OFDM with BICM-ID and Rotated MPSK
Constellations and Signal Space Diversity
Nauman F. Kiyani and Jos H. Weber
IRCTR/CWPC, Wireless and Mobile Communications Group,
Faculty of Electrical Engineering, Mathematics and Computer Science,
Delft University of Technology, Delft, the Netherlands.
Email(s): {n.f.kiyani, j.h.weber}@tudelft.nl
Abstract— Co-ordinate interleaving (also known as signal space
diversity (SSD)) with the rotation of signal constellation of multilevel modulation schemes with bit interleaved coded modulation
(BICM) is known to provide good performance for single carrier
systems. This paper studies the extension of SSD with recursive
systematic convolutional (RSC) coded BICM and constellation rotation in an orthogonal frequency division multiplexing (OFDM)
system. Iterative demodulation and decoding (ID) is employed
at the receiver. It is shown that for a signal constellation and
labeling using SSD on a frequency-selective fading channel, a well
considered choice of the rotation angle leads to a significant gain
over the conventional BICM-ID OFDM system. Furthermore,
the choice of the rotation angle for the system is found to be
dependent upon the symbol labeling and the number of iterations
performed between the demodulator and the decoder.
I. I NTRODUCTION
Fading causes significant performance degradation in wireless digital communication systems. It is well known that
an efficient way to mitigate fading is to apply diversity
techniques. Any diversity scheme/technique (e.g, time, frequency, and space) tries to provide statistically independent
copies of the transmitted sequence at the receiver for more
reliable detection. Orthogonal frequency division multiplexing
(OFDM) has been employed for a variety of applications,
e.g., cellular radio, digital audio and video broadcasting, and
wireless local area network (WLAN) systems such as IEEE
802.11, and HIPERLAN etc. OFDM is a block modulation
technique in which the data symbols are transmitted in parallel
on orthogonal subcarriers. Individual subcarriers experience
flat fading as the bandwidth of the subcarriers is small as
compared to the coherence bandwidth of the channel.
Signal space diversity (SSD) can provide performance improvement over fading channels by increasing the diversity
order of a communication system [1]. SSD with bit interleaved
coded modulation (BICM) has been shown to improve the
performance of a single carrier system in flat Rayleigh fading
channels, e.g., in [2], [3] and the references therein. In [2],
low density parity check (LDPC) codes are employed with
BICM coupled with rotated M-ary phase shift keying (MPSK)
constellations and SSD. It was shown in [2], that the LDPCcoded BICM coupled with SSD and optimally rotated MPSK
constellations outperforms conventional LDPC-coded BICM.
In [3], iterative demodulation and decoding of BICM coupled
with SSD and rotated MPSK constellations was presented.
1-4244-1370-2/07/$25.00 ©2007 IEEE
Application of space diversity has also been considered in
uncoded OFDM systems [4] and the references therein.
In this paper, we propose an iterative demodulation and
decoding technique using practically sized recursive systematic convolutional (RSC) codes with different code rates and
rotated multi-level modulation schemes for an OFDM system
over a frequency-selective Rayleigh fading channel. The effect
of different signal mappings, the code rate, and the number
of iterations on the choice of the rotation angle is presented.
It is shown that the proposed system with SSD and rotated
MPSK constellations provides degrees of freedom which are
not available in the conventional system. Furthermore, the
proposed system with a well considered choice of the rotation
angle leads to a substantial gain over the conventional BICMID OFDM system. Different signal constellation labeling for
MPSK constellations under various rotation angles are considered.
The paper is organized as follows. Section II briefly outlines
the main blocks of the system model. Section III presents the
effect of constellation rotation, mapping and code rate on the
system performance. The performance curves of the proposed
system are also presented. Lastly, in Section IV conclusions
are presented.
II. S YSTEM M ODEL
Figure 1, shows a generic block diagram of a system
employing SSD with BICM-ID, OFDM and rotation of signal
constellation (hereafter referred to as SSD-ID OFDM). The
encoded bits, which are punctured to a desirable rate, are interleaved using a pseudo-random interleaver, represented by π,
and are mapped onto symbols using rotated constellations with
pre-defined signal labeling. A clockwise rotation over the an
θ
= {sl = e(2π(l/M )−θ)j :
angle θ leads to the constellation SM
l = 0, 1, . . . , M − 1}, where the energy has been constrained
to unity. The symbol mapper can be represented by the oneθ
, s = ℘(b),
to-one mapping function ℘ : {0, 1}log2 M → SM
where, b = (b1 , · · · , blog2 M ), bj ∈ {0, 1} represents the binary
θ
consisting of M
sequence and s is chosen from the set SM
complex signal points. Consider an OFDM system with N
subcarriers. Each group of N log2 M encoded bits are mapped
to an OFDM symbol. The in-phase (I) and quadrature phase
(Q) components of the mapped symbols are then separately
interleaved. The coordinate interleaving being employed is to
p
Q
pR
Serial- to
Parallel
converter &
Interleaver
T
Data Source
Encoder
Ȇ
I
IDFT
Cyclic Prefix
& Serial
Converter
D/A
Symbol
Mapper
cos( wc t )
h
Data
Estimate
SISO
Decoder
Ȇ-1
Parallel-toSerial
converter &
De-interleaver
Symbol to
bit
demapper
DFT
Remove
Prefix Serialto-Parallel
converter
A/D
LPF
cos( wc t )
Ȇ
Fig. 1.
System Model
make the I and Q channels uncorrelated implying that the
I and Q channels experience independent fades. A single
coordinate interleaver or a delay line introducing a delay that
exceeds the coherence time of the channel would also, have the
same effect and can be used instead of the two interleavers [2].
Let X represent the rotated and symbol interleaved sequence
passed to the inverse discrete Fourier transform (IDFT) block,
which is efficiently implemented by using the IFFT algorithm.
The IDFT yields the OFDM symbol consisting of the sequence
x[n] = x[0], · · · , x[N − 1] of length N, where
N −1
1 X[i]ej2πni/N , 0 ≤ n ≤ N − 1.
x[n] = √
N i=0
(1)
The sequence in (1) corresponds to the samples of the multicarrier signal: i.e., the multicarrier signal consists of linearly
modulated subchannels. The MPSK interleaved symbols X[i]
are modulated by the carrier frequency. The cyclic prefix
is then added to the OFDM symbol and the resulting time
samples are ordered by the parallel-to-serial converter and
passed through to a D/A converter, resulting in the baseband
OFDM signal x(t), which is then up converted to frequency
fc .
Consider a frequency-selective fading channel h =
[h0 h1 · · · hL−1 ]T with L channel taps. Each tap is assumed to
be statistically independent and is modeled as a circularly symmetric complex Gaussian random variable. It is assumed that
the fading is quasi-static, i.e., the fading coefficients remain
constant within one OFDM symbol and change independently
from one OFDM symbol to the next. The transmitted signal
is filtered by the channel impulse response and corrupted
by additive noise. The cyclic prefix Lcp converts the linear
convolution of the transmitted signal and the L-tap channel
into a circular convolution. Assuming that the discrete Fourier
transform (DFT) and cyclic prefix removal at the receiver
with coherent detection are carrier out error-free; OFDM converts the frequency-selective fading channel into N flat subchannels. The received signal, therefore, can be represented
as
r = xH + n,
(2)
where, n = [n0 , . . . , nN −1 ] is a DFT processed complex
white Gaussian noise vector with independent components
having two-sided spectral density of N0 /2. The noise vector is
white and Gaussian with same correlation matrix because the
DFT matrix is unitary. OFDM yields a diagonally equivalent
channel matrix H = diag(H0 , . . . , HN −1 ), where
Hi =
L−1
hl e−j2πil/N , 0 ≤ i ≤ N − 1,
(3)
l=0
is the frequency response of the underlying FIR channel channel evaluated at the DFT grid. In equation (3), hl represents
the channel gain of the lth tap.
A. Symbol-to-bit De-mapper
A serial concatenation of a soft-input soft-output (SISO)
demodulator and a SISO decoder is employed in the receiver
(as shown in Figure 1) to approach the maximum likelihood
receiver performance of joint demodulation and decoding. The
extrinsic information is passed between the SISO demodulator
(indicated as “symbol-to-bit de-mapper”) and the SISO decoder. The SISO decoder is used to generate the a posteriori
bit probabilities for information and coded bits.
Assuming perfect CSI, the soft demodulator computes the
log likelihood ratios (LLR) at iteration ‘q’ of bi,m which is
the mth bit of the ith received symbol denoted by Λq (bi,m ),
as in [3]


log2 M


q−1
(b
)
Ω
i,k
sl,k
+
(−1)
.
Λq (bi,m ) = max
E
i,l
0 

2
l∈Sm
k=1,k=m


log2 M

q−1
(bi,k ) 
sl,k Ω
− max
+
(−1)
.
E
.
i,l
1 

2
l∈Sm
k=1,k=m
(4)
0
1
In (4), Sm
= {l : sl,m = 0} and Sm
= {l : sl,m = 1}, sl,k
th
th
θ
denotes the k bit in the l symbol in the symbol set SM
,
q−1
(bi,k ) is the interleaved extrinsic LLR-value of bit bi,k
Ω
calculated by the SISO decoder in the previous iteration, and
Ei,l is defined as
2 2 1 I
ri − HiI sIl + riQ − HiQ sQ
Ei,l = −
.
(5)
l N0
−1
10
Gray
Natural
st
1 Iteration
2nd Iteration
3rd Iteration
4th Iteration
5th Iteration
−2
10
BER
In the first iteration Ω0 (bi,k ) = 0; ∀i, k, is assumed. On the
subsequent passes (i.e., iterations q ≥ 2), the extrinsic information of the bits Ωq−1 (bi,k ) is used as a priori information
by the symbol to bit de-mapper. Since we have bit interleaving,
we may assume that the probabilities of the bits that compose
the symbol are independent [5]. From (4), it is clear that when
recalculating the bit metrics for one bit, we only need to use
the a priori probabilities of the other bits in the same channel
symbol.
−3
10
B. Iterative SISO Decoder
−4
10
0
10
20
30
40
°
50
60
70
80
90
θ
Fig. 2. BER performance of QPSK SSD-ID OFDM over different rotation
angles. 16-state, rate 12 RSC codes are used with Gray and Natural signal
constellation labeling in a frequency-selective fading channel at Eb /N0 = 5
dB.
−1
10
Gray
Natural
1st Iteration
nd
2 Iteration
rd
3 Iteration
4th Iteration
5th Iteration
BER
The SISO decoder [6] is used for convolutional decoding
and to generate the extrinsic LLR values of the coded bits for
iterative demodulation and decoding. The a priori probability
of the information bits is unavailable and is not used in
the entire process. The extrinsic information passed from the
symbol to bit de-mapper is de-interleaved and used to calculate
the extrinsic information of the coded bits by the SISO
decoder. The regenerated bit metrics are interleaved and passed
to the symbol to bit de-mapper and we iterate the demodulation
and decoding. The final output is the hard decision based
on the extrinsic bit probability of the information bits which
is also the total a posteriori probability. We also employ in
our simulations punctured convolutional codes in which all
the transmitted bits are a subset of a lower rate code. This
permits an increase in the code redundancy by incremental
transmission, and can be combined with ARQ schemes.
−2
10
III. R ESULTS
Rate 12 and 23 , 16-state RSC codes with generator polynomials [37, 21]8 of length 2600 are used in an SSD-ID OFDM
system in a frequency-selective fading channel. The code is
punctured to a desirable rate according to the puncture pattern
given in Table I. . The systems are simulated with N = 64
TABLE I
P UNCTURE PATTERNS FOR THE 16- STATE PUNCTURED CONVOLUTIONAL
CODES
Code Rate (R)
1
2
2
3
−3
10
0
10
20
30
40
θ°
50
60
70
80
90
Fig. 3. BER performance of QPSK SSD-ID OFDM over different rotation
angles. 16-state, rate 23 RSC codes are used with Gray and Natural signal
constellation labeling in a frequency-selective fading channel at Eb /N0 = 5
dB.
[7]
Puncturing Pattern
1111
1111
1111
1x1x
subcarriers and a channel model with root mean square (RMS)
delay spread of about 50 nsec, a bandwidth of 50 MHz
and a 29-tap power delay profile having an exponentially
decaying fading characteristic with each ray being assumed
to be independently Rayleigh faded. Perfect channel state
information is assumed at the receiver.
A. Effect of Constellation Rotation
The effect of the constellation rotation can be visualized
by fixing the Eb /N0 -value and observing a performance
parameter, e.g., bit error rate (BER). Figure 2 shows the
BER as a function of various rotation angles for QPSK
modulation scheme at Eb /N0 = 5 dB. The figure shows
that the system performance, i.e., BER is dependent upon
the choice of the rotation angle. The system shows best
performance at 0◦ and 15◦ for Gray and Natural labeling,
respectively, when 5 iterations are performed between the
demodulator and the decoder. Rotation angles having better
BER performance change after the first iteration for both
mappings. No further improvement in performance can be
obtained with Gray labeled constellation after 2 iterations.
B. Effect of Code Rate
The effect of a change in the code rate on the system
characteristics and performance is shown in Figure 3. Rate
2
3 RSC codes are used. The figure elucidate that signal
constellation rotation is immune to the change in the code
rate. The BER performance degrades due to the fact that the
transmitted bits are punctured to a lower rate by removing the
parity bits and this lowers the error correcting capability but it
0
10
BICM−ID (Gray)
BICM−ID (Nat.)
°
SSD−ID (Gray, 0 )
−1
10
SSD−ID (Nat., 15°)
st
1 Iteration
5th Iteration
−2
10
−3
BER
10
−4
10
−5
10
−6
10
−7
10
4
5
6
7
8
9
10
E /N [dB]
b
0
Fig. 4. BER performance of SSD-ID OFDM and BICM-ID OFDM with
QPSK signal constellation. Gray and Natural labeling are used. 16-state RSC
codes of length 2600 and rate 12 are employed in a frequency-selective fading
channel.
has no effect on the characteristic features of the system, i.e.,
SSD and signal constellation rotation.
C. Performance Curves
Figure 4 shows the comparative performance gain of SSDID OFDM over BICM-ID OFDM with QPSK signal constellation. Figure 4 shows the performance of QPSK signal constellation with Gray and Natural mapping. Gray labeled BICMID OFDM does not show any performance gain after the 1st
iteration. SSD-ID OFDM outperforms BICM-ID OFDM over
all iterations and all mappings, e.g., the Natural labeled rotated
SSD-ID OFDM has a performance improvement of 2.7 dB at
a BER of 3.04 × 10−5 at the 5th iteration as shown in Figure
4.
IV. C ONCLUSIONS
In this paper, we have investigated the performance of RSC
codes over frequency-selective fading channels using SSD and
OFDM. The performance of the proposed scheme is compared
with the BICM-ID by employing rotated MPSK constellations
and different symbol to bit mappings. It is shown that a well
considered choice of rotation angle and signal constellation
labeling can provide significant performance improvement
over the conventional BICM-ID OFDM system. The choice
of the rotation angle is seen to be dependent on the signal
constellation mapping and quite interestingly on the number
of iterations performed at the receiver and is immune to the
change in the code-rate.
ACKNOWLEDGEMENT
The authors would like to thank Z. Irahhauten for his
comments and suggestions on channel characterization. This
work was supported by STW under McAT project DTC.6438.
R EFERENCES
[1] J. Boutros and E. Viterbo, “Signal space diversity: a power and
bandwidth-efficient technique for the Rayleigh fading channel,” IEEE
Trans. Information Theory, vol. 44, no. 4, pp. 1453–1467, July 1998.
[2] N. F. Kiyani, U. H. Rizvi, J. H. Weber, and G. J. M. Janssen, “Optimized
rotations for LDPC-coded MPSK constellations with signal space diversity,” Proc. IEEE Wireless Comm. and Networking Conf., pp. 677–681,
Mar. 2007.
[3] N. F. Kiyani and J. H. Weber, “Iterative demodulation and decoding for
rotated MPSK constellations with convolutional coding and signal space
diversity,” IEEE 66th Vehicular Technology Conf., Oct. 2007.
[4] Z. Liu, Y. Xin, and G. B. Giannakis, “Space-time-frequency coded
OFDM over frequency-selective fading channels,” IEEE Trans. on Signal
Process., vol. 50, no. 10, pp. 2465–2476, Oct. 2002.
[5] A. Stefanov and T. M. Duman, “Turbo coded modulation for systems with
transmit and receive antenna diversity over block fading channels:system
model, decoding approaches and practical considerations,” IEEE Journal
on Selected Areas in Comm., vol. 19, no. 5, pp. 958–968, May 2001.
[6] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “A soft-input softouput APP module for iterative decoding of concatenated codes,” IEEE
Comm. Letters, no. 1, pp. 22–24, Jan. 1997.
[7] J. Hagenauer, “Rate compatible punctured convolutional codes (RCPC
codes) and their applications,” IEEE Trans. on Comm., vol. 36, no. 4, pp.
389–400, April 1988.