Performance of a Rotated QPSK Based System in a ...

Performance of a Rotated QPSK Based System in a Fading Channel Subject to Estimation Errors Waslon Terllizzie Ara´ujo Lopes and Marcelo Sampaio de Alencar Abstract — This paper presents a performance analysis of a modified QPSK scheme for transmission over fading channels. Interleaving and rotation of the signal constellation are applied with the objective of improving its performance. It is shown that the proposed system outperforms the original scheme even in the presence of channel estimation errors. Index Terms — Fading channels, transmission diversity, wireless communications, channel estimation errors.

I. I NTRODUCTION ADING causes significant degradation in the performance of digital wireless communications systems. Unlike the additive Gaussian channel, the wireless channel suffers from attenuation due to destructive addition of multi-paths in the propagation media and due to interference from other users [1]. In order to minimize the effects of fading, some systems use a resource called diversity which consists, basically, in providing replicas of the transmitted signals at the receiver. Examples of diversity techniques are: temporal diversity, frequency diversity and antenna diversity. Another way to increase the system diversity is to introduce redundancy by combining rotation and interleaving of the constellation symbols before the modulation process [2], [3], [4], [5], [6]. This recent technique has been called “modulation diversity”. In a recent paper [7], the authors have shown that a considerable performance gain, in terms of the bit error probability, can be achieved by selecting the rotation angle, for a QPSK constellation, when the channel is modeled by Rayleigh fading. However, it has been assumed ideal channel state information (ideal CSI) at the receiver, which is a very restrictive assumption for practical systems where channel estimation errors should be considered. This paper presents the performance of the system proposed in [7] in the absence of ideal channel state information, i.e., considering the presence of channel estimation errors. This paper is organized as follow: In Section II, the modulation diversity technique for fading channels is presented. The basic principles of the rotation of the QPSK constellation for performance improvement are introduced in Section III. The presence of channel estimation errors is treated in Section IV. Section V includes the numerical results and discussions, while Section VI is devoted to conclusions. II. T HE M ODULATION D IVERSITY T ECHNIQUE The key point to increase the modulation diversity is to apply a certain amount of rotation to a classical signal constellation in such way that any two points achieve the maximum Laborat´orio de Comunicac¸ o˜ es – LABCOM, Departamento de Engenharia El´etrica – DEE, Universidade Federal da Para´ıba – UFPB, 58.109-970 - Campina Grande, PB, Brasil, Phone: +55 83 3101410 Fax: +55 83 3101418 waslon,malencar @dee.ufpb.br




number of distinct components [4]. Fig. 1 illustrates this idea for a QPSK scheme. In fact if it is supposed that a deep fade hits only one of the components of the transmitted signal vector, then one can see that the “compressed” constellation in Figure 1(b) (empty circles) offers more protection against the effects of noise, since no two points collapse together as would happen in Fig. 1(a). A component interleaver/deinterleaver is required to assume that the in-phase and quadrature components of the received symbol are affected by independent fading.

(a)

(b)

Fig. 1. The basic idea of performance improvement using a rotated constellation.

An interesting feature of the rotation operation is that the rotated signal set has exactly the same performance of the non-rotated one when used over a pure additive white Gaussian noise (AWGN). This occurs due to the fact that the Euclidean distance between the symbols, for rotated and nonrotated QPSK constellations, is the same in the two cases.

III. ROTATING T HE QPSK C ONSTELLATION For a frequency-nonselective slowly fading channel, coded modulation combined with interleaving showed good performance improvement of digital communications systems. Interleaving destroys fading correlation, adding diversity to the coded scheme [5], [6]. In [5] the author used a new form of interleaving, where each coordinate is interleaved independently to enhance the system performance. In [7] the key idea was to analyze the influence of rotating the QPSK constellation, by a constant phase , in the system performance. The block diagram of the modified QPSK scheme is shown in Fig. 2. 

Consider a conventional QPSK scheme. The transmitted

xn

cn

Baseband modulator

an

S/P

bn

Interleaver 1 cos(wc t)

s(t) sen(wc t)

yn

Interleaver 2

(a) Modulator.

cos(wc t) sen(wc t)

Decision

r(t)

Baseband Demodulator

Deinterleaver 1

cn

Deinterleaver 2

(b) Demodulator. Fig. 2. Block diagram of the modified QPSK scheme.

Q

signal is given by


     *

.

where

 
!#"%$&
('*)+*,-

/

 
01  2$-354
('*)+ , 

I

 6 /  87:9

with equal probability,


=@?  ?  =!6 elsewhere,

+ , is the carrier frequency and is the carrier amplitude. When the signals in phase (I channel) and quadrature (Q channel) are interleaved independently, a diversity gain can be obtained because the fading in one channel is independent from the other channel. From Equation 1 the sequence A  CB is / independent of the sequence A CB . In this case the system cannot take advantage of the previous described diversity unless some kind of redundancy between the two quadrature channels is introduced. Introducing redundancy in the QPSK scheme can be achieved by rotating its signal constellation by a constant phase , as show in Fig. 3. After the rotation and the interleaving, the transmitted signal becomes 


DE

 F *

 . L *

1M

θ

(1)

 
D0H  2#"$I
J'K)+ , 

1G NOP
D12$-3Q4F
('K)+, 6

(2)

where R is an integer representing the time delay, in number of symbols, introduced by the interleaving between the S and T components and /     I"%$   $354 (3) /     $-354 .  #"$ G are the rotated symbol M components. It is important to note that the rotation does not affect the system spectral efficiency for the rotated system, that also transmits two bits in one symbol interval. The digital communication channel is assumed to be frequency-nonselective slowly fading with a multiplicative factor representing the effect of fading and an additive term representing the AWGN channel. It is also assumed that the system is unaffected by intersymbol interference. Thus, the received signal is then written as U2
DVXWV
D
 .ZY
 (4) 







where WV
D is modeled as zero-mean complex Gaussian process.

IV. T HE P RESENCE OF C HANNEL E STIMATION E RRORS T HE S YSTEM


DE W
 .
D 6

0

'

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Phase rotation (degrees)

Fig. 4. The bit error probability for the modified QPSK scheme as a function of the phase rotation ( . Ideal channel state information (CSI) was considered 1

6

(6)

Original QPSK scheme (no CSI) Original QPSK scheme (CSI) Rotated QPSK scheme (no CSI) Rotated QPSK scheme (CSI)

0.1

Bit Error Probability

 

0.001

0.0001



Var
D 

0.01

(5)

where
 represents the channel estimation error. In this work,
 is modeled as a complex Gaussian random variable having zero-mean. Regardless of which method is used for estimation, the variance of
D can be directly obtained from the Cram´er-Rao Bound [8], which establishes that the variance of the estimation error is 

Eb/No = 10 dB Eb/No = 15 dB Eb/No = 20 dB

IN

If channel estimation errors are considered, the receiver is not able to obtain the correct value of the fading coefficient W in the decoding process, thus the estimation 
D of the fading coefficient used by the decoder can be written as 

0.1

Bit error probability

If coherent demodulation is used the fading coefficients can be modeled, after phase elimination, as real random variables W 0 with Rayleigh distribution and unit second moment

  9  . The independence of the fading samples represents the situation where the components of the transmitted points are perfectly interleaved. An undesirable effect of the component interleaving is the fact that it produces nonconstant envelope transmitted signals [4].

0.01

0.001

0.0001





where is the number of pilot symbols and is the symbol energy. As expected, it can be seen from the previous equation, that the variance of the estimation error decreases when the signal-to-noise ratio increases. After channel estimation, the decoder selects, for each received signal, the symbol constellation 
 that minimizes the metric      U2
D 

D  (7)

1e-05

1e-06 0

5

10

15 Eb/No (dB)

20

25

30

Fig. 5. The bit error probability for the rotated QPSK scheme and the nonrotated QPSK scheme as a function of )+*-,/.10 . Two cases were considered: Ideal CSI and absence of CSI

V. S IMULATIONS R ESULTS This section presents the simulations used to find out the rotation phase which produces the best performance, in terms of bit error rate, considering no channel estimation errors (ideal CSI). Next, this rotation is applied to the scheme considering the estimation errors and its performance is assessed by simulations. In order to determine the phase rotation that achieves the best performance, the transmission system in Fig. 2 was sim) ' . Fig. 4 shows the sysulated for varying from zero to  tem performance, measured in terms of bit error probability, for    9 = , !9 and ' = dB. It can be seen that the optimum performance is obtained for approximately equal to ) #" for the three curves presented. Considering this optimum  phase rotation, Fig. 5 compares the performance of the original QPSK scheme (  = ) and its rotated version for $%  varying from zero to 30 dB. When  = , the performance of the system reduces to that of a conventional QPSK scheme. It can be noted that a considerable performance improvement is obtained compared to the conventional QPSK scheme, which  & ' can reach 8 dB at a bit error probability of 9 = . 



Fig. 5 also shows the performance of the proposed system in the absence of ideal channel estimation. The dotted lines represents the bit error probability for the original system and the rotated system. It is important to note that, in both cases, the estimation errors increase the bit error probability. However, the rotated system is more robust than the original scheme in terms of performance decrease. For example, at a bit error  probability of 9 =  , the channel errors produce a performance decrease of 1 dB, for rotated system, while this drop can reach 5 dB for the non-rotated system. VI. C ONCLUSIONS







This paper presented the performance analysis of a rotated based system in the absence of ideal channel state information. Based on the simulation results, the effects of the presence of channel estimation errors were obtained. As can be inferred from the simulation results the optimum performance ) 2" is obtained for a rotation angle approximately equal to  for the three curves presented. At this rotation phase a considerable performance improvement, over the original QPSK 

scheme, is obtained. ACKNOWLEDGMENTS The authors would like to express their thanks to CNPq for the financial support to this research and to J. F. Galdino for the helpful comments. R EFERENCES [1] V. Tarokh, N. Seshadri and A. R. Calderbank. “Space-Time Codes for High Data Rate Wireless Communication: Performance Criterion an Code Construction”. IEEE Transactions on Information Theory, vol. 44, no. 2, pp. 744–765, March 1998. [2] K. J. Kerpez. “Constellations for Good Diversity Performance”. IEEE Transactions on Communications, vol. 41, no. 9, pp. 1412–1421, September 1993. [3] S. B. Slimane. “An Improved PSK Scheme for Fading Channels”. IEEE Transactions on Vehicular Technology, vol. 47, no. 2, pp. 703–710, May 1998. [4] J. Boutros and E. Viterbo. “Signal Space Diversity: A Power- and Bandwidth-Efficient Diversity Technique for the Rayleigh Fading Channel”. IEEE Transactions on Information Theory, vol. 44, no. 4, pp. 1453– 1467, July 1998. [5] B. D. Jeli˘ci´c and S. Roy. “Design of Trellis Coded QAM for Flat Fading and AWGN Channels”. IEEE Transactions on Vehicular Technology, vol. 44, pp. 192–201, February 1995. [6] D. Divsalar and M. K. Simon. “The Design of Trellis Coded MPSK for Fading Channels: Performance Criteria”. IEEE Transactions on Communications, vol. 36, no. 9, pp. 1004–1012, 1988. [7] W. T. A. Lopes and M. S. Alencar. “Space-Time Coding Performance Improvement Using a Rotated Constellation”. In XVIII Simp´osio Brasileiro de Telecomunicac¸ o˜ es (SBT’2000), Gramado - RS, Brasil, Setembro 2000. [8] V. Tarokh, A. Naguib, N. Seshadri and A. R. Calderbank. “Space-Time Codes for High Data Rate Wireless Communication: Performance Criteria in the Presence of Channel Estimation Errors, Mobility and Multiple Paths”. IEEE Transactions on Communications, vol. 47, no. 2, pp. 199– 207, February 1999.