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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: peerapol@mut.ac.th

Sutat SuwannajanDepartment of Electronic Engineering

Mahanakorn University of Technology

Nong-Chok, Bangkok, Thailand 10530

Email: sutats@hotmail.com

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.

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