Impact of the Estimation Errors and Doppler Effect on the Modulation Diversity Technique Waslon Terllizzie A. Lopes∗ , Francisco Madeiro† , Juraci F. Galdino‡ and Marcelo S. Alencar§ ∗ Faculdade
´ AREA1, Salvador, BA, Brazil Cat´olica de Pernambuco, Recife, PE, Brazil ‡ Instituto Militar de Engenharia, Rio de Janeiro, RJ, Brazil § Universidade Federal de Campina Grande, Campina Grande, PB, Brazil E-mails:
[email protected],
[email protected],
[email protected] and
[email protected] † Universidade
Abstract— The performance of wireless communications systems can be significantly improved using the modulation diversity technique, which is basically based on the combination of a suitable choice of the reference angle of an MPSK constellation with independent interleaving of the symbol components. This technique presents good performance assuming the absence of estimation errors for channels characterized by flat fading. In this paper, the performance of this technique is analyzed taking into account the effects of channel estimation errors. It is shown, by simulation, that the efficiency of this technique is maintained even under this assumption. Additionally, the impact of the Doppler effect on the system performance is treated and a tradeoff between the interleaving depth and the error probability is achieved.
I. I NTRODUCTION Fading, caused by multipath in wireless communication channels, can significantly degrade the performance of digital communication systems. In effect, many techniques have been proposed to improve the performance of those systems. Among them, one can mention: diversity techniques [1]–[3] and coded modulation schemes [4], [5]. The diversity techniques consist, basically, of providing replicas of the transmitted signals to the receiver. Typical examples of diversity techniques are: time diversity, frequency diversity and spatial diversity [2]. Another type of diversity has been recently proposed and it is based on the combination of a suitable choice of the reference angle of an MPSK constellation with the independent interleaving of the symbol components before transmission [6]–[9]. In this work, this technique will be called modulation diversity. Assuming that the channel is subject to Rayleigh fading, it was shown in [9] that the modulation diversity technique leads to a performance gain, in terms of the bit error probability, when the reference angle of the QPSK constellation is suitably chosen. However, in [9] the results were obtained considering a time-uncorrelated channel (which corresponds to infinite Doppler frequency) and the absence of channel estimation errors, which is a restrictive assumption in practice. The effect of channel estimation errors on the performance of systems that use modulation diversity was analyzed in [10], where the Cramer-Rao Bound [11] was used to obtain the variance of the estimation error. The analysis presented in [10] did not address how the decision errors and the Doppler frequency
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influenciates the channel estimation and its corresponding effect on the system performance. In this work, the scheme proposed in [7] is analyzed taking into account the presence of errors in the estimation of the channel impulse response (IR). More specifically, the LMS (Least Mean Square) algorithm and a first order PLL (PhaseLock Loop) are used to track the amplitude and phase of the communication channel, respectively. Additionally, another important aspect in the system performance, the amount of correlation between the fading coefficients is treated and a trade-off between the system performance and the interleaving depth is established. The remaining of this paper is organized as follows. Section II presents the system model and the basic principles of the modulation diversity applied to fading channels. The estimation algorithms are described in Section III. In Section IV, simulations results are presented and discussed. Finally, Section V presents the conclusions. II. T HE S YSTEM M ODEL The QPSK modulation can be seen as two binary PSK modulation schemes in parallel – one in phase and another in quadrature. The two corresponding signals are orthogonal and can be separated at the receiver. In this scheme, the transmitted signal is given by s(t) =
+∞ n=−∞
an cos(ωc t) +
+∞
bn sin(ωc t),
(1)
n=−∞
where an , bn ∈ {±1} and ωc is carrier frequency. It can be seen from Equation 1 that the information transmitted on one component is independent of the information transmitted on the other component. Moreover, the transmission of these signals over independent fading channels can introduce a diversity gain when there is redundancy between the two components. In the modulation diversity technique the introduction of redundancy in a QPSK scheme can be obtained by combining the judicious choice of the reference angle θ of the signal constellation, with the independent interleaving of the symbol components [7]. Considering this rotated constellation, the
transmitted signal can be rewritten as +∞
s(t) =
xn cos(ωc t) +
n=−∞
+∞
yn−k sin(ωc t),
(2)
n=−∞
where k is an integer which represents the time delay (expressed in number of symbols) introduced by the interleaving between the I and Q components and xn = an cos θ − bn sin θ
yn = an sin θ + bn cos θ (3)
and
are the new QPSK symbols. The block diagram of the transmitter that implements this task is presented in Fig. 1. xn
an Baseband modulator
Input bits
S/P
bn
Interleaver
s(t)
cos(wc t) sen(wc t)
yn
At the receiver (Fig. 1), r(t) is baseband converted, and the obtained signal rn (t) (low-pass equivalent) for each signaling interval can be expressed as rn (t) = |αn (t)|ejφn (t) sn (t) + ηn (t), nTs ≤ t ≤ (n + 1)Ts , (6) where η(t) represents the complex white Gaussian noise, |αn (t)| denotes the amplitude of the channel IR for time t, φn (t) represents the phase shift, sn (t) denotes the low-pass equivalent of the transmitted signal s(t) and Ts is the signaling interval. After the phase compensation (multiplication of rn (t) by e−jφn (t) ), the received vector in the n-th signaling interval, denoted by r˜ n , can be expressed as r˜ n = αn sn + η n ,
(7)
where sn is the complex-valued representation of the transmitted signal in the signaling interval nTs , given by sn = xn + jyn−k .
Transmitter |α(t)| e
j φ(t)
(8)
The elements of the complex vector η n are independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and variance N0 /2. At the receiver, after deinterleaving (Fig. 1) the received vector becomes
Channel η(t)
Receiver PLL
r n = [αn xn + Re{η n }] + j[αn+k yn + Im{η n+k }], Baseband demodulator
Detector
Deinterleaver Output bits
Compensation of the phase fading
^φ (t)
cos(w c t) sen(wc t)
r(t)
|^ α(t) | LMS
Fig. 1.
Block diagram of the simulated system.
Assuming that the communications channel is characterized by fast fading, the received signal, denoted by r(t), is given by r(t) = α(t)s(t) + η(t), (4) where η(t) represents the additive noise, modeled as a complex white Gaussian process, with zero mean and variance N0 /2 by dimension. The multiplicative factor α(t) is modeled as a wide sense stationary Gaussian process whose autocorrelation function is given by 1 (5) RA (τ ) = E{α∗ (t)α(t + τ )} = J0 (2πfD τ ), 2 where J0 (·) is the zero-order Bessel function, τ is the time interval between fading samples and fD is the maximum Doppler frequency [12].
(9)
where Re{ηn } and Im{ηn+k } represent the real and imaginary parts of the complex noise η in signaling intervals nTs and (n + k)Ts , respectively. Assuming the transmission of equiprobable symbols, the optimum detector, based on the estimates of |αn |, computes the Euclidean distance between the received signal and each constellation vector (multiplied by estimates |αn | and |αn+k |) and choses the closest one to r n as the received symbol. Considering that the receiver is able to estimate without error the actual values of |α(t)| and φ(t) and that αn (t) and αn+k (t) are uncorrelated, it was shown in [9] that the system bit error probability is minimized for θ = 27◦ . However, when there is correlation between αn and αn+k , the performance presented in [9] can be achieved using an interleaving depth k corresponding to the points where the autocorrelation curves (Equation 5) are zero. This technique has low complexity compared to the original QPSK scheme (without rotation or component interleaving) because it only requires the addition of interleavers to the transmitter (the rotation can be done by a direct mapping of the input bits on the rotated symbols). On the other hand, the receiver complexity will be increased due to the fact that the channel estimators use a larger number of decision regions. For example, considering a QPSK constellation the presented method has a total of 4 × 4 = 16 decision regions, while a conventional scheme has only four decision regions.
A. The Amplitude Estimator The LMS algorithm is used in order to obtain the amplitude estimates of the channel impulse response. In the LMS algorithm, the updating process of the estimate α ˆ (n + 1) is given by [11] (10) α ˆ (n + 1) = α ˆ (n) + µs(n)e∗ (n), where µ is the step-size parameter of the algorithm and e(n) = rn − α ˆ n sˆ(n) is the error signal. During the training stage sˆ(n) = s(n) and after the training the estimate of this signal is provided by the detector. B. The Phase Estimator The estimation of the channel phase is obtained by using a recursive filter. More precisely, a first order PLL is used. In this scheme the phase updating process is given by ˆ + 1) = φ(n) ˆ φ(n + κuφ (n),
(11)
where the constant κ is the step (or gain) of the recursive filter and uφ (n) represents a phase error detector, given by [13] ˆ
uφ (n) = Im[e−j φ(n) s∗n rn ].
(12)
The objective of the PLL is to maximize the phase likelihood function, this could be obtained when the output of the phase error detector is zero. In spite of its simplicity, this phase detection scheme achieves satisfactory results as can be seen in Section IV. A detailed description of this algorithm can be found in [13]. IV. S IMULATION R ESULTS In this section, simulation results concerning the modulation diversity technique are presented. A. Performance over Time-Uncorrelated Channel and Perfect Channel Estimation Assuming time-uncorrelated channel and perfect channel estimation, the system performance (in terms of bit error rate) depends on the rotation angle θ. According to reference [10], the optimum rotation angle is around 27◦ . Moreover, the worst performance is obtained for θ = 0◦ . This occurs because there is no redundancy between the quadrature and in-phase components of the transmitted symbol. Fig. 2 presents the performance curves for the modulation diversity technique as a function of the signal-to-noise ratio.
The figure presents two curves – one related to the reference constellation and other corresponding to the reference constellation rotated by the optimum angle. It can be seen that the rotated constellation outperforms the reference constellation. For example, considering that the bit error rate equals 10−4 , the rotated constellation presents a gain of 14 dB when compared to the reference QPSK constellation. 100
Bit error rate
III. E STIMATION A LGORITHMS In the modulation diversity technique, the estimates of the amplitude and phase of the communication channel are used in distinct points at the receiver, as can be seen in Fig. 1. In particular, the phase estimate is used to compensate the phase shift produced by the channel. This operation has fundamental importance on the performance of the proposed scheme because phase errors affect the correct deinterleaving of the transmitted signals and the overall system performance is degraded. Thus, in this work, the channel estimator is composed of two distinct schemes to detect the phase and the amplitude of the channel. These schemes are described as follows.
10
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10−7
Conventional QPSK (theoretical) θ=0° θ=27°
0
5
10
15 20 Eb/N0 (dB)
25
30
35
Fig. 2. Bit error rate for the proposed system as a function of the signal-tonoise ratio considering the QPSK constellation and perfect channel estimation.
Considering the reference constellation (θ = 0◦ ), the performance of the presented system equals that of a conventional QPSK (that is, without rotation and interleaving), whose bit error probability (Pb ) is also shown in Fig. 2 and is given by [12] Eb /N0 1 Pb = . (13) 1− 2 1 + Eb /N0 B. Impact of the Interleaving Depth on the System Performance From Equation 5, one can see that the smaller the maximum Doppler frequency (fD ), the larger the interleaving depth required to reduce the correlation between the channel samples in order to obtain the maximum performance of the modulation diversity technique. Based on this, a second set of simulations has been used to verify the impact of the channel correlation on the system performance. Fig. 3 shows bit error rate curves for the QPSK constellation considering two cases: the reference (θ = 0◦ ) and the rotated (θ = 27◦ ) constellations. The curves were plotted as a function of the interleaving depth considering perfect channel estimation and fD equals 100 Hz and a transmission rate of 24.3 kbauds. One can note from Equation 5 that the autocorrelation curve presents its first null in the time corresponding to 94 symbols interval. However, based on Fig. 3, an expressive gain, in terms of the bit error rate, can be achieved using an interleaving depth of only 53 symbols, which corresponds to 60% of channel correlation. Thus, considering fD = 100 Hz, the interleaving depth of 53 symbols is a trade-off between the bit error rate and the processing delay for the proposed system.
Bit error rate
10
0
10
−1
10
−2
10
−3
TABLE I S TEPS OF LMS (µ) AND PLL(κ).
θ=0°, Eb/N0=10 dB θ=27°, Eb/N0=10 dB θ=0°, Eb/N0=20 dB θ=27°, Eb/N0=20 dB θ=0°, Eb/N0=30 dB θ=27°, Eb/N0=30 dB
θ=
fD = 50 Hz 0.5 0.8 0.25 0.6
µ κ µ κ
0◦
θ = 27◦
fD = 100 Hz 0.5 0.9 0.4 0.7
10−4
10
−5
10
−6
10
20
30
40
50
60
70
80
90
100
rotated system (θ = 27◦ ) outperforms the reference scheme (θ = 0◦ ). The system performance curves considering the absence of channel estimation errors (that is perfect estimation) were also included in these figures for comparison purposes.
Interleaving depth (k)
100
Fig. 3. Bit error rate of the proposed system as a function of the interleaving depth (k, expressed in symbol intervals) considering the QPSK constellation and fD = 100 Hz.
This is an important result, since it establishes that a good performance could be obtained using an interleaving depth corresponding to a time interval with only 60% of channel correlation. It is important to mention that a larger interleaving depth causes a long processing delay and large memory requirements of the system. As mentioned before, for θ = 0◦ the system performance does not depend on the interleaving depth because there is no redundancy between the quadrature and in-phase components of the transmitted symbols. These curves were also included in Fig. 3 for comparison purposes.
Bit error rate
10
θ=0° (LMS+PLL) θ=27° (LMS+PLL) θ=0° (Perfect estimation) θ=27° (Perfect estimation)
−1
10−2
10
−3
10−4
0
5
10
15 Eb/N0 (dB)
20
25
30
Fig. 4. Bit error rate of the proposed system as a function of the signalto-noise ratio (Eb /N0 ) considering the QPSK constellation and fD = 50 Hz.
C. Impact of the Estimation Errors on the System Performance 100
θ=0° (LMS+PLL) θ=27° (LMS+PLL) θ=0° (Perfect estimation) θ=27° (Perfect estimation)
10−1 Bit error rate
In order to verify the impact of the channel estimation errors on the performance of the modulation diversity technique, a set of simulations was performed using the channel estimators described in Section III, considering the QPSK constellation. Two cases were analyzed: the reference constellation (θ = 0◦ ) and the optimum rotated constellation (θ = 27◦ ). In all simulations reported in this section the transmission rate was 24.3 kbauds and a minimum of 104 channel realizations was done for each bit error rate investigated. Assuming blocks with 250 symbols, an amount of 5 × 106 bits was used. In order to avoid the propagation of decision errors, the blocks were divided in 50 training symbols and 200 information symbols. The step-size parameters of the estimation were established by computer simulation for each value of fD , in order to reduce the bit error rate of the system. The following strategy was used: to determine the step µ of the LMS Algorithm the perfect phase estimation was assumed and to determine the PLL step κ the perfect amplitude estimation was assumed. The steps are presented in Table I considering the reference constellation (θ = 0◦ ) and the rotated one (θ = 27◦ ). The simulation results concerning the impact of channel estimation errors on the system performance are presented in Figs. 4 and 5. These curves were obtained considering an interleaving depth of 100 symbols and fD equal to 50 Hz and 100 Hz, respectively. In these cases, one can see that the
10
−2
10
−3
0
5
10
15 Eb/N0 (dB)
20
25
30
Fig. 5. Bit error rate of the proposed system as a function of the signalto-noise ratio (Eb /N0 ) considering the QPSK constellation and fD = 100 Hz.
Qualitatively, the results presented in Figs. 4 and 5 are similar. They show the effect of the bit error rate floor, which is a typical behavior for fast fading channels as a consequence of estimation errors of the channel IR. As can be observed in these figures, the error floor increases as fD increases. This occurs due to the fact that, for a fixed transmission rate, the channel variations are faster for higher values of fD . As a consequence, the estimation errors increase as fD increases. The
error floor can be lowered using more robust and sophisticated filtering schemes, such as Kalman Filtering [11]. However, in all cases, the rotated scheme (θ = 27◦ ) outperforms the non-rotated one (θ = 0◦ ) and the error floor is significantly minimized. For example, considering fD = 50 Hz, the error floor decreases from 7 × 10−3 to 5 × 10−4 . In this case (Fig. 5), the rotated scheme outperforms the non-rotated one even with perfect channel estimation for Eb /N0 ranging from 6 dB to 25 dB. D. Improving Image VQ for a Rayleigh Fading Channel Vector quantization (VQ) [14] has been widely used in image coding systems. However, it is highly sensitive to channel errors, which may lead to very annoying blocking artifacts in the reconstructed images. This section considers VQ-based image transmission over a Rayleigh fading channel. Results regarding the traditional Lena image (256 × 256 pixels) are presented. VQ with dimension 16 (corresponding to image blocks of 4 × 4 pixels) and codebook size 256 is used. The corresponding code rate is 0.5 bits per pixel. Channel estimation errors and temporal channel correlation are considered. The simulations involving modulation diversity consist on using a QPSK scheme with a constellation rotation θ = 27◦ . The transmission system has an interleaving depth of 50 symbols. For a transmission rate of 24.3 kbauds, Doppler frequencies (fD ) of 50 Hz, 100 Hz and 150 Hz are considered. Fig. 6 shows examples of typical images considering a channel signal-to-noise ratio Eb /N0 = 16 dB and fD = 50 Hz, 100 Hz and 150 Hz. For each value of fD , the image with better quality (with smaller number of blocking artifacts) corresponds to the one obtained by using a transmission with modulation diversity. The benefits of MD in terms of reducing the reconstructed image degradation may be observed for each value of fD .
(a) θ = 0◦ , fD = 50 Hz.
(b) θ = 0◦ , fD = 100 Hz.
(c) θ = 0◦ , fD = 150 Hz.
(d) θ = 27◦ , fD = 50 Hz.
(e) θ = 27◦ , fD = 100 Hz.
(f) θ = 27◦ , fD = 150 Hz.
Fig. 6. Reconstructed Lena image after transmission for a Rayleigh fading channel, considering Eb /N0 = 16 dB.
V. C ONCLUSION This work presented a performance analysis of the modulation diversity technique applied to wireless communication channels. This technique combines a suitable choice of the
reference angle of a PSK constellation with the component interleaving before the symbol transmission. The performance analysis was carried out considering a fast fading channel and use of the LMS and PLL estimation algorithms to track the the amplitude and phase of the wireless channel, respectively. Assuming the absence of estimation errors, the optimum rotation angle was obtained. It was shown that the use of rotated constellation instead of the reference constellation improves the system performance in terms of bit error rate. It was also shown that the performance improvement is kept even in the presence of channel estimation errors. Moreover, the impact of the Doppler effect on the system performance was evaluated and a trade-off between the bit error probability and the interleaving depth was reached. It was pointed out that an interleaving depth corresponding to 60% of channel correlation constitutes an adequate choice considering the following aspects: system performance, processing delay and memory requirements. Additionally, this paper also presented results from the transmission of vector quantized images over fading channels. It was shown, by simulation, that the use of the modulation diversity technique leads to better reconstructed images. As future works, the authors will investigate the performance of modulation diversity over multipath scenarios and frequency-selective channels. R EFERENCES [1] V. M. DaSilva and E. S. Sousa, “Fading-resistant modulation using several transmitter antennas,” IEEE Transactions on Communications, vol. 45, no. 10, pp. 1236–1244, October 1997. [2] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Transactions on Information Theory, vol. 44, no. 2, pp. 744–765, March 1998. [3] J. H. Winters and R. D. Gitlin, “The impact of antenna diversity on the capacity of wireless communications systems,” IEEE Transactions on Communications, vol. 42, no. 2/3/4, pp. 1740–1751, February/March/April 1994. [4] N. Seshadri and C. W. Sundberg, “Multilevel trellis coded modulations for the Rayleigh fading channel,” IEEE Transactions on Communications, vol. 41, no. 9, pp. 1300–1310, September 1993. [5] J. Wu and S. Lin, “Multilevel trellis MPSK modulation codes for the Rayleigh fading channel,” IEEE Transactions on Communications, vol. 41, no. 9, pp. 1311–1318, September 1993. [6] K. J. Kerpez, “Constellations for good diversity performance,” IEEE Transactions on Communications, vol. 41, no. 9, pp. 1412–1421, September 1993. [7] S. B. Slimane, “An improved PSK scheme for fading channels,” IEEE Transactions on Vehicular Technology, vol. 47, no. 2, pp. 703–710, May 1998. [8] 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. [9] W. T. A. Lopes and M. S. Alencar, “Space-Time coding performance improvement using a rotated constellation,” in Anais do XVIII Simp´osio Brasileiro de Telecomunicac¸o˜ es (SBrT’2000), Gramado, RS, Brasil, Setembro 2000. [10] ——, “Performance of a rotated QPSK based system in a fading channel subject to estimation errors,” in Proceedings of The IEEE International Microwave and Optoelectronics Conference (IMOC’2001), Bel´em, PA, Brasil, Agosto 2001, pp. 27–30. [11] S. S. Haykin, Adaptive Filter Theory. Prentice Hall, 1991. [12] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1989. [13] P. Koufalas, “State variable approach to carrier phase recovery and fine automatic gain control on flat fading channels,” Master’s Thesis, University of South Australia, 1996. [14] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston, MA: Kluwer Academic Publishers, 1992.
