Channel Capacity of Multi User TR-MIMO-UWB Communications System Tran Ha Vu† , Nguyen Thanh Hieu∗ , Ho Duc Tam Linh† , Nguyen Thuy Dung and Le Van Tuan †Faculty of Electronics and Telecommunications, Hue University ∗School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 University of Engineering and Technology, Vietnam National University, Hanoi Email:
[email protected],
[email protected]
response (CIR) of an environment where a base station (BS) is forwarded to any user. MU-MIMO UWB TR system has been proposed as a hot research direction [5], [6], [11] and [12].In [7], there are results about channel capacity of MU-SISO UWB TR and MU-MISO UWB TR systems (without correlation among antennas), however, that of MU-MIMO UWB TR system still have not been presented yet. In this paper, we will investigate the operation capability of this system in different noisy environments with correlation among antennas. Especially, we achieve an interesting result that the transmit correlation affect achieved capacity more significantly than receive correlation does. When the correlation factor among receive antennas is too high (approximately 0.9), if we increase a number of receive antennas then the achieved capacity is virtually unchanged. Meanwhile, with correlation factor among transmit antennas is higher than 0.7 (MR = 2), if we increase a number of transmit antennas the channel capacity will be decreased. In addition, the UWB system in combination with MIMO and TR operates better in highly noisy environment, reduces the influence of multi-path fading effect and provides user with stable traffic even when the noise of environment changes. The rest of article is organized as follow: part II is MUMIMO UWB TR system description, part III is channel capacity analysis of the proposed model, part IV is numerical results, part V is conclusion of problems.
Abstract—Ultra wideband (UWB) system using multi antennas and time reversal (TR) possesses capability of multiuser system. Each user with long channel response signature is exploited for multiple access problems. If UWB channels are independent, the multi antenna UWB system shows both capacity and diversity gain. However, due to correlation between antennas, the performance of system degrades. In this paper, the channel capacity of multi user multi antenna TR-UWB system and the difference between the impact of transmit correlation and that of receive correlation on the system are derived. Channel capacity of system is significantly improved when number of using antennas increases in a acceptable range of the correlation coefficient. If the correlation of antennas is considered, at high correlation, the capacity reduces up to 60% in comparison with capacity if without correlation. Keywords– UWB, TR, channel capacity, multiple access, correlation.
I. I NTRODUCTION Ultra Wideband (UWB) is an attractive topic in recent years because of its capability of high-speed communication in short distance [1], [2]. UWB comes thoroughly to solve the problem of bandwidth limit in wireless environment. However, it is realized that channels in reality are multi-path fading channels [3], [4], so issues about affecting quality of transmission in UWB system serving multi-user is really complex. One can resolve this problem by combining Multi-User Multi-Input Multi-Output (MU-MIMO) UWB system and Time-Reversal (TR) technique to improve transmission rate and minimize the influences of channel which decrease the quality of MU UWB system [5], [6], [7]. MIMO technique is well-known with its extremely good anti-multipath-fading-effect capability using multi transmit and receive antennas to increase diversity and also help the system improve its transmission rate significantly. MIMOUWB is a combination with high efficiency [5]. Time-Reversal (TR) technique has been used in acoustic, medical and especially submarine communication applications [8], [9]. Its advantage is decreasing bad effects caused by the channel such as Inter-Symbol Interference (ISI), Inter-User Interference (IUI), etc., without the need of using highly complex equalizers at transmitter and receiver [10], [9]. In a TR system, multipleaccess mechanism is based on the unique of channel impulse
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II. S YSTEM DESCRIPTION A. Correlation channel model MU-MIMO UWB TR system only operates when it anticipates information of CIRs which is forwarded to each user. Therefore, first of all, users will send an impulse to BS to get the CIRs’ information of the communication environment. The Time-Reversal Mirror (TRM) module of BS records and stores received information which is used for processing transmitted signals. Multiple-Access (MA) technique is based on TR technique which uses the unique of CIRs that BS is forwarded to each antenna or user. Thus, the MA-TR technique operates efficiently in the environment in which CIRs are various. Proposed MU-MIMO UWB TR model includes MT transmit
22
1st
1st
X1
a1
In the above formulas, CIRs are statistical independent parameters. Therefore, the correlation between them is equivalent. However, there is correlation among CIRs of transmit antennas and receive antennas. A problem about creating the correlation is given. If we fix transmit and receive correlation matrices; Kronecker model can be used to solve this problem [13],[14]. We continuously calculate, so that
aN
e = R1/2 HR1/2 , H Rx Tx
a1 M T ! th
XN
Y1
M r ! th
X1
where RRx and RT x are correlation matrices of transmit and receive antennas, respectively. RT x is the transmit correlation matrix with dimension MT × MT which is represented as
YN
aN
XN RT x Fig. 1.
MU-MIMO UWB TR system model
antennas at BS and MR receive antennas at user for each of N user. Let H denote the received CIRs matrix of global channel from transmitted impulses of users, which is expressed as: T
T
T
H = [H(1) H(2) . . . H(N ) ]T .
(1)
αlij δ(t − τlij ),
(n) RRx
(n)
(n)
(n)
s − kT σ T
,
0
=
(n)
1
ρRx 1 (n) ρRx .. .
(n)
ρRx (n)2 ρRx .. . (n)MR −1
ρRx
(n) e H =
(4)
(n)
(n)
,
(7)
(N )
...
RRx
(n)MR −2
ρRx
(n)2
ρRx (n) ρRx 1 .. .
... ... ... .. .
(n)MR −3
ρRx
...
(n)M −1
ρRx R (n)M −2 ρRx R (n)M −3 ρRx R .. . 1
, (9)
where is the correlation coefficient of receive antennas e = for n − th user . Thus, we can rewrite as following H (1) e (2) (N ) (n) e e e [H H . . . H ] , in which H can be expressed as:
(3)
where hij [k] is the k−th tap of CIR with the length of L, δ[·] is the Dirac pulse function. For each downlink, we assume that there are independent circular symmetric complex Gaussian (CSCG) random variables with zero mean and variance E[|hij [k]|2 ] = e
T −3 ρM Tx
(n) ρRx
i = 1, . . . , MR , j = 1, . . . , MR , n = 1, . . . , N where αlij and τlij are amplitude and delay of the l − th tap respectively. (n) The discrete time form of hij is represented as (n)
T −2 ρM Tx
T −1 ρM Tx MT −2 ρT x T −3 ρM Tx .. . 1
(n)
l=0
hij = [hij [0], hij [1], . . . , hij [L − 1]],
T −1 ρM Tx
... ... ... .. . ...
where RRx with n = 1, . . . , N is the receive correlation matrix with dimension MR × MR for n−th user.
(n)
(n)
ρT x ρ2T x .. .
ρ2T x ρT x 1 .. .
0
where hij (1 ≤ n ≤ N ) is CIR between j − th the transmit antenna and the i − th receive antenna of the n − th user. It can be shown as hij =
=
ρT x 1 ρT x .. .
1
where ρT x is the correlation coefficient of transmit antennas. RRx is the block-diagonal receive correlation matrix with dimension N × N as follow (1) RRx 0 ... 0 (2) 0 0 RRx . . . RRx = . (8) .. .. .. , . .. . .
Matrix H(n) (1 ≤ n ≤ N ) is CIRs matrix of environment which BS forward to user n-th. It can be written as (n) (n) (n) h11 h12 ... h1MT (n) (n) (n) h22 ... h2MT h21 , H= (2) .. .. .. . . ... . (n) (n) (n) hMR 1 hMR 2 . . . hMR MT
l−1 X
(6)
e (n) H 11 e (n) H 21 .. .
e (n) H MR 1
e (n) H 12 e (n) H 22 .. .
e (n) H MR 2
B. Received signal model
... ... .. . ...
e (n) H 1MT e (n) H 2MT .. .
e (n) H MR M T
,
(10)
When the BS received impulses from users which hold CIRs information, the block Time-Reversal (TR) Mirror will use the CIRs information to create waveforms which is used to communicate to the corresponding the i − th antenna of the n−th user. Let G is TR Mirrors’ s matrix, which is expressed as:
(5)
with 0 ≤ k ≤ L − 1, Ts is the sampling time of the system. σT is the delay spread of the channel.
23
e (N ) ], e = [G e (1) G e (2) . . . G G
In this time, we will consider received signal. After we create correlation for each CIR, we can rewrite formula (11) as follow:
(11)
e (n) is based on matrix H e (n) transposed which can be where G written as:
e (n) G
=
hn∗ ij
(n)∗ e h11 (n)∗ e h21 .. . (n)∗ e hM R 1
(n)∗ e h12 (n)∗ e h22 .. . (n)∗ e hM R 2
... ... .. . ...
(n)∗ e h1MT (n)∗ e h2MT .. . (n)∗ e hM R M T
hnij .
.
Y (m) = am
MR MT 2L−2 N X X X X
n=1 i=1 j=1 l=1
We separate according to power components of signal, the inter-symbol interference (ISI), the inter-antenna interference (IAI), the inter-user interference (IUI) and the noise, as follow:
(12)
MR MT
Y (m)
=
am
XX
i=1 j=1
is complex conjugate of In which, If input of system is X = [X1 , X2 , . . . , XN ] and each of [X1 , X2 , . . . , XN ] is independent complex random variables with zero-mean and variance of θ, then output of system is given by
+ am
(m) (m) (e hij ∗ geij )[L − 1]Xm [L − 1]
(2L−2) MR MT X XX
i=1 j=1
l=0
(13) + am
e H)+ e n e , where Y = [Y (1) , Y (2) , . . . , Y (N ) ]T , or Y = X∗(G∗ (1) (2) e = [[e e is white Gaussian noise n n ,n e ,...,n e(N ) ]T ]. n sequence with zero mean and variance σ 2 . Thus, received signal at user m(1 ≤ m ≤ N ) is can be written as: Y (m) =
MR X MT N X X
n=1 i=1 j=1
(n)
(m)
(Xn ∗ geij ∗ e hij ) + n e(m) .
X
XX X X
i=1 ′ j=1 i =1 j =1 ′ ′ i 6=i j 6=j
Y
(m)
= am
n=1 i=1 j=1 l=1
(m) (n) (e hij ∗ geij )[l]Xn [l] + am n e(m) .
(m) (m) (e hij ∗ ge ′ ′ )[l]Xm [l] i j
l=0
XX X X
i =1 i=1 j =1 j=1 ′
′
(IAI)
′
(m) (m ) ′ (e hij ∗ ge ′ ′ )[l]Xm [l] i j
(IU I) (noise)
(19)
Inter-antenna interference (IAI) component is the noise component among antennas in the same user. In this calculation, we consider values am = 1. According to the formula (19), we calculate the power of signal, ISI, IAI and IUI as follow:
(14)
PSig (m) = θ| (15)
MR X MT X i=1 j=1
PISI (m) = θ|
(m) (m) (e hij ∗ geij )[L − 1]|2 ,
(2L−2) MR MT X XX l=0
i=1 j=1
(m) (m) (e hij ∗ geij )[l]|2 ,
2 (2L−2) MR MR MT MT X X X X X (m) (m) e , ( h ∗ g e PIAI (m) = θ )[l] ij i′ j ′ i=1 ′ j=1 l=0 ′ j =1 i ′ =1 ′ i 6=i j 6=j
(16)
PIU I (m) = θ
N X
m′ =1 m′ 6=m
SINR(signal-to-interference plus noise ratio) is used to evaluate the quality of signal at each user which is calculated at user in multi-user network accordingly to the formula as PSig (m) , PISI (m) + PIAI (m) + PIU I (m) + σ 2
X
+ am n e(m)
Due to extremely short-pulse and ultra wideband property, received signal of UWB has particularity that the path delay is far higher pulse width. Thus, if multi-antenna is used at BS and user, UWB system will take full advantage of multipath-gain to increase capacity and decrease ISI, IUI. We will consider system capacity when using multi-antenna at users. The system capacity is calculated as follow:
SIN R(m) =
(ISI)
(2L−2) MR MR MT MT
N X ′
III. C APACITY OF SYSTEM
C = log2 (1 + SIN R).
′
m =1 ′ m 6=m
The m − th receiver (user m) only performs a one-tap gain adjustment am to the received signal to recover the signal, ending up with Y m is given as follow: MR MT 2L−2 N X X X X
(m) (m) (e hij ∗ geij )[l]Xm [l]
(signal)
(2L−2) MR MR MT MT
+ am
l=0
e ∗ X) ∗ H e +n e, Y = (G
(m) (n) (e hij ∗ geij )[l]Xn [l] + am n e(m) . (18)
(20)
(21)
(22)
2 X X X X X (m) ′ (m ) e . )[l] ( h ∗ g e ′ ′ ij i j l=0 i′ =1 i=1 j ′ =1 j=1
(2L−2) MR MR MT MT
(23)
Based on the formulas of capacity, it is obvious that capacity belongs to the number of transmit and receive antennas, the number of users and correlation coefficient. The number of antennas and user is directly proportional the capacity, vice versa, the correlation coefficient is inversely proportional to the capacity.
(17)
where PISI , PIAI and PIU I is the power of inter-symbol interference, inter-antenna interference and inter-user interference respectively.
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IV. N UMERICAL RESULTS
0.5
To evaluate the system clearly, we carry out simulation. In this part, numerical results will describe quality of system and system capacity when apply both TR and MU-MIMO techniques simultaneously to the UWB technology. Similar to SINR, SNR (signal-to-noise ratio) parameter is used to estimate the quality of system influenced by white Gaussian noise sequence with zero mean and variance σ 2 , is calculated according to the following formula: "L−1 # X (n) P 2 |e hij | . SN R = 2 E (24) σ
r
M =4 w/o corr r
0.4
M =8 w/o corr
Channel Capacity (bps/Hz)
r
M =1 w corr
0.35
r
M =4 w corr r
0.3
M =8 w corr r
0.25 0.2 0.15 0.1 0.05
l=0
0 −10
We use the following parameters for the simulation: Parameter Environment Length of CIRs (L) Sampling time of the system (TS ) Delay spread of channel (σT ) Number of users (N) Number of transmit antennas(MT ) Number of receive antennas ( Mr )
M =1 w/o corr
0.45
Values Part II.A 403 0.2 × 10−6 125TS 5 3 [1 4 8]
−5
Fig. 3.
0
5 SNR(dB)
10
15
20
Capacity of a random channel (MT =3).
(n)
and correlation coefficients (ρT x = 0.4 and ρRx = 0.3(n = 1, . . . , N )) are used to simulate each channel. From result of capacity, we observe that the capacity increase versus SNR and number of antenna. Channel capacity will be more stable at high SNR values. It is suitable for all theories and reality.
TABLE I S IMULATION PARAMETERS
0.4
First of all, we use simulation parameters at table.1 to survey the SINR and capacity of any user in different noisy condition, particularly . Fig. 2, 3 indicate the affection of correlation among antennas at BS and each user. We assume that ρT x = 0.4 and (n) ρRx = 0.3(n = 1, . . . , N ) . In Fig. 2, with SN R = −5dB; SINR at a user using 1 antenna will approximate −12dB (without correlation) and −14dB (with correlation). In Fig. 3, with SN R = 5dB; channel capacity at a user using 4 recieve antennas approximate 0.375 bps/Hz (without correlation) and 0.2 bps/Hz (with correlation).
Channel Capacity (bps/Hz)
0.35 0.3 0.25 0.2 0.15 M =1 r
0.1
Mr=2 M =4
0.05
r
M =8 r
0 −10
−5
−2
Average Effective SINR (dB)
−4
Fig. 4.
0
5 SNR(dB)
10
15
20
Capacity of CM1 channel (MT =3).
−6
The capacity as a function of the transmit and receive correlation coefficient (ρT x and ρRx respectively) is shown in Fig. 5, and 6. We can easily see from these figures that; when ρT x and ρRx is increased, the capacity is decreased. The transmit correlation affects achieved capacity more significantly than receive correlation does. When ρRx is too high (e.g. ρRx = 0.9), if we increase the number of receive antennas then capacity virtually unchanged. Meanwhile, with high ρT x (e.g. ρT x > 0.6), if we increase the number of transmit antennas then the channel capacity will be decreased. Arcording to above results, we must base on the situation of the correlation among transmit antennas to decide how many antennas are used at BS to achieve optimal capacity. With the more transmit antennas at BS, the lower correlation coefficient among transmit antennas is required.
−8 −10 M =1 w/o corr r
−12
M =4 w/o corr
−14
Mr=8 w/o corr
−16
M =4 w corr
r
Mr=1 w corr r
Mr=8 w corr −18 −10
Fig. 2.
−5
0
5 SNR(dB)
10
15
20
Effective SINR of random channel (MT =3).
In Fig. 4, we continuously study achieved capacity in real environment. The IEEE 802.15.4.a CM1 channel model [4]
25
V. CONCLUSION
[3] D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp. 1247–1257, 2002. [4] A. F. Molisch, “Ieee 802.15.4a channel model final report,” IEEE, New York, Tech. Rep., 2004. [Online]. Available: http://www.ieee802.org/15/pub/04/15-04-0662-02-004a-channelmodel-final-report-r1.pdf [5] H. T. Nguyen, I. Z. Kovacs, and P. C. F. Eggers, “A time reversal transmission approach for multiuser uwb communications,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3216–3224, 2006. [6] R. C. Qiu, “A theory of time-reversed impulse multiple-input multipleoutput (mimo) for ultra-wideband (uwb) communications,” in Proc. IEEE 2006 Int Ultra-Wideband Conf, 2006, pp. 587–592. [7] F. Han, Y.-H. Yang, B. Wang, Y. Wu, and L. K.J.R., “Time-reversal division multiple access in multi-path channels,” Global Telecommunications Conference 2011, pp. 1–5, 2011. [8] M. Fink, “Time reversal of ultrasonic fields. i. basic principles,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 39, no. 5, pp. 555–566, 1992. [9] P. Derode, A. Roux and M. Fink, “Robust acoustic time reversal with high-order multiple scattering,” Phys. Rev. Lett., vol. 75, 1995. [10] T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with mmse equalizer to uwb communications,” in Proc. IEEE Global Telecommunications Conf. GLOBECOM ’04, vol. 5, 2004, pp. 3123–3127. [11] C. Zhou, N. Guo, and R. Caiming Qiu, “Time-reversed ultra-wideband (uwb) multiple input multiple output (mimo) based on measured spatial channels,” IEEE Trans. Veh. Technol., vol. 58, no. 6, pp. 2884–2898, 2009. [12] H. Nguyen, Z. Zhao, F. Zheng, and T. Kaiser, “Preequalizer design for spatial multiplexing simo-uwb tr systems,” IEEE Trans. Veh. Technol., vol. 59, no. 8, pp. 3798–3805, 2010. [13] T. K. Nguyen, H. Nguyen, F. Zheng, and T. Kaiser, “Spatial correlation in the broadcast mu-mimo uwb system using a pre-equalizer and time reversal pre-filter,” in Proc. 4th Int Signal Processing and Communication Systems (ICSPCS) Conf, 2010, pp. 1–6. [14] R. Paulraj, A. Nabar and D. Gore, Introduction to Space-Time WirelessCommunications. Cambridge University Press, 2003.
To support multi users in UWB systems, MU-MIMO UWB TR model which is the combination between MU-MIMO UWB system and TR technique has been proposed. Using TR technique helps decrease complexity structure of receiver at users and provide a suitable multiple-access scheme. Particularly, this paper has clearly indicated the difference between the impact of transmit correlation and that of receive correlation on the system. There is an interesting result that the transmit antenna correlation makes a stronger effect on channel capacity than the receive antenna correlation does. This is an important conclusion in design the user’s receiver due to their compact size. 0.4 MT=3 Mr=2 M =3 M =4 T
0.35
r
MT=3 Mr=8 M =5 M =2 T
Capacity (bps/Hz)
0.3
r
MT=5 Mr=4 M =5 M =8 T
0.25
r
0.2
0.15
0.1
0.05 0.1
0.2
0.3
0.4
0.5 ρ
0.6
0.7
0.8
0.9
T
x
Fig. 5.
The impact of the transmit correlation
0.4 MT=3 Mr=2 M =3 M =4 T
0.35
r
MT=3 Mr=8 M =5 M =2 T
Capacity (bps/Hz)
0.3
r
MT=5 Mr=4 M =5 M =8 T
0.25
r
0.2
0.15
0.1
0.05 0.1
0.2
0.3
0.4
0.5 ρ
0.6
0.7
0.8
0.9
R
x
Fig. 6.
The impact of the receive correlation
R EFERENCES [1] M. Z. Win and R. A. Scholtz, “Impulse radio: how it works,” IEEE Commun. Lett., vol. 2, no. 2, pp. 36–38, 1998. [2] S. Wood and D. R. Aiello, Essentials of UWB. Cambridge University Press, 2008.
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