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Antenna Selection for Time Reversal MIMO UWB

Systems

Hieu Nguyen, Feng Zheng and Thomas Kaiser

Institute of Communication Technology

Leibniz University of Hannover

Hannover, Germany

Email: {hieu.nguyen, feng.zheng, thomas.kaiser}@ikt.uni-hannover.de

Abstract—An antenna selection scheme for Multi-Input Multi-

Output (MIMO) Ultra-Wide Band (UWB) communication system

with Time Reversal (TR) is investigated in this paper. The system

exploits spatial multiplexing (SM) scheme to achieve high data

rate. In order to cope with long delay spread of the UWB

channel, time reversal technique is adopted. TR can mitigate not

only the Interference InterSymbol (ISI) but also Multi Stream

Interference (MSI) caused by transmitting several data streams

simultaneously. Antenna selection algorithm allows to reduce

the number of transmit antenna by using the Channel State

Information (CSI), which is already available for time reversal.

Simulation results show that the algorithm can considerably

improve the BER performance of the system when the number

of diversity branch is not so large.

I. INTRODUCTION

An Ultra Wideband (UWB) communication system, whose

relative bandwidth is usually defined as greater than 25 %, has

become a promising candidate for high-data rate and short

range communication systems. It has attracted great interest

from both academic and industrial aspects recently [1], [2], [3].

However, due to the wide bandwidth property, UWB systems

suffer from a very long delay spread by multipath effect [4],

[5], [6], [7], [8]. Time Reversal (TR) technique has showed its

potential in dealing with the ISI problems in UWB [9], [10]. In

TR, the time reversed channel impulse response (CIR) is taken

as a filter at the transmitter side. This process leads to a very

narrow focus of power at the receiver at one specialized time

instant and one specialized space position if the CIRs between

any two communication-pairs at different spatial points are de-

correlated.

The space-time focusing feature is also beneficial in a

MIMO spatial multiplexing scheme [11], [12], [13]. The

potential of a MIMO UWB system using spatial multiplex-

ing scheme is considered in [14], where the matched filter

plays a role of passive time reversal filter and the Maximum

Likelihood (ML) detector is used to deal with the MSI but

ignores the ISI. Several studies have applied TR technique

to improve the performance of the beamforming multiple

antennas systems [15], [16], [17], [18], [19]. In [20], TR is

used for a multi-antenna multi-user system, where TR pre-

filter has taken its spatial focusing advantage to mitigate the

multi user interference (MUI), nevertheless, it suffers a Bit

Error Rate (BER) curve floor in high SNR regime and also

when the number of users grows high.

In this paper, TR is proposed to cope with both MSI and

ISI in MIMO spatial multiplexing as a low cost, low power

and low complexity receiver solution. Time reversal filter has

naturally improve the signal power to MSI ratio. When the

number of transmit antenna is large, TR-MIMO-UWB system

can exploit transmit diversity and its performance approaches

the performance in AWGN channel. However, the more spatial

diversity branch the higher hardware cost. Antenna selection

algorithms have been used to reduce the number of transmit

antenna [21], [22]. In this paper, antenna selection scheme

is proposed by using the channel state information, which

is already available for time reversal. When the number of

transmit antenna is not large, the selection algorithm based on

maximizing Signal-to-Interference Ratio (SIR) improves the

performance of TR-MIMO-UWB considerably.

II. SYSTEM DESCRIPTION

A. SISO UWB system and Time Reversal

Let us consider the impulse UWB system using Binary

Pulse Amplitude Modulation (BPAM) with pulse shaping

according to desired power spectrum density. The transmitted

signal is

?Eb

where dk={±1} is the transmit data, Ebis the bit energy, p(t)

is the desired pulse shape, and T denotes the symbol duration.

The pulse shaped signal is transmitted over a multipath

channel with its channel impulse response (CIR) h(t).

x(t) =

+∞

∑

k=−∞

dkp(t −kT),

(1)

h(t) =

L

∑

l=1

αlδ(t −τl),

(2)

where αlis the amplitude, τlis the delay of the l-th tap and

L is the maximum delay spread.

p*(t) p(t)

d={+1,-1}

h(t)

()dt

Fig. 1. Block diagram of SISO UWB system.

The received signal after transmission over multipath chan-

nel is

y(t) = x(t)⊗h(t)+v(t),

(3)

978-1-4244-2517-4/09/$20.00 ©2009 IEEE

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where v(t) is the additive noise, and ⊗ denotes the convolution

operation.

The receiver is a simple integrate-and-dump receiver with

sampling rate is 1/T.

Time reversal technique is proposed to combat the ISI

problem caused by multipath channel [18], [20]. In the TR

technique, the reverse of the CIR is used at the transmitter

side as a pre-matched filter h(−t) to generate the encoded data

c(t) = h(−t)⊗x(t). Now, the received signal is expressed as

y(t)=

h(t)⊗[h(−t)⊗x(t)]+v(t)

=

R(t)⊗x(t)+v(t)

where R(t)=h(t)⊗h(−t) is the autocorrelation of the impulse

response of multipath channel. The power of channel is

focused within a very narrow time duration and the TR signal

can achieve high focusing gain.

The performance of a TR system can approach the perfor-

mance of matched filter bound (MFB) [13]. If the power of

channel is normalized to 1, this performance is close to the

performance of BPSK system over AWGN channel

??2Eb

where

Q(x) ?

√2π

B. MIMO-TR-UWB using Spatial Multiplexing

In the MIMO spatial multiplexing scheme, the MIMO chan-

nel degrees of freedom are exploited to achieve the high data

rate without expanding the bandwidth or adopting high order

of constellation map. In the SM system, several streams of data

are transmitted over several transmit antennas at the same time.

The channel capacity can be increased proportionally to the

number of antennas. However, the SM scheme has a limitation

in performance caused by MSI.

,

(4)

Pe= Q

N0

?

,

(5)

1

?−∞

x

e

−t2

2 dt.

x1

p*(t)

p*(t)

TR

filter

1

1

MN

h11

h1N

hM1

hMN

xN

()dt

()dt

Fig. 2.Block diagram of TR-MIMO-UWB system

In this paper, the TR scheme is used in the SM-UWB system

in order to cope with the ISI and MSI problems. The UWB-

TR-MIMO system is illustrated in Fig. 2 with M transmit and

N receive antennas.

We have M × N multipath channel realizations between

transmit and receive sides. We assume that the maximum

length of each channel realization is L. The channel impulse

response between transmit antenna j and receive antenna i is

hi,j(t) =

L

∑

l=1

αi,j

lδ(t −τi,j

l),

i = 1...N, j = 1...M

(6)

or the discrete time form in reverse order

hi,j= [hi,j[L−1],hi,j[L−2],...,hi,j[0]]T.

The entire channel between transmitter and receiver is

⎛

⎜

h1,M

h2,N

with dimension N×ML.

If CIR of all channels hijare known at the transmitter side,

time reverse of them are used to filter the transmit data. We

construct filtering matrix based on the time reverse of the

channels. This matrix is an ML×N(2L−1) matrix given by

⎛

⎜

¯H1,M

¯H2,N

where each sub-matrix¯Hi,jis an L×(2L−1) Toeplitz matrix

defined by

(7)

H =

⎜

⎜

⎝

h1,1

h1,2

...

h2,1

h2,2

...

...

...

...

...

hN,1

hN,2

...

hN,M

⎞

⎟

⎟

⎟

⎠

T

,

(8)

¯H =

⎜

⎜

⎝

¯H1,1

¯H1,2

...

¯H2,1

¯H2,2

...

...

...

...

...

¯HN,1

¯HN,2

...

¯HN,M

⎞

⎟

⎟

⎟

⎠,

(9)

¯Hi,j=

⎛

⎜

⎜

⎜

⎝

hi,j[0]

0

...

0

...

hi,j[L−1]

...

...

0

0

...

...

...

...

0

0

...

hi,j[0]

...

...

hi,j[L−1]

...

hi,j[0]

hi,j[L−1]

⎞

⎟

(10)

⎟

⎟

⎠.

We define the equivalent channel response isˆH=H¯H which

has dimension N×N(2L−1)

⎛

⎜

ˆh1,N

ˆh2,N

Each elementˆhi,jis an (2L−1)×1 auto-correlation or cross-

correlation vector. The peak value of auto-correlation is at L-th

index.

Let us consider a block K+2L−1 symbols of transmit data

at the j-th antenna. We assume that channel does not change

within this block. This block of data can be represented by an

(2L−1)×K Hankel matrix

⎛

⎜

ˆH =

⎜

⎜

⎝

ˆh1,1

ˆh1,2

...

ˆh2,1

ˆh2,2

...

...

...

...

...

ˆhN,1

ˆhN,2

...

ˆhN,N

⎞

⎟

⎟

⎟

⎠

T

.

(11)

Xj=

⎜

⎝

xj[k−2L+2]

...

xj[k−1]

xj[k]

xj[k−2L+3]

...

xj[k]

xj[k+1]

...

xj[k−2L+K+1]

...

xj[k+K−2]

xj[k+K−1]

...

...

...

⎞

⎟

⎟

⎠.

(12)

The entire transmit signal matrix is

X =

1

√M[X1,X2,...,XN]T,

(13)

with dimension N(2L−1)×K. The scale factor 1/√M keeps

the total transmit power of signal as the same as it in the SISO

case.

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The additive Gaussian noise at receive antenna i-th is a K×1

vector

vi= [vi[k],vi[k+1],...,vi[k+K−1]]T,

and the entire noise matrix is an N×K matrix

V = [v1,v2,...,vN]T.

(14)

(15)

The matrix of received signal with TR pre-filtering is

Y

=

=

H¯HX+V

ˆHX+V

,

(16)

where each row of the received matrix Y is

yi= [yi[k],yi[k+1],...,yi[k+K−1]],

Replacing Eq. (8), (9) and (12) to Eq. (16), the received

signal at i-th receiver can be expressed by

i = 1,...,N.

(17)

yi

=

M

∑

j=1

hT

i,j¯Hi,jXi+

+

M

∑

j=1

N

∑

k=1

k?=i

hT

i,j¯Hk,jXk+vT

i.

(18)

The first part of the received signal is the desired data stream

with summation of auto-correlation of M channels. The second

part is the interference from other streams. In the second part,

the equivalent channel is the cross-correlation of channels,

which is small generally in comparison with the former. The

temporal focusing still remains for TR-MIMO-UWB system

based on the SM scheme. However, we note that the ISI still

exists in the first part of signal.

III. ANTENNA SELECTION SCHEME

It is difficult to derive the exact performance of the TR-

MIMO-UWB system. However, the approximation of the

performance of the TR-MIMO-UWB system can be derived

from Eq. (18).

The first part of received signal has desired signal and

ISI components. Because of focusing property of TR, we

can neglect the contribution of ISI parts. Similarly, the first

part related to auto-correlation function of channels is domi-

nant signal while the second part related to cross-correlation

function has minor contribution. We can consider TR-MIMO-

UWB system as a system with M diversity branches. When

M is large, the average probability of bit error is bounded by

¯Pe≤¯Ne

M

∏

i=1

1

min/4M,

1+ρd2

(19)

where ρ is SNR, ¯Ne and dmin are the number of nearest

neighbors and minimum distance of signal constellation [13].

If increase the number of transmit antenna M, the BER

performance approaches the BER of AWGN channel. When

M → ∞, Eq. (19) becomes

¯Pe≤¯Nee−ρd2

min

4 .

(20)

The large number of transmit antenna with high order of

diversity charges highly cost of hardware. Antenna selection

algorithm is proposed to choose an optimal subset of transmit

antenna. We assume that N streams of data are transmitted

over Mt antennas instead of all M antennas. There are

possible subsets of transmit antenna. The criterion for subset

selection is based on maximize the signal to interference ratio

(SIR) defined by

???∑M

?M

Mt

?

SIR =

j=1hT

i,j¯Hi,j

???

2

F

????∑M

j=1∑N

k=1

k?=i

hT

i,j¯Hk,j

????

2

F

,

(21)

where ?•?Fdenotes the Frobenius norm. This ratio is calcu-

lated from equivalent channel matrix at transmitter side.

In here, it is should be noted that the number of transmit

antenna M or Mtis not necessary to be greater than the number

of data stream N.

IV. NUMERICAL RESULTS

Simulation is conducted to verify the performance of pro-

posed antenna selection scheme for MIMO-TR-UWB. Binary

data is modulated to BPAM signal taking values from {±1}.

This data is pulse shaped by Gaussian pulse:

?

w

p(t) =

1−4π

?t −tc

?2?

e−2π(t−tc

w)2,

(22)

where w is a parameter corresponding to pulse width, and tc

is a time shifting of the pulse. In our simulation, w=1ns, and

tc= w/2. The symbol duration T is 5ns corresponding to data

rate 200 Mbps per data stream. We assume that the signal is

transmitted over UWB channels and perfectly synchronized

at receiver. In this paper, the IEEE 802.15.3a CM4 channel

model [7] is used for each channel in simulation.

−180−120−60060 120180

−0.5

0

0.5

1

ˆh11(t)(ns)

Amplitude

−180−120−60060120180

−0.5

0

0.5

1

ˆh12(t)(ns)

Amplitude

−180−120−60060120180

−0.5

0

0.5

1

ˆh21(t)(ns)

Amplitude

−180−120−600 60120180

−0.5

0

0.5

1

ˆh22(t)(ns)

Amplitude

Fig. 3. Impulse response of 2 × 2 MIMO-TR-UWB equivalent channel with

pulse shaping.

In Fig. 3, the equivalent CIR for a 2 × 2 TR-MIMO-UWB

system is shown. It should be noted that the signal transmitted

over the desired channel is focused into a narrow time instant,

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05 101520 24

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/N0 (dB)

Average BER

1x2

2x2

3x2

4x2

5x2

6x2

AWGN

Fig. 4. BER performance of x × 2 TR-MIMO-UWB.

051015 2024

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/N0 (dB)

Average BER

1x3

2x3

3x3

4x3

5x3

6x3

AWGN

Fig. 5.BER performance of x × 3 TR-MIMO-UWB.

46810 12 1416 18202224

10

−3

10

−2

10

−1

Eb/N0 (dB)

Average BER

with selection

w/o selection

1x3

1x4

1x2

Fig. 6.

antennas).

BER performance of 1 × x TR-MIMO-UWB (selection from 2

46810 12 1416 1820 22 24

10

−4

10

−3

10

−2

10

−1

Eb/N0 (dB)

Average BER

with selection

w/o selection

2x3

2x4

Fig. 7.

antennas).

BER performance of 2 × x TR-MIMO-UWB (selection from 4

while MSI channel spreads the signal. This property is useful

for ISI mitigation when the signal is transmitted over long

delay spread channel.

Fig. 4 and 5 consequently show the transmit diversity

property of time reversal in spatial multiplexing UWB systems.

There are two systems with the number of parallel data stream

is N = 2 and 3 corresponding to the number of receive antenna.

The number of transmit antenna is M = 1,...,6, respectively.

When the number of transmit antenna increases, the BER

performance of two systems converge to the performance of

system over the AWGN channel. These results verify the

performance has been shown in Eq. (20). If the same number

of transmit antenna is used, the performance of x × 3 TR-

MIMO-UWB system is worse than the performance of x × 2

TR-MIMO-UWB system. The reason is that the increasing of

MSI.

The performance of TR-MIMO-UWB system with antenna

selection scheme are shown in Fig. 6 and 7. In Fig. 6, the

performance of systems with 2,3 and 4 data streams transmit-

ted over 1 transmit antenna are illustrated. Antenna selection

algorithm chooses 1 transmit antenna from set of 2 antennas.

Fig. 7 shows the performance of systems select 2 antennas

from set of 4 antennas to transmit 3 and 4 data streams. With

selection scheme, the BER performance of these systems is

considerably improved. For example, 2 × 3 TR-MIMO-UWB

system with selection from 4 transmit antennas can improve

about 5dB SNR at the average BER=10−3. However, if the

number of transmit antenna is large, the effect of diversity

is dominant, the transmit antenna selection algorithm is not

efficient.

V. CONCLUSION

Spatial multiplexing UWB using time reversal can achieve

the transmit diversity to combat the ISI and the MSI. However,

the large number of transmit antenna requires lots of expen-

sive hardware. In this paper, an antenna selection technique

is proposed to reduce the number of transmit antenna for

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the TR-MIMO-UWB system. The performance of system is

considerably improved when the number of transmit antenna is

less than the number of data stream. If the number of transmit

antenna is large, the diversity is dominant and the selection

scheme is no longer efficient.

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