In this contribution the number of activated MIMO layers and the number of bits per symbol are jointly optimized under the constraint of a given fixed data throughput and integrity. In general, non-frequency selective MIMO links have attracted a lot of research and have reached a state of maturity. By contrast, frequency selective MIMO links require substantial further research, where spatio-temporal vector coding (STVC) introduced by Raleigh seems to be an appropriate candidate for broadband transmission channels. In analogy to bit-interleaved coded irregular modulation, a broadband MIMO-BICM scheme is introduced, where different signal constellations and mappings are used within a single codeword. Extrinsic Information Transfer (EXIT) charts are used for analyzing and optimizing the convergence behaviour of the iterative demapping and decoding. Our results show that in order to achieve the best bit-error rate, not necessarily all MIMO layers have to be activated

Ahrens, Andreas

Benavente Peces, César

English

Archivo Digital UPM

Adaptive Filter Theory.

Bit-Interleaved Coded Irregular Modulation.”

Bit-Interleaved Coded Modulation,”

Design, Analysis, and Performance Evaluation for BICM-ID withsquare QAM Constellations in Rayleigh Fading Channels.”

Digital Communications.

Diversity and Multiplexing: A Fundamental Tradeoff in Multiple-Antenna Channels.”

Modulation-Mode and Power Assignment

Modulation-Mode and Power Assignment in SVD-equalized

Modulation-Mode Assignment for SVD-Aided and BICM-Assisted

Multivariate Modulation and Coding for Wireless Communication.”

Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate,”

Spatio-Temporal Coding for Wireless Communication.”

Wireless Communications over MIMO Channels – Applications to CDMA and Multiple Antenna Systems.

EXIT chart analysis of iteratively detected and SVD-assisted broadband MIMO-BICM schemes

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

EXIT Chart Analysis of Iteratively Detected and
SVD-assisted Broadband MIMO-BICM Schemes
Ce´sar Benavente-Peces
Universidad Polite´cnica de Madrid
E.U.I.T de Telecomunicacio´n
Ctra. Valencia. km. 7, 28031 Madrid, Spain
Email: cesar.benavente@upm.es
Andreas Ahrens
Hochschule Wismar, University of Technology, Business and Design
Department of Electrical Engineering and Computer Science
Philipp-Mu¨ller-Straße 14, 23966 Wismar, Germany
Email: andreas.ahrens@hs-wismar.de
Abstract—In this contribution the number of activated MIMO
layers and the number of bits per symbol are jointly optimized
under the constraint of a given ﬁxed data throughput and
integrity. In general, non-frequency selective MIMO links have
attracted a lot of research and have reached a state of maturity.
By contrast, frequency selective MIMO links require substantial
further research, where spatio-temporal vector coding (STVC)
introduced by RALEIGH seems to be an appropriate candidate for
broadband transmission channels. In analogy to bit-interleaved
coded irregular modulation, a broadband MIMO-BICM scheme
is introduced, where different signal constellations and mappings
are used within a single codeword. Extrinsic Information Trans-
fer (EXIT) charts are used for analyzing and optimizing the
convergence behaviour of the iterative demapping and decoding.
Our results show that in order to achieve the best bit-error rate,
not necessarily all MIMO layers have to be activated.
I. INTRODUCTION
Iterative demapping and decoding aided bit-interleaved
coded modulation (BICM-ID) was designed for bandwidth
efﬁcient transmission over fading channels [1], [2]. The BICM
philosophy has been extended by using different signal con-
stellations and bit-to-symbol mapping arrangements within
a single codeword, leading to the concept of bit-interleaved
coded irregular modulation (BICIM) schemes, offering an
improved link adaptation capability and an increased design
freedom [3]. Since the capacity of multiple-input multiple-
output (MIMO) systems increases linearly with the mini-
mum number of antennas at both, the transmitter as well
as the receiver side, MIMO-BICM schemes have attracted
substantial attention and can be considered as an essential
part of increasing both the achievable capacity and integrity
of future generations of wireless systems [4], [5]. However,
non-frequency selective MIMO links have reached a state of
maturity [6]. By contrast, frequency selective MIMO links
require substantial further research, where spatio-temporal
vector coding (STVC), introduced by RALEIGH, seems to be
an appropriate candidate for broadband MIMO transmission
channels [7], [8]. In general, the choice of the number of
bits per symbol and the number of activated MIMO layers
combined with powerful error correcting codes offer a certain
degree of design freedom, which substantially affects the
performance of MIMO systems. Against this background, the
novel contribution of this paper is that we jointly optimize the
number of activated MIMO layers and the number of bits per
symbol combined with powerful error correcting codes under
the constraint of a given ﬁxed data throughput and integrity.
The remaining part of this contribution is organized as
follows: Section II introduces our system model, while the
proposed uncoded solutions are discussed in Section III. In
Section IV the channel encoded MIMO system is intro-
duced. The associated performance results are presented and
interpreted in Section V. Finally, Section VI provides our
concluding remarks.
II. BROADBAND MIMO SYSTEM MODEL
When considering a frequency selective SDM (spatial divi-
sion multiplexing) MIMO link, composed of nT transmit and
nR receive antennas, the block-oriented system is modelled by
u = H · c + w . (1)
In (1), c is the (NT × 1) transmitted signal vector containing
the complex input symbols transmitted over nT transmit
antennas in K consecutive time slots, i. e., NT = K nT.
This vector can be decomposed into nT antenna-speciﬁc signal
vectors cμ according to
c =
(
cT1 , . . . , c
T
μ , . . . , c
T
nT
)T
. (2)
In (2), the (K × 1) antenna-speciﬁc signal vector cμ trans-
mitted by the transmit antenna μ (with μ = 1, . . . , nT) is
modelled by
cμ = (c1μ, . . . , ck μ, . . . , cK μ)
T
. (3)
The (NR × 1) received signal vector u, deﬁned in (1), can
again be decomposed into nR antenna-speciﬁc signal vectors
uν (with ν = 1, . . . , nR) of the length K + Lc, i. e., NR =
(K + Lc)nR, and results in
u =
(
uT1 , . . . ,u
T
ν , . . . ,u
T
nR
)T
. (4)
By taking the (Lc+1) non-zero elements of the resulting sym-
bol rate sampled overall channel impulse response between the
μth transmit and νth receive antenna into account, the antenna-
speciﬁc received vector uν has to be extended by Lc elements,
compared to the transmitted antenna-speciﬁc signal vector cμ
deﬁned in (3). The ((K + Lc)× 1) signal vector uν received
by the antenna ν (with ν = 1, . . . , nR) can be constructed,
including the extension through the multipath propagation, as
follows
uν =
(
u1 ν , u2 ν , . . . , u(K+Lc) ν
)T
. (5)
Similarly, in (1) the (NR × 1) noise vector w results in
w =
(
wT1 , . . . ,w
T
ν , . . . ,w
T
nR
)T
. (6)
The vector w of the additive, white Gaussian noise (AWGN)
is assumed to have a variance of U2R for both the real and
imaginary parts and can still be decomposed into nR antenna-
speciﬁc signal vectors wν (with ν = 1, . . . , nR) according
to
wν =
(
w1 ν , w2 ν , . . . , w(K+Lc) ν
)T
. (7)
Finally, the (NR×NT) system matrix H of the block-oriented
system model, introduced in (1), results in
H =
⎡
⎢⎣
H1 1 . . . H1nT
.
.
.
.
.
.
.
.
.
HnR 1 · · · HnR nT
⎤
⎥⎦ , (8)
and consists of nR nT single-input single-output (SISO) chan-
nel matrices Hν μ (with ν = 1, . . . , nR and μ = 1, . . . , nT).
The channel convolution matrix Hν μ between the μth transmit
and the νth receive antenna is obtained by taking the (Lc +1)
non-zero elements of the resulting symbol rate sampled overall
impulse response into account [9]. Throughout this paper, it is
assumed that the (Lc + 1) uncorrelated channel coefﬁcients,
between the μth transmit and νth receive antenna have the
same averaged power and undergo a Rayleigh distribution.
Furthermore, a block fading channel model is applied, i. e.,
the channel is assumed to be time invariant for the duration
of one SDM MIMO data vector.
The interference, introduced by the off-diagonal elements of
the channel matrix H, requires appropriate signal processing
strategies. A popular technique is based on the singular-value
decomposition (SVD) [10] of the system matrix H, which can
be written as H = S ·V ·DH, where S and DH are unitary
matrices and V is a real-valued diagonal matrix of the positive
square roots of the eigenvalues of the matrix HH H sorted in
descending order1. The SDM MIMO data vector c is now
multiplied by the matrix D before transmission. In turn, the
receiver multiplies the received vector u by the matrix SH.
Thereby neither the transmit power nor the noise power is
enhanced. The overall transmission relationship is deﬁned as
y = SH (H ·D · c + w) = V · c + w˜ . (9)
As a consequence of the processing in (9), the channel matrix
H is transformed into independent, non-interfering layers
having unequal gains [9].
1The transpose and conjugate transpose (Hermitian) of D are denoted by
DT and DH, respectively.
III. QUALITY CRITERIA
In general, the quality of data transmission can be infor-
mally assessed by using the signal-to-noise ratio (SNR) at the
detector’s input deﬁned by the half vertical eye opening and
the noise power per quadrature component according to
 =
(Half vertical eye opening)2
Noise Power
=
(UA)
2
(UR)
2 , (10)
which is often used as a quality parameter [6]. The relationship
between the signal-to-noise ratio  = U2A/U2R and the bit-
error probability evaluated for AWGN channels and M -ary
Quadrature Amplitude Modulation (QAM) is given by [11]
PBER =
2
log2(M)
(
1− 1√
M
)
erfc
(√

2
)
. (11)
When applying the proposed system structure, the SVD-based
equalization leads to different eye openings per activated
MIMO layer  (with  = 1, 2, · · · , L) at the time k (with
k = 1, 2, · · · ,K) within the SDM MIMO signal vector
according to
U
(,k)
A =
√
ξ,k · Us  , (12)
where Us  denotes the half-level transmit amplitude assuming
M-ary QAM and
√
ξ,k represents the corresponding positive
square roots of the eigenvalues of the matrix HH H. Together
with the noise power per quadrature component, the SNR per
MIMO layer  at the time k becomes
(,k) =
(
U
(,k)
A
)2
U2R
= ξ,k
(Us )
2
U2R
. (13)
Using the parallel transmission over L ≤ min(nT, nR)
MIMO layers, the overall mean transmit power becomes
Ps =
∑L
=1 Ps , where the number of readily separable layers
is limited by min(nT, nR). Considering QAM constellations,
the average transmit power Ps  per MIMO layer  may be
expressed as Ps  = 2/3U2s  (M − 1) [11]. Together with
(13), the layer-speciﬁc SNR at the time k results in
(,k) = ξ,k
3
2 (M − 1)
Ps 
U2R
. (14)
In order to transmit at a ﬁxed data rate while maintaining
the best possible integrity, i. e., bit-error rate, an appropriate
number of MIMO layers L ≤ min(nT, nR) has to be used,
which depends on the speciﬁc transmission mode, as detailed
in Table I. However, in order to avoid any signalling over-
head, ﬁxed transmission modes are used in this contribution
regardless of the channel quality [9].
IV. CODED MIMO SYSTEM
The channel encoded transmitter structure is depicted in
Fig. 1. The encoder employs a half-rate nonrecursive, non-
systematic convolutional (NSC) code using the generator poly-
nomials (7, 5) in octal notation. The uncoded information is
organized in blocks of Ni bits, consisting of at least 3000 bits,
depending on the speciﬁc QAM constellation used. Each data
TABLE I
Investigated transmission modes
throughput layer 1 layer 2 layer 3 layer 4
8 bit/s/Hz 256 0 0 0
8 bit/s/Hz 64 4 0 0
8 bit/s/Hz 16 16 0 0
8 bit/s/Hz 16 4 4 0
8 bit/s/Hz 4 4 4 4
block i is encoded and results in the block b consisting of
Nb = 2Ni + 4 encoded bits, including 2 termination bits.
The encoded bits are interleaved using a random interleaver
and stored in the vector b˜. The encoded and interleaved
bits are then mapped to the MIMO layers. The task of the
multiplexer and buffer block of Fig. 1 is to divide the vector
of encoded and interleaved information bits b˜ into subvectors
(b˜1,k, b˜2,k, · · · , b˜L,k), each consisting of 8 bits according to
the chosen transmission mode (Table I). The individual binary
data vectors b˜,k are then mapped to the QAM symbols c,k
according to the speciﬁc mapper used [12], [4]. The iterative
demodulator structure is shown in Fig. 2. When using the
iteration index ν, the ﬁrst iteration of ν = 1 commences
with the soft-demapper delivering the Nb log-likelihood ratios
(LLRs) L(ν=1)2 (b˜) of the encoded and interleaved information
bits, whose de-interleaved version L(ν=1)a,1 (b) represents the
input of the convolutional decoder as depicted in Fig. 2 [13],
[4]. This channel decoder provides the estimates L(ν=1)1 (i) of
the original uncoded information bits as well as the LLRs of
the Nb NSC-encoded bits in the form of
L
(ν=1)
1 (b) = L
(ν=1)
a,1 (b) + L
(ν=1)
e,1 (b) . (15)
As seen in Fig. 2 and (15), the LLRs of the NSC-encoded bits
consist of the receiver’s input signal itself plus the extrinsic
information L(ν=1)e,1 (b), which is generated by subtracting
L
(ν=1)
a,1 (b) from L
(ν=1)
1 (b). The appropriately ordered, i. e.
interleaved extrinsic LLRs are fed back as a priori information
L
(ν=2)
a,2 (b˜) to the soft demapper of Fig. 2 for the second
iteration.
V. RESULTS
In this contribution ﬁxed transmission modes are used re-
gardless of the channel quality. Assuming predeﬁned transmis-
sion modes, a ﬁxed data rate can be guaranteed. The obtained
i b b˜
c1,k
c2,k
cL,k
encoder
∏
Mapper
Mapper
Mapper
b˜1,k
b˜2,k
b˜L,kM
UX
a
n
d
Bu
ffe
r
Fig. 1. The channel-encoded MIMO transmitter structure
y1,k
y2,k
yL,k S
of
tD
em
ap
pe
r
L
(ν)
2 (b˜)
∏−1
∏
L
(ν)
a,1 (b)
decoder
L
(ν)
1 (i)
L
(ν)
1 (b)L
(ν−1)
e,1 (b)L
(ν)
a,2 (b˜)
Fig. 2. Iterative demodulator structure
uncoded BER curves are depicted in Fig. 3 for the different
QAM constellation sizes of Table I, when transmitting at
a bandwidth efﬁciency of 8 bit/s/Hz. Assuming a uniform
distribution of the transmit power over the number of activated
MIMO layers, it turns out that not all MIMO layers have to
be activated in order to achieve the best BERs.
More explicitly, our goal is to ﬁnd that speciﬁc combination
of the QAM mode and the number of MIMO layers, which
gives the best possible BER performance at a given ﬁxed
bit/s/Hz bandwidth efﬁciency. However, the lowest BERs can
only be achieved by using bit auction procedures leading to a
high signalling overhead [9].
Using the half-rate constraint-length Kcl = 3 NSC code, the
BER performance is analyzed for an effective user throughput
of 4 bit/s/Hz. The BER investigations using the NSC code are
based on the best uncoded schemes of Table I. The information
word length is 3000 bits and a random interleaver is applied.
In addition to the number of bits per symbol and the number
of activated MIMO layers, the achievable performance of
the iterative decoder is substantially affected by the speciﬁc
mapping of the bits to both the QAM symbols as well as to
the MIMO layers. While the employment of the classic Gray-
mapping is appropriate in the absence of a priori information,
the availability of a priori information in iterative receivers
10 15 20 25
10−8
10−6
10−4
10−2
100
10 · log10(Es/N0) (in dB)→
bi
t-e
rro
rr
a
te
→
(256, 0, 0, 0) QAM
(64, 4, 0, 0) QAM
(16, 16, 0, 0) QAM
(16, 4, 4, 0) QAM
(4, 4, 4, 4) QAM
Fig. 3. BER when using the transmission modes introduced in Table I and
transmitting 8 bit/s/Hz over frequency-selective channels with Lc = 1 (two-
path channel model)
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ex
tri
ns
ic
de
m
ap
pe
ro
u
tp
ut
→
a-priori input→
Layer 1
Layer 2
Layer 3
Fig. 4. Layer-speciﬁc transfer characteristic when using anti-Gray mapping
and the (16, 4, 4, 0) transmission mode over frequency-selective MIMO links
(10 log10(Es/N0) = 2 dB, two-path channel model (solid line, Lc = 1),
ﬁve-path channel model (dashed line, Lc = 4))
requires an exhaustive search for ﬁnding the best non-Gray –
synonymously also referred to as anti-Gray – mapping scheme
[2].
Activating only an appropriate number of MIMO layers,
as determined by analyzing the uncoded transmission, the
layer-speciﬁc transfer characteristics are depicted in Fig. 4.
However, taking frequency selective MIMO links rather than
non-frequency selective MIMO links into account, large delay-
spreads seems to be highly beneﬁcial and lead to further
degree’s of design freedom as investigated in [7] or [8].
The results, depicted in Fig. 4, highlight that at low SNR
an increased delay spread is only beneﬁcial for MIMO layers
carrying a low number of bits, whereas the characteristic of the
strongest MIMO layers remains nearly unchanged. Therefore,
in combination with an appropriate number of MIMO layers,
as determined by optimizing the uncoded transmission scheme,
only minor improvements can be expected. The corresponding
BER curves are shown in Fig. 5 and conﬁrm the EXIT charts
results. The best uncoded solutions seems also to be useful in
the coded scenario.
VI. CONCLUSION
The choice of the number of bits per symbol and the number
of MIMO layers combined with powerful error correcting
codes substantially affects the performance of a MIMO sys-
tem. Analyzing the uncoded system, it turns out that not all
MIMO layers have to be activated in order to achieve the
best BERs. Considering the coded system, the choice of the
mapping strategies combined with the appropriate number of
activated MIMO layers and transmitted bits per symbol offers
a certain degree of design freedom, which substantially affects
the performance of MIMO-BICM systems. As demonstrated
2 4 6 8 10
10−4
10−3
10−2
10−1
10 · log10(Es/N0) (in dB)→
bi
t-e
rro
rr
a
te
→
(4, 4, 4, 4) QAM, 3 Iter.
(16, 4, 4, 0) QAM, 3 Iter.
Fig. 5. BER for an effective user throughput of 4 bit/s/Hz and anti-Gray
mapping in combination with different transmission modes (two-path channel
model (solid line, Lc = 1), ﬁve-path channel model (dashed line, Lc = 4))
and the half-rate NSC code with the generator polynomials of (7, 5) in octal
notation
along this work, it is showed that activating an appropriate
number of MIMO layers seems to be a promising solution for
minimizing the overall BER characteristic.
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EXIT chart analysis of iteratively detected and SVD-assisted broadband MIMO-BICM schemes

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