Wavelet-based multicarrier CDMA system and its corresponding multi-user detection. A. Muayyadi and M.A. Abu-Rgheff. Abstract: A novel digital transmission ...
Wavelet-based multicarrier CDMA system and its corresponding multi-user detection A. Muayyadi and M.A. Abu-Rgheff Abstract: A novel digital transmission system based on wavelet signalling has been proposed and analysed. The basic system consists of a bank of filters derived using the quadrature constrained least square (QCLS) algorithm and satisfying the perfect reconstructed quadrature mirror filter (PR-QMF) theory. A multicarrier CDMA system is constructed using this technique, where a user’s data is spread in the frequency domain and constraints are imposed to ensure channel orthogonality. The system exploits the localisation properties of the wavelets in time and frequency to develop data channels that have compact PSD characteristics and are resilient in an environment exposed to fading. Practical implementation of the wavelet transmission system using a bank of polyphase filters is considered. Simulation results are presented which show the wavelet system can better combat ISI and multiple access interference than an FFT-based multicarrier modulation system when both systems operate in identical fading environments. A wavelet detection scheme based on the MMSE criterion and then made adaptive using the LMS algorithm is theoretically analysed and software simulated. The proposed wavelet detector is near–far resistant and shows better BER performance compared with a decorrelator, but with higher computation costs.
1
Introduction
Wireless connectivity of mobile devices and notebook computers is the hallmark feature of the next generation of network infrastructure. A major challenge to the traffic capacity of such systems is the complex characteristics of the mobile channel encompassing multipath propagation, ‘near–far’ effects, and multiple access interference. Significant research effort has been directed at the design of vital building blocks of mobile wireless networks. The main purpose of this effort is to develop spectrum-efficient modulation, and to achieve low cost, reduced complexity receivers and their signal processing solutions. An attractive transmission scheme that is capable of providing such features is a combination of (code-division multiple access, CDMA) and multicarrier (orthogonal frequency-division multiplexing, OFDM) modulation [1, 2]. Conventionally, multicarrier modulation is generated using fast Fourier transform (FFT) devices but the subcarrier orthogonality and synchronisation are very sensitive to signal phase offset and frequency errors. The FFT processing generates many sidelobes, and the first sidelobe is only 13 dB lower than the main lobe component, so that any destruction of orthogonality generates a high level of intercarrier interference. Wavelet pulses have discrete time and frequency representations in contrast to rectangular pulses, which are discrete in time but ideally have unlimited response in the frequency domain. The use of wavelet signalling was proposed in 1994 [3] such that data symbols linearly r IEE, 2003 IEE Proceedings online no. 20030715 doi:10.1049/ip-com:20030715 Paper first received 14th November 2002 and in revised form 21st May 2003 The authors are with the Mobile Communication Networks Research Group, Department of Communication & Electronic Engineering, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK IEE Proc.-Commun., Vol. 150, No. 6, December 2003
modulate the envelope of the scaling and wavelet functions. The authors in [3] argued that wavelet signalling fulfils Nyquist’s first criterion on ISI removal. Multicarrier modulation using wavelet pulses can be implemented in three ways: by using a dyadic wavelet transform [4, 5] that applies a two-band discrete wavelet transform to produce logarithmic band division; by the wavelet packet [4, 5] generating a multiband transmission system using the full dyadic tree; and by the M-band wavelets [6] scheme dividing the available spectrum into M equal bands. The M-band wavelets system is simple and flexible with a higher degree of freedom compared with the other two types of wavelet systems [4]. In the M-band wavelets system, it is possible that only one low-pass prototype filter with the desired characteristics (stop-band, ripple, etc.) is designed and its frequency response is shifted by modulation with a cosine function to generate the higher subbands. The M-band transmission system, used in this project, comprises a filter bank of perfect reconstructed quadrature mirror filter (PR-QMF) [7–9] and the prototype filter is derived using a quadratic constrained least square (QCLS) algorithm [4, 10, 11]. In this paper, we extend our work presented in [12] to include the theoretical analysis of the wavelet transmission system, the group detection based on the adaptive minimum mean square error (MMSE) receiver [2], and frequency diversity techniques. 2
Wavelet transmission system
The transmission system comprises M branches, each consisting of an up-sampler followed by a synthesis filter whose impulse response, derived from the wavelet function’s orthogonality conditions, generates a specific wavelet pulse. The block diagram of the baseband wavelet processing is shown in Fig. 1. The transmitted signal is generated as follows. A single data symbol is copied into the M branches. In the mth 445
X
M
F0 (z)
G0(z)
M
X
M
F1 (z)
G1(z)
M
X
ck,0 X
ak(n)
0ck,0 Σ
ck,1
X
sk(n)
M
GM −1(z)
X M −1 ck,M −1
ck,M −1
Fig. 1
Wavelet multicarrier CDMA transmission system
branch, the symbol is multiplied by the mth chip, ck[m], upsampled by M factor and then filtered using synthesis filter Fm(z). The transmitted signal of the kth user, sk (z), is given by: sk ðzÞ ¼ FðzÞ C k ak ðzM Þ
ð1Þ
where F(z) ¼ [F0(z)F1(z)yFM1(z)] is a row vector representing the transmitter synthesis filters in which Fm(z) is the wavelet generator filter of the mth branch, Ck is the user’s signature code which is the kth column vector of the code matrix: 2 3 c1; 0 cK1; 0 c0; 0 6 c0; 1 c1; 1 cK1; 1 7 6 7 ð2Þ C ¼6 . 7 4 .. 5 c0; M1
c1; M1
cK1; M1
rk ðzÞ ¼ H k ðzÞsk ðzÞ þ nðzÞ
ð3Þ
where Hk(z) is the channel carrying the kth user signal and the term n(z) is the AWGN with a two-sided power spectral density of N0/2. Wavelet multi-user detection
3.1 Theoretical analysis of the wavelet detection system in a multi-user channel Consider K active users (KrM) sharing the AWGN channel. The received signal, r(z), is given by: rðzÞ ¼
K 1 X
H k ðzÞsk ðzÞ þ nðzÞ
ð4Þ
k¼0
where Hk(z) is the kth column vector of channel matrix H, and each element of the vector represents the channel amplitude scaling for each subband in use. Let the inner product of the spreading sequence and the channel amplitude scaling be Ek where: Ek ¼ C k H k
ð5Þ
is the kth column vector of the M K matrix E, and Ek also represents the effective spreading sequence of the kth user’s signal at the output of the channel. Let bk be the kth row of 446
the K M matrix b where bk,m ¼ ak,mck,m is the subband gain factor (which depends on the combining strategy employed) multiplied by the code chip. We have shown in the Appendix (Section 10.1) that the decision variables for all K users, y(z), are given by: yðzÞ ¼ bEaðzÞ þ nðzÞ
ð6Þ
T
where a(z) ¼ [a0(z) a1(z) a2(z) y aK1(z)] , n(z) is the filtered Gaussian noise, and y(z) ¼ [y0(z) y1(z) y2(z) y yK1 (z)]T. The signal, y(z), can be split up into three terms, identified as the desired signal that is given by Diag/bES, the multiple access interference (MAI) that is given by [bEDiag/bES], and the channel noise n. Consequently, we have: y ¼ DiaghbEia þ ½bE DiaghbEi a |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} desired signal
and ak is the kth user’s input symbol. Throughout the paper, bold characters are used to indicate vector and matrix variables. The basic wavelet receiver, shown in Fig. 1, consists of M parallel branches; each consists of an analysis filter followed by a down-sampler. Each of the analysis filters is a matched counterpart of the corresponding synthesis filter. Furthermore, the two filter banks must satisfy the biorthogonality constraint [13]. The receiver filter bank is G(z) ¼ [G0(z)G1(z)yGM1(z)]T. Consider the transmission through the additive white Gaussian noise (AWGN) channel. The kth user received signal, rk(z), is given by:
3
y(n)
1ck,1 Σ
channel
FM −1(z)
M
r(n)
þ
MAI signal
n |{z}
ð7Þ
filtered noise
It can easily be shown that for a single-user channel, the MAI term becomes zero.
3.2 Optimisation of the wavelet multi-user detection algorithm The detection scheme described in the previous section optimises the system performance in the AWGN channel. However, as the user roams within the wireless network, the mobility generates multipath transmission that disturbs the channel’s orthogonality causing degradation in the system’s performance. We propose a two-stage optimisation technique, which is described in this section. The first stage is based on the fact that signal energy is spread in the frequency domain, so a frequency diversity scheme (minimum mean square error (MMSE) combining technique) is used to combat the fading effects of the mobile channel. The second stage uses a MMSE multi-user detector to eliminate the MAI interference. Let the received signal at the output of the downsamplers be xk(z), where xk ðzÞ ¼ ½xk;0 xk;1 xk;M1 T The minimum mean square error criterion chooses the M 1 vector a which minimises: 2 Eð½ak aH k xk Þ
ð8Þ
where the superscript H denotes hermitian transposition. The estimated data signal is given as: ^ak ¼ sgnðaH k xk Þ
ð9Þ
Using the Wiener–Hopf equation in the adaptive filter theory [1], we have shown in the Appendix (Section 10.2) that: aopt ¼ ½EðHH H Þ þ r2 I 1 H k
ð10Þ
IEE Proc.-Commun., Vol. 150, No. 6, December 2003
Equation (10) gives the optimum coefficients aopt that minimise the mean square error between the estimated data signal and the desired data signal and it is similar to the expression given in [2] when the same assumptions are considered. So far the received signal is processed using frequency diversity and MMSE combining to combat variations in the kth user channel. The output signal is still distorted by the multiple access interference but the distortion can be minimised using an MMSE multi-user detection (MUD) filter. Applying the MMSE criterion again in the wavelet multiuser detection, the problem is then simplified to choosing the K K matrix x that minimises Eð½a xH y 2 Þ
4 Practical implementation of the wavelet signalling system In practice, using the identity in [4], the up-samplers, synthesis filters and summing operator can be replaced by polyphase filters plus a multiplexer. The block diagram of a practical wavelet baseband transmitter is shown in Fig. 3. The practical receiver consists of a bank of polyphase filters to replace the analysis filters and the down-samplers as shown in Fig. 4.
1
X ak(n)
ck,1
The optimum filter coefficients xopt are given in the Appendix (Section 10.3). Now let c ¼ bE
xopt ¼ ½cc þ r b c
PM −1(z)
ck,M−1
Fig. 3 2 2 1
Wavelet multicarrier CDMA transmitter (implementation)
ð14Þ
Equation (14) gives the filter coefficients that optimise the performance of the wavelet group detection. The optimised receiver is shown in Fig. 4 below.
x0(n)
P 0(z)
0
P 1(z)
1
r(n)
xðn þ 1Þ ¼ xðnÞ þ myðnÞ½aH ðnÞ yH ðnÞxðnÞ
r(n)
wavelet demod
MF User k
MMSE comb yk(n)
ck,1
MMSE comb
0 1
xM −1(n) M−1
PM−1(z)
y0(n)
y1(n)
MUD
y ′/0(n)
X ck,M −1
K−1 yM −1(n)
Fig. 4 Wavelet multicarrier CDMA optimised receiver (implementation)
The design of the synthesis and analysis filters are based on the following equations:
p 1 1 N kp kþ nþ ð1Þ fk ½n ¼ 2 p½n cos M 2 2 2 4 ð16Þ
p 1 1 N kp kþ gk ½n ¼ 2 p½n cos nþ þ ð1Þ M 2 2 2 4
X k,0 z−1
ð17Þ X
y′k (n)
k,k X
DC z−1
+
where p(n) is the prototype filter derived using the QCLS algorithm. In the z-domain, this filter becomes P ðzÞ ¼
yK−1(n) DC: decision circuit TSG: training sequence generator X MF: matched filter
X
N 1 X
pðnÞzn
ð18Þ
n¼0
k,K−1
where N is the filter length. The polyphase components, Pm(z), of the prototype filter come from the following equation:
z−1
+
X u
+
− +
P ðzÞ ¼P0 ðzM Þ þ z1 P1 ðzM Þ þ þ zðM1Þ PM1 ðzM Þ
TSG
P ðzÞ ¼ Fig. 2
x1(n)
ð15Þ
The value of the step size m restrains the rate of convergence (ROC) of the LMS algorithm and modifies the minimum square error (MSE). Incorrect step size causes the system to lose convergence. The optimum value of m is chosen to compromise between ROC and the MSE. The block diagram of the adaptive LMS wavelet receiver is shown in Fig. 2.
+
X C
3.3 Adaptive wavelet LMS multi-user detection The LMS adaptive algorithm used in this project applies the following recursive expression on the multiuser axis:
X ck,0
DEMUX
X
sk(n) MUX
M−1
X
ð13Þ
Thus:
y0(n)
P 1(z) ^ C −1
ð12Þ
^ a ¼ sgnðxH yÞ
H
P 0(z)
ck,0
ð11Þ
and the estimated data is given as:
0
X
Wavelet multicarrier CDMA adaptive LMS receiver
IEE Proc.-Commun., Vol. 150, No. 6, December 2003
M 1 X
zm Pm ðzM Þ
ð19Þ
m¼0
447
From (18) and (19), the polyphase filters Pm(z) can be written as:
0
N
pðm þ lMÞz‘
‘¼0
The modulation matrix (C1) in Fig. 3 and the corresponding demodulation matrix (C) in Fig. 4 are given in [4, 10]. 5
−5
ð20Þ
Complexity considerations
The cost of the modulation–demodulation in the wavelet transmission system is analysed and found to be higher by an additional M(g+1) operations per symbol (where g ¼ N/M is the overlap factor) than if the processing is achieved by the conventional FFT processor. The cost of computing the coefficients of various detectors considered in this paper is shown in Fig. 5. Clearly the complexity of the decorrelator is less than the complexity of the proposed MMSE detector. However the BER performance of the decorrelator is inferior to the MMSE detector when data is transmitted through an AWGN or multipath channel.
normalised power, dB
Pm ðzÞ ¼
M 1 X
−25
0.4 0.6 normalised frequency a
0.8
1.0
0
0.1
0.2 0.3 normalised frequency b
0.4
0.5
−10
−30 normalised power, dB
Decor
4.5
MMSE−Comb 4.0 3.5
−40 −50 −60
3.0
−70
2.5
−80 −90
2.0
0
5
10
15
20
25
30
35
number of users
Fig. 5
0.2
0
MMSE−MUD MMSE−Comb−MUD
1.5
0
−20
5.0
log10(computations)
−15
−20
5.5
6
−10
Cost of computing the detector coefficients
Fig. 6 Channel frequency transfer function of 16-channel multicarrier CDMA system a FFT processing b Wavelet processing
Simulation results 0.10
448
FFT system wavelet system
0.08
0.06 PSD
The power spectral density (PSD) of a 16-channel multicarrier system generated by FFT is shown in Fig. 6a and the PSD of a similar system produced by wavelet processing is shown in Fig. 6b. The side components are about 13 dB below the main component in FFT and about 80 dB below the main component in the wavelet system. The PSDs of an individual channel produced by the two systems are compared in Fig. 7. The energy of the FFT channel is spread over many sidelobes but the spectrum of the wavelet channel is more compact. The peak to average power ratio (PAPR) was assessed using a single-user 16-channel multicarrier transmission system generating FFT-based and wavelet-based signals. It was found that the PAPR in the wavelet-based signal is 0.79 dB higher than in the FFT-based signal. Furthermore, 6.25% of the measured instantaneous powers in the FFT-based signal are found to be higher than the average power compared to only 2% for the wavelet-based signal. The variance of the instantaneous power is 9.28 for the wavelet-based signal and a maximum of 15.00 for the FFT signal.
0.04
0.02
0
0
0.1
0.2 0.3 normalised frequency
0.4
0.5
Fig. 7 Signal PSD for a single-channel multicarrier CDMA system using FFT processing and wavelet processing
The Rayleigh fading channel used for the simulation consists of four users, each with four rays with delays [0, 1/M, 2/M, and 3/M] symbol intervals where M denotes IEE Proc.-Commun., Vol. 150, No. 6, December 2003
the number of carriers. The fading factors are chosen from four independent Rayleigh random sources with different variances. The BER performance of the fading channel was evaluated using conventional matched filter detectors in order to have a perception of the performance of the proposed system compared with the conventional FFT system. A 16-channel MC-CDMA wavelet-based and FFTbased systems are simulated with Rayleigh fading sources with variances [0.8, 06, 0.4 and 0.2]. The BER performance for both systems is shown in Fig. 8.
0.5 0.4
0.3 0.2
0.1 0
100
0
0.1
0.2
FB−MC−CDMA WB−MC−CDMA
0.3
0.4
0.5
a
10−1
10−2 BER
0.5 0.4
10−3 0.3 10−4
10−5
0.2
0
2
4
6 Eb /N0, dB
8
10
0.1
12
0
Fig. 8 BER performance under multipath channel for both FFT and wavelet-based systems
IEE Proc.-Commun., Vol. 150, No. 6, December 2003
0.1
0.2
0.3
0.4
0.5
b
Fig. 9
A main source of distortion in a mobile channel in an urban area is due to intersymbol interference (ISI). A 32band wavelet system and 32-carrier FFT system were tested for ISI by simulating a single user transmission in the multipath channel with amplitude fading factors drawn from Rayleigh sources with variances [1, 0.8, 0.5, 0.2]. The simulation parameters are given in the Appendix (Section 10.4). A second ISI simulation was performed for both wavelet and FFT systems with the same parameters, apart from choosing non-integer delays of [0, 1.2/32, 2.5/32, 4.3/ 32] symbol duration but are not displayed as explained below. The PSDs of the ISI signal for the wavelet and FFT transmissions in Fig. 9 are shown with very little changes in the PSDs when the delays were non-integer. Simulation of 2000 symbols resulted in a signal-to-ISI power ratio of 9 dB for the wavelet system and 7 dB for the FFT system whether the delays were integer or non-integer. The mathematical expression for the MAI generated by a number of users sharing a wavelet transmission channel at the output of a matched filter is given by (7). This equation is used to compute the MAI in a fading channel where the fading factor is selected from a Rayleigh random source with mean value of 1.45 and variance 0.43. The calculated values and the simulated results are compared in Fig. 10a when the subbands are independent. A similar curve is plotted in Fig. 10b for a Rayleigh fading channel with mean of 1.25 and variance 0.43 but with partially correlated subbands. The correlation values are uniformly distributed between 0.3 and 1. In both simulations, the minimum value of the fading random variable is kept greater than zero to avoid deep fading. A 32-band wavelet transmission multiple access system was considered and the BER at the output of the MMSE
0
PSD of the ISI signal under Rayleigh fading
a FFT system b Wavelet system
and decorrelator (decor) detectors were simulated. The MMSE detector was optimised along the band axis only (MMSE-comb), and along the user axis only (MMSEMUD), and finally it is optimised along both the band and the user axis simultaneously (MMSE-comb-MUD) and made adaptive on the user axis (LMS-MUD). The BER performance of MMSE and the decorrelator detectors in the multipath fading channel and with varying number of users is shown in Fig. 11a and with varying Eb/N0 is shown in Fig. 11b. In both cases, the variances of the Rayleigh random sources are [1, 0.8, 0.5, and 0.2]. 7
Conclusions
A new digital transmission system that is a combination of CDMA and multicarrier modulation based on wavelet processing has been proposed and analysed. The basic system consists of a bank of filters satisfying the PR-QMF criterion and derived using the QCLS algorithm. The superiority of the proposed scheme is demonstrated by being more resilient in a mobile transmission environment as shown in Fig. 6 and offers BER performance improvement compared with the conventional system. For example, when Eb/N0 ¼ 8 dB, the BER for the waveletbased system was 0.0048 and for the FFT-based system was 0.019 as shown in Fig. 9. Furthermore, when the proposed system is stationary, there will be no interference on adjacent channels since these channels are orthogonal. However, when transmission fading is considered, the proposed system is slightly more resistive to ISI than the FFT channel. Simulation has shown 449
20 calc sim
10−1
16 10−2 BER
SIR, dB
12
8
MMSE−comb MMSE−mud decorr MMSE−comb−MUD LMS−MUD
10−4
4
0
10−3
0
2
4
6 8 10 number of users a
12
14
10−5
16
5
10
15 20 number of users a
25
30
30 calc sim
10−1
25
20
10−2
15
BER
SIR, dB
MMSE−comb MMSE−MUD decorr MMSE−comb−MUD LMS−MUD
10−3
10 10−4
5
0
0
2
4
6 8 10 number of users b
12
14
Fig. 10 Signal-to-interference ratio for multi-user system with MF detection under Rayleigh fading channel a Independent subbands b With partial correlation between subbands
about a 2-dB improvement in the signal-to-ISI power ratio compared to FFT-based modulation when both systems operated in identical fading environments. The proposed system has the potential for greater bandwidth efficiency as it generates more compact channels and does not require the cyclic prefix (guard time) nor the pilot tone to achieve synchronisation. Instead, the time synchronisation is performed based on the correlation properties of the wavelets [14]. The proposed system shows robustness against narrowband interference compared with the FFT-based system as in [12]. The proposed system is more robust against interchannel interference than the FFT-based system since the processing at the transmitter and receiver are independent among subbands while they are interrelated in the FFT-based system. Moreover, further processing of the signal in multiuser detection in the proposed system enhances the performance of the detection process better than in the FFT-based system. The proposed joint detection scheme uses frequency diversity to combine the energy of various bands and applies the MMSE criterion in multi-user detection and is then made adaptive using the LMS algorithm. The adaptive MMSE detector is near–far resistant and shows better BER performance compared with the decorrelator. However, the complexity of the proposed detection scheme is higher than the complexity of the decorrelator. 450
10−5
16
0
10
5
15
Eb /N0 b
Fig. 11 BER of wavelet multicarrier CDMA system with MUD under multipath channel a Different number of users b Various Eb/N0
8
Acknowledgement
The authors would like to thank the University of Plymouth and STT Telkom (Indonesia) for providing the financial support for this research. 9
References
1 Hara, S., and Prasad, R.: ‘An overview of multi-carrier CDMA’, IEEE Commun. Mag., 1997, 35, pp. 126–133 2 Yee, N., and Linnartz, J.P.M.G.: ‘Wiener filtering for multi-carrier CDMA’. Proc. IEEE/ICCC Conf. on Personal indoor mobile radio communications (PIMRC) and Wireless computer networks (WCN), The Hague, Netherlands, 19–23 Sept. 1994, vol. 4, pp. 1344–1347 3 Daneshgaran, F., and Mondin, M.: ‘Bandwidth efficient modulation with wavelets’, Electron. Lett., July 1994, 30, (1), pp. 1200–1202 4 Strang, G., and Nguyen, T.: ‘Wavelets and filter banks’ (WellesleyCambridge Press, 1997) 5 Vetterly, M., and Kovacevic, J.: ‘Wavelets and sub-band coding’ (Prentice Hall, 1995) 6 Wornell, G.W.: ‘Emerging applications of multirate signal processing and wavelets in digital communications’, Proc. IEEE, April 1996, 84, (4), pp. 586–603 7 Hetling, K.J., Saulnier, G.J., and Das, P.: ‘Optimised filter design for PR-QMF based spread spectrum communications’, Proc. IEEE International Conf. on Communications, Seattle, WA, USA , June 1995, vol. 3, pp. 1350–1354 8 Hetling, K.J., Saulnier, G.J., and Das, P.: ‘PR-QMF based codes for multipath/ multiuser communications’, Proc. IEEE Globecom Conf., Singapore, 13–17 Nov. 1995 9 Vaidyanathan, P.P.: ‘Multirate systems and filter banks’ (Prentice Hall, 1992) IEE Proc.-Commun., Vol. 150, No. 6, December 2003
10 Nguyen, T.: ‘Digital filter banks design quadratic-constrained formation’, IEEE Trans Signal Process., 1995, 43, pp. 2103–2108 11 Nguyen, T., and Vaidyanathan, P.P.: ‘Structures for M-channel perfect-reconstruction FIR QMF banks which yield linear-phase analysis filters’, IEEE Trans. Acoust. Speech Signal Process., 1990, 38, pp. 433–446 12 Muayyadi, A., and Abu-Rgheff, M.N.A.: ‘A wavelet-based MCCDMA cellular systems’. Proc. IEEE Sixth International Symp. on Spread spectrum techniques & applications, NJ, USA, Sept. 2000, pp. 145–149 13 Steffen, P., Heller, P., Gopinath, R.A., and Burrus, C.S.: ‘Theory of regular M-band wavelet bases’, IEEE Trans. Signal Process., 1993, 41, pp. 3497–3511 14 Muayyadi, A., and Abu-Rgheff, M.A.: ‘Wavelet-based synchronisation in multi-carrier CDMA systems’, Proc. PREP-2003 Conf., Exeter, UK, pp. 1–2 15 Haykin, S.: ‘Adaptive filter theory’ (Prentice Hall, NJ, USA, 1996, 3rd edn.)
where bk is the kth row of matrix b 2 3 b0;1 b0;M1 b0;0 .. 6 7 6 b1;0 7 . 6 7 b¼6 7 . 4 .. 5 bK1;0 bK1;M1
ð30Þ
which is a K M matrix of bk,m, and bk,m ¼ ak,mck,m is the subband gain factor (which depends on the combining strategy employed) multiplied by the code chip. In general for all K users these decision variables are: yðzÞ ¼ bxðzÞ ð31Þ The filtered noise term becomes
10 Appendix
nðzÞ ¼ bGðzÞnðz1=M Þ and the column vector is
10.1 Derivation of (6) Substituting for sk(z) from (1) into (4), we obtain: rðzÞ ¼
K 1 X
FðzÞðC k H k Þak ðzM Þ þ nðzÞ
xðzÞ ¼ ½x0 ðzÞ x1 ðzÞ xK1 ðzÞ T : ð21Þ
k¼0
where Hk is the kth column vector of the channel matrix H, and 2 3 h1;0 hK1;0 h0;0 6 h0;1 h1;1 hK1;1 7 6 7 ð22Þ H¼6 . 7 4 .. 5 h0;M 1 h1;M1 hK1;M1 Equation (21) can be simplified to the following equation: rðzÞ ¼
K 1 X
FðzÞEðzÞak ðzM Þ
ð23Þ
k¼0
where Ek is given by (5). For a flat-faded channel, Ek is constant across each of the subbands. E¼ 2 6 6 6 6 4
c0;0 h0;0 c0;1 h0;1 .. .
c1;0 h1;0 c1;1 h1;1
c0;M1 h0;M1
c1;M1 h1;M1
3
cK1;0 hK1;0 cK1;1 hK1;1
7 7 7 7 5
cK1;M1 hK1;M1 ð24Þ
The signal at the output of the mth branch of the kth user receiver is: xðzÞ ¼ GðzÞrðz1=M Þ
ð25Þ
T
where x (z) ¼ [w0 (z) w1 (z)?wM1(z)] . Substituting for r(z), we have: xðzÞ ¼
K 1 X
ð32Þ
Substituting for x(z), the decision variables become: yðzÞ ¼ bEaðzÞ þ xðzÞ ð33Þ
10.2 Derivation of (10) Let the received signal at the output of the down-samplers be defined by vector xk(z) such that ð34Þ xk ðzÞ ¼ C k xðzÞ Substituting for x(z) from (15), we obtain: xk ðzÞ ¼ C k ½EðzÞaðzÞ þ NðzÞ
ð35Þ
Using the Wiener–Hopf equation in adaptive filter theory [15], the optimum condition is achieved when the filter coefficients (ak) satisfy the following equation aopt ¼ R1 XX RXa
ð36Þ
where RXX is the autocorrelation of the input signal x and RXa is the cross-correlation between the input signal and the data signal a. Substituting for xk(z) in (35), we obtain RXX ¼ EðEaaH E H Þ þ EðNN H Þ
ð37Þ
The noise signal, nm(z), at the output of the analysis filter Gm(z) in the mth branch is nm ðzÞ ¼ Gm ðzÞnðz1=M Þ
ð38Þ
The variance of this noise is
h o i Eðv2m Þ ¼ E½G2m ðzÞn2 ðz1=M Þ ¼ E G2m ðoÞn2 ð39Þ M Simplifying further this equation and taking the approximation Z1 o s2 ð40Þ do Eðv2m Þ ¼ G2m ðoÞn2 M M 0
GðzÞFðzÞE k ak ðzÞ þ N ðzÞ
ð26Þ
in matrix form, the variance is
k¼0
EðNN H Þ ¼ E½G 2 ðzÞn2 ðz1=M Þ ¼
where the filtered noise N(z) is given by: NðzÞ ¼ GðzÞnðz1=M Þ
ð27Þ
Using the biorthogonality property, (26) can be simplified to: xðzÞ ¼ EðzÞaðzÞ þ NðzÞ
ð28Þ
The decision variable in the kth user’s receiver is written as follows: yk ðzÞ ¼ bk xðzÞ IEE Proc.-Commun., Vol. 150, No. 6, December 2003
ð29Þ
s2 M I ¼ r2 I M
ð41Þ
The autocorrelation matrix of the input signal x can be shown to be RXX ¼ ½EðHH H Þ þ s2 I The cross-correlation RXa is RXa ¼ Eðxak Þ
ð42Þ ð43Þ
Substituting for x in (43) and simplifying, RXa is given by ð44Þ RXa ¼ H k 451
Substituting RXX and RXa into (36), we obtain H
2
1
aopt ¼ ½EðHH Þ þ s I H k
Substituting Ryy and Rya into (46), we obtain: ð45Þ
xopt ¼ ðbEE H bH þ r2 b2 Þ1
ðbEÞ
ð52Þ
Now using (13), (52) can be rewritten as
10.3 Optimum filter coefficients Applying the MMSE algorithm to wavelet multi-user detection, the optimum filter coefficients are given by adaptive filter theory and satisfy the following equation xopt ¼ R1 yy Rya
ð46Þ
The autocorrelation of the input signal y is Ryy ¼ EðbEaaT E T bT Þ þ E ðnnT Þ
ð47Þ
where n(z) is defined as nðzÞ ¼ bGðzÞnðz1=M Þ
ð48Þ
Further simplification reduces (47) to Ryy ¼ bE 2 bT þ r2 b2
ð49Þ
Now the cross-correlation between the input signal y and the desired data a is given by Ray ¼ EðyaT Þ
ð50Þ
Substituting for y, (50) becomes Ray ¼ EðbEaaT ÞbE
452
ð51Þ
xopt ¼ ½ccH þ r2 b2 1 c
ð53Þ
Equation (53) gives the optimum filter coefficients that minimise the mean square error between the estimated data signal and the desired data signal and consequently optimise the performance of the wavelet group detection.
10.4 Simulation parameters Channel bandwidth ¼ 5 MHz Number of subchannels, M ¼ 32 Subchannel bandwidth ¼ 5 MHz/32 ¼ 0.1563 MHz Symbol period ¼ 1/0.1563 ¼ 6.4 ms The delay spread is chosen to be much less than the symbol period, i.e. delay spread ¼ 1.6 ms Thus the coherence bandwidth for each subchannel is 1/1.6 ms ¼ 0.625 MHz Therefore since the coherence bandwidth is greater than the subchannel bandwidth, the channel is subjected to frequency non-selective (flat) fading.
IEE Proc.-Commun., Vol. 150, No. 6, December 2003
