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Linear pre-coding performance in measured

very-large MIMO channels

Xiang Gao, Ove Edfors, Fredrik Rusek, and Fredrik Tufvesson

Department of Electrical and Information Technology, Lund University, Lund, Sweden

Email: {xiang.gao, ove.edfors, fredrik.rusek, fredrik.tufvesson}@eit.lth.se

Abstract—Wireless communication using very-large multiple-

input multiple-output (MIMO) antennas is a new research field,

where base stations are equipped with a very large number

of antennas as compared to previously considered systems.

In theory, as the number of antennas increases, propagation

properties that were random before start to become deter-

ministic. Theoretical investigations with independent identically

distributed (i.i.d.) complex Gaussian (Rayleigh fading) channels

and unlimited number of antennas have been done, but in

practice we need to know what benefits we can get from very

large, but limited, number of antenna elements in realistic

propagation environments. In this study we evaluate properties

of measured residential-area channels, where the base station

is equipped with 128 antenna ports. An important property to

consider is the orthogonality between channels to different users,

since this property tells us how advanced multi-user MIMO (MU-

MIMO) pre-coding schemes we need in the downlink. We show

that orthogonality improves with increasing number of antennas,

but for two single-antenna users there is very little improvement

beyond 20 antennas. We also evaluate sum-rate performance for

two linear pre-coding schemes, zero-forcing (ZF) and minimum

mean squared error (MMSE), as a function of the number of

base station antennas. Already at 20 base station antennas these

linear pre-coding schemes reach 98% of the optimal dirty-paper

coding (DPC) capacity for the measured channels.

I. INTRODUCTION

Multiple-antenna (MIMO) technology for wireless com-

munications is becoming mature and has been incorporated

into many advanced standards such as HSPA and LTE [1].

Basically the more antennas the transceivers are equipped

with, the better performance can be obtained in terms of

data rate and link reliability. The price to pay is increased

complexity of hardware and signal processing at both ends.

In classical point-to-point single-user MIMO systems (SU-

MIMO), the multiplexing gain may disappear when the signal

power is low, relative to interference and noise, or in propaga-

tion environments with dominating line-of-sight or insufficient

scatterers. SU-MIMO systems also require complex and ex-

pensive multiple-antenna terminals. Practical size limitations

on terminals also limit the number of antennas that can be

used and thereby the achievable multiplexing gains.

To overcome these drawbacks of SU-MIMO, multi-user

MIMO (MU-MIMO) with single-antenna terminals and an un-

limited number of base station antennas is investigated in [2].

This approach involves MU-MIMO operation with an infinite

number of base station antennas in a multi-cell environment.

It is shown that all the effects of uncorrelated noise and fast

fading disappear, as does the intra-cell interference, and the

only remaining impediment is the inter-cell interference due

to pilot contamination. All of these motivate entirely new

theoretical research on signal processing, coding and network

design for such very-large MIMO systems. The vision put

forward in [2] is that the base station array would consist of a

very large number of small active antenna units, each of which

uses extremely low power.

The assumption of an unlimited number of base station

antennas in [2] greatly simplifies the theoretical analysis. In a

practical system, however, the number of antennas cannot be

arbitrarily large due to physical constraints. From a feasibility

point of view, it is reasonable to ask how large the antenna

array should be. The answer depends on the propagation

environment, but in general, the asymptotic results of random

matrix theory can be observed even for relatively small di-

mensions.

The analysis in [2] assumes that inner products between

propagation vectors of different users grow at a lesser rate than

inner products of propagation vectors with themselves, i.e.,

the user channels are asymptotically orthogonal. Experimental

work is clearly of great importance to investigate the range of

validity of this assumption. Therefore, in the initial phase of

this new research in wireless communications, we study how

well the measurements resemble the theoretical results and

what benefits we can obtain at very-large, but limited, number

of base-station antennas.

In the present paper, we study the linear pre-coding per-

formance in measured very-large MIMO downlink channels.

We consider a single-cell environment in which a base station

with a very-large antenna array serves a number of single-

antenna users simultaneously. The interference between cells

and pilot contamination issues are therefore not addressed

in this paper. Channel measurements were done with a 128-

antenna base station in a residential area. To the best of the

authors’ knowledge, there are no other studies performed on

this type of systems, with this high a number of antennas.

In Sec. II we describe our system model and define a

number of measures. In Sec. III we describe the measurement

setup and the residential-area environment where the mea-

surements are performed. As a basis for our comparison of

systems with more or less base station antennas we describe

a number of pre-coding schemes in Sec. IV – both the linear

zero-forcing (ZF) and minimum mean squared error (MMSE)

pre-coders, as well as the optimal dirty-paper coding (DPC).

System performance is then evaluated in Sec. V for different

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Fig. 1.

station and two single-antenna users.

System model of a MU-MIMO system with an M-antenna base

number of antennas and we show how the low-complex linear

pre-coders perform in relation to the optimal DPC scheme.

Finally, we summarize our contributions and draw conclusions

in Sec. VI.

II. SYSTEM DESCRIPTION

We consider the downlink of a single-cell MU-MIMO

system: the base station is equipped with M antennas, and

serves K single-antenna users. The total transmit power is

constrained to an average of Pt. The composite received K×1

vector y at the users can be described as

y =√ρHz + n,

(1)

where H is a composite K ×M channel matrix, z is the

transmitted vector across the M antennas, and n is a noise

vector with unit variance. The variable ρ contains the transmit

energy and channel energy so that the total power in H is K

and z satisfies E??z?2?= 1. The M ×1 transmit vector z

x. Through pre-coding at the transmit side we have

contains a pre-coded version of the K×1 data symbol vector

z = Ux,

(2)

where U is a M × K pre-coding matrix including power

allocation to the data symbols. The vector x comprises data

symbols from an alphabet X, and each entry has unit average

energy, i.e. E?|xk|2?= 1, k = 1,2,...,K. Taken together,

on U: Tr

UHU

=1, where Tr(·) is the trace-operator and

(·)Hdenotes the Hermitian transpose.

To facilitate analytical derivation of pre-coders and their

performance, we will assume that the number of users is K=

2. The input-output relation of the channel for this two-user

case is shown in Fig. 1. The Gram matrix associated with H

can be expressed as

?

where g denotes the power imbalance between the two user

channels, and δ is a factor measuring the correlation between

the two channels. Since we can permute the rows of H at

will, we can without loss of generality assume that 0≤g <1.

The correlation between the channels to the two users can be

the energy constraints on x and z yield an energy constraint

?

?

G ? HHH=

1 + g

δ∗

δ

1 − g

?

,

(3)

Fig. 2.

specific houses.

Overview of the residential measurement area. The numbers indicate

expressed as |δ|/?1−g2. Further, we require |δ| <

III. MEASUREMENT SCENARIO

?1−g2

in order to have a positive definite matrix G.

The channel measurements were carried out in a residential

area north of Lund city center, Sweden. Fig. 2 shows an

overview of the measurement area, where the numbers indicate

specific houses. The measurements were originally performed

with the aim of studying channel properties for residential

femto-cell systems [3]. However, the large receive array with

128 antenna ports also enables this study of very-large MIMO

channels. The receive antenna array was placed upstairs in

house 63, which is shown at street level in Fig. 3. This array

is a cylindrical patch array having 16 dual polarized antennas

in each circle and 4 such circles stacked on top of each other,

giving in total 128 antenna ports. The left part of Fig. 4 shows

this large antenna array. The diameter is 29.4 cm and the height

is 28.3 cm. The distance between adjacent antennas is about

6 cm, half a wavelength at the 2.6 GHz carrier frequency

used. The transmit antenna array was placed indoors and

outdoors at different positions, therefore indoor-to-outdoor-to-

indoor and outdoor-to-indoor channels were measured. The

right part of Fig. 4 shows the transmit array which consists

of a planar patch array having 2 rows of 8 dual polarized

antennas, giving in total 32 antenna ports. The outdoor-to-

indoor channels are selected for very-large MIMO study, as we

consider the scenario in which the users are outdoors around

the base station. The outdoor measurement positions were (to

the west of) houses 29, 33, 37, 41, 43, 45, 47, 49, 51, and

53, respectively. The measurement data was recorded with the

RUSK LUND channel sounder at a center frequency of 2.6

GHz and a signal bandwidth of 50 MHz. At each measurement

position, the transmit antenna was moved along a 5-10 m

straight line parallel to the direction of antenna array back-

plane.

For this very-large MIMO study, we extract the measure-

ment data to form MU-MIMO channels. The first antenna

in the 32-antenna transmit array is selected to represent a

single-antenna user terminal. Through all the measurement

positions, we can have several different users. In this paper we

concentrate on the two-user case, where two different positions

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Fig. 3.

receive antenna array was positioned.

Street level view of the measurement area, house 63, in which the

Fig. 4.Receive antenna array (left) and transmit antenna array (right).

are selected randomly. The receive antenna array positioned in

house 63 represents an indoor base station.

IV. PRE-CODING SCHEMES

In this section we derive closed form expressions for DPC

capacity and linear pre-coding sum rates for two-user case,

using the system model in Sec. II.

A. Dirty-paper coding

The optimal sum rate in the downlink of a MU-MIMO

system can be achieved by the interference pre-subtraction

coding technique called dirty-paper coding (DPC), as long as

the transmitter has perfect side information about the additive

interference at the receiver [4]. The optimal DPC capacity for

the two-user case is given as

CDPC= max

P1,P2log2det

?

I + ρHHPH

?

,

(4)

where P is a 2×2 diagonal matrix for power allocation with P1

and P2on its main diagonal. The DPC capacity is maximized

by optimizing over the power allocation under constraint that

P1+P2= 1. By substituting the Gram matrix G in (3) into

(4), we find the optimal power allocation as

?1

1,

(P1)opt

DPC=

2+

g

ρ(1−g2−|δ|2),

|δ|2≤ δth

|δ|2> δth,

(5)

where δth= 1−g2−2g/ρ. The corresponding DPC capacity

becomes

⎧

⎩

If |δ|2is higher than a certain threshold δth, all power will be

allocated to the user with the stronger channel, and the DPC

CDPC=

⎨

log2

?

1 +ρ +

ρ2(1−g2−|δ|2)

4(1−g2−|δ|2)

2+4g2

?

,|δ|2≤δth

|δ|2>δth.

log2[1+ρ(1+g)],

(6)

capacity becomes the same as the single-user transmission rate

CSU= log2[1 + ρ(1 + g)].

(7)

Although optimal sum rate can be achieved, DPC is far

too complex to be implemented in practice. We hence take

the optimal DPC capacity as a benchmark for the sum rates

achieved by the linear pre-coding schemes, ZF and MMSE,

which are of more practical interest.

B. Linear pre-coding schemes

The pre-coding matrix U can be decomposed as

U =

1

√γW

√P,

(8)

where W represents a particular linear pre-coding algorithm,

P is the power allocation matrix, and γ is used to normal-

ize the total transmit power in z to unity. Therefore, from

Tr

UHU

=1, the power normalization factor γ should be

?

ZF pre-coding scheme. ZF pre-coding eliminates the inter-

ference by transmitting the signals towards the intended user

with nulls in the “direction” of other users. The ZF pre-coder

is given as

WZF= H†,

where H†= HH?

channel matrix H. Using ZF pre-coding, the signal model

becomes

y =

γPx + n.

??

γ = Tr

PWHW

?

.

(9)

(10)

HHH?−1

is the pseudoinverse of the

?ρ

(11)

Since perfect nulling makes this scheme interference free, the

sum rate can be calculated as

?

subject to P1+ P2=1. By substituting the ZF pre-coder and

power allocation matrix into (9), and letting P2= 1−P1we

obtain the normalization factor

γ =1 + g − 2P1g

1 − g2− |δ|2.

Inserting this γ into (12), we find the optimal power allocation

⎧

⎩

⎧

⎪

The ZF interference cancellation has significant signal

power penalty if the two user channels are highly correlated.

From (15) we can see that the capacity goes to zero when the

CZF= max

P1,P2

2

i=1

log2

?

1 +ρPi

γ

?

,

(12)

(13)

(P1)opt

ZF=

⎨

1

2

1,

?

1 + g +

2g(1−g2)

ρ(1−g2−|δ|2)+2g2

?

, |δ|2≤ δth

|δ|2> δth.

(14)

The resulting sum rate for ZF pre-coding becomes

?(2+ρ(1−g2−|δ|2))

log2

1 +

CZF=

⎪

⎪

⎪

⎩

⎨

log2

2

4(1−g2)

ρ(1−g2−|δ|2)

1−g

?

,

,

|δ|2≤ δth

|δ|2> δth.

??

(15)

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channel correlation is high (low orthogonality), i.e. when |δ|

approaches

MMSE pre-coding scheme. MMSE pre-coding can trade

interference suppression against signal power efficiency. The

optimal MMSE pre-coder is given by [5]

WMMSE= HH?

where α = K/ρ, or in our case, α = 2/ρ. At high SNRs

(α small) the MMSE pre-coder approaches the ZF pre-coder,

while at low SNRs (α large) the MMSE pre-coder approaches

the matched filter (MF) pre-coder. In [6], the power allocation,

i.e. the matrix P, is also considered in minimizing the mean

square error, but in this paper we optimize P to get the

maximal sum rate.

The equivalent signal model of MMSE pre-coding scheme

can be written as

?ρ

where the normalization factor γ is

2P1g?α2−1+g2+|δ|2?

+(1+g+α)2(1−g)−(1+g+2α)|δ|2

[(1+g+α)(1−g−α)−|δ|2]2

Therefore, the two signal branches can be expressed by

parameters ρ, g, δ and power allocations P1and P2. We can

calculate the SINR and then obtain the sum rate of the MMSE

pre-coding scheme, subject to P1+P2=1, as

?1−g2.

HHH+ αI

?−1

,

(16)

y =

γG(G + αI)−1√Px + n,

(17)

γ =

[(1+g+α)(1−g−α)−|δ|2]2

.

(18)

CMMSE= max

P1,P2

2

?

i=1

log2(1 + SINRi),

(19)

where

SINR1=

ρP1

?(1+g)(1−g+α)−|δ|2?2

ρP2α2|δ|2+γ [(1+g+α)(1−g+α)−|δ|2]2

(20)

and

SINR2=

ρP2

?(1−g)(1+g+α)−|δ|2?2

ρP1α2|δ|2+γ [(1+g+α)(1−g+α)−|δ|2]2. (21)

Closed form expressions of optimal power allocation and

maximized sum rate can be reached but are far too long and

complicated to be given here, but in the case of g=0, a simple

expression of the sum rate is obtained as

?

V. PERFORMANCE COMPARISON

CMMSE|g=0= 2log2

1 +ρ

2

?

1 −

ρ

ρ + 2|δ|2

??

.

(22)

Using the closed form sum rate expressions above, we first

study how the DPC capacity and linear pre-coding sum rates

are affected by correlation and power imbalance between user

channels. We then let the number of the base station antenna

grow large, both for measured channels and simulated i.i.d.

Gaussian channels, to see to what degree a realistic propaga-

tion environment decorrelates the user channels. Finally, we

00.1 0.20.3 0.40.50.60.70.80.9

0

1

2

3

4

5

6

|δ|2

Sum Rate [bps/Hz]

DPC

ZF

MMSE

SU

Fig. 5.

channel correlation-related factor |δ|2when ρ = 10 dB, g = 0.3.

Sum rates for DPC, ZF, MMSE and single user transmission versus

compare the linear pre-coding sum rates with the DPC capacity

as the number of antennas increases.

A. Numerical evaluation

It can be seen from the expressions above that if the channel

correlation approaches to zero, i.e. |δ|≈0, ZF and MMSE pre-

coding sum rates become equal to the DPC capacity,

??2 + ρ?1 − g2??2

If the channel correlation grows very high, i.e. |δ| approaches

?1−g2, signal power would only be transmitted over the

capacity. In that case, CMMSE and CDPC become equal to

single user transmission rate in (7), while CZFtends to zero.

Fig. 5 shows the DPC capacity, linear pre-coding sum

rates and single-user transmission rate as functions of the

correlation-related factor |δ|2for ρ = 10 dB and g = 0.3.

We can see that the gap between DPC capacity and linear

pre-coding sum rates becomes smaller when |δ|2decreases.

Eventually the linear pre-coding sum rates are the same as

the DPC capacity when |δ|2= 0, i.e, when the two user

channels are orthogonal. When the channel correlation grows

high, ZF capacity decreases rapidly to zero and the DPC

capacity decreases to single-user capacity. It is interesting to

notice that MMSE sum rate decreases first and in fact becomes

lower than the single-user capacity, but then increases after

|δ|2reaches a certain value, e.g. around 0.7-0.8 in this figure.

By investigating the power allocation for MMSE, we find the

power is only transmitted to the stronger user channel when

|δ|2> 0.7, hence, as the correlation gets higher, the MMSE

pre-coding eventually approaches the single-user transmission.

The channel power imbalance factor g also has an effect

on the capacity. Basically, as g grows, the channel power

difference becomes large and thus the channel correlation

|δ|/?1−g2grows. Consequently, the ZF sum rate decreases

first and then both become the same as single-user capacity.

Furthermore, the DPC capacity and linear pre-coding sum

rates are low when g is large. Hence, in order to have higher

CZF,MMSE,DPC= log2

4(1 − g2)

?

.

(23)

stronger user channel, and the other user would get zero

rapidly while the MMSE sum rate and DPC capacity decrease

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capacity, users with small channel power differences should be

served at the same time according to some grouping strategies.

B. Measured channels

As the number of base station antennas M increases,

one hopes that the two user channels become less and less

correlated. The ideal scenario would be that if the two users

are spatially separated enough, the channels could be approx-

imately orthogonal, i.e. |δ|2approaches zero. In that case the

DPC capacity could be achieved by linear pre-coding schemes.

Here we verify whether it is true or not that the correlation

decreases as M goes large for measured physical channels.

Simulated i.i.d. Gaussian channels with the same dimension

and channel power imbalance as the measured channels are

used as reference.

We select one representative group of user positions - the

two users are positioned outside house 49 and 53 respectively.

Fig. 6 plots the average channel correlation as a function of

M. For each M, the averaging is performed over time and

frequency, and also over different groups of antennas since

the cylindrical structure of the array may cause receive power

imbalances over the antennas. From Fig. 6 we can see that the

channel correlation is higher in the measured channels than in

the i.i.d. Gaussian channels. This is because the two user posi-

tions are close and probably have common scatterers that make

the channels similar [3]. However, in both i.i.d. Gaussian and

measured channels, the average channel correlation decreases

as M increases. This suggests that the very-large array can

decorrelate the user channels. Then we compare the linear

pre-coding sum rates with the DPC capacity. Fig. 7 shows the

ratio of average linear pre-coding sum rates and average DPC

capacity as M grows. The transmit SNR here is set to 20

dB and the total transmit power is kept unchanged. With the

increase of M, the ratios for ZF and MMSE in both i.i.d. and

measured channels is close to one.

We notice that the channel correlation in Fig. 6 decreases

fast as the number of base station antennas increases from

2 to 8. Correspondingly in Fig. 7 the ratios of linear pre-

coding sum rates and DPC capacity grow very rapidly as the

number of antennas increases. When the number of antennas

increases to more than 20, the channel correlation as well

as the sum rate ratios saturate. At 20 base station antennas

the linear pre-coding sum rates already reach 98% of DPC

capacity. This shows that the optimal DPC capacity can be

achieved by linear pre-coding schemes at a relatively limited

number of base station antennas.

VI. SUMMARY AND CONCLUSIONS

In this paper, linear pre-coding performance is studied for

measured very-large MIMO downlink channels. We find that

the user channels, in the studied residential-area propagation

environment, can be decorrelated by using reasonably large

antenna arrays at the base station. With linear pre-coding,

sum rates as high as 98% of DPC capacity were achieved for

two single-antenna users already at 20 base station antennas.

This shows that even in realistic propagation environments

28 14 2026 32

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Base Station Antennas

Channel Correlation

Measured Channel

IID Gaussian Channel

Fig. 6.

number of base station antennas. The two users are outside house 49 and 53

respectively.

Average channel correlation |δ|2/?1−g2?

as a function of the

28 1420 2632

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Number of Base Station Antennas

Ratio of Average Sum Rates

ZF, Measured Channel

MMSE, Measured Channel

ZF, IID Gaussian Channel

MMSE, IID Gaussian Channel

Fig. 7.

function of the number of base station antennas. The two users are outside

house 49 and 53 respectively.

Ratio of average linear pre-coding sum rate and DPC capacity as a

and with a relatively limited number of antennas, we can see

clear benefits with using an excessive number of base station

antennas.

ACKNOWLEDGEMENT

The authors would like to acknowledge the support from

ELLIIT – an Excellence Center at Link¨ oping-Lund in Infor-

mation Technology.

REFERENCES

[1] E. Dahlman, S. Parkvall, J. Sk¨ old, and P. Beming, 3G Evolution HSPA

and LTE for Mobile Broadband.

[2] T. Marzetta, “Noncooperative cellular wireless with unlimited numbers of

base station antennas,” IEEE Transactions on Wireless Communications,

vol. 9, no. 11, pp. 3590–3600, Nov 2010.

[3] X. Gao, A. Alayon Glazunov, J. Weng, C. Fang, J. Zhang, and F. Tufves-

son, “Channel measurement and characterization of interference between

residential femto-cell systems,” in Proceedings of the 5th European

Conference on Antennas and Propagation (EUCAP), Rome, Italy, April

2011, pp. 3769–3773.

[4] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless

Communications. UK: Cambridge University Press, 2003.

[5] C. Peel, B. Hochwald, and A. Swindlehurst, “A vector-perturbation

technique for near-capacity multiantenna multiuser communication-part

i: channel inversion and regularization,” IEEE Transactions on Commu-

nications, vol. 53, no. 1, pp. 195–202, Jan 2005.

[6] S. Shi and M. Schubert, “MMSE transmit optimization for multi-user

multi-antenna systems,” in IEEE International Conference on Acoustics,

Speech, and Signal Processing, March 2005, pp. iii/409–iii/412.

Academic Press, 2008.