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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
Genetic Algorithm based Equalization for Direct Sequence Ultra-Wideband Communications Systems Nazmat Surajudeen-Bakinde, Xu Zhu, Jingbo Gao and Asoke K. Nandi Department of Electrical Engineering and Electronics University of Liverpool, Liverpool, UK. {nazmat.surajudeen-bakinde, xuzhu, jgao and a.nandi}@liverpool.ac.uk
Abstract— We propose a genetic algorithm (GA) based equalization approach for direct sequence Ultra-wideband (DS-UWB) wireless communications, where GA is combined with a RAKE receiver to combat the inter-symbol interference (ISI) due to the frequency selective nature of UWB channels for high data rate transmission. Simulation results show that the proposed GA based structure significantly outperforms the RAKE receiver. It also provides a close bit error rate (BER) performance to the optimal maximum likelihood detection (MLD) approach, while requiring a much lower computational complexity. Index Terms— Genetic algorithm, Ultra-wideband, Maximum likelihood detection, RAKE receivers.
I. INTRODUCTION Ultra-wideband (UWB) is a promising radio technology for networks delivering extremely high data rates of 110Mbps to 480Mbps at distance of 2m to 10m respectively in an unlicensed frequency spectrum of 3.1GHz to 10.6GHz [1]. UWB systems transmit signals that demonstrate extremely low-power-spectral-density and occupy very wide bandwidths (greater than 25% of their center frequency) [2]. In an impulse-based DS-UWB system, the transmitted data bit is spread over multiple consecutive pulses of very low power density and ultra-short duration. This introduces resolvable multipath components having differential delays in the order of nanoseconds. Thus, the performance of a DS-UWB system is significantly degraded by the inter-chip interference (ICI) and inter-symbol interference (ISI) due to multipath propagation [3]. In a frequency-selective fading channel, a RAKE receiver can be used to exploit multipath diversity by combining constructively the monocycles received from the resolvable paths [4]. Maximum ratio combining (MRC)-RAKE is optimum when the disturbance to the desired signal is sourced only from additive white Gaussian noise (AWGN), therefore it has low computational complexity. However the presence of multipath fading, ISI, and/or narrowband interference (NBI) degrade the system performance severely [5]. The maximum likelihood detection (MLD) is optimal in such a frequency selective channel environment as UWB channel but its computational complexity grows exponentially with the constellation size and the number of RAKE fingers. The genetic algorithm (GA) is of much lower computational complexity compared to the MLD approach. This work was supported by the Commonwealth Scholarship Commission, the University of Liverpool,UK and the University of Ilorin,Nigeria.
The high computational complexity of MLD motivates research into suboptimal receivers with reduced complexity such as linear and non-linear equalizers. In [6], performance of non-linear frequency domain equalization schemes viz. decision feedback equalization (DFE) and iterative DFE for DS-UWB systems were studied. Eslami et al in [7] presented the performance of joint rake and MMSE equalizer receiver for UWB communications systems. The performance of the GA based equalization algorithm at small and moderate number of RAKE fingers is better than the performance of these suboptimal receivers at the same number of RAKE fingers and moderate number of equalizer taps. The comparison of the GA and linear MMSE equalizer is presented in our next paper. GA is a well studied and effective search technique used in lots of work in CDMA communications systems as can be found in [8] and [9] . GA has also been applied to UWB communications systems in [10] and [11]. In [10], a GAbased iterative finger selection scheme which depends on the direct evaluation of the objective function was proposed. UWB pulse design method was carried out in [11] using the GA optimization. However, to the best of our knowledge, no work has been done on using GA for channel equalization in DSUWB communications systems. In this paper, we propose an approach using GA for channel equalization in DS-UWB wireless communications, where GA is combined with a RAKE receiver to combat the ISI due to the frequency selective nature of UWB channels for high data rate transmission. Simulation results show that the proposed GA based structure significantly outperforms the RAKE receiver. It also provides a close bit error rate (BER) performance to the optimal MLD approach, while requiring a much lower computational complexity. The impact of the number of RAKE fingers on the algorithm and the speed of convergence in terms of the BER against the number of generations are also presented. Section II is the system model. The RAKE receiver is in section III. The proposed GA-based channel equalization approach and the computational complexity of the RAKE-GA in comparison to the RAKE-MLD are presented in Section IV. Section V presents the simulation results and section VI draws the conclusion.
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
II. SYSTEM MODEL
III. RAKE RECEIVER
A. Transmit Signal The transmit pulse vTR (t), is generated by using the ternary orthogonal code sequence of the form given by vTR (t) =
N c −1
bi g (t − iTc )
(1)
i=0
where bi is the ith component of the spreading code, Tc is the chip width, g (t) represents the transmitted monocycle waveform which is normalized to have unit energy and Nc is the length of the spreading code sequence. The transmit signal for the DS-UWB can be expressed as x (t) =
Ec
∞
dk vTR (t − kTf )
(2)
k=−∞
where Ec is the energy per transmitted pulse, dk ∈ {±1} is the kth transmit symbol, Tf is the interval of one symbol or frame time and each frame is subdivided into Nc equally spaced chips giving Tf = Nc Tc .
A typical RAKE receiver is composed of several correlators followed by a linear combiner, as shown in Fig. 1. The signal received at the RAKE receiver is correlated with delayed versions of the reference pulse multiplied by the tap weights, the output signals are then combined linearly. The performance of a RAKE receiver depends on the path selection approach and the combining method [4]. MRC, which was used for the finger weight estimation and the selective RAKE (SRAKE) combiner in [13], is employed in this paper. In the path selection part, L fingers select the received signal at the delay times τf l (l = 0, ..., L − 1) corresponding to L strongest paths. Each correlator correlates the received signal with the reference waveform at the delay times τf l , and integrates over symbol duration Tf . We assume perfect channel state information (CSI) is available at the receiver. Also perfect chip synchronization between the transmitter and the receiver is assumed. The correlator’s output for the kth desired symbol is given by ykf l
ˆ =
(k+1)Tf +τf l
kTf +τf l
r (t) vTR (t − kTf − τf l ) dt (5)
(5) in vector notation is expressed as shown in (6) B. Channel Model The UWB channel model derived from the Saleh-Valenzuela model with a couple of slight modifications is used in this paper. A log-normal distribution rather than a Rayleigh distribution for the multipath gain magnitude is used because the log-normal distribution fits the measurement data better. In addition, independent fading is assumed for each cluster as well as each ray within the cluster. Therefore, the multipath model which consists of the discrete time impulse responses can be expressed in a simpler form as h (t) =
L tot −1
hl δ (t − τl )
(6) yk = Es dk h + ik + nk T where yk = ykf 0 , ...ykf L−1 , Es = Nc Ec which is the enT T ergy per symbol, h = [hf 0 , ...hf L−1 ] , ik = ifk 0 , ...ifk L−1
with ifk L denoting ISI of the kth symbol for the lth correT with nfk L being the noise lator and nk = nfk 0 , ...nfk L−1 component of the kth symbol for the lth correlator. L is the number of RAKE fingers. The SRAKE receiver output with MRC technique is expressed as ˜ T yk d˜k = γ
(3)
l=0
(7)
T
where Ltot is the total number of paths, τl (= lTc ) is the delay of the lth path component and hl is the lth path gain [12].
where γ ˜ = [˜ γ0 , ..., γ˜L−1 ] is the finger weights of the RAKE ˆ f l where receiver estimated from the channel taps, γ˜l = h T ˆ ˆ ˆ hf l = hf 0 , ...hf L−1 . The hard estimates in the MRC-RAKE receiver is obtained from the decision device which is given as
C. Receive Signal The receive signal r (t), which is the convolution of the transmit signal in (2) with the discrete time channel impulse responses given in (3) and the addition of noise is shown in (4) as
dˆk = sign(d˜k )
(8)
IV. GENETIC ALGORITHMS IN DS - UWB COMMUNICATIONS SYSTEMS
r (t) = x (t) ∗ h (t) + n (t) L ∞ tot −1 dk vTR hl (t − kTf − τl ) + n (t) = Ec k=−∞
A. Genetic Algorithm (4)
l=0
where n (t) is the additive white Gaussian noise (AWGN) with zero mean and a variance of σ 2 , ∗ denotes the convolution operator.
The GA works on the Darwinian principle of natural selection called "survival of the fittest". GA possesses an intrinsic flexibility and freedom to choose desirable optima according to design specifications. GA presumes that the potential solution of any problem is an individual and can be represented by a set of parameters. These parameters are regarded as the genes of a chromosome and can be structured by a string of values
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
in binary form. A fitness value is used to reflect the degree of goodness of the chromosome for the problem which would be highly related with its objective value. Throughout a genetic evolution, a fitter chromosome has a tendency to yield good quality offspring which means a better solution to any problem. In a practical GA application, a population pool of chromosomes has to be installed and these can be randomly set initially. In each cycle of genetic operation, termed as evolving process, a subsequent generation, is created from the chromosomes in the current population. This can only succeed if a group of these chromosomes, generally called "parents" or a collection term "mating pool" is selected [14] . B. Fitness Function Evaluation The GA searches only a specified number of the possible solutions and refine them through the selection, reproduction, crossover and mutation operations. The GA minimizes the fitness function in terms of the distance measure criteria. The size of the available possible solutions is M, 2M but only (M, P ) is utilized by the GA where P is the population size being refined by the GA operators G times, that is the number of generations such that P × G ≤ 2M . P is spread by the same ternary code sequence of length Nc used in spreading the transmitted signal to obtain sk = (1, M × Nc ) which is then convolved with the discrete time channel impulse ˆ already defined in (7). response, h The correlator’s output expressed in vector notation is given by yGAk =
ˆ Es sk h.
(9)
A good initial guess is critical in GA optimization. The GA performance without RAKE provides a very bad initial guess and so the initial guess of the GA is obtained from the soft estimates of the RAKE receiver output d˜k , from (7). The fitness function for the GA is now expressed as given in 10. J=
P T ˜ γ (yk − yGAk ) .
(10)
k=1
When the whole search space of possible solutions are utilized, we have the MLD scheme. The output, d˜k of the RAKE receiver in (7) is also used as the input to the MLD. The MLD detector searches through all the possible solutions of data bits, M, 2M and the one close in distance to the transmitted data based on the distance measure criteria is chosen. The cost function for the RAKE-MLD is also (10) but with P = 2M thereby making the MLD to spend longer simulation time and be more computationally complex. C. RAKE-GA in DS-UWB The proposed GA based equalization approach is carried out as follows. After evaluating the fitness of individuals within the population (M, P ) specified from M, 2M according to (10), the bitstring population type was used and population sizes of P = 50 and 100 were considered. The number of variables for the fitness function is the number of symbols
per block which is M and it is used in generating the initial population such that (P × G) ≤ 2M where G is the number of generations used as the stopping criteria for the algorithm. • Proportional fitness scaling was used to convert the raw fitness score returned by the objective function to values in a range that is suitable for the selection function. It makes the expectation proportional to the raw fitness scores. This is advantageous when the raw scores are in good range. When the objective values vary a little, all individuals have approximately the same chance of reproduction. • Stochastic selection now chooses parents for the next generation based on their scaled values from the fitness scaling function. It lays out a line in which each parent corresponds to a section of the line of length proportional to its expectation. A certain elites are now chosen which are guaranteed to survive to the next generation. • Scattered crossover combines two parents to form a child for the next generation. It creates a random binary vector, then selects the genes where the vector is a 1 from the first parent, and the genes where the vector is a 0 from the second parent, and combines the genes to form the child. • Gaussian mutation was used and it adds a random number from a Gaussian distribution with mean zero to each vector entry of an individual. The variance of this distribution can be controlled with the scale and shrink parameters. The scale parameter determines the variance at the first generation, that is it controls the standard deviation of the mutation and it is given by scale ∗ (1 − 0), where scale = 0 ∼ 10. The shrink parameter controls how the variance shrinks as generations go by, that is it controls the rate at which the average amount of mutation decreases and the variance at the kth generation k where G is given by vark = vark−1 1 − shrink. G shrink = −1 ∼ 3, G = number of generations. If the shrink parameter is 0, the variance is constant, if the shrink is 1, the variance shrinks to 0 linearly as the last generation is reached and a negative value of shrink causes the variance to grow. • Hybrid function specifies another minimization function that runs after the GA terminates in order to improve the value of the fitness function. It uses the final point from the genetic algorithm as its initial point. The unconstrained minimization function was used in this work. • Stopping criteria determines what causes the algorithm to terminate. Generations G was used in terminating the algorithm . • Vectorize makes the algorithm to run faster if the fitness function is vectorized. The fitness function is called once and it computes the fitness function for all individuals in the current population at once [15]. Fig. 1 is the block diagram of the RAKE-GA for DS-UWB systems. D. Computational Complexity The symbolic computational complexities in terms of complex valued floating point multiplication and addition are car-
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
ykf0
r(t)
³
vTR(t-WWf0)
. . . . . .
J~ f 0
¦
~ dk
GA
dˆ k
fL-1
yk ³
J~fL1
vTR(t-WWfL-1) Fig. 1.
RAKE-GA for DS-UWB TABLE I COMPUTATIONAL COMPLEXITY (M = 10, P = 100, G = 10) Receiver RAKE-GA
RAKE-MLD
L 5 10 15 20 10
Normalized 1 1.2 1.3 1.5 4.2
ried out for the RAKE, RAKE-GA and RAKE-MLD receivers. The order of complexities of the RAKE receiver is O (LM ), it 2 the RAKE-GA is O [GP (LM + logP )] + LM for
and for the RAKE-MLD we have O M 2M (LM + log2) where L is the number of RAKE fingers, M is the number of symbols per block, Nc is the length of the ternary orthogonal code sequence. G is the number of generations and P is the population size for the RAKE-GA only. The computational complexities of the receivers depend on the derivation of the finger weights, the fitness function evaluation and the demodulation of the signal. Table I shows the complexities of the RAKE-GA at L = 10, 15, 20 and the RAKE-MLD at L = 10 which were normalized to the RAKE-GA complexity at L = 5. The RAKE-MLD at L = 10 is more than four times more complex than RAKE-GA at L = 5 and almost four times more complex with the same number of fingers when L = 10. The RAKEMLD is still more complex even at L = 10 than RAKE-GA at L = 15, 20 being more than three times more complex at L = 15 and almost three times more complex at L = 20. The RAKE-GA at L = 10, 15,20 are not even twice as complex as at L = 5. The RAKE-GA is less complex than the RAKEMLD with performance very close. V. SIMULATION RESULTS A. Simulation Setup The simulation for the RAKE, RAKE-GA and RAKEMLD receivers were carried out using BPSK modulation at a transmission rate of Rb = 250Mbps with symbol duration or frame length of Tf = 4ns. Each packet consists of 1000 symbols. A ternary code length of Nc = 24 is used for spreading with the result of the chip width, Tc = 0.167ns. The simulated IEEE 802.15.3a UWB multipath channel model [12] with perfect CSI for a single user scenario is employed for
simulation. The channel model 3 (CM3) which is a non-lineof-sight (NLOS) environment with a distance of 4 ∼ 10m, mean excess delay of 14.18ns and RMS delay spread of 14.28ns is considered in this work. The number of RAKE fingers used are L = 5, 10, 15, 20. For the proposed RAKE-GA approach, the population size P = 50 and 100 while the number of generations is G = 1 ∼ 20. The proportional scaling is employed for the scaling of the fitness values before selection. The crossover of 0.85 is used with elite count of 0.05. The Gaussian mutation values are shrink = 1.0 and scale = 0.75. In addition, the unconstrained minimization hybrid function was employed to improve the value of the fitness function. B. Performance Evaluation Fig. 2 shows the RAKE-GA at a population of P = 100 and generation of G = 10 compared with the RAKE-MLD and RAKE receivers all at L = 10. The RAKE receiver encountered error floor because the number of RAKE fingers was too small for it to capture a large signal energy and also RAKE receiver with MRC weight estimation cannot remove ISI. The proposed RAKE-GA and RAKE-MLD were able to remove the ISI symbol by symbol using the distance measure criteria since the RAKE soft estimates output was used as the input to the two receivers. The RAKE, RAKE-GA and RAKE-MLD were of the same BER at very low SNR values. The RAKE-GA and RAKE-MLD performed better than the RAKE receiver at the same RAKE fingers at moderate to high SNR values though at higher complexities. The RAKE-MLD was of a bit lower BER at moderate to high SNR values and the difference in performance was highly insignificant. Fig. 3 is a plot of BER against SNR for RAKE-GA at P = 100, G = 10 at values of L = 5, 10, 15, 20. This shows the impact of the number of RAKE fingers on the performance of the scheme. This BER performance improvement is as a result of increase in the number of RAKE fingers. The RAKE-GA at L = 5 was of higher BER to the system when L = 10, 15, 20 where they were almost of the same BER at all SNR values. Fig 4 shows the impact of the number of generations in the BER performance, where G = 1 ∼ 20 for P = 100 and G = 2 ∼ 20 for P = 50 both at L = 10 to show the speed of convergence of the algorithm. The algorithm at G = 1 ∼ 10 for P = 100 gave better BER generally than at G = 2 ∼ 20 for P = 50. However, at the same values of P G for the two scenarios, it was discovered that the case with lower P value gives a lower BER but at a little higher complexity. For example at P G = 500 for the two cases, the BER was 3.2 × 10−3 at G = 5 and P = 100 with a little lower complexity than at G = 10 and P = 50 having a BER of 1.98×10−4 both with L = 10. It can thus be concluded that the GA with a relatively large population size achieves a lower steady state BER than the case with only half the population size at a cost of slightly more generations. The decrease in BER was gradual from G = 1 ∼ 8 until when G = 8 and there was no significant difference in the BER up to when G = 10 for P = 100. When P = 50, the BER decreases gradually from G = 2 ∼ 8 until when G = 10 ∼ 20 when
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
there was no difference in BER values anymore The RAKEGA at P = 50 is of a little lower complexity than the one with P = 100 both at the same values of G and L.
RAKE,RAKE−GA and RAKE−MLD for CM3 with L=10
0
10
RAKE RAKE−GA,P=100,G=10 RAKE−MLD
−1
10
VI. CONCLUSION
−2
10
BER
We have proposed a GA based channel equalization scheme in DS-UWB wireless communications using proportional scaling, Gaussian mutation and hybrid function. Our simulation results have shown that the proposed GA based scheme significantly outperforms the RAKE receiver and also gives a very close BER performance to the optimal MLD approach at a much lower computational complexity. With a moderate number of RAKE fingers, RAKE-GA can achieve a good BER performance. A further increase in the number of RAKE fingers has little effect on the performance. GA with a relatively large population size achieves a lower steady state BER than the case with only half the population size at a cost of slightly more generations.
−3
10
−4
10
−5
10
−6
10
Fig. 2.
0
5
10
15 Eb/No(dB)
20
25
30
BER vs. SNR for all receivers
R EFERENCES RAKE−GA for CM3 with P=100 and G=10
0
10
RAKE−GA,L=5 RAKE−GA,L=10 RAKE−GA,L=15 RAKE−GA,L=20
−1
10
−2
BER
10
−3
10
−4
10
−5
10
−6
10
Fig. 3.
0
5
10
15 Eb/No(dB)
20
25
30
BER vs. SNR for RAKE-GA
RAKE−GA for CM3 with L=10,SNR=20dB
−1
10
P=50 P=100
−2
10
BER
[1] V. S. Somayazulu, J. R. Foerster, and S. Roy, “Design challenges for very high data rate UWB systems,” in Conf Rec Asilomar Conf Signals Syst Comput, Nov. 2002, pp. 717–721. [2] C. R. Nassar, F. Zhu, and Z. Wu, “Direct sequence spreading UWB systems: Frequency domain processing for enhanced performance and throughput,” in IEEE International Conference on Communications, May 2003, pp. 2180–2186. [3] R. Liu and J. Elmirghani, “Performance of impulse radio direct sequence ultra-wideband system with variable-length spreading sequences,” IET Commun., vol. 1, no. 4, pp. 597–603, Aug. 2007. [4] W. Siriwongpairat and K. Liu, Ultra-Wideband Communications Systems Multiband OFDM APPROACH. John Wiley & sons,Inc., 2008. [5] H. Sato and T. Ohtsuki, “Computational complexity and performance of RAKE receivers with channel estimation for DS-UWB,” IEICE Trans Fund Electron Commun Comput Sci, vol. 88, no. 9, pp. 2318–2326, Sept. 2005. [6] P. Kaligineedi and V. K. Bhargava, “Frequency-domain equalization techniques for ds-uwb systems,” in Global Telecommunications Conference, Nov. 2006, pp. 1–6. [7] M. Eslami and X. Dong, “Rake-mmse-equalizer performance for UWB,” IEEE Communications Letters, vol. 9, no. 6, pp. 502–504, June 2005. [8] C.Erguin and K.Hacioglu, “Multiuser detection using a genetic algorithm in CDMA communications systems,” IEEE Trans. Commun, vol. 48, no. 8, pp. 1374–1383, Aug. 2000. [9] K.Yen and L.Hanzo, “Genetic algorithm assisted joint multiuser symbol detection and fading channel estimation for synchronous CDMA systems,” IEEE Journal on Selected Areas of Communications, vol. 19, no. 6, pp. 985–998, June 2001. [10] S. Gezici, M. Chiang, H. V. Poor, and H. Kobayashi, “A genetic algorithm based finger selection scheme for UWB MMSE rake receivers,” in IEEE Int. Conf. Ultra-Wideband Conf. Proc., Sept. 2005, pp. 164–169. [11] M. Wang, S. Yang, and S. Wu, “A GA-based UWB pulse waveform design method,” Digital Signal Process Rev J, vol. 18, no. 1, pp. 65–74, Jan. 2008. [12] J. Foerster, “Channel modelling sub-committee report final,” IEEE P802.15-02/490r1-SG3a, Tech. Rep., Feb. 2003. [13] D. Cassioli, M. Z. Win, F. Vatalaro, and A. F. Molisch, “Performance of low-complexity rake reception in a realistic UWB channel,” in IEEE Int Conf Commun, May 2002, pp. 763–767. [14] K. Man, K. Tang, and S. Kwong, Genetic algorithms: concepts and designs. Spriger-Verlag London Limited, 1999. [15] MATLAB, “Genetic algorithm and direct search toolbox 2 user’s guide,” The Mathworks,Inc, Tech. Rep., 2007.
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Fig. 4.
0
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10 No of Generations
Convergence speed of RAKE-GA
15
20
Genetic Algorithm based Equalization for Direct Sequence Ultra-Wideband Communications Systems Nazmat Surajudeen-Bakinde, Xu Zhu, Jingbo Gao and Asoke K. Nandi Department of Electrical Engineering and Electronics University of Liverpool, Liverpool, UK. {nazmat.surajudeen-bakinde, xuzhu, jgao and a.nandi}@liverpool.ac.uk
Abstract— We propose a genetic algorithm (GA) based equalization approach for direct sequence Ultra-wideband (DS-UWB) wireless communications, where GA is combined with a RAKE receiver to combat the inter-symbol interference (ISI) due to the frequency selective nature of UWB channels for high data rate transmission. Simulation results show that the proposed GA based structure significantly outperforms the RAKE receiver. It also provides a close bit error rate (BER) performance to the optimal maximum likelihood detection (MLD) approach, while requiring a much lower computational complexity. Index Terms— Genetic algorithm, Ultra-wideband, Maximum likelihood detection, RAKE receivers.
I. INTRODUCTION Ultra-wideband (UWB) is a promising radio technology for networks delivering extremely high data rates of 110Mbps to 480Mbps at distance of 2m to 10m respectively in an unlicensed frequency spectrum of 3.1GHz to 10.6GHz [1]. UWB systems transmit signals that demonstrate extremely low-power-spectral-density and occupy very wide bandwidths (greater than 25% of their center frequency) [2]. In an impulse-based DS-UWB system, the transmitted data bit is spread over multiple consecutive pulses of very low power density and ultra-short duration. This introduces resolvable multipath components having differential delays in the order of nanoseconds. Thus, the performance of a DS-UWB system is significantly degraded by the inter-chip interference (ICI) and inter-symbol interference (ISI) due to multipath propagation [3]. In a frequency-selective fading channel, a RAKE receiver can be used to exploit multipath diversity by combining constructively the monocycles received from the resolvable paths [4]. Maximum ratio combining (MRC)-RAKE is optimum when the disturbance to the desired signal is sourced only from additive white Gaussian noise (AWGN), therefore it has low computational complexity. However the presence of multipath fading, ISI, and/or narrowband interference (NBI) degrade the system performance severely [5]. The maximum likelihood detection (MLD) is optimal in such a frequency selective channel environment as UWB channel but its computational complexity grows exponentially with the constellation size and the number of RAKE fingers. The genetic algorithm (GA) is of much lower computational complexity compared to the MLD approach. This work was supported by the Commonwealth Scholarship Commission, the University of Liverpool,UK and the University of Ilorin,Nigeria.
The high computational complexity of MLD motivates research into suboptimal receivers with reduced complexity such as linear and non-linear equalizers. In [6], performance of non-linear frequency domain equalization schemes viz. decision feedback equalization (DFE) and iterative DFE for DS-UWB systems were studied. Eslami et al in [7] presented the performance of joint rake and MMSE equalizer receiver for UWB communications systems. The performance of the GA based equalization algorithm at small and moderate number of RAKE fingers is better than the performance of these suboptimal receivers at the same number of RAKE fingers and moderate number of equalizer taps. The comparison of the GA and linear MMSE equalizer is presented in our next paper. GA is a well studied and effective search technique used in lots of work in CDMA communications systems as can be found in [8] and [9] . GA has also been applied to UWB communications systems in [10] and [11]. In [10], a GAbased iterative finger selection scheme which depends on the direct evaluation of the objective function was proposed. UWB pulse design method was carried out in [11] using the GA optimization. However, to the best of our knowledge, no work has been done on using GA for channel equalization in DSUWB communications systems. In this paper, we propose an approach using GA for channel equalization in DS-UWB wireless communications, where GA is combined with a RAKE receiver to combat the ISI due to the frequency selective nature of UWB channels for high data rate transmission. Simulation results show that the proposed GA based structure significantly outperforms the RAKE receiver. It also provides a close bit error rate (BER) performance to the optimal MLD approach, while requiring a much lower computational complexity. The impact of the number of RAKE fingers on the algorithm and the speed of convergence in terms of the BER against the number of generations are also presented. Section II is the system model. The RAKE receiver is in section III. The proposed GA-based channel equalization approach and the computational complexity of the RAKE-GA in comparison to the RAKE-MLD are presented in Section IV. Section V presents the simulation results and section VI draws the conclusion.
978-1-4244-2948-6/09/$25.00 ©2009 Crown
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
II. SYSTEM MODEL
III. RAKE RECEIVER
A. Transmit Signal The transmit pulse vTR (t), is generated by using the ternary orthogonal code sequence of the form given by vTR (t) =
N c −1
bi g (t − iTc )
(1)
i=0
where bi is the ith component of the spreading code, Tc is the chip width, g (t) represents the transmitted monocycle waveform which is normalized to have unit energy and Nc is the length of the spreading code sequence. The transmit signal for the DS-UWB can be expressed as x (t) =
Ec
∞
dk vTR (t − kTf )
(2)
k=−∞
where Ec is the energy per transmitted pulse, dk ∈ {±1} is the kth transmit symbol, Tf is the interval of one symbol or frame time and each frame is subdivided into Nc equally spaced chips giving Tf = Nc Tc .
A typical RAKE receiver is composed of several correlators followed by a linear combiner, as shown in Fig. 1. The signal received at the RAKE receiver is correlated with delayed versions of the reference pulse multiplied by the tap weights, the output signals are then combined linearly. The performance of a RAKE receiver depends on the path selection approach and the combining method [4]. MRC, which was used for the finger weight estimation and the selective RAKE (SRAKE) combiner in [13], is employed in this paper. In the path selection part, L fingers select the received signal at the delay times τf l (l = 0, ..., L − 1) corresponding to L strongest paths. Each correlator correlates the received signal with the reference waveform at the delay times τf l , and integrates over symbol duration Tf . We assume perfect channel state information (CSI) is available at the receiver. Also perfect chip synchronization between the transmitter and the receiver is assumed. The correlator’s output for the kth desired symbol is given by ykf l
ˆ =
(k+1)Tf +τf l
kTf +τf l
r (t) vTR (t − kTf − τf l ) dt (5)
(5) in vector notation is expressed as shown in (6) B. Channel Model The UWB channel model derived from the Saleh-Valenzuela model with a couple of slight modifications is used in this paper. A log-normal distribution rather than a Rayleigh distribution for the multipath gain magnitude is used because the log-normal distribution fits the measurement data better. In addition, independent fading is assumed for each cluster as well as each ray within the cluster. Therefore, the multipath model which consists of the discrete time impulse responses can be expressed in a simpler form as h (t) =
L tot −1
hl δ (t − τl )
(6) yk = Es dk h + ik + nk T where yk = ykf 0 , ...ykf L−1 , Es = Nc Ec which is the enT T ergy per symbol, h = [hf 0 , ...hf L−1 ] , ik = ifk 0 , ...ifk L−1
with ifk L denoting ISI of the kth symbol for the lth correT with nfk L being the noise lator and nk = nfk 0 , ...nfk L−1 component of the kth symbol for the lth correlator. L is the number of RAKE fingers. The SRAKE receiver output with MRC technique is expressed as ˜ T yk d˜k = γ
(3)
l=0
(7)
T
where Ltot is the total number of paths, τl (= lTc ) is the delay of the lth path component and hl is the lth path gain [12].
where γ ˜ = [˜ γ0 , ..., γ˜L−1 ] is the finger weights of the RAKE ˆ f l where receiver estimated from the channel taps, γ˜l = h T ˆ ˆ ˆ hf l = hf 0 , ...hf L−1 . The hard estimates in the MRC-RAKE receiver is obtained from the decision device which is given as
C. Receive Signal The receive signal r (t), which is the convolution of the transmit signal in (2) with the discrete time channel impulse responses given in (3) and the addition of noise is shown in (4) as
dˆk = sign(d˜k )
(8)
IV. GENETIC ALGORITHMS IN DS - UWB COMMUNICATIONS SYSTEMS
r (t) = x (t) ∗ h (t) + n (t) L ∞ tot −1 dk vTR hl (t − kTf − τl ) + n (t) = Ec k=−∞
A. Genetic Algorithm (4)
l=0
where n (t) is the additive white Gaussian noise (AWGN) with zero mean and a variance of σ 2 , ∗ denotes the convolution operator.
The GA works on the Darwinian principle of natural selection called "survival of the fittest". GA possesses an intrinsic flexibility and freedom to choose desirable optima according to design specifications. GA presumes that the potential solution of any problem is an individual and can be represented by a set of parameters. These parameters are regarded as the genes of a chromosome and can be structured by a string of values
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
in binary form. A fitness value is used to reflect the degree of goodness of the chromosome for the problem which would be highly related with its objective value. Throughout a genetic evolution, a fitter chromosome has a tendency to yield good quality offspring which means a better solution to any problem. In a practical GA application, a population pool of chromosomes has to be installed and these can be randomly set initially. In each cycle of genetic operation, termed as evolving process, a subsequent generation, is created from the chromosomes in the current population. This can only succeed if a group of these chromosomes, generally called "parents" or a collection term "mating pool" is selected [14] . B. Fitness Function Evaluation The GA searches only a specified number of the possible solutions and refine them through the selection, reproduction, crossover and mutation operations. The GA minimizes the fitness function in terms of the distance measure criteria. The size of the available possible solutions is M, 2M but only (M, P ) is utilized by the GA where P is the population size being refined by the GA operators G times, that is the number of generations such that P × G ≤ 2M . P is spread by the same ternary code sequence of length Nc used in spreading the transmitted signal to obtain sk = (1, M × Nc ) which is then convolved with the discrete time channel impulse ˆ already defined in (7). response, h The correlator’s output expressed in vector notation is given by yGAk =
ˆ Es sk h.
(9)
A good initial guess is critical in GA optimization. The GA performance without RAKE provides a very bad initial guess and so the initial guess of the GA is obtained from the soft estimates of the RAKE receiver output d˜k , from (7). The fitness function for the GA is now expressed as given in 10. J=
P T ˜ γ (yk − yGAk ) .
(10)
k=1
When the whole search space of possible solutions are utilized, we have the MLD scheme. The output, d˜k of the RAKE receiver in (7) is also used as the input to the MLD. The MLD detector searches through all the possible solutions of data bits, M, 2M and the one close in distance to the transmitted data based on the distance measure criteria is chosen. The cost function for the RAKE-MLD is also (10) but with P = 2M thereby making the MLD to spend longer simulation time and be more computationally complex. C. RAKE-GA in DS-UWB The proposed GA based equalization approach is carried out as follows. After evaluating the fitness of individuals within the population (M, P ) specified from M, 2M according to (10), the bitstring population type was used and population sizes of P = 50 and 100 were considered. The number of variables for the fitness function is the number of symbols
per block which is M and it is used in generating the initial population such that (P × G) ≤ 2M where G is the number of generations used as the stopping criteria for the algorithm. • Proportional fitness scaling was used to convert the raw fitness score returned by the objective function to values in a range that is suitable for the selection function. It makes the expectation proportional to the raw fitness scores. This is advantageous when the raw scores are in good range. When the objective values vary a little, all individuals have approximately the same chance of reproduction. • Stochastic selection now chooses parents for the next generation based on their scaled values from the fitness scaling function. It lays out a line in which each parent corresponds to a section of the line of length proportional to its expectation. A certain elites are now chosen which are guaranteed to survive to the next generation. • Scattered crossover combines two parents to form a child for the next generation. It creates a random binary vector, then selects the genes where the vector is a 1 from the first parent, and the genes where the vector is a 0 from the second parent, and combines the genes to form the child. • Gaussian mutation was used and it adds a random number from a Gaussian distribution with mean zero to each vector entry of an individual. The variance of this distribution can be controlled with the scale and shrink parameters. The scale parameter determines the variance at the first generation, that is it controls the standard deviation of the mutation and it is given by scale ∗ (1 − 0), where scale = 0 ∼ 10. The shrink parameter controls how the variance shrinks as generations go by, that is it controls the rate at which the average amount of mutation decreases and the variance at the kth generation k where G is given by vark = vark−1 1 − shrink. G shrink = −1 ∼ 3, G = number of generations. If the shrink parameter is 0, the variance is constant, if the shrink is 1, the variance shrinks to 0 linearly as the last generation is reached and a negative value of shrink causes the variance to grow. • Hybrid function specifies another minimization function that runs after the GA terminates in order to improve the value of the fitness function. It uses the final point from the genetic algorithm as its initial point. The unconstrained minimization function was used in this work. • Stopping criteria determines what causes the algorithm to terminate. Generations G was used in terminating the algorithm . • Vectorize makes the algorithm to run faster if the fitness function is vectorized. The fitness function is called once and it computes the fitness function for all individuals in the current population at once [15]. Fig. 1 is the block diagram of the RAKE-GA for DS-UWB systems. D. Computational Complexity The symbolic computational complexities in terms of complex valued floating point multiplication and addition are car-
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
ykf0
r(t)
³
vTR(t-WWf0)
. . . . . .
J~ f 0
¦
~ dk
GA
dˆ k
fL-1
yk ³
J~fL1
vTR(t-WWfL-1) Fig. 1.
RAKE-GA for DS-UWB TABLE I COMPUTATIONAL COMPLEXITY (M = 10, P = 100, G = 10) Receiver RAKE-GA
RAKE-MLD
L 5 10 15 20 10
Normalized 1 1.2 1.3 1.5 4.2
ried out for the RAKE, RAKE-GA and RAKE-MLD receivers. The order of complexities of the RAKE receiver is O (LM ), it 2 the RAKE-GA is O [GP (LM + logP )] + LM for
and for the RAKE-MLD we have O M 2M (LM + log2) where L is the number of RAKE fingers, M is the number of symbols per block, Nc is the length of the ternary orthogonal code sequence. G is the number of generations and P is the population size for the RAKE-GA only. The computational complexities of the receivers depend on the derivation of the finger weights, the fitness function evaluation and the demodulation of the signal. Table I shows the complexities of the RAKE-GA at L = 10, 15, 20 and the RAKE-MLD at L = 10 which were normalized to the RAKE-GA complexity at L = 5. The RAKE-MLD at L = 10 is more than four times more complex than RAKE-GA at L = 5 and almost four times more complex with the same number of fingers when L = 10. The RAKEMLD is still more complex even at L = 10 than RAKE-GA at L = 15, 20 being more than three times more complex at L = 15 and almost three times more complex at L = 20. The RAKE-GA at L = 10, 15,20 are not even twice as complex as at L = 5. The RAKE-GA is less complex than the RAKEMLD with performance very close. V. SIMULATION RESULTS A. Simulation Setup The simulation for the RAKE, RAKE-GA and RAKEMLD receivers were carried out using BPSK modulation at a transmission rate of Rb = 250Mbps with symbol duration or frame length of Tf = 4ns. Each packet consists of 1000 symbols. A ternary code length of Nc = 24 is used for spreading with the result of the chip width, Tc = 0.167ns. The simulated IEEE 802.15.3a UWB multipath channel model [12] with perfect CSI for a single user scenario is employed for
simulation. The channel model 3 (CM3) which is a non-lineof-sight (NLOS) environment with a distance of 4 ∼ 10m, mean excess delay of 14.18ns and RMS delay spread of 14.28ns is considered in this work. The number of RAKE fingers used are L = 5, 10, 15, 20. For the proposed RAKE-GA approach, the population size P = 50 and 100 while the number of generations is G = 1 ∼ 20. The proportional scaling is employed for the scaling of the fitness values before selection. The crossover of 0.85 is used with elite count of 0.05. The Gaussian mutation values are shrink = 1.0 and scale = 0.75. In addition, the unconstrained minimization hybrid function was employed to improve the value of the fitness function. B. Performance Evaluation Fig. 2 shows the RAKE-GA at a population of P = 100 and generation of G = 10 compared with the RAKE-MLD and RAKE receivers all at L = 10. The RAKE receiver encountered error floor because the number of RAKE fingers was too small for it to capture a large signal energy and also RAKE receiver with MRC weight estimation cannot remove ISI. The proposed RAKE-GA and RAKE-MLD were able to remove the ISI symbol by symbol using the distance measure criteria since the RAKE soft estimates output was used as the input to the two receivers. The RAKE, RAKE-GA and RAKE-MLD were of the same BER at very low SNR values. The RAKE-GA and RAKE-MLD performed better than the RAKE receiver at the same RAKE fingers at moderate to high SNR values though at higher complexities. The RAKE-MLD was of a bit lower BER at moderate to high SNR values and the difference in performance was highly insignificant. Fig. 3 is a plot of BER against SNR for RAKE-GA at P = 100, G = 10 at values of L = 5, 10, 15, 20. This shows the impact of the number of RAKE fingers on the performance of the scheme. This BER performance improvement is as a result of increase in the number of RAKE fingers. The RAKE-GA at L = 5 was of higher BER to the system when L = 10, 15, 20 where they were almost of the same BER at all SNR values. Fig 4 shows the impact of the number of generations in the BER performance, where G = 1 ∼ 20 for P = 100 and G = 2 ∼ 20 for P = 50 both at L = 10 to show the speed of convergence of the algorithm. The algorithm at G = 1 ∼ 10 for P = 100 gave better BER generally than at G = 2 ∼ 20 for P = 50. However, at the same values of P G for the two scenarios, it was discovered that the case with lower P value gives a lower BER but at a little higher complexity. For example at P G = 500 for the two cases, the BER was 3.2 × 10−3 at G = 5 and P = 100 with a little lower complexity than at G = 10 and P = 50 having a BER of 1.98×10−4 both with L = 10. It can thus be concluded that the GA with a relatively large population size achieves a lower steady state BER than the case with only half the population size at a cost of slightly more generations. The decrease in BER was gradual from G = 1 ∼ 8 until when G = 8 and there was no significant difference in the BER up to when G = 10 for P = 100. When P = 50, the BER decreases gradually from G = 2 ∼ 8 until when G = 10 ∼ 20 when
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
there was no difference in BER values anymore The RAKEGA at P = 50 is of a little lower complexity than the one with P = 100 both at the same values of G and L.
RAKE,RAKE−GA and RAKE−MLD for CM3 with L=10
0
10
RAKE RAKE−GA,P=100,G=10 RAKE−MLD
−1
10
VI. CONCLUSION
−2
10
BER
We have proposed a GA based channel equalization scheme in DS-UWB wireless communications using proportional scaling, Gaussian mutation and hybrid function. Our simulation results have shown that the proposed GA based scheme significantly outperforms the RAKE receiver and also gives a very close BER performance to the optimal MLD approach at a much lower computational complexity. With a moderate number of RAKE fingers, RAKE-GA can achieve a good BER performance. A further increase in the number of RAKE fingers has little effect on the performance. GA with a relatively large population size achieves a lower steady state BER than the case with only half the population size at a cost of slightly more generations.
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Fig. 2.
0
5
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15 Eb/No(dB)
20
25
30
BER vs. SNR for all receivers
R EFERENCES RAKE−GA for CM3 with P=100 and G=10
0
10
RAKE−GA,L=5 RAKE−GA,L=10 RAKE−GA,L=15 RAKE−GA,L=20
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BER
10
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Fig. 3.
0
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15 Eb/No(dB)
20
25
30
BER vs. SNR for RAKE-GA
RAKE−GA for CM3 with L=10,SNR=20dB
−1
10
P=50 P=100
−2
10
BER
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Fig. 4.
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Convergence speed of RAKE-GA
15
20
