An Ordered Successive Interference
Cancellation Scheme in UWB MIMO Systems
Jinyoung An and Sangchoon Kim
ABSTRACT⎯In this letter, an ordered successive
interference cancellation (OSIC) scheme is applied for
multiple-input multiple-output (MIMO) detection in ultrawideband (UWB) communication systems. The error rate
expression of an OSIC receiver on a log-normal multipath
fading channel is theoretically derived in a closed form
solution. Its bit error rate performance is analytically
compared with that of a zero forcing receiver in the UWB
MIMO detection scheme followed by RAKE combining.
Keywords⎯Ultra-wideband (UWB), MIMO, zero forcing
(ZF), ordered successive interference cancellation (OSIC).
I. Introduction
Ultra-wideband (UWB) radio transmission technology has
attracted enormous interests for applications requiring high
data rates over short ranges. To increase the transmission data
rate through spatial multiplexing (SM) and extend system RF
coverage, multiple-input multiple-output (MIMO) techniques
can be employed in UWB systems for certain applications
such as high-definition video transmission [1]. In [1], the error
performance of UWB MIMO systems over indoor wireless
multipath channels was analyzed. A zero forcing (ZF) receiver
is applied to separate the spatially multiplexed data on a pathby-path basis. The zero-forced resolvable paths are then
combined by a RAKE. This detection structure is called a ZFRAKE. Also, the minimum mean square error (MMSE)
detector for MIMO systems on flat Rayleigh channels was
considered in [2] and [3].
Manuscript received April 5, 2009; revised April 29, 2009; accepted May 11, 2009.
This work was supported by Korea Research Foundation Grant funded by the Korean
Government (MOEHRD, Basic Research Promotion Fund) (KRF-2008-331-D00382).
Jinyoung An (phone: +82 51 200 5589, email: sirius0422@nate.com) and Sangchoon Kim
(phone: +82 51 200 7795, email: sckim@dau.ac.kr) are with the Department of Electronics
Engineering, Dong-A University, Busan, Rep. of Korea.
doi:10.4218/etrij.09.0209.0140
472
Jinyoung An et al.
© 2009
Performance of the ZF detector is known to be similar to that
of the MMSE receiver at high signal-to-noise ratios (SNRs).
This work focuses on the analysis of a ZF related scheme. In
this letter, we present the error performance of a ZF scheme
with an ordered successive interference cancellation (OSIC) for
UWB SM MIMO detection in a log-normal fading channel.
We analytically investigate the theoretic bit error rate (BER)
performance of the ZF-OSIC architecture followed by RAKE
combining, called a ZF-OSIC-RAKE, in an UWB SM MIMO
receiver. The diversity order that the ZF-OSIC-RAKE offers
for detecting the n-th transmitted data stream is obtained for
UWB MIMO systems. It is also seen that the ZF-OSIC-RAKE
scheme outperforms the ZF-RAKE.
II. System Model
A multiple antenna UWB communication system with NT
transmit and NR receive antennas ( N R ≥ N T ) is considered.
Independent NT data streams modulated with the pulseamplitude of a short-duration UWB pulse are sent on different
transmit antennas. To avoid severe intersymbol interference,
we assume that the pulse repetition interval is sufficiently large
compared with the channel delay spread. We also assume that
the signals put through the multipath channel, and only L
resolvable paths for the symbol-by-symbol detection are
exploited by the receiver at each receive antenna. The received
signal of the l-th propagation path at the m-th receive antenna
denoted by xm(l), l = 0,1, " , L − 1 , can be obtained by passing
through a UWB pulse matched filter. The discrete-time
T
received signal vector, x(l ) = ⎡⎣ x1 (l ) x2 (l ) " xN R (l ) ⎤⎦ , at
the matched filter output for the l-th path can be written as
x(l ) = Es H (l ) a + w (l ),
(1)
ETRI Journal, Volume 31, Number 4, August 2009
where Es is the average symbol energy. Here, the information
symbol vector over NT transmit antennas is denoted by
a = [a1 a2 " aNT ] T , where an is an information symbol from the
n-th transmit antenna. The received noise vector is represented
for i = 1, 2, ", N T
FZF,i (l ) = H i (l ) +
ki (l ) = arg min fi( j ) (l )
yki (l ) (l ) = f
(l ) xi (l )
xi+1 (l ) = xi (l ) − h ki (l ) (l ) ⋅ Quan ⎡⎣ yki (l ) (l ) ⎤⎦
T
by w (l ) = ⎡⎣ w1 (l ) w2 (l ) " wN R (l ) ⎤⎦ , where wm(l) is a real
zero-mean white Gaussian noise of variance N0/2. For the l-th
delayed path, an NR×NT discrete-time channel matrix H(l) can
be expressed as H (l ) = ⎡⎣h1 (l ) h 2 (l ) " h NT (l ) ⎤⎦ where the column
T
vector hn(l) is given as h n (l ) = ⎡⎣ h1n (l ) h2 n (l ) " hN R n (l ) ⎤⎦ . Here,
2
j
( ki ( l ))
i
end
H i+1 (l ) = H iki ( l ) (l )
T
⎛
⎞
Permutate the elements of y (l ) ⎜ = ⎡ yk1 (l ) (l ) yk2 (l ) (l ) " ykN (l ) (l ) ⎤ ⎟
⎦ ⎠
R
⎝ ⎣
according to the original sequence ( 1, 2, ", N T ) .
Fig. 1. ZF-OSIC algorithm for each path in ZF-OSIC-RAKE.
hmn(l), m =1, 2, " , N R , n = 1, 2," , N T , is the channel coefficient
of the l-th path for the signal from the n-th transmit antenna to
the m-th receive antenna. It is given by hmn (l ) = ζ mn (l ) g mn (l )
[4], where ζ mn (l ) ∈ { ± 1 } with equal probability is a discrete
random variable (RV) representing pulse-phase inversion, and
gmn(l) is the fading magnitude term with a log-normal
distribution.
( Hi (l ), FZF,i (l ), xi (l ) ) denotes the specific variables in the i-th
detection step. Initial values are set to be H1 (l ) = H (l ),
FZF,1 (l ) = FZF (l ), x1 (l ) = x(l ); fi( j ) (l ) and Quan [ i ] indicate
the row j of FZF,i (l )(= H i (l ) + ) and quantization operation,
respectively; and H i +1 (l ) = H iki (l ) (l ) depicts the nulling of
column ki(l) of the channel matrix Hi(l).
To obtain the single zero-forced signal of the l-th path signal
in OSIC step i, the row of FZF, i(l) with the minimal norm is
III. Ordered Successive Interference Cancellation
chosen. After getting the zero-forced signal vector associated
with each multipath component, we reorder the elements of the
In the ZF-RAKE examined in [1], ZF detection is
zero-forced vector according to the original sequence of the
performed by the filter matrix F ZF(l)=H(l)+, where
transmitted data streams. Then, RAKE combining of the zero+
( ⋅) denotes the pseudoinverse. The output signal vector, forced signals is accomplished to detect each data stream and
T
y (l ) = ⎡⎣ y1 (l ) y2 (l ) " y NR (l ) ⎤⎦ , for the l-th path of a leads to obtaining a DV for each transmitted data stream. The
particular bit is given by y(l)=FZF(l)x(l). The decision variable DV of the RAKE combiner output for a particular bit of the
(DV) of the MRC output for a particular bit of the n-th n-th transmitted data stream can be written as (2).
The instantaneous SNR γ n of the DV of the RAKE
transmitted data stream can be written as
combiner output for a particular bit of the n-th data stream is
L −1
L −1
zn = ∑ l = 0 α n2 (l ) yn (l ),
(2)
given by γ n = 2λ ∑ l = 0 α n2 (l ) with λ = Es N 0 . By
−
1
where α n2 (l ) = 1 υn (l ) and υn (l ) = ⎡⎣ H(l ) H H(l ) ⎤⎦ . Here, following an analysis similar to that in [1], we obtain the
nn
H
(⋅) is the conjugate transpose. The diversity order of an quadratic form, α n2 (l ) = h n (l ) H S n (l ) h n (l ) , where Sn(l) is a
(NR, NT, L) MIMO system based on the ZF-RAKE scheme is NR×NR non-negative Hermitian matrix constructed from
given by the parameter DZF=L(NR–NT+1).
h n +1 (l ), h n + 2 (l ), " , h NT (l ) . Then, γ n can be expressed as
In a ZF-RAKE, a bank of separate filters has been γ = 2λ ∑ L NR λ ( n ) q ( n ) 2 = 2λ γ ′ , where q ( n ) and λ ( n ) ,
i
i
n
n
i
i
i =1
considered to estimate the NT data substreams. However, the respectively, are the zero-mean unit-variance Gaussian RV and
output of one of the filters can be exploited to help the eigenvalue of the matrix S = diag S (0) S (1)" S ( L −1) .
[ n
]
n
n
n
operation of the others. In the OSIC technique [5], the signal
Thus, the eigenvalues of Sn(l) with N R − N T + n are equal
with the highest post-detection SNR is first chosen for
to 1, and the others with NT–n eigenvalues are 0. Hence, the
processing and then cancelled from the overall received signal
matrix Sn has LNR eigenvalues, among which the L(NR–NT+n)
vector. This reduces the burden of inter-channel interference on
eigenvalues
are equal to 1, and the other L(NT–n) eigenvalues
the receivers of the remaining data substreams. To apply OSIC
2
to the ZF-RAKE architecture for multipath channels, the OSIC are equal to 0; therefore, γ n′ = ∑ L ( N R − NT + n ) qi( n ) . The
i =0
procedure for each path is performed prior to RAKE
variable γ n′ is a central chi-square distributed RV with
combining. The zero-forced signal is first obtained in the
(n)
detection step of each propagation path signal. Then, the zero- DZF-OSIC = L ( N R − N T + n) degrees of freedom, which has a
forced signals are combined to detect each substream. This probability density function (PDF) of
detection scheme is called a ZF-OSIC-RAKE. The OSIC
(3)
fγ n′ (t ) = 0.5κ Γ(κ ) t κ −1e− t 2 ,
algorithm for the l-th path signal is shown in Fig. 1. Here,
(
ETRI Journal, Volume 31, Number 4, August 2009
)
Jinyoung An et al.
473
n)
where κ = 0.5 D (ZF-OSIC
, and Γ(⋅) is the gamma function
e dτ , b > 0 . Thus, by averaging
0
the conditional BER P (t ) = Q
( n)
r
(
2λ t
)
over the PDF
fγ n′ (t ) , the average BER for a particular bit of the n-th
transmitted data stream over log-normal fading channels can be
calculated as
P (t )
(1 12 ) e
+ (1 6 ) e
.
(5)
By using (3) and (5) in (4), the average BER for the ZF-OSICRAKE expressed in a closed form can be obtained as
n)
Pb(,ZF-OSIC
(κ − 1)!
(κ − 1)!
. (6)
+
κ
κ
12 ( 2λ + 1) Γ (κ ) 6 ( (8 / 3)λ + 1) Γ (κ )
Here, (⋅)! is the factorial defined as s ! = s ⋅ ( s − 1) " 2 ⋅1 .
The average BER over all NT data streams can be given by
n)
BERZF-OSIC = (1 N T ) ∑ n =T1 Pb(,ZF-OSIC
. Note that to obtain the
N
theoretical performance of the ZF-RAKE, the same formula as
that of ZF-OSIC-RAKE can be used except κ = 0.5 DZF .
IV. Simulation Results
10-2
10-3
10-3
0
To derive the average BER in a closed form, a tightly
approximated expression of Q-function such as Q( β )
2
2
(1 12)e − β 2 + (1 6)e − 2 β 3 [2] is employed. Then, the
conditional BER, given λ = Es N 0 , can be approximated as
− 4λ t 3
10-1
-2
10
10-4
(4)
0
−λt
100
-1
10-4
∞
n)
Pb(,ZF-OSIC
= ∫ Pr( n ) (t ) fγ n′ (t ) dt .
(n)
r
100
10
ZF-OSIC (5, 4, 4)
ZF (6, 4, 4)
ZF-OSIC (6, 4, 4)
ZF (4, 4, 4)
ZF-OSIC (4, 4, 4)
ZF (5, 4, 4)
BER
defined as Γ(b) = ∫ τ
ZF-OSIC (4, 3, 4)
ZF (4, 3, 8)
ZF-OSIC (4, 3, 8)
ZF (4, 3, 2)
ZF-OSIC (4, 3, 2)
ZF (4, 3, 4)
b −1 −τ
BER
∞
2
4
6
8
SNR per bit (dB)
(a)
10
12
0
2
4
6
8
SNR per bit (dB)
10
12
(b)
Fig. 2. BER vs. SNR for different values of (a) L and (b) NR.
power gain due to the increase of diversity order in each
increasing cancellation step. Also, multipath combining yields
enhanced performance in both ZF-OSIC-RAKE and ZFRAKE. In an (NR, 4, 4) case, Fig. 2(b) shows the analytical and
simulated results of BER as a function of SNR per bit for
various NR. The increase in NR increases the diversity order of
the ZF-OSIC-RAKE receiver as well as that of the ZF-RAKE.
V. Conclusion
The error performance of a UWB MIMO system over indoor
log-normal fading channels has been analyzed. The UWB
MIMO receiver employs ZF-OSIC detectors and then a RAKE
temporal combiner. It has been shown that the ZF-OSIC for
detecting the n-th transmitted data stream in an (NR, NT, L)
MIMO system achieves a diversity order of L(NR–NT+n). The
ZF-OSIC offers better performance than the ZF receiver.
Consider the binary PAM scheme (Es=Eb). In the simulations,
the SNR per bit in decibels is defined as ηb =
2λ (dB) + 10 log10 L ( N R − N T + 1) for (NR, NT, L) systems. References
The log-normal fading amplitude g(l) can be expressed as
[1] H. Liu, R.C. Qiu, and Z. Tian, “Error Performance of Pulse-Based
g (l ) = eψ (l ) , where ψ (l ) is a Gaussian RV with mean μψ (l )
Ultra-Wideband MIMO Systems over Indoor Wireless
and variance σψ2 . We assume that the standard deviation of
Channels,” IEEE Trans. Wireless Commun., vol. 4, no. 6, Nov.
20 log10 g (l ) = ψ (l ) ( 20 log10 e ) is 5 dB. To satisfy
2005, pp. 2939-2944.
E[ g (l ) 2 ] = e− ρ l representing the average power of path l,
[2] N. Kim, Y. Lee, and H. Park, “Performance Analysis of MIMO
μψ (l ) = −σψ2 − ρ l 2 is required, where σψ is obtained as
System with Linear MMSE Receiver,” IEEE Trans. Wireless
σψ = 5 ( 20 log10 e ) . Here, the power decay factor ρ = 0 is
Commun., vol. 7, no. 11, Nov. 2008, pp. 4474-4478.
used. The receiver is assumed to have perfect knowledge of the
[3] A. Zanella, M. Chiani, and M.Z. Win, “MMSE Reception and
channel fading coefficients of L resolvable paths. In the figures,
Successive Interference Cancellation for MIMO Systems with
theoretic and simulated BER curves of the UWB MIMO
High Spectral Efficiency,” IEEE Trans. Wireless Commun., vol. 4,
system are shown with lines and markers, respectively.
no. 3, May 2005, pp. 1244-1253
For a (4, 3, L) MIMO system, Fig. 2(a) shows the analytical
[4] A.F. Molish, J.R. Foerster, and M. Pendergrass, “Channel Models
and simulated BERs of the ZF-RAKE and ZF-OSIC-RAKE
for Ultra-Wideband Personal Area Network,” IEEE Wireless
Commun., vol. 10, no. 6, Dec. 2003, pp. 14-21.
as a function of SNR per bit in decibels. The analytical results
[5] P.W. Wolniansky et al., “V-BLAST: An Architecture for
are similar to the simulated BER performance and demonstrate
Realizing Very High Data Rates Over the Rich-Scattering
the performance boost in using the ZF-OSIC-RAKE over the
Wireless Channel,” Proc. IEEE ISSSE, Sept. 1998, pp. 295-300.
ZF-RAKE. This boost in BER performance comes from a
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