Log-Likelihood Ratio based Detection Ordering for
the V-BLAST
Sang Wu Kim
Dept. of Electrical and Computer Engineering
Iowa State University
Ames, Iowa 50011
sangkim@ieee.org
Abstract- We propose a new detection ordering for the
V-BLAST. The main idea is to detect and cancel sub-streams
in order of the magnitude of log-likelihood ratio (LLR), i.e. the
symbol with the largest magnitude of LLR is detected first. The
motivation is that the reliability of data decision increases with
increasing magnitude of LLR. As a result, the error propagation
associated with a wrong decision and the resulting error probability
for the remaining sub-streams can be minimized. It is shown that
the proposed LLR-based ordering significantly outperforms the
conventional SNR-based ordering. Simplified LLR-based ordering
and envelope-based ordering that require a much less computation,
but provide a performance virtually identical to the LLR-based
ordering, are also proposed.
Index Terms- V-BLAST, detection ordering, log-likelihood
ratio, Rayleigh fading channel.
I. I NTRODUCTION
The use of multiple antennas at transmitter and receiver
has been shown to provide high spectral efficiencies [1],[2].
Current transmission schemes using multiple antennas typically fall into two categories: data rate maximization [3],[4]
or diversity maximization [5],[6], and some efforts looking
at various tradeoffs between multiplexing and diversity have
begun recently [7]. The first approach, shown in Fig.1, is to
perform spatial multiplexing by sending many independent
signals through multiple transmit antennas. One such approach
is the vertical BLAST (Bell Laboratories Layered Space-Time)
or V-BLAST [3],[4] which splits a single user’s data stream
into multiple sub-streams and uses an array of transmitting
antennas to simultaneously transmit parallel sub-streams in the
same frequency band.
A popular decoding approach, known as nulling and cancelling, gives a reasonable tradeoff between complexity and
performance, but its performance is affected by the order
in which the sub-streams are detected and cancelled [4].
The original method detects the sub-stream that presents
the maximum signal-to-noise ratio (SNR) and then cancels
its contribution from the received signal. Its corresponding
channel matrix is then zeroed. For the remaining symbols,
the above process is repeated by detecting the next strongest,
and so on.
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In this paper we propose a new detection ordering for
the V-BLAST. The proposed scheme is to detect and cancel
sub-streams in order of the magnitude of log-likelihood ratio
(LLR). The motivation is that the reliability of data decision
increases with increasing magnitude of LLR. As a result, the
error propagation associated with a wrong decision and the
resulting error probability for the remaining sub-streams can
be reduced. We show that the magnitude of LLR depends
on the SNR as well as the instantaneous noise. As such,
the performance is governed by the peak, as opposed to
the mean, channel condition. It is shown that the proposed
LLR-based ordering, exploiting the instantaneous noise, significantly outperforms the conventional SNR-based ordering.
The BER with the LLR-based ordering that employs four Tx
and Rx antenna pairs is shown to be virtually identical to
that with the SNR-based ordering that employs eight Tx and
Rx antenna pairs, thereby providing a significant saving in
implementation complexity. We also present simplified LLRbased ordering and envelope-based ordering that require a
much less computation, but provide a performance virtually
identical to the LLR-based ordering.
This paper is organized as follows. Section II describes the
system model, and Section III presents a brief review of VBLAST. Section IV describes the proposed ordering for binary
signaling and it is extended to M -ary signaling in Section V.
Section VI presents the simulation results and Section VII
contains some concluding remarks.
II. S YSTEM M ODEL
The block diagram of the V-BLAST system is illustrated in
Figure 1, where NT transmit antennas send a vector symbol of
size NT over a rich-scattering wireless channel to NR (≥ NT )
receive antennas at each symbol time. At the transmitter end,
the data stream is partitioned into NT substreams, and each
substream is encoded and sent through a different transmit
antenna. Each receive antenna receives the signals from all
NT transmit antennas.
At each time, the received signal can be written as
y = Hx + n
(1)
where H = [h1 h2 ...hNT ] with hi = [hi1 , hi2 , ..., hiNR ]T
is the NR xNT channel matrix whose elements are i.i.d.
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complex Gaussian with mean zero and unit variance, x =
[x1 , x2 , ..., xNT ]T is the transmitted signal vector, y =
[y1 , y2 , ..., yNR ]T is the received signal vector, and n =
[n1 , n2 , ..., nNR ]T is assumed to be an i.i.d. complex Gaussian
noise vector with each component having a mean equal to zero
and a variance equal to N0 /2 per dimension. The modulation
is chosen to be binary phase-shift
keying
(BPSK) with coher√
√
ent detection: {xi } are + Es or − Es with probability 1/2,
where Es is the transmit energy per symbol. An extension to
an M -ary signaling will be discussed in Section V.
III. V-BLAST:R EVIEW
In this section, we briefly review the V-BLAST detection
algorithm. The reader is referred to [3],[4] for more details.
The V-BLAST detection algorithm consists of three parts:
interference nulling, interference cancellation, ordering.
Nulling
Let
T
W = (H H H)−1 H H = [w1 w2 ...wNT ] ,
(2)
where wi is a 1xNR nulling vector, and y(0) = y, H (0) = H.
Then,
(0)
W y(0) = W H
(3)
x + W n
=I
or
yi = wi y(0) = xi + wi n
(4)
where wi n is a complex Gaussian random variable with mean
zero and variance ||wi ||2 N0 /2 per dimension. We get x̂i by
slicing yi , i.e. x̂i = Q(yi ), where Q(·) denotes the quantization
(slicing) operation appropriate to√the constellation in use. For
example,
for BPSK, Q(y) = + Es if y ≥ 0 and Q(y) =
√
− Es if y < 0.
Cancelling
Assuming x̂i = xi , we cancel xi from from y(0) and
generate a modified received vector y(1)
y(1) = y(0) − x̂i hi
...
hi−1
hi+1
... hNT ]
V. E XTENSION TO M - ARY S IGNALING
In this section we consider an M -ary signals such that xi
in (4) is now in the set {s1 , s2 , ..., sM }. Let
(6)
x̂i = arg max P (xi = sm |yi )
Ordering
When symbol cancellation is employed, the order in which
the components of x are detected is important to the overall
system performance [4]. It follows from (4) that the signal-tonoise ratio (SNR) for xi is
SN Ri
=
=
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|xi |2
E[||wi n||2 ]
Es
||wi ||2 N0
IV. P ROPOSED O RDERING
In this section, we propose a new detection ordering based
on the LLR. From (4) the LLR Λi for xi is given by
√
P (xi = + Es |yi )
√
(9)
Λi = ln
P (x = − Es |yi )
√ i
4 Es Re{yi }
=
·
(10)
N0
||wi ||2
assuming that xi is equally probable. The sign of Λi is the
hard decision value, and the magnitude of Λi is the reliability
of hard decision. In general, the bit error probability Pe,i =
P (x̂i = xi ) and the LLR Λi is related by
1
Pe,i =
(11)
1 + e|Λi |
A derivation of (11) is provided in Appendix A. Since Pe,i
decreases with increasing |Λi |, the proposed detection ordering
is to detect the component of x that provides the largest |Λi |
first, where
√
4 Es |Re{yi }|
·
(12)
|Λi | =
N
||wi ||2
√0
4 Es |xi + Re{wi n}|
=
(13)
N0 ||wi ||2
Replacing wi n in (13) by its mean, which is zero, and
taking the absolute value of Λi yields 4Es /(||wi ||2 N0 ), which
is proportional to the SNR for xi . Therefore, selecting the
component of x providing the largest SNR or equivalently
the smallest ||wi ||2 (assuming that the average noise power is
a constant, N0 /2, across all Rx antennas) takes the average
noise power into account. However, the proposed ordering
exploits the noise term wi n by selecting the component of
x for which signs of xi and wi n are are identical, thereby
providing a larger LLR magnitude, i.e. a higher reliability. As
a result, the performance is governed by the peak, as opposed
to the mean, channel condition.
(5)
and a modified channel matrix
H (1) = [ h1
The original method calculates ||wi ||2 for all i, and detects
and cancels the component of x that minimizes ||wi ||2 , i.e.
that maximizes the SN Ri .
sm
(14)
be the maximum a posteriori (MAP) decision for the ith
symbol, and
P (xi = x̂i |yi )
Λi,m = ln
(15)
P (xi = sm |yi )
be
Mthe LLR. Then, it follows from (15) and the equality
m=1 P (xi = sm |yi ) = 1 that the conditional probability
of symbol error given yi is
(7)
(8)
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P (xi = x̂i |yi )
=
=
1 − P (xi = x̂i |yi )
1
1 − M
−Λi,m
m=1 e
(16)
(17)
0-7803-7974-8/03/$17.00 © 2003 IEEE
Since the conditional
of symbol error decreases
M probability
−Λi,m
e
,
we
propose cancelling in
with decreasing
m=1
M
order of m=1 e−Λi,m , i.e. cancelling the symbol minimizing
M
−Λi,m
first.
m=1 e
Assuming that each symbol is transmitted with equal probability, (15) can be expressed by
Λi,m = ln
P (yi |xi = x̂i )
P (yi |xi = sm )
(18)
where
2
2
1
P (yi |xi = sm ) =
e−|yi −sm | /(||Wi || N0 )
2
π||wi || N0
SNR-based Ordering
Since the SNR for symbol xi is proportional to
|xi |2 /||wi ||2 , the SNR-based ordering proceeds in order of
|xi |2 /||wi ||2 . For equi-energy signaling |xi |2 is a constant, so
the detection proceeds in order of 1/||wi ||2 .
Maximum Likelihood Decoding
(19)
Substituting (19) in (18) yields
Λi,m = (|yi − sm |2 − |yi − x̂i |2 )/(||wi ||2 N0 ).
This will be referred to as the envelope-based ordering,
because |yi | is the envelope of yi (t).
(20)
The optimum decoding method that minimizes the error
probability is maximum likelihood (ML) decoding where the
receiver compares all possible combinations of symbols which
could have been transmitted with what is observed:
x̂ = arg min ||y − Hx||
x
Simplified LLR-based Ordering
In this subsection, we present a simple but suboptimum
ordering that requires a less computation. Note that
P (xi = x̂i |yi )
=
1 − M
1
m=1
e−Λi,m
1
1 + e−Λi,m∗
1
1 + eΛi,m∗
(21)
≥ 1−
(22)
=
(23)
where
Λi,m∗
=
min Λi,m
sm =xˆi
(24)
)
min (|yi − sm |2 − |yi − x̂i |2 )/(||wi ||2 N0(25)
=
sm =xˆi
is the difference between the shortest distance and the second
shortest distance from yi normalized by ||wi ||2 N0 . Since the
lower bound on the error probability decreases monotonically
with Λi,m∗ , we propose cancelling in order of Λi,m∗ . This will
be referred to as simplified LLR-based ordering.
Envelope-based Ordering
For equi-energy signaling such as MPSK or MFSK, |sm |2 =
|xi |2 for all m and i, and thus (25) can be further simplified
as
Λi,m∗ = min 2Re{yi (x̂i − sm )}/(||wi ||2 N0 .)
sm =xˆi
(26)
Then, it follows from the relation
Re{xy} ≤ |x| · |y|
(27)
that we obtain
Λi,m∗
≤ 2 min |yi | · |x̂i − sm |/(||wi ||2 N0 )
sm =xˆi
=
2|yi |dmin /(||wi ||2 N0 )
(28)
(29)
where dmin is the minimum Euclidean distance. Since
|/||wi ||2 is much simpler to calculate than (25) or
|y
iM
−Λi,m
, we propose cancelling in order of |yi |/||wi ||2 .
m=1 e
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(30)
However, the complexity of ML decoding becomes prohibitive
when many antennas or high-order modulations are used.
VI. S IMULATION R ESULTS AND D ISCUSSION
In this section, we present several simulation results in
Rayleigh flat fading channels. Figure 2 shows the average
bit error rate (BER) versus Ēb /N0 per receive antenna for
BPSK with conventional SNR-based ordering and proposed
LLR-based ordering. We find that the proposed LLR-based
ordering provides a power gain of 5dB over the conventional
SNR-based ordering for NT = NR = 4 and the power gain
increases with increasing number of antennas. Also, the BER
with the proposed LLR-ordering that employs four Tx and
Rx antenna pairs (4x4) is virtually identical to that with the
conventional SNR-ordering that employs eight Tx and Rx
antenna pairs (8x8), thereby providing a significant saving in
implementation complexity.
Figure 3 is a plot of average BER versus Ēb /N0 per receive
antenna for different number of receive antennas, NR . We find
that as NR increases the slope becomes steeper for both LLRand SNR-based ordering and their average BERs converge.
This is due to an increase of diversity order with increasing
NR .
Fig. 4 shows the average bit error rate (BER) versus
Ēb /N0 per receive antenna for QPSK with several detection ordering rules. We find that simplified LLR-based ordering and envelope-based ordering, which require a much
less computation than the LLR-based ordering, provide a
performance virtually identical to the LLR-based ordering.
In fact, the envelope-based ordering performs slightly better
than the LLR-based ordering. This is because the LLR-based
ordering that minimizes P (x̂i = xi ) does not necessarily
cancels the component causing the highest interference for
the remaining symbols, i.e. maxi ||hi ||2 in (5). Also, shown is
the performance with ML decoding, which allows extraction
of some diversity gain. Note that this benefit comes at the cost
of receiver complexity.
Fig.5 is a plot of average BER versus Ēb /N0 per receive
antenna for different constellation sizes. We find that the power
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gain provided by the LLR-based ordering over the SNR-based
ordering decreases
as the constellation size M increases. This
M
is because m=1 e−Λi,m in (17), representing the unreliability
measure for the LLR-based ordering, increases with increasing
M , whereas the SNR, reliability measure for SNR-based
ordering, does not depend on M . Also, we note that BPSK
provides a lower average BER than QPSK. This is because
the ordering for QPSK is updated for each symbol (two
bits), whereas for BPSK the ordering is updated for each bit.
Therefore, BPSK should provide a lower average BER than
QPSK.
where x̂ is the detector output. If Λ(R) > 0, i.e. x̂ = 1, then
since P (A, B|R) ≤ P (A|R) or P (B|R),
Pe (R) ≤ P (x = −1|R) + P (x̂ = −1|R)
=P (Λ(R)<0)=0
=
1
.
1 + eΛ(R)
(37)
Also,
Pe (R) = 1 − P (x̂ = x|R)
=
VII. C ONCLUSION
In this paper we proposed a new detection ordering for the
V-BLAST. The main idea is to detect and cancel sub-streams
in order of the magnitude of log-likelihood ratio (LLR). The
motivation is that the reliability of data decision increases with
increasing magnitude of LLR. As a result, the error propagation associated with a wrong decision and the resulting error
probability for the remaining sub-streams can be minimized.
We showed that the proposed LLR-based ordering significantly
outperforms the conventional SNR-based ordering. The BER
with the LLR-based ordering that employs four Tx and Rx
antenna pairs (4x4) is shown to be virtually identical to that
with the conventional SNR-based ordering that employs eight
Tx and Rx antenna pairs (8x8), thereby providing a significant
saving in implementation complexity. We also presented simplified LLR-based ordering and envelope-based ordering that
require a much less computation, but provide a performance
virtually identical to the LLR-based ordering.
(36)
(38)
1 − P (x̂ = 1, x = 1|R)
− P (x̂ = −1, x = −1|R)
≥ 1 − P (x = 1|R) − P (x̂ = −1|R)
(39)
(40)
=P (Λ(R)<0)=0
=
=
1
1−
1 + e−Λ(R)
1
.
1 + eΛ(R)
(41)
(42)
Since, for Λ(R) > 0, Pe (R) is upper and lower bounded by
the same quantity, then
Pe (R) =
1
.
1 + eΛ(R)
(43)
If Λ(R) < 0, we can similarly show that
Pe (R) =
1
.
1 + e−Λ(R)
(44)
Pe (R) =
1
.
1 + e|Λ(R)|
(45)
As a result,
A PPENDIX A
In this appendix, we show that the bit error probability,
Pe (R), with the MAP decision for a given observation R is
given by
1
(31)
Pe (R) =
1 + e|Λ(R)|
where
P (x = +1|R)
(32)
Λ(R) = ln
P (x = −1|R)
is the log-likelihood ratio (LLR). Moreover, the relationship
in (31) is true for any binary signals in any channel.
Proof:
It follows from (32) and P (x = +1|R) + P (x = −1|R) = 1
that
1
P (x = +1|R) =
(33)
1 + e−Λ(R)
and
1
P (x = −1|R) =
.
(34)
1 + eΛ(R)
By definition,
Pe (R) = P (x̂ = x|R)
R EFERENCES
[1] G.J.Foschini, ”Layered space-time architecture for wireless communication in a fading environment using multiple antennas,” Bell Labs
Technical Journal, Vol.1, No.2, pp.41-59, Autumn 1996.
[2] G.J.Foschini and M.J.Gans, “On the limits of wireless communications
in a fading environment when using multiple antennas ,” Wireless
Personal Communications, vol.6, pp.311-335, 1998.
[3] G.D.Golden, G.J.Foschini, R.A.Valenzuela, and P.W.Wolniansky, “Detection algorithm and initial laboratory results using the V-BLAST
space-time communication architecture ,” Electronics Letters, vol.35,
no.1, pp.14-15, 1999.
[4] G.J.Foschini, G.D.Golden, P.W.Wolniansky, and R.A.Valenzuela, “Simplified processing for wireless communication at high spectral efficiency
,” IEEE J. Selec. Areas Commun.-Wireless Commun. Series, vol.17,
pp.1841-1852, 1999.
[5] S.M.Alamouti, ”A simple transmit diversity technique for wireless
communications,” IEEE J. Selec. Areas Commun., pp.1451-1458, Oct.
1998.
[6] V.Tarokh, N.Seshadri, and A.R.Calderbank, ”Space-time codes for high
data rate wireless communication: Performance criterion and code
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[7] L.Zheng and D.Tse, “Diversity and multiplexing: A fundamental tradeoff
in multiple antenna channels ,” IEEE Tr. Infor. Theory, submitted for
publication.
(35)
= P (x̂ = 1, x = −1|R) + P (x̂ = −1, x = 1|R)
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yŸ
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Block diagram of the V-BLAST system.
Fig. 4.
Average BER versus E¯b /N0 per receive antenna with several
detection orderings: QPSK, NT = NR = 4.
Fig. 2. Average BER versus E¯b /N0 per receive antenna with the SNR-based
ordering and the LLR-based ordering: BPSK.
Fig. 5.
Average BER versus E¯b /N0 per receive antenna for various
constellations: NT = NR = 4.
Fig. 3. Average BER versus E¯b /N0 per receive antenna with the SNR-based
ordering and the LLR-based ordering: BPSK.
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