# On Detection Issues in the SC-based Uplink of a MU-MIMO System with a Large Number of BS Antennas

###### Abstract

This paper deals with \SC/\FDE within a \MU-\MIMO system where a large number of \BS antennas is adopted. In this context, either linear or reduced-complexity iterative \DF detection techniques are considered. Regarding performance evaluation by simulation, appropriate semi-analytical methods are proposed.

This paper includes a detailed evaluation of \BER performances for uncoded 4-Quadrature Amplitude Modulation (4-QAM) schemes and a \MU-\MIMO channel with uncorrelated Rayleigh fading. The accuracy of performance results obtained through the semi-analytical simulation methods is assessed by means of parallel conventional Monte Carlo simulations, under the assumptions of perfect power control and perfect channel estimation. The performance results are discussed in detail, with the help of selected performance bounds. We emphasize that a moderately large number of \BS antennas is enough to closely approximate the \SIMO \MFB performance, especially when using the suggested low-complexity iterative \DF technique, which does not require matrix inversion operations. We also emphasize the achievable ”massive \MIMO” effects, even for strongly reduced-complexity linear detection techniques, provided that the number of BS antennas is much higher than the number of antennas which are jointly employed in the terminals of the multiple autonomous users.

BERBit Error Rate \newacronymBSBase Station \newacronymCIRChannel Impulse Response \newacronymCFRChannel Frequency Response \newacronymCPCyclic-Prefix \newacronymDFDecision-Feedback \newacronymFDEFrequency Domain Equalization \newacronymFFTFast Fourier Transform \newacronymIDFTInverse Discrete Fourier Transform \newacronymISIInter-Symbol Interference \newacronymLDBLinear Detection Bound \newacronymMFMatched Filter \newacronymMFBMatched-Filter Bound \newacronymMLMaximum Likelihood \newacronymMLDMaximum-Likelihood Detection \newacronymMLDBMaximum-Likelihood Detection Bound \newacronymMIMOMulti-Input Multi-Output \newacronymMRCMaximal Ratio Combining \newacronymMUMulti-User \newacronymMUIMulti-User Interference \newacronymMSIMulti-Stream Interference \newacronymMUDMultiUser Detection \newacronymMTMobile Terminal \newacronymMMSEMinimum Mean-Squared Error \newacronymOFDMOrthogonal Frequency Division Multiplexing \newacronymQAMQuadrature Amplitude Modulation \newacronymSINRSignal-to-Interference-plus-Noise-Ratio \newacronymSCSingle Carrier \newacronymSUSingle User \newacronymTXTransmitter \newacronymRXReceiver \newacronymSIMOSingle-Input Multi-Output \newacronymZFZero Forcing

Broadband wireless communications; MU-MIMO systems; massive MIMO; SC/FDE; linear detection; iterative DF detection; performance evaluation.

## I Introduction

\CP-assisted block transmission schemes were proposed and developed, in the last two decades, for broadband wireless systems, which have to deal with strongly frequency-selective fading channel conditions. These schemes take advantage of current low-cost, flexible, \FFT-based signal processing technology, with both \OFDM and \SC/\FDE alternative choices [1, 2, 3]. Mixed air interface solutions, with \OFDM for the downlink and \SC/\FDE for the uplink, as proposed in [2], are now widely accepted; the main reason for replacing \OFDM by \SC/\FDE, with regard to uplink transmission, is the lower envelope fluctuation of the transmitted signals when data symbols are directly defined in the time domain, leading to reduced power amplification problems at the mobile terminals.

The development of \MIMO technologies has been crucial for the ”success story” of broadband wireless communications in the last two decades. Through spatial multiplexing schemes, following and extending ideas early presented in [4], \MIMO systems are currently able to provide very high bandwidth efficiencies and a reliable radiotransmission at data rates beyond Gigabit/s. Appropriate \MIMO detection schemes, offering a range of performance/complexity tradeoffs [5] - and also joint iterative detection and decoding schemes [6], have been essential for the technological improvements in this area. In the last decade, \MU-\MIMO systems have been successfully implemented and introduced in several broadband communication standards [7]; in such ”space division mutiple access” systems, the more antennas the BS is equipped with, the more users can simultaneously communicate in the same time-frequency resource.

Recently, the adoption of \MU-\MIMO systems with a very large number of antennas in the BS, much larger than the number of mobile terminal (MT) antennas in its cell, was proposed in [8]. This ”massive \MIMO” approach has been shown to be recommendable for several reasons [8, 9, 10]: simple linear processing for \MIMO detection/precoding (uplink/downlink), namely when using \OFDM for broadband block transmission, becomes nearly optimal; both \MUI/\MSI effects and fast fading effects of multipath propagation tend to disappear; both power efficiency and bandwidth efficiency become substantially increased.

This paper deals with \SC/\FDE for the uplink of a \MU-\MIMO system where the BS is constrained to adopt low-complexity detection techniques but can be equipped with a large number of receiver antennas. In this context, either a linear detection or a reduced-complexity iterative \DF detection are considered. As to the linear detection alternative, we include both the optimum \MMSE [11] and the quite simple \MF detection cases. The iterative \DF detection alternative, which resorts to joint cancellation of estimated \MUI/\MSI and \ISI, does not involve channel decoding,differently from the iterative receiver technique of [6]; it can be regarded as an extension to the multi-input context of the reduced-complexity iterative receiver techniques previously considered for \SIMO systems by the authors (see [12, 13, 14] and the references therein).

Regarding performance evaluation by simulation, appropriate semi-analytical methods are proposed, combining simulated channel realizations and analytical computations of BER performance which are conditional on those channel realizations; selected analytical and semi-analytical performance bounds and a simple characterization of ”massive \MIMO” effects are also provided. This paper shows and discusses a set of numerical performance results. The main conclusions of the paper are presented in the final section.

## Ii System Model

### Ii-a SC/FDE for MU-MIMO Uplink Block Transmission

We consider here a \CP-assisted \SC/\FDE block transmission, within a \MU-\MIMO system with TX antennas and RX antennas; for example, but not necessarily, one antenna per \MT. We assume, in the th TX antenna () a length- block of time-domain data symbols in accordance with the corresponding binary data block and the selected 4-QAM constellation under a Gray mapping rule. The insertion of a length- \CP, long enough to cope with the time-dispersive effects of multipath propagation, is also assumed.

By using the frequency-domain version of the time-domain data block , given by , we can describe the frequency-domain transmission rule as follows, for any subchannel :

(1) |

where is the ”input vector”, is the Gaussian noise vector and , denotes the channel matrix with entries , concerning a given channel realization, and is the resulting, frequency-domain, ”output vector” .

As to a given \MIMO channel realization, it should be noted that the \CFR , concerning the antenna pair , is the DFT of the \CIR , where for . Regarding a statistical channel model - which encompasses all possible channel realizations -, let us assume that and for . By also assuming, for any , a constant

(2) |

(of course, with for ), the average bit energy at each \BS antenna is given by

(3) |

where , and .

### Ii-B Linear Detection Techniques

An appropriate linear detector can be implemented by resorting to frequency-domain processing. After CP removal, a DFT operation leads to the required set of length- inputs to the frequency-domain detector ( given by (1)); it works, for each , as shown in Fig. 1(a), leading to a set of length- outputs .

When , possibly with , either an \MMSE, frequency-domain, optimum linear detection or a reduced-complexity, frequency-domain, linear detection can be considered. In all cases, the detection matrix, for each subchannel () can be written as

(4) |

where is the conjugate transpose of the estimated \MU-\MIMO channel matrix and is a selected matrix, possibly depending on . Therefore, at the output of the frequency-domain linear detector (see Fig. 1(a)).

It should be noted that the component of is given by : this means that the factor provides \MRC procedures, one per \MT antenna, all of them based on an appropriate \MF for each component of the length- received vector at subchannel .

For a \MMSE detection - the optimum linear detection - or a \ZF detection [5, 11], an inversion of each matrix is required.

A reduced-complexity linear detection can be achieved by using an diagonal matrix . The easiest implementation corresponds to adopting an identity matrix . Of course, and when , which means an ”\MF detection”, actually not requiring a matrix inversion.

For a given channel realization and a given detection matrix , which depends on the estimated channel realization , the output of the frequency-domain detector is given by

(5) |

where and .

With \SC/\FDE (time-domain data symbols), an \IDFT is required for each vector. The th component of the resulting length- vector can be written as

(6) |

with .

Therefore, can be written as

(7) |

where is a diagonal matrix with entries given by .

When is written as

(8) | |||

the four terms in the right-hand side of eq. (8) are concerned, respectively, to ”signal”, \ISI, \MUI/\MSI and ”Gaussian noise”, at subchannel .

### Ii-C Low-Complexity Iterative DF Technique

A low-complexity iterative \DF technique can be easily devised having in mind eq. (7). This frequency-domain nonlinear detection technique combines the use of a linear detector and, for all iterations after the initial iteration (i.e., for ), a cancellation of residual \MUI - and residual \MSI, when some users adopt several TX antennas for spatial multiplexing purposes - as well as residual \ISI; such cancellation is based on the estimated data block which is provided by the preceding iteration and fed back to the frequency-domain detector. The output of this frequency-domain detector, for iteration , is as follows:

(9) |

(for , where - with denoting the detection matrix employed in iteration - and the entries of the diagonal matrix are given by . Of course, .

The implementation of this iterative \DF technique is especially simple when for any , i.e., when a linear \MF detector is adopted as shown in Fig. 1 (b) for all iterations; the matrix inversion which is inherent to more sophisticated linear detectors is then avoided. On the other hand, a slightly improved performance can be achieved through an increased implementation complexity, by feeding back vectors of soft (instead of hard) time-domain symbol decisions for interference cancellation.

## Iii Evaluation of the Achievable Detection Performances

### Iii-a Semi-analytical Performance Evaluation

Regarding evaluation of detection performances by simulation, simple semi-analytical methods are presented here, for the detection techniques of subsecs II-B and II-C, both combining simulated channel realizations and analytical computations of \BER performance which are conditional on those channel realizations. In all cases, the conditional \BER values are directly computed by resorting to a \SINR, under the assumption that the ”interference” has a quasi-Gaussian nature. These ratios are simply derived in accordance with the channel realization (). Of course, for concluding the \BER computation in each case - involving random generation of a large number of channel realizations and conditional \BER computations - a complementary averaging operation over the set of channel realizations is required.

When using a linear detection technique (sec. II-B), it is easy to conclude, having in mind (8), that the ”signal-to-interference-plus-noise” ratio concerning the th input of the MU-MIMO system is given by

(10) |

where , and with .

For 4-QAM transmission, the resulting () - conditional on the channel realization - is given by

(11) |

(where is the Gaussian error function) with as computed above, and .

### Iii-B Reference MMSE Performance and SIMO Performance Bounds

When adopting an ”\MMSE detector” and a perfect channel estimation is assumed, [11]. It can be shown that the resulting - which can be used for computing , and then \BER - can be written as , with as defined in subsec. II-B.

SIMO/LDB | |
---|---|

SIMO/MFB | |

SIMO/AWGN/MFB |

Successively improved performance bounds can be obtained as follows (see Table I): also under , with the same but , which corresponds to a \SIMO/\LDB; under , with and the same , by suppressing the resulting first term (\ISI) in the denominator of eq. (10), which corresponds to a \SIMO/\MFB; when replacing the fading channel by an AWGN channel for each antenna pair in the \SIMO/\MFB, which corresponds to a \SIMO/AWGN/\MFB.

It should be noted that the BER curve for the \SIMO/AWGN/\MFB actually corresponds to the achievable BER performance (against White Gaussian Noise) in a \SIMO system with single-path propagation for all antenna pairs, provided that an \MF detection, under ideal channel estimation, is adopted:

(13) |

### Iii-C Massive MIMO effects

When , both the \MUI/\MSI effects and the effects of multipath propagation (fading, \ISI) tend to disappear: consequently, the \BER performances for the \MU-\MIMO Rayleigh fading channel become very close to those concerning a \SIMO channel with single-path propagation for all TX/RX antenna pairs. The achievable performances under a ”truly massive” \MU-\MIMO implementation can be analytically derived as explained in the following.

Entries of are i.i.d. Gaussian-distributed random variables with zero mean and variance . Therefore, and

(14) | |||

Consequently, for ,

(15) |

and

(16) |

Therefore,

(17) |

(by assuming that ). When ,

which implies that , i.e. a BER performance closely approximating the \SIMO/AWGN/MFB (eq. (13)).

When , it should also be noted - having in mind the equations (15) and (16) - that the \MMSE linear detection becomes practically equivalent to a \MF linear detection, since

(19) |

with (assuming a perfect channel estimation). Of course, the corresponding matrix is then , leading to (). Therefore, the resulting \SINR’s are as expected, under a perfect channel estimation . Clearly, when , the \MUI/\MSI effects as well as both the fading and the ISI effects of multipath propagation become vanishingly small, leading to a close approximation to the \SIMO/AWGN/\MFB reference performance.

## Iv Numerical Results and Discussion

The set of performance results which are presented here are concerned to \SC/\FDE uplink block transmission, with and in a \MU-\MIMO Rayleigh fading channel. Perfect channel estimation and perfect power control are assumed. The fading effects regarding the several TX/RX antenna pairs are supposed to be uncorrelated, with independent zero-mean complex Gaussian coefficients assumed to have variances , ( for ).

The accuracy of performance results obtained through the semi-analytical simulation methods of sec. III was assessed by means of parallel conventional Monte Carlo simulations (involving an error counting procedure), which correspond to the superposed dots in the several \BER performance curves of Figs. 2 and 3, concerning the \MU-\MIMO system.

Fig. 2 shows the simulated \BER performances for an \SC/\FDE-based \MU-\MIMO uplink and three possibilities regarding for , when using two linear detection techniques: optimum (\MMSE) detection; reduced-complexity (\MF) detection. In each subfigure, for the sake of comparisons, we also include the \SIMO performance bounds of Table I. In the simulation results concerning each subfigure of Fig. 2, the five BER performance curves are ordered, from the worst to the best, as follows: \MU-\MIMO with reduced-complexity (\MF) linear detection; \MU-\MIMO with \MMSE detection; \SIMO/\LDB (); \SIMO/\MFB (); \SIMO/AWGN/\MFB () [practically superposed to the \SIMO/\MFB curve]. These results clearly show that the performance degradation which is inherent to the reduced-complexity linear detection technique (\MF) - as compared with the \MMSE linear detection - can be made quite small, by increasing significantly; they also show that, under highly increased values, the ”\MUI/\MSI-free” \SIMO (multipath) performance and the ultimate bound - the ”\MUI/\MSI-free and \ISI fading-free” \SIMO (single-path) performance - can be closely approximated, even when adopting the reduced-complexity linear detection. This figure emphasizes a ”massive \MIMO” effect when , which leads to \BER performances very close to the ultimate ”\MUI/\MSI-free and \ISI fading-free” \SIMO (single-path) performance bound (the \SIMO/AWGN/\MFB of Table I).

Fig. 3 shows the simulated BER performances for an \SC/\FDE-based \MU-\MIMO uplink and three possibilities regarding for , when using the reduced-complexity iterative \DF detection technique of Fig. 1 (b), which does not require matrix inversions. By comparing these results to the results of Fig. 2, also for , we can conclude that, whenever , the reduced-complexity \DF technique of Fig. 1 (b) is able to provide BER performances which are better than those of the \MMSE linear detector [while avoiding the inversion of a matrix ], closely approximating the (practically identical) \SIMO/\MFB and the \SIMO/AWGN/\MFB reference performances after a small number of iterations.

It should be noted that the numerical results reported above, for TX antennas, are compatible with an uplink transmission scenario involving up to users (for example, users with one TX antenna per user); a specific scenario with TX antennas, involving six users, is depicted in Fig. 4.

## V Conclusions

This paper was dedicated to the uplink detection and performance evaluation for a \MU-\MIMO system with \SC/\FDE transmission, when adopting a large number of antennas and low-complexity detection techniques at the BS. With the help of selected numerical performance results, discussed in detail in Section IV, we show that a moderately large number of \BS antennas (say, ) is enough to closely approximate the \SIMO/\MFB performance - and also the \SIMO/AWGN/\MFB performance, expressed as -, especially when using the suggested low-complexity iterative \DF technique, which does not require matrix inversion. We also emphasize the ”massive \MIMO” effects provided by a number of BS antennas much higher than the number of antennas which are jointly employed in the terminals of the multiple autonomous users, even when strongly reduced-complexity linear detection techniques - such as the so-called ”\MF detection -, are adopted.

The accuracy of performance results obtained by semi-analytical means, much less time-consuming than conventional, ’error counting’-based, Monte Carlo simulations - was also demonstrated. The proposed performance evaluation method can be very useful for rapidly knowing ”how many antennas do we need in the \BS?”, for a given number of antennas jointly employed in the user terminals.

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