mimo systems with mutual coupling_ how many antennas to pack into fixed-length arrays

7/29/2019 MIMO Systems With Mutual Coupling_ How Many Antennas to Pack Into Fixed-length Arrays

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MIMO Systems with Mutual Coupling: How Many

Antennas to Pack into Fixed-Length Arrays?

Shuo Shen, Matthew R. McKay and Ross D. MurchDepartment of Electronic and Computer Engineering

The Hong Kong University of Science and Technology

Clear Water Bay, Kowloon, Hong Kong

Email: [email protected], [email protected], [email protected]

AbstractFor multiple-input multiple-output wireless systems,we investigate the important question of how many antennasto place in fixed-length arrays, accounting for both spatialcorrelation and mutual coupling at the transmitter and receiver.We show that there is an optimal antenna configuration yieldingthe highest capacity which depends strongly on the array lengthand the transmission wavelength, but does not depend stronglyon the signal to noise ratio (SNR). Moreover, we show thatignoring the effect of mutual coupling gives misleading results,

yielding unbounded capacity growth. As another key finding, wedemonstrate the surprising result that if the optimal antenna con-figuration is employed, then further optimizing the transmissionbased on the channel statistics gives very little benefit over simpleequal-power spatial multiplexing. By deriving an expression forthe capacity at low SNR, we provide a straightforward methodfor estimating the optimal number of antennas.

Index TermsMIMO, correlation, mutual coupling, capacity

I. INTRODUCTION

Multiple-input multiple-output (MIMO) technology is a key

candidate for enhancing the capabilities of modern wireless

communication systems. Two of the most important factors

which govern the performance of practical MIMO systems arespatial correlation and antenna mutual coupling [1]. Spatial

correlation occurs due to a lack of scattering in the environ-

ment, and its effect on MIMO performance has been studied

extensively in recent years. Mutual coupling, on the other

hand, is a physical phenomenon which occurs when the anten-

nas becomes sufficiently close to one another [2]. Compared

with spatial correlation, the effect of mutual coupling has

received far less attention. Prior work dealing with mutual

coupling is presented in [36], where it was demonstrated

that mutual coupling among closely-placed antenna elements

leads to a lower correlation, and may in fact deliver increased

capacity by providing a higher angle diversity. Further results

were presented in [7] and [8], which investigated the effect ofmutual coupling on the capacity of MIMO systems. In [7], the

formulation is based on using antenna patterns in which the

other ports are open, whereas in [8] the formulation is based

on patterns in which the other ports are shorted. While both

formulations are correct we prefer to use the formulation in

[7]. This is because when we calculate the patterns with the

other ports open the other antennas are effectively invisible

[9] and the mutual coupling effects are fully incorporated into

the coupling matrix itself. Therefore we can simply use the

antenna patterns of the individual antennas themselves.

In this paper we consider the following key issue which

arises in the design of MIMO systems: How many antennas

should be employed within the transmit and receive arrays of

a fixed length, in order to maximize capacity? This important

practical question has yet to be adequately addressed. Some

related work has been presented in [10] and [11], however the

effects of mutual coupling were not explicitly accounted for,and a heuristic approach was employed to capture the inherent

power loss. In this paper, we consider the rigorous mutual

coupling model advocated in [7], and aim to gain practical

insights into the channel capacity under various conditions.

We first present simulations to characterize how the MIMO

capacity with spatial correlation and mutual coupling varies

with the number of antennas for uniform linear arraysconstrained to a fixed length. Assuming that the transmitter

has no knowledge of the channel and therefore employs

equal-power spatial-multiplexing transmission, we find that

the capacity increases with up to a point opt, and thendecreases beyond that point. In contrast, if mutual coupling

is ignored, the capacity increases monotonically with andtherefore gives misleading results, particularly beyond opt.We also investigate the effect of transmitter optimization,

assuming that the channel statistics are known at the trans-

mitter. Our key finding is that whilst transmitter optimization

leads to noticeable capacity improvements beyond opt, for opt the benefit is almost negligible, despite demandinga significantly increased complexity. This demonstrates that

for MIMO systems with fixed length arrays, it is extremely

important to optimize the number of antennas; however, if this

is done correctly, then further optimization of the transmitted

signals based on the channel statistics is not required.

Whilst obtaining an analytical solution for opt as afunction of the system parameters is difficult in general, we

show that this problem can be simplified by considering the

low signal to noise ratio (SNR) regime. To this end, we first

derive a simple expression for the low SNR capacity, adopting

the general framework from [12, 13]. We then demonstrate

that the antenna optimization problem can be re-posed into

one involving the minimum required normalized energy per

information bit, which admits a very simple expression and is

trivial to evaluate.

ISITA2010, Taichung, Taiwan, October 17-20, 2010

9781424460175/10/$26.00 c 2010 IEEE531


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I I . SYSTEM MODEL

Dipole antennas are assumed to be placed in an array of

fixed total length . The length of each dipole is a halfwavelength. The same antenna arrays are used at both the

transmitter and receiver. Letting and denote the numberof transmit and receive antennas respectively, the 1received signal vector is given by

r = Hx+ n, (1)

where x is the 1 transmitted symbol vector, n is an1 noise vector with independent zero-mean unit-variancecomplex Gaussian entries, and H is the channel matrix. We

model H stochastically, taking into account the joint effects

of spatial correlation and mutual coupling, as follows

H = CR1

2HS1

2C . (2)

Here, H is an matrix of independent zero-meanunit-variance complex Gaussian variables, R is the correlation matrix at the receiver side, and S is the cor-relation matrix at the transmitter side. Assuming a uniformly

distributed angular spread, from Jakes model, each entry ofthe receiver correlation matrix is given by

R = 0(2/) , = 1,...,, (3)

where is the wavelength, 0() is the zeroth-order Besselfunction, and is the distance separating the th and thantenna elements. Assuming that the antenna elements are

placed uniformly in the array, =

1. The transmit

correlation matrix S is obtained similarly. The matrixC and matrix C reflect the mutual coupling effectsat the receive and transmit sides respectively, and are defined

according to [7]

C = Z(Z + Z)1, C = (Z +Z)

1 . (4)

Here, Z is an impedance matrix at the trans-mitter end with diagonal elements containing the self

impedance terms and off-diagonal entries containing the mu-

tual impedance terms, and Z is a diagonal matrix with

diagonal entries equal to the conjugate of the diagonal entries

of Z. For a dipole array, Z is well-known from standard

antenna theory (e.g., see [14, Ch. 8]). Similar to Z, Z is the

impedance matrix at the receiver end, and Z is a diagonal

matrix with diagonal entries equal to the conjugate of the

diagonal entries ofZ.

The factor in (7) is used for normalization purposes. Toestablish this constant, we note that if the receiving antenna

elements are uncoupled, then C is diagonal with entries

[C] =(Z)

(Z) + (Z)=

(Z)11(Z)11 + (Z)11

= .

(5)

Hence, it makes sense to normalize the receiver mutual cou-

pling matrix C by dividing by . Similarly, we introducea normalization factor for the transmitter,

=1

(Z)11 + (Z)11. (6)

The channel matrix can then be written as

H =CR

1

2HS1

2C

, (7)

and the normalization factor =1

.

III. MIMO CHANNEL CAPACITY WITH MUTUAL

COUPLING AND CORRELATION

In this section, we investigate the capacity of the MIMOchannel, taking into account both spatial correlation and mu-

tual coupling effects. We also present analytical results for

the low SNR regime, which lead to a simple method for

optimizing the number of antennas to place in the transmit

and receive arrays.

A. Mutual Coupling and Optimal Number of Antennas

In this section, we ask the question: If the transmitter and

receiver have the same length- antenna array, and are eachequipped with antennas, then what is the optimal numberof antennas to maximize capacity?

We start by assuming that H is known perfectly at the

receiver (e.g., through the use of standard channel estimation

techniques), but the transmitter either has no knowledge ofH

or chooses not to use this knowledge. In this case, it is common

to assume that the input x is zero-mean complex Gaussian with

covariance Q = I, which transmits independent equal-powersignals from each antenna. In this case, the ergodic capacity1

(in bits/s/Hz) is given by

iid =

[log2 det

(I+

SNR

HH

)], (8)

where SNR is the average received SNR per element in the

absence of mutual coupling. The expectation is taken with

respect to the matrix-variate distribution ofH. For our channel

model (7), closed-form solutions can be obtained for this

expectation based on the results in [15], with some minor

modifications. The equations however, involve considerable

notation and are not reproduced here.

Fig. 1 shows the capacity as a function of , for atotal array length of 1 wavelength, i.e., = . The curvemarked correlation and coupling is based on (8) and our

channel model (7). The important observation from the figure

is that if we explicitly take into account both the correlation

effects of the channel and the mutual coupling effects of the

antennas, then the capacity increases monotonically with up to a certain point, = 4, and then begins to decrease

monotonically with . Fig. 2 shows a similar phenomenon foran array of length = 2; however, in this case the turningpoint is increased to = 6. The results shown in Figs. 1and 2 are given for a moderately high SNR of 20 dB. Fig.3 demonstrates the corresponding results for a low SNR of

0 dB, considering an array length of = . We observe thesame general behavior as for the high SNR results and, perhaps

surprisingly, we see the same turning point = 4 as shown in

1Strictly speaking, this is the mutual information rather than the channelcapacity, under the assumption of independent equal-power Gaussian inputs.

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Fig. 1. This implies that optimal number of antennas to employ

does not have a strong dependence on the operating SNR, but

as expected has a strong dependence on the relative antenna

spacing. This, in turn, suggests that we may gain more insights

into the optimal antenna configuration by focusing on these

asymptotic regimes, in which case the capacity expression (8)

simplifies into more intuitive forms. This will be explored in

more detail in Section IV, where we focus primarily on the

low SNR regime.

B. Comparison with Correlation-Only Model

Whilst much previous work has investigated the effect of

spatial correlation on the capacity of MIMO systems (see e.g.,

[15, 16]), the effect of mutual coupling is typically not consid-

ered. Thus, it is of interest to study the difference in capacity

between such correlation-only models, and the capacity with

mutual coupling. The correlation-only model is obtained from

(7) by substituting C = I and C = I, in which casethe channel then reduces to the now-standard representation

H = R1

2HS1

2 . The corresponding curves are shown in

Figs. 13, labeled as correlation but no coupling. From thesecurves, we observe some very interesting behavior. First, for

the high SNR scenarios in Figs. 1 and 2, the correlation-only

curve closely follows the corresponding correlation-coupling

curve up to the turning point, indicating that mutual coupling

has relatively little impact if the SNR is high and the number

of antennas is small. However, beyond the turning point, the

behavior is markedly different. In particular, the correlation-

only curve, whilst having a deflection at the turning point,

still continues to increase monotonically as the number of

antennas are further increased. Thus, beyond the turning point,

considering a correlation-only gives incorrect results, implying

that the capacity grows unbounded with . Similar trends arealso seen for the low SNR scenario in Fig. 3, however in

this case the correlation-only model overestimates the capacity

even for below the turning point.

The difference in behavior observed by ignoring mutual

coupling can be explained by comparing the average channel

gains [tr(HH)] for spatially correlated channels, with andwithout mutual coupling. We know that mutual coupling will

yield a loss in transmission power, which in our model is

reflected by a loss in the average channel gain. This is shown

in Fig. 4, where the average channel gain with mutual coupling

is seen to increase with up to the turning point ( = 4), andthen decreases monotonically. If mutual coupling is neglected

however, the average channel gain grows unbounded.These results, in general, indicate the importance of con-

sidering both channel effects (spatial correlation) and antenna

effects (mutual coupling) to accurately predict the capacity of

MIMO channels.

C. Benefit of Transmitter Optimization

So far, we have assumed that the transmitter employs

uncorrelated equal-power Gaussian inputs. If the transmitter

has knowledge of the channel distribution however, then it

0 5 10 155

10

15

20

25

30

No. of antennas

Capacity(bits/s/Hz)

correlation and coupling

correlation but no coupling

correlation and coupling with power allocation

Fig. 1. Comparison of ergodic MIMO Capacity with (i) mutual coupling andequal power allocation, (ii) no mutual coupling and equal power allocation,and (iii) mutual coupling and optimal power allocation. Results are shown forSNR = 20 dB and = 1.

0 5 10 155

10

15

20

25

30

35

40

No. of antennas

Capacity(b

its/s/Hz)




Fig. 2. Comparison of ergodic MIMO capacity with (i) mutual coupling andequal power allocation, (ii) no mutual coupling and equal power allocation,(iii) mutual coupling and optimal power allocation. Results are shown forSNR = 20 dB and = 2.

can use this knowledge to optimize its transmission. In this

case, it is well-known that the ergodic capacity is given by

opt = maxQ

[log2 det

(I+

SNR

HQH

)], (9)

where the optimization is over the input covariance matrix

Q = [xx]

1

[x2]

, which is normalized to satisfy tr[Q] = .

A closed-form solution to this problem is intractable, however

iterative solutions have been proposed [17]. In general, the

ergodic capacity (9) will be higher than that achieved with

equal-power inputs in (8), however the complexity is also

increased significantly. Moreover, this complexity difference

increases with the number of antennas. Here, we aim to

investigate the capacity benefit obtained through transmitteroptimization, taking into account both mutual coupling and

spatial correlation.

The ergodic capacity curves based on (9) are plotted on

Figs. 13, labeled as correlation and coupling with power

allocation. These results were generated using the iterative

algorithm from [17]. On all three figures we see the inter-

esting behavior that, up to the respective turning points, the

optimization gives very little benefit. This is true for both high

and low SNR. Moreover, even with transmitter optimization,

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0 5 10 150

1

2

3

4

5

6

7

No. of antennas

Capacity(bits/s/Hz)




Fig. 3. Comparison of ergodic MIMO capacity with (i) mutual coupling andequal power allocation, (ii) no mutual coupling and equal power allocation,and (iii) mutual coupling and optimal power allocation. Results are shown forSNR = 0 dB and = 1.

we observe the same exact turning points as for the equal-

power case, beyond which the capacity begins to decrease

monotonically. Thus, the main benefit of optimization is only

seen to be a more gradual degradation with . This can beexplained by noting that as increases, mutual coupling andcorrelation become more severe, and the transmitter can ex-

ploit this information to more effectively steer or beamform

the signals along stronger channel directions. The expense paid

for this more gradual degradation in capacity is a much higher

complexity, which increases significantly with .These results are quite unexpected and highlight the im-

portance of selecting the optimal number of antennas. In

particular, they imply that if the system is optimized with

respect to the number of antennas, then there may be very little

to be gained from covariance optimization of the input signals.

It is therefore of interest to further investigate analytical

methods for establishing this optimal antenna configuration,

for a given antenna array of a given length. In Section IV we

will do this by appealing to the low SNR regime.

D. Comparison with Previous Models

Previous contributions have shown that for a MIMO system

with fixed length antenna arrays and only correlation, the

capacity converges to a limit as goes to infinity [10, 11].In their work, the following approximation was used to model

the effect of fixed length arrays:

=

[log2 det

(I+

SNR

2HH

)]. (10)

This modification, which further scales the transmitted powerby 1/, ensures that the total received power does not divergeas increases, and provides a straightforward approximationfor capturing the power loss effect caused by mutual coupling.

However, we have found that whist this model yields the same

turning point as the complete correlation-coupling model in

(7), beyond this turning point the approximation is no longer

accurate. We also mention that the alternative mutual coupling

model presented in [8] (based on shorted ports), seems to yield

a different turning point altogether.

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

No. of antennas

ChannelGain

no mutual coupling

mutual coupling

Fig. 4. Comparison of channel gain [tr(HH)] for (i) correlation andmutual coupling, (ii) correlation but no mutual coupling. Results shown for = 1.

IV. ANALYTICAL CHARACTERIZATION AT LOW-SNR

From the results above, we have observed that there is a

capacity turning point as the number of antennas increase, and

that this turning point depends primarily on the total length of

the antenna array (relative to the wavelength), but not the SNR.Thus, to gain more insights into the joint effect of correlation

and mutual coupling, here we focus on the low SNR regime.

A. Low-SNR Capacity with Mutual Coupling and Correlation

A general framework for studying the low SNR capacity

was presented in Verdus pioneering paper [12], and further

elaborated for MIMO systems in [13]. It was shown that at

low SNR the capacity could be expressed as a linear function

of the normalized energy per information bit, 0

, as follows

C

(0

) 0 log2

0

0min

(11)

where 0min

denotes the minimum required 0

for reliable

communications, and 0 is referred to as the wideband slope.Based on this, we have the following key result:

Proposition 1: For a MIMO system with mutual coupling

and correlation, the capacity at low SNR, as a function of 0

,

is given by (11) with

0 min

= log 2

22

tr()tr()(12)

and

0 =2

tr(2)

tr2()+

tr(2)

tr2()

(13)

where = CRC and = C

SC.

Proof: See the Appendix.

Fig. 5 compares the exact capacity, based on Monte-Carlo

simulation, with the linear approximation based on Proposition

1. As evident from the figure, the analytical approximation is

quite accurate. Note that the results are shown as a function of

0, where is the received energy per information bit in the

absence of mutual coupling, which is given by = ,with defined in (19).

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B. Comparison with Correlation-Only Model

We now investigate the implications of ignoring mutual

coupling at low SNR. We require the following corollary:

Corollary 1: For a MIMO system with correlation but no

mutual coupling, the capacity at low SNR, as a function of0

, is given by (11) with

0 min = log

21

(14)

and

0 =2

(S)

+ (R)

. (15)

Here, () is the matrix dispersion function, defined for an matrix X as (X) = tr(X2)/.

Proof: Follows by setting C = I and C = I inProposition 1, and using tr(R) = and tr(S) = .

The result in Corollary 1 was derived previously in [13].

Comparing (12) and (14), we see that whilst correlation by

itself has no effect on 0

min

, if both mutual coupling and

correlation are considered, then 0min

has a strong depen-

dence through the matrices and . In particular, mutual

coupling has the negative effect of increasing the required0min

by a factor of , as given in (21). This behavior is

illustrated in Fig. 6, which plots 0min

as a function of the

number of antennas, considering = = . The curvesare based on (12) and (14). We clearly see that if mutual

coupling is ignored, then 0min

decreases monotonically as

more antennas are added. With mutual coupling, however, the

behavior is markedly differentdecreasing to a point = 4,and then increasing beyond that. This is explained intuitively

by noting that for 4, the additional multiplexing gain

afforded by adding more antennas outweighs the power lossdue to mutual coupling, as the number of antennas increase.

Beyond this, however, the power loss due to mutual coupling

has a more dominant effect (i.e., due to the antennas being

packed so closely together), which leads to a net loss in

overall capacity. Importantly, this turning point is consistent

with that found for the exact capacity in Figs. 1 and 3. These

results again highlight the importance of considering a joint

correlationcoupling model.

Fig. 7 plots the wideband slope 0 versus the number ofantennas, comparing the scenarios with and without mutual

coupling. Again, = = is considered, and thecurves are based on (13) and (15). We see that in contrast

to the 0min, mutual coupling has relatively little effect on0. Perhaps surprisingly, the wideband slope is underestimatedif mutual coupling is ignored. The most important point,

however, is that we see the same general trend with and

without mutual coupling, and that the familiar turning point

= 4 occurs for both curves.

C. Estimating the Optimal Number of Antennas

Whilst it is very difficult to find the optimal number of

antennas to maximize the capacity, even at low SNR, based on

2 1 0 1 2 30

1

2

3

4

5

6

7

8

Eb

r/N0

(dB)

Capacity(bits/s/Hz)

N=1 exact

N=1 appx

N=4 exact

N=4 appx

Fig. 5. Comparison of exact ergodic MIMO capacity with mutual couplingand correlation, and the analytical low SNR approximation. Results are shownfor = 1 and = 4 , with = 1.

1 2 3 4 5 6 7 8 9 1012

10

8

6

4

2

0

No. of antennas

Eb/N

0min(dB)



Fig. 6. Comparison of the minimum required energy per information bit

0minfor (i) correlation and mutual coupling, and (ii) correlation but no

mutual coupling. Results are shown for = 1.

Fig. 7 (and also on other numerical experiments not shown),

we propose a new simple design method for estimating the

optimal solution. The method is to consider a correlation-only model, and to choose the number of antennas which

maximizes 0. For the case of = = , this leadsto the very simple optimization problem:

opt = arg min

tr(R2)

2. (16)

Whilst a closed-form solution is not forthcoming, this is

trivially evaluated numerically for any given value of and ,based on (3). The results obtained through this optimization

have been validated experimentally, and have been found to

accurately estimate the optimal capacity-maximizing antenna

configuration.

V. CONCLUDING REMARKS

We have investigated the capacity of MIMO wireless sys-

tems where dipole antennas are placed in a fixed-length linear

array. We have demonstrated that it is critical to account for

the joint effects of spatial correlation and mutual coupling, and

that ignoring the effects of mutual coupling gives misleading

results. Another important and unexpected finding is that if the

number of antennas is properly chosen, then further optimizing

the transmitted signals based on the channel statistics yields

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1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

4

No. of antennas

Slope



Fig. 7. Comparison of the wideband slope 0 in bits/s/Hz/(3dB) for (i)correlation and mutual coupling, and (ii) correlation but no mutual coupling.Results are shown for = 1.

very little additional gainsimply employing equal-power

spatial multiplexing techniques yield almost the same capacity.

We also identified a simple method for estimating the optimal

antenna configuration for a given array length and transmission

wavelength, based on capacity expressions which we derived

for the low SNR regime.

APPENDIX

We start by noting that, due to the effect of mutual coupling,

there is a power loss at the transmitter and the receiver. Thus,

whilst the channel gain [tr(HH)] without mutual couplingis , with mutual coupling it becomes less. It is thereforeconvenient to define a normalized matrix H1, satisfying

[tr(H1H1)] = , (17)

in order to separate the distinct effects of power loss (due to

coupling) and correlation. From (7), this can be written as

H1 = 0CR1

2HS1

2C, (18)

where 0 is an appropriate normalization factor, and thechannel matrix H becomes H =

0H1. Define as the

average power loss, which is given by

=[tr(HH)]

[tr(H1H1)]

=20

. (19)

From (17) and (18),

20 =

[tr(CRCHC

SCH

)]

=

tr()tr()

. (20)

Substituting =1

and (20) into (19) yields

=tr()tr()

22. (21)

Now, from [12, 13], 0min

and 0 can be computed from thefirst and second derivatives of (SNR) at SNR = 0 via

0 min

=

[tr(H1H1)]

1

C(0)(22)

and

0 =2[C(0)]2

[C(0)]log 2 . (23)

Using the properties dd

log det [I+ A]=0 = tr(A) andd2

d2log det [I+ A]=0 = tr(A

2), we have

0 min=

log 2

[tr(H1H1)](24)

and

0 =2

(H1H1)

. (25)

From (21) and (24) we obtain (12). From (18) and (25) we

obtain (13).

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