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A Blind Adaptive Step-Size Time-Domain Receiver
for MC-CDMA Systems
Peerapol YuvapoositanonCentre of Electronic System Design and
Signal Processing (CESdSP)
Department of Electronic Engineering
Mahanakorn University of Technology
Nong-Chok, Bangkok, Thailand 10530
Email: [email protected]
Sutat SuwannajanDepartment of Electronic Engineering
Mahanakorn University of Technology
Nong-Chok, Bangkok, Thailand 10530
Email: [email protected]
AbstractA blind adaptive step-size time-domain receiver formulti-carrier code-division multiple access (MC-CDMA) systemsis presented. Adjustment rules for the receiver tap-weight as wellas its step-size are based upon the stochastic approximation ofthe constant modulus (CM) criterion. The ability of the proposed
receiver to detect the desired user in multipath fading channelsat full load is assessed. Sensitivity to the various initial valuesof the step-size and the adaptation rates of the algorithm is alsoinvestigated.
Keywords: Multi-carrier CDMA, Adaptive step-size, ConstantModulus Algorithm.
I. INTRODUCTION
Multi-carrier Code-Division Multiple Access (MC-CDMA)
has long gained considerable interest to be an air interface
scheme which has the potential to deliver the demanding
100 Mb/s - 1 Gb/s data rate as required in the fourth
generation (4G) mobile communications systems [1]. MC-
CDMA emerged originally as an ingenious way to combineOrthogonal Frequency Division Multiplexing (OFDM) and
CDMA and reap advantageous aspects from both multiplexing
and multiple access schemes [2]. MC-CDMA has been studied
extensively and been found favourable in providing frequency
diversity [3], bandwidth efficiency [4] and robustness for
frequency-selective Rayleigh fading downlink channels [5].
By inheriting the multi-carrier legacy of OFDM, each MC-
CDMA subcarrier expects to see an individual frequency-
nonselective fading channel. One-tap frequency-domain
equalisers are responsible in handling the magnitude and phase
equalisation for such channels [1]. But unless the number of
subcarriers is large enough or guard interval is inserted, the
decent frequency-nonselective fading channels can easily turninto harsh frequency-selective ones [6], [4]. In this case, low
complexity adaptive receivers are usually introduced to handle
with the dispersive received signals. However, the precious
bandwidth is shared unnecessarily to provide either training
signals or channel state information (CSI) to the receiver.
To minimise bandwidth wasting, variants of the constant
modulus algorithm (CMA) are proposed for blind multiuser
detectors for MC-CDMA systems [7], [8], [9], [10]. Recently,
a blind MMSE receiver using the constrained minimum output
energy (CMOE) cost function is investigated in [11]. These
receivers are however constructed in the frequency-domain
which inevitably incurs the burden of including the Discrete
or Fast Fourier Transform (DFT/FFT) operations.
We propose in this paper a blind adaptive time-domainreceiver for MC-CDMA systems which removes the necessity
of training information sending from the transmitter as well
as the inclusion of the DFT or FFT operations at the receiver
end. The step-size and tap-weight of the proposed receiver
are updated based upon the rule of minimising the CM
cost function [12]. Simulations confirm the applicability of
the proposed algorithm for a multipath fading MC-CDMA
channel in full load situation. Insensitivity of the algorithm
to initial settings of the step-size as well as the rate of step-
size adaptation are also shown.
For notation, we use bold lower case for vectors, bold upper
case for matrices, denotes the convolution operation, ()T fortransposition and E{} for the statistical expectation operator.
II. SIGNAL MODEL
The MC-CDMA system initially proposed by [2] provides
the model of transmitted MC-CDMA signal for user m as
sm(t) =1Ts
N1
k=0
Cm[k]am[i]ej(2fct+2kLt/Ts), (1)
where Cm[k] is the frequency-domain spreading code for userm at subcarrier k, am[i] is the data symbol for user m at bit iwith symbol interval Ts, and fc is the carrier frequency. Thespreading code Cm[k] has the spreading gain N.
The continuous-time transmitted signal sm(t) in (1) can
be represented in the discrete-time version by assigning thesampling time to be Ts/N and also applied to all M users.Therefore, the discrete-time version of transmitted signals
from all users s[n] is then
s[n] =M1
m
sm[n] =1N
M1
m
N1
k=0
Cm[k]am[i]ej2kn/N,
(2)
where N denotes the number of subcarriers which is generallydesigned to be equal to the spreading gain of Cm[k] [2].
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The modulated signal s[n] is propagated through an FIRchannel model with the channel impulse response {h[n]}, n {0, , Lh 1}. Hence, the received signal r[n] at a des-ignated receiver is the result of a convolutional operation
between the transmitted signal s[n] and h[n], i.e., r[n] =s[n] h[n]. At the receiver, the discrete-time received signalis therefore
r[n] = 1N
M1
m
N1
k=0
Hm[k]Cm[k]am[i]ej2kn/N + g[n],
(3)
where g[n] is an additive white Gaussian noise (AWGN) withvariance 2g .
We are able to represent r[n] as a function of the combinedchannel-spreading code impulse response by rearranging (3)
into
r[n] =
M1
m=0
am[i]tm[n] + g[n], (4)
where tm[n] =1N
N1k=0 Hm[k]Cm[k]e
j2kn/N represents
the combined channel-spreading code response [13]. From (4),the chip-level received signal r[n] can then be written in thesymbol-level one in the form of vector-matrix notation as
r[i] = Ta[i] + g[i], (5)
where
r[i] = [r[iN], r[iN + 1], , r[(i + 1)N 1]]T,a[i] = [a0[i], a1[i], , aM1[i]]T,T = [t0[i] t1[i] tM1[i]],
tm[i] = [tm[iN], tm[iN + 1] , tm[(i + 1)N 1]T,g[i] = [g[iN], g[iN + 1], , g[(i + 1)N 1]]T.
Note that r[i] collects r[n] via the serial-to-parallel (S/P) op-eration in order to feed the symbol-level receiver as described
in the next section.
III. DEVELOPMENT OF A BLIND ADAPTIVE STE P-SIZE
RECEIVER
We develop the time-domain adaptive MC-CDMA receiver
based on the fact that the transmitted and received MC-CDMA
signals can be theoretically interpreted as Direct-sequence
(DS)-CDMA ones [13]. The function of the receiver fl[i] istherefore to concurrently despread, demodulate and equalise
the received signal r[n] to give al[i]: the symbol estimate ofthe lth user. At the receiver, all updates are performed every
symbol i and the symbol-spaced input for the receiver is r[i].The symbol estimate al[i] is derived from the soft-decision
output zl[i] by
al[i] = sgn(zl[i]) = sgnfTl [i]r[i]
(6)
where sgn() denotes the sign function of decision device. Theblock diagram of the proposed receiver structure is shown in
Fig. 11.
1For the sake of notation simplicity and yet without loss of generality, weshall from now on drop the subscript l from fl .
Fig. 1. Linear adaptive AS-CMA receiver structure for MC-CDMA systems.Note that the receiver f[i] is updated at every symbol.
A. Update Algorithm
We consider the adaptation of both receiver tap-weight and
step-size based on the constant modulus (CM) property of
the transmitted symbols. The CM cost is denoted by J =E{(z2[i])2} where representing the dispersion coefficientand is defined by the type of modulation of transmitted
symbols [14]. The CMA algorithm involves finding f[i] whichuses a stochastic gradient minimisation of J with respect to
equaliser tap-weight f, i.e., Jf
f=f[i]
= 0. The update equation
of f[i] at the MC-CDMA symbol time i is given by
f[i + 1] = f[i] [i](z2[i] )z[i]r[i]. (7)Using adaptive step-size derivation scheme of [15], the blind
mode step-size is updated in order to minimising the CM cost
J with respect to the step-size , i.e., J
=[i]
= 0 which
gives
[i + 1] =
[i] (z2[i] )z[i]rT[i]Y[i]+
, (8)
where denotes the adaptation parameter and [ ]+
denotes
truncation to lower and upper step-size limits. Y[i] represents
the derivative off[i]
=[i]
and its update equation is given
by
Y[i] =I [i](3z2[i] )r[i]rT[i]Y[i](z2[i])z[i]r[i].
(9)
Equations (7), (8) and (9) constitute the adaptive step-size
CMA (AS-CMA) time-domain receiver for MC-CDMA sys-
tems which is identical to the AS-CMA algorithm of the DS-
CDMA receiver [12].
IV. UNBIASED MEA N SQUARED ERROR
Since there is no prior information such as phase and
amplitude of the desired user transmission available at the
receiver, the blind estimates are subject to experience the
gain ambiguity [16]. For meaningful performance evaluation,the conditionally unbiased mean squared error (UMSE) is
introduced as a measuring function as
UMSE = E{(z[i]/q al[i])2}, (10)where q denotes the gain factor of the estimate z[i] providedthat z[i] = qal[i] + g[i] and g[i] is the representative of thefiltered interference plus noise signal. The UMSE will be used
as a performance index for both blind and non-blind receivers
in the next section.
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V. SIMULATIONS
We considered a downlink MC-CDMA system with four
active users without additive guard interval or cyclic prefix
(CP). The spreading gain for all users were N = 4. Walsh-Hadamard codes cm[n] {+ 1Nc ,
1Nc
}, for n = 0, . . . , 3and m = 0, . . . , 3, were generated and then fed into theFast Fourier Transform (FFT) operation to give the frequency-
domain spreading codes Cm[k]. Since downlink transmissionwas adopted, all users propagated through the same four-raymultipath raised cosine channel and no near-far situation is
observed. We assumed without loss of generality that the
location of the delay of the dominant path was known at
the receiver. The rectangular waveform was used as the pulse
shaping filter. The AWGN was assumed to SNR=20 dB at the
symbol level.
We considered the performance comparison in detecting the
desired user transmission of three algorithms for adaptive MC-
CDMA receivers, i.e., the Least Mean Square (LMS) algorithm
[17], the standard CMA algorithm [18] and the proposed AS-
CMA algorithm. The Minimum Mean Squared Error (MMSE)
receiver is used to provide the UMSE bound as described inSection IV and is denoted as [13]
fMMSE =TTT + 2gI
1T. (11)
Choice of initialisation is crucial to assist all blind algo-
rithms, i.e., CMA and AS-CMA receivers, in avoiding local
minima and converge with high probability to the desired
solution [14], [19]. Initialising f[0] of both blind receiverswith the time-domain spreading code of the desired (first) user
c0 = [c0[0], c0[1], c0[2], c0[3]]T is a reasonable choice due tothe availability of such code at the receiver.
We started by fixing the adaptation parameter to 1103and examined the behaviours of the AS-CMA receiver at
different settings of the initial step-size [0]. Fig. 2 showsrespectively three unbiased mean-squared error trajectories of
different initialised step-sizes [0] = {1105, 1103, 1102} as compared to those of other receivers. Setting [0] toohigh may result in divergence of the algorithm. Each plot was
averaged over 100 Monte Carlo runs for 250 transmitted sym-bols. The UMSE trajectories of the standard CMA receivers
are also plotted for = {1104, 1103, 1102}. Withthe 1,000-fold [0] variation, all trajectories of the step-sizesfor the proposed AS-CMA receiver in Fig. 3 show an approx-
imately identical behaviour in convergence. As compared to
CMA, the AS-CMA receiver tends to converge faster given
that an initial step-size for AS-CMA being equal to a fixed
one for CMA. AS-CMA is able to adapt its step-size in orderto minimise the CM cost while the speed of CMA depends
heavily on its predetermined step-size. As compared to LMS,
the convergence speed of AS-CMA is expected to be slower
due to its non-linear cost function. However, it is shown that
the steady-state UMSE performance of the AS-CMA coincides
with that of LMS given the virtue that no training information
is required for AS-CMA.
After symbol i = 100, all trajectories stay at the same level.It is noticed that the algorithm converges slightly faster at
0 50 100 150 200
102
101
100
Number of Symbols
UMSE
Mean Squared Error Trajectories
CMA, =1 102
CMA, =1 103
CMA, =1 104
ASCMA, [0]=1 105
ASCMA, [0]=1 10
3
LMS, =1 101
ASCMA, [0]=1 102
MMSE
Fig. 2. UMSE trajectories of the AS-CMA receiver for different [0] with = 1 103 as compared to CMA, LMS and MMSE receivers. Despitea 100-fold variation in [0] setting, comparable convergence behaviours arenoticed. Slightly faster convergence rate is shown for larger setting of [0].
0 50 100 150 200 25010
5
104
103
102
101
100
Number of Symbols
[n]
Stepsize Trajectories
[0]=1e2
[0]=1e3
[0]=1e5
Fig. 3. The trajectories of step-sizes [n] of AS-CMA receiver for different[0] with = 1 103. All trajectories converge to the same location atapproximately 0.1.
larger initial setting of step-sizes. A similar result has also
been observed for the non-blind adaptive step-size LMS (AS-
LMS) algorithm shown in [20]. Fig. 3 shows the trajectories
of step-sizes of AS-CMA. Clearly, different trajectories with
each initialisation is barely distinguishable since all trajectoriesgo to the same location. It also suggests that the steady-state
step-size to be [n]n= 0.1.
We shall now proceed to examine the performance of the
AS-CMA receiver at different s. Fig. 4 and Fig. 5 showtrajectories of UMSE and [n] respectively at different settingsof = {1 104, 1 103, 1 102} while [0] was set at1 105. Clearly, changing does affect the convergencespeed of the algorithm. Larger s give faster convergencespeed than smaller ones but with the penalty of noisy steady-
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0 50 100 150 200
102
101
100
Number of Symbols
UMSE
Mean Squared Error Trajectories
ASCMA, =1 104
ASCMA, =1 103
ASCMA, =1 102
LMS, =1 101
MMSE
Fig. 4. UMSE trajectories of the AS-CMA receiver for different s with[0] = 1 102 as compared to CMA, LMS and MMSE receivers. Aspredicted, varying affects convergence speed of the receiver. Larger sgive faster convergence speed than smaller ones but with the penalty of noisysteady-state UMSE.
0 50 100 150 200 25010
5
104
103
102
101
100
Number of Symbols
[n]
Stepsize Trajectories
=1e2
=1e3
=1e4
Fig. 5. The trajectories of step-sizes [n] of AS-CMA for different with[0] = 1 105. Notice the comparable steady-state locations of eachtrajectory of[n] which associated with a 100-fold variation ofs.
state UMSE. However, each [n] still converges in the vicinityof 0.1 despite a 100-fold difference of s. Although uniqueglobal convergence of AS-CMA receiver has not yet been
proven, the consistency in convergence of different adaptation
rates suggests the insensitivity of adaptation rate of the AS-CMA receiver for MC-CDMA systems.
V I . CONCLUSION
In this paper, a blind adaptive step-size time-domain receiver
for MC-CDMA is introduced. The step-size and tap-weight
of the proposed receiver are updated based upon the rule
of minimising the CM cost function. It is shown that the
receiver can be employed for a downlink frequency-selective
fading MC-CDMA channel. Simulation results suggest that
the proposed AS-CMA receiver is relatively insensitive to the
1,000-fold variation of initial step-sizes and 100-fold adapta-
tion rate settings of the AS-CMA algorithm. The performance
comparison has shown consistency of AS-CMA with the
existing non-blind time-domain receivers.
REFERENCES
[1] S. Hara and R. Prasad, Multicarrier Techniques for 4G Mobile Com-
munications, Artech House universal personal communications series,Boston, 2003.
[2] N. Yee, J. P. Linnartz, and G. Fettweis, Multicarrier cdma in indoorwireless networks, in Proc. Int. Symp. Personal Indoor and Mobile
Radio Communications (PIMRC), Yokohama, Japan, 1993, pp. 109113.[3] D. N. Kalofonos, M. Stojanovic, and J. G. Proakis, Performance
of adaptive MC-CDMA detectors in rapidly fading rayleigh channels,IEEE Trans. Wireless Commun., vol. 2, no. 2, pp. 229239, MArch 2003.
[4] X. Gui and T.S. Ng, Performance of asynchronous orthogonal mul-ticarrier CDMA system in frequency selective fading channel, IEEETrans. Commun., vol. 47, no. 7, pp. 10841091, July 1999.
[5] S. Hara and R. Prasad, Design and performance of multicarrier CDMAsystem in frequency-selective rayleigh fading channels, IEEE Trans. onVehic. Technol., vol. 48, no. 5, pp. 15841595, September 1999.
[6] S. L. Miller and B. J. Rainbolt, MMSE detection of multicarrierCDMA, IEEE J. Select. Areas Commun., vol. 18, no. 11, pp. 23562362, November 2000.
[7] J. Mguez and L. Castedo, Blid multiuser interference cancellation inmulticarrier CDMA: a linearly constrained constant modulus approach,in Proc. Int. Symp. Personal Indoor and Mobile Radio Communications(PIMRC), Boston, MA, September 1998, vol. 2, pp. 523 527.
[8] D.-J. Kim, J.-E. Kim, and C.-E. Kang, The new approach to mitigateMAI in MC-CDMA systems, in IEEE VTS 50th Vehicular TechnologyConference, Fall., Amsterdam, September 1999, vol. 1, pp. 171 175.
[9] D. Darsena, G. Gelli, L. Paura, and F. Verde, Blind multiuser detectionfor MC-CDMA systems, in Proc. Asilomar Conf. on Signals, Systemsand Computers, Pacific Grove, CA, November 2002, vol. 2, pp. 1419 1423.
[10] S.-J.Chern, C.-Y. Chang, and H.-C. Liu, Multiuser wavelet based MC-CDMA receiver with linerly constrained constant modulus IQRD-RLSalgorithm, in Proc. Int. Symp. Circuits and Systems (ISCAS), 2002,vol. 1, pp. I193 I196.
[11] H. Cheng and S. C. Chan, Blind linear MMSE receivers for MC-CDMA systems, IEEE trans. Circuits and Systems-I, vol. 54, no. 2,
pp. 367376, February 2007.[12] P. Yuvapoositanon and J. A. Chambers, Adaptive step-size constant
modulus algorithm for DS-CDMA receivers in nonstationary environ-ments, Signal Processing, vol. 82, pp. 311315, 2002.
[13] S. Nahm and W. Sung, Time-domain equalization for the orthogonalmulti-carrier cdma systems, in Proc. Global TelecommunicationsConference (GLOBECOM), London, UK, 1996, pp. 15831587.
[14] C. R. Johnson Jr., P. Schniter, I. Fijalkow, L. Tong, J. D. Behm, M. G.Larimore, D. R. Brown, R. A. Casas, T. J. Endres, S. Lambotharan,A. Touzni, H. H. Zeng, M. Green, and J. R. Treichler, The core of FSE-CMA behavior theory, in Unsupervised Adaptive Filtering, S. Haykin,Ed., vol. 2, pp. 13112. John Wiley & Sons, New York, 2000.
[15] A. Benveniste, M. Metivier, and P. Priouret, Adaptive Algorithms andStochastic Approximations, Volume 22 of Applications of Mathematic,Springer-Verlag, New York, 1990.
[16] P. Schniter, Blind Estimation without Priors: Performance, Convergence,and Efficient Implementation, Ph.D. thesis, Cornell University, Ithaca,
NY, 2000.[17] S. Haykin, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs N.J,
1986.[18] D. N. Godard, Self-recovering equalization and carrier tracking in two-
dimensional data communication systems, IEEE Trans. Commun., vol.28, no. 11, pp. 18671875, 1980.
[19] P. Yuvapoositanon, Blind Adaptive Techniques for Direct-Sequence Code Division Multiple Access Receivers, Ph.D. thesis, Imperial College,University of London, England, 2002.
[20] V. J. Matthews and Z. Xie, A stochastic gradient adaptive filter withgradient adaptive step size, IEEE Trans. Signal Processing, vol. 41,no. 6, pp. 20752087, 1993.