[hi00 hi01hi10 hi11

]︸ ︷︷ ︸

Hi

[SiN0

SiN1

]+

[N i

N0

N iN1

].

Hence, the outputs of a FDE using the MMSE criterion, forthe ith sub-carrier, are obtained as[

SiN0

SiN1

]= Hi†

(HiHi† + σ2I2

)−1 [ RiN0

RiN1

]. (2)

From the result in (2), it can be seen that, since there is nocoupling between signals transmitted at different frequencies,

392

Fig. 2. Schematics diagram for a SC-FDMA LTE uplink scenario receiver.[H00 H01

H10 H11

]=

[D†MN 0NM

0NM D†MN

] [FM 0M

0M FM

] [Hcp

00 Hcp01

Hcp10 Hcp

11

] [F†M 0M

0M F†M

] [DMN 0MN

0MN DMN

](1)

the FDE can operate by inverting N matrices of size 2 × 2,instead of the much more costly task of inverting one matrixof size 2N × 2N in the time domain.

Following the receiver processing chain in Fig. 2, each oneof the outputs of the FDE is sent back to the time domain andconverted from parallel to serial, resulting in ~si. This signal isdemodulated resulting in log-likelihood ratios (LLRs) for eachtransmitted bit, deinterleaved, and the results from all layersare combined. This results, at the output of the rate rematcher,in a detected vector ~c that is sent to a turbo decoder.

Due to its low-complexity and ease of analysis, FDE hasbeen widely studied and implemented for SC-FDMA re-ceivers [2]. However, as discussed before, FDE leads to noise-enhancement effects which, for some channels, can result inunsatisfactory performance.

III. TWO-STAGE DETECTOR

A schematics diagram for the the two-stage detector pro-posed here is shown in Fig. 3. This systems differs from theconventional detector, depicted in Fig. 2, for its linear FDEsuppresses only the ISI, leaving the task of eliminating theMAI to be performed by a sphere detector in the time domain.

This scheme was envisaged based on the observation that,for small to medium bandwidths, most of the energy at thereceived signal is concentrated in the first tap. For suchscenarios, the channel’s frequency response is almost flat andthe ISI is small. Hence, a linear filter to remove only the ISIdoes not incur a large noise enhancement.

Prior to determining the linear filter to be used in the firststage, a target channel Htg is defined. This channel is equal tothe channel matrix Hcp, with the exception of the coefficientscorresponding to echoes, which are set to zero. Then, theMMSE linear filter Ω should be chosen in order to minimize:

E

∣∣∣∣∣(

Ω

([H00 H01

H10 H11

] [~SN0

~SN1

]+

[~N0

~N1

])

−[

Htg00 Htg

01

Htg10 Htg

11

] [~SN0

~SN1

])2∣∣∣∣∣,

(3)

where all matrices Hij and Htgij are obtained according to (1).

It is shown in [7] that the filter Ω that minimizes (3) can beobtained by multiplying the target channel by the MMSE filterfor the whole channel. I.e., the liner filter is given by

Ω = HtgΨ, (4)

where Ψ corresponds to the MMSE solution

Ψ = H†(HH† + σ2I

)−1.

Following, the filtered signal is sent back to the time domain,resulting in ~νN0 and ~νN1 for the 2×2 MIMO scenario depictedin Fig. 3, which are to be processed by a sphere detector.

The choice of sphere detector in this work is a tuple searchdetector (TSD) with matched candidate determination [8],applying a sorted QR decomposition with MMSE bias reduc-tion [9]. This sphere detection, shown in [10] to be able todeliver close to optimal performance, while requiring a smallcomputational effort, works as follows. First, for the ith timeinstant, the NRx×NTx channel coefficients corresponding toeach pair of transmitting and receiving antennas are groupedinto a matrix Htg(i). Then, a sorted QR decomposition isapplied to an extended channel matrix,

Hext =

[Htg(i)σ2I

]=

[Q Q3

Q2 Q4

] [R0

].

This extension ensures that the variance of the noise at thereceiver is taken into account, reducing the number of close-to-singular diagonal entries in R, and thus the detectioncomplexity [11]. Following, for the ith time instant, a vectorcontaining the detected signals from each transmitting antenna,νN0(i) and νN1(i), is multiplied by Q†, resulting in

~y = Q†[νN0(i)νN1(i)

].

Since Q has unitary energy, this procedure does not alter thestatistics of the noise. Finally, the sphere detector searches forthe ML solution

min~x||~y −R~x|| , (5)

taking advantage of the upper triangular property of R. Parallelto the search for the ML solution in (5), the sphere detectionalgorithm stores the statistics of other possible transmitted sig-nals. These statistics are used, together with the ML solution,to determine LLRs for each transmitted bit2.

The same way as in the conventional system, the LLRs atthe output of the sphere detector are converted from parallel

2For a more detailed description of the tuple sphere detector used in thiswork, the reader is referred to [11].

393

Fig. 3. Schematics diagram for the proposed two-stage processing receiver.

to serial and deinterleaved3. The results from all layers arecombined, resulting, at the output of the rate rematcher, in adetected vector ~c to be sent to a turbo decoder.

A. Adequacy of sphere detection for filtered signals.

In the second stage of the proposed receiver, the signal atthe ith time instant is processed by the sphere detector as if ithad been transmitted over a target channel Htg . This clearlydiffers from the real scenario where a signal is transmittedover a channel H and filtered in the frequency domain byΩ. Hence, the successful application of a sphere detectionscheme for this scenario depends on the appropriateness ofthe following approximation

Ω

(H

[~SN0

~SN1

]+

[~NN0

~NN1

])≈ Htg

[~SN0

~SN1

]+

[~NN0

~NN1

],

which, using (4), can be shown to be equivalent to

HtgH†(HH† + σ2I

)−1H ≈ Htg, (6a)

HtgH†(HH† + σ2I

)−1≈ I. (6b)

Clearly, (6a) is equivalent to assuming that the FDE hassuccessfully removed all the ISI. To verify this, a thousand4 × 4 MIMO channels were generated according to theguidelines set for the LTE standard in [12] using the EPA delayprofile at 3 MHz and a SNR of 20 dB. It is shown in Fig. 4 thepercentage of the total transmitted power received in the formof echoes, before and after FDE. For this scenario, the residualISI, i.e., after the FDE, accounts for less than one percent ofthe total energy. Also, (6b) is equivalent to assuming that theFDE does not significantly alter the statistics of the noise. Toverify this, the autocorrelation of the noise at the output of theFDE for a random 4×4 MIMO EPA channel with a bandwidthof 3MHz is plotted as a function of the frequency-space τdistance4. As one can see, there is a significant autocorrelationat 128, 256, and 384 for the classical FDE. These valuescorrespond to multiples of the chosen value of N = 128. Forthe FDE used in the two-stage processor, on the other hand,no noticeable autocorrelation exists.

Making use of (6a), and (6b), the two-stage system proposedhere does not need to use a whitening noise filter at theoutput of the FDE, as done in [7], resulting in a significantcomputational complexity reduction.

3Note that, since the sphere detector returns LLRs, it acts as both detectorand demodulator. Hence, as shown in Fig. 3, there is no need for a separatedemodulator block as in the case of the conventional system depicted in Fig. 2.

4Frequency and space are combined since the vectors ~RNi in (2) arestacked on top of each other.

Fig. 4. Histogram showing the percentage of the total transmitted powerreceived in the form of echoes (or ISI) for a thousand 4×4 MIMO channels,assuming an EPA delay profile, 3 MHz bandwidth, and a SNR of 20 dB.

Fig. 5. Correlation between noise samples as a function of the time-spacedistance τ for the two-stage FDE as well as for the conventional FDE.

IV. SIMULATION RESULTS

To assess the improvement in performance obtained by thepresented two-stage detector, over the classical FDE for theuplink scenario of the LTE Advanced standard, Monte Carlosimulations were carried out. For all simulations, a CQI of 10is assumed, resulting in a code rate of 0.455, and a modulationchoice of 64-QAM. The signal is transmitted over a 4 × 4MIMO system using a bandwidth of 1.4 MHz. The subcarriermapping operation maps 60 subcarriers into 128 availablefrequencies in a localized fashion. Finally, three power delayprofiles are used, extended pedestrian A (EPA), extendedvehicular A (EVA) and extended typical urban (ETU).

A thousand transport blocks were transmitted by eachMIMO layer, according to the processing steps depicted inFig. 1. For each transport block the channel is assumedconstant (block-fading). These transport blocks are receivedaccording to the processing steps depicted in Fig. 2 for theclassical FDE detector, or the processing steps depicted inFig. 3 for the proposed two-stage detector. The obtained block

394

Fig. 6. Block error rate curves for FDE and Two-Stage detectors for theLTE uplink scenario over 4× 4 MIMO channels, assuming a CQI of 10, anEPA delay profile, and a 1.4 MHz bandwidth.

Fig. 7. Block error rate curves for FDE and Two-Stage detectors for theLTE uplink scenario over 4× 4 MIMO channels, assuming a CQI of 10, anEVA delay profile, and a 1.4 MHz bandwidth.

error statistics are presented in Figs. 6, 7, and 8 for the EPA,EVA, and ETU delay profiles, respectively.

As it can be seen, for all scenarios, the proposed two-stagedetector results in significant performance gains. Nominally,at a block error rate (BLER) of 10−1, gains in performance of7 dB, 4 dB, and 1 dB were obtained for the EPA, EVA, andETU scenarios, respectively. Further simulations, not shownhere due to space constraints, showed that significant gainsin performance can still be obtained for the EPA profile forbandwidths up to 5 MHz, and for bandwidths up to 3 MHzfor the EVA profile. For the ETU profile, the proposed schemedid not result in significant gains for higher bandwidths.

V. CONCLUSIONS

In this work, a two-stage detector was proposed to replaceFDE detectors normally employed in the LTE Advanceduplink scenario for MIMO channels. It was shown, via MonteCarlo simulations, that the proposed method results in a sig-nificant performance gain for small to average bandwidths inrealistic scenarios, allowing a very positive trade-off between

Fig. 8. Block error rate curves for FDE and Two-Stage detectors for theLTE uplink scenario over 4× 4 MIMO channels, assuming a CQI of 10, anETU delay profile, and a 1.4 MHz bandwidth.

performance and computational complexity.For large bandwidths the proposed method can still result

in large gains in performance, as long as multiple users arebeing served, by partitioning the total bandwidth in sub-bandsand assigning different bands for each users.

ACKNOWLEDGMENT

The authors would like to acknowledge the financial supportfrom NEC Corporation.

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