A Decision Feedback CP-Assisted CDMA SchemeUsing Despreading Before Equalization

Dah-Chung Chang, Ruo-Yu Wu, and Yung-Fang ChenDepartment of Communication Engineering, National Central University, Taiwan

E-mail: {dcchang, 975203010, yfchen}@cc.ncu.edu.tw

Abstract—We study a cyclic prefix (CP) assisted method usedfor the uplink of code division multiple access (CDMA) systemsin which the code despreading is performed in front of thefrequency domain equalizer (FDE). The main advantage of thenew despreading scheme is that the required points of discreteFourier transform in the FDE are significantly reduced comparedto the conventional CDMA scheme. Although the CDMA systemcan enhance system capacity as the number of users increases,the multiuser interference (MUI) needs to be mitigated. In ourwork, a minimum mean square error (MMSE) MUI cancellationmethod in a combination of decision feedback equalization isproposed. Simulation results show that the proposed schemeachieves the similar performance to the conventional CP-CDMAsystem, and hence, the new MUI cancellation method essentiallyeliminates most of MUI.

Index Terms—CDMA, Single Carrier, Cyclic Prefix, OFDM,Frequency Domain Equalization, Decision Feedback, MultiuserInterference.

I. INTRODUCTION

Direct sequence code division multiple access (DS-CDMA)is the key single carrier (SC) wireless access technologyadopted in the 3G communication system. Two kinds ofreceivers are typically used in a DS-CDMA system: Rakereceiver and time-domain equalization receiver. One of themain drawback is that for the capacity of combating wirelessmultipath channels, the DS-CDMA system is not as goodas orthogonal frequency division multiplexing (OFDM). Theperformance of DS-CDMA receivers depends on the propertyof the channel and the traffic load. Although OFDM has beenwell recognized as the new technology for 4G and beyond,the inherent property of high peak to average power ratio(PAPR) of OFDM signals limits the use of uplink becauseof a stringent requirement of low power [1]. Hence, the SCbased system such as SC frequency division multiple access(FDMA) or SC-FDMA [2] [3] is considered as the uplink in3GPP long term evolution advanced (LTE-A) technology.

The capacity of CDMA can be improved as the numberof users increases. However, the multiuser interference (MUI)will arise as the number of users increases, and hence, MUImitigation becomes an important issue in DS-CDMA systems.A lot of research can be found in the literature [4] [5]. ThoseMUI cancellation methods mainly include minimum meansquare error and zero-forcing receiver [6], subtractive inter-ference cancellation, and more generally, parallel interferencecancellation (PIC) [7], and successive interference cancellation

0This work was supported by the National Science Council of Taiwan undercontracts NSC99-2221-E-008-039 and NSC99-2221-E-008-053.

(SIC) [8]. In recent years, a new method was proposed touse 2-dimensional orthogonal variable spreading factor (2D-OVSF) codes combined with chip-interleaved coding for DS-CDMA uplink transmission [9]. The method intended toreduce the complexity of canceling MUI.

One new advanced version regarding DS-CDMA is the SCcyclic prefix (CP) assisted DS-CDMA (CP-CDMA) whichhas began its study for the application of broadband cellularsystem [10]. As inserting a CP portion prior to the transmittedsymbol block, the relationship between the transmitted signaland the channel impulse response can be transformed tocircular convolution from the linear convolution under settinga proper length of CP. Hence, CP-CDMA has better capa-bility of combating multipath channels than the conventionalDS-CDMA receivers through the one-tap per subcarrier fastFourier transform (FFT) based equalizer. However, a large FFTsize due to the chip rate operation and MUI cancellation stillattract continuous study on CP-CDMA.

In a conventional CP-CDMA receiver, the frequency domainequalizer (FDE) is performed before the code spreader trans-forming chip based samples to symbol based samples [10].In this paper, we study a new CP-CDMA receiver in wherethe code despreader is placed in front of the FDE. In the newreceiver, the FDE operates at symbol rate rather than at chiprate as the conventional CP-CDMA receiver does. In addition,the FFT size in the new receiver becomes smaller. However,the MUI effect can be more complicated. Some past worksregarding the new receiver can be found in the literature [11][12]. They proposed different block spread schemes to elimi-nate MUI. Their works need to limit the length of the channeldelay spread such that it is usually rather small compared tothe delay spread dealt with in typical OFDM systems. In ourwork, we are concerned with a long channel delay spread thatis usually considered in an OFDM system, e.g., one-quarterof an OFDM symbol block. We analyze the desired signalof the new receiver and utilize the minimum mean squareerror (MMSE) receiver to remove the residual intersymbolinterference (ISI). To mitigate MUI, we propose a decisionfeedback MUI canceler to achieve a satisfying performance.Simulations show that the new receiver with the proposedalgorithm has similar bit error rate (BER) performance to theconventional receiver as well as a rather low complexity inimplementing the FDE.

978-1-61284-233-2/11/$26.00 ©2011 IEEE

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AddCyclicPrefix

AddCyclicPrefix

Channel

Channel

(1)c

( )Pc

Rx

(1)d

( )Pd

(1)s

( )Ps( )Pcps

(1)cps

(1)h

( )Ph

Tx

Tx

(a) Transmitter

RemoveCyclicPrefix

DespreadingMMSEequalizer

Detection

AWGN z

Rx

FFTcNL -point

IFFTcNL -point

(b) Conventional Receiver

RemoveCyclicPrefix

DespreadingMMSEequalizer Detection

AWGN z

Rx

IFFTN-point

FFTN-pointcpr r y Y

G

D d d

(c) Proposed Receiver

Fig. 1: System Model

II. PROPOSED CP-CDMA SYSTEM MODEL

In Fig. 1, we depict a CP-CDMA system model which isdivided into transmitter and receiver parts. The CP-CDMAreceiver realizes the FDE by FFT and IFFT. As shown in Fig.1 (b) is the conventional receiver and Fig. 1 (c) is the proposedreceiver. We notice that the FFT sizes in these two receiversare different. Let N be the number of a symbol block andLc the length of the spreading code. The FFT size in theconventional receiver is NLc since the FDE operates at chiprate while that in the proposed receiver is N since the FDEoperates at symbol rate. Hence, compared to the conventionalreceiver, the proposed receiver does not require a large FFTsize, and thus, significantly reduces the complexity of the FDE.

Fig. 1 (a) is the P-user CP-CDMA transmitter, in whichthe transmitted symbol d

(u)n is spreaded by the code c(u)

where u is the user index with 1 ≤ u ≤ P and nis the symbol index with n = 0, 1, · · · , N − 1. Denoteby d(u) = [d(u)

0 , d(u)1 , d

(u)2 , · · · , d

(u)N−1]

T the input symbolblock for user u. The spreading code for user u is c(u) =[c(u)

0 , c(u)1 , · · · , c

(u)Lc−1], where c

(u)i ∈ (1,−1) with 0 ≤ i ≤

Lc−1. Each symbol is divided into Lc chips. After spreading,the spreaded block becomes

s(u) = [d(u)0 c(u) , d

(u)1 c(u) , · · · , d

(u)N−1c

(u) ]T

= [s(u)o , s

(u)1 , s

(u)2 , · · · , s

(u)NLc−1]

T . (1)

In order to avoid multipath channel effect leading to ISI,a CP of length Ng is added. We denote the channel impulseresponse by h(u) = [h(u)

0 , h(u)1 , · · · , h

(u)L−1]

T , where L is thetap number of the multipath channel at chip rate and L < Ng .Suppose the channel response can be treated non-time-varyingwithin a symbol block. Without incurring the synchronizationissue for simplicity, the received signal after removing the CP

can be expressed as

r =P∑

u=1

h(u) ⊗ s(u) + z (2)

where ⊗ denotes circular convolution and z is the additivewhite Gaussian noise (AWGN). Here, assume s(m) is thedesired user’s signal and s(u), u �= m, are other users’ signals.From (2), we have

r = h(m) ⊗ s(m) +P∑

u=1u�=m

h(u) ⊗ s(u) + z (3)

where h(m) is the channel impulse response of the desireduser. Notice that the first term at the right-hand side of (3)is the desired receiving signal. After despreading for the mthuser, the despreaded signal is

y(m)n =

1Lc

(n+1)Lc−1∑t=nLc

rtc(m)t

=1Lc

(n+1)Lc−1∑t=nLc

(h(m)t ⊗ s

(m)t )c(m)

t

︸ ︷︷ ︸(1)

+1Lc

P∑u=1u�=m

(n+1)Lc∑t=nLc

(h(u)t ⊗ s

(u)t )c(m)

t

︸ ︷︷ ︸(2)

+zn (4)

where the first term is the mth user’s signal, the second termis the MUI signal, and the third term zn,

zn =1Lc

(n+1)Lc−1∑t=nLc

ztc(m)t (5)

is also an additive white Gaussian noise (AWGN).

III. PROPOSED CP-CDMA SYSTEM

A. The Desired Signal

Let us begin with the first term of (4) which can be writtenas

y′(m)n =

1Lc

Lc−1∑t=0

L−1∑j=0

h(m)[nLc+t−j]L

s(m)j c

(m)t (6)

where [·]L denotes [·] modulo L operation. Note that let Lh

denote the length of channel impulse response at symbol rate,then L = LcLh. Changing the variable j = lLc + k withl = 0, 1, ..., Lh − 1 and k = 0, 1, ..., Lc − 1, and using the

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relationship s(m)lLc+k = d

(m)l c

(m)k , (6) becomes

y′(m)n =

1Lc

Lc−1∑t=0

Lh−1∑l=0

Lc−1∑k=0

h(m)[nLc+l−lLc−k]L

s(m)lLc+kc

(m)t

=Lh−1∑l=0

1Lc

Lc−1∑t=0

Lc−1∑k=0

c(m)t c

(m)k h

(m)[(n−l)Lc+(t−k)]L

︸ ︷︷ ︸M

(m)n−l

d(m)l (7)

Consider the term M(m)n−l as denoted in (7), we can divide the

result obtained from the summation over t into two parts:(i) When t = k,

M(m)n−l =

1Lc

Lc−1∑t=0

c(m)2

t h(m)[(n−l)Lc]L

= h(m)[n−l]Lh

(8)

where c(m)2

t = 1 and h(m)[n−l]Lh

can be treated as the equivalentchannel impulse response at symbol rate.

(ii) When t �= k,

M(m)n−l =

1Lc

Lc−1∑t=0t�=k

Lc−1∑k=0

c(m)t c

(m)k h

(m)[(n−l)Lc+(t−k)]L

≈h

(m)[n−l]Lh

Lc

Lc−1∑t=0t�=k

Lc−1∑k=0

c(m)t c

(m)k (9)

where the approximation is true when assuming the channel isalmost time-invariant for every chips during a symbol interval.As we choose that c

(m)t approaches a white sequence and Lc

is large enough, the result obtained from (9) is close to zero.Combining (7), (8), and (9), the DFT output of y

′(m)n is

Y′(m)k = DFTN{

Lh−1∑l=0

h(m)[n−l]Lh

d(m)l +

Lh−1∑l=0

M(m)n−ld

(m)l }

= DFTN{h(m)n ⊗ d(m)

n + w(m)n (h, c, d)}

= H(m)k D

(m)k + W

(m)k (h, c, d) (10)

with k = 0, 1, · · · , N − 1 where H(m)k = DFTN{h(m)

n },D

(m)k = DFTN{d(m)

n } , and W(m)k (h, c, d) =

DFTN{w(m)n (h, c, d)} which accounts for a nonzero

value of M(m)n−l obtained from (9). Note that the value of

W(m)k (h, c, d) is complicated and related to channel response,

spreading code, and transmitted data. However, it can be asmall variation in practical application as we explained above.For simplicity, we treat it an additional noise at the receiverin this paper.

B. MUI Removal

From (10), we can obtain D(m)k from the desired signal

Y′(m)k by using least square error (LSE) or MMSE methods

once the channel response is given. However, the ideal desiredsignal can be only obtained when the MUI signal, i.e., thesecond term of (4) is properly removed. Note that from (3) the

MUI term can be canceled as the channel impulse responseh(u) and the transmitted signal s(u) can be estimated. Takingadvantage of the residual MUI being able to be further reducedafter passing through the despreading operation, we proposean MUI removal algorithm in front of the code despreader asfollows.

First, we merely apply the despreading to roughly reducethe effect of MUI and noises, then a frequency domain MMSEequalizer is used to yield the intial estimate of d(u), denotedas d(u). From (4) and (10), we can obtain the DFT outputsignal of y

(m)n as follows:

Y(m)k = H

(m)k D

(m)k + Z

(m)k,MUI + Z

(m)k (11)

where Z(m)k,MUI is the DFT output of the second term in (4)

and Z(m)k is simply treated as a new AWGN in the frequency

domain, Z(m)k = W

(m)k (h, c, d)+DFTN{zn}. From (11), we

can find the coefficients of the MMSE equalizer by giving

G(u)1,k =

H(u)∗k∣∣∣H(u)

k

∣∣∣2 + σ2w

σ2d

(12)

where σ2w = σ2

MUI + σ2z is the noise power in total, σ2

MUI =E[Z(m)H

k,MUIZ(m)k,MUI], σ2

z = E[Z(m)Hk Z

(m)k ], and σ2

d is the signalpower. Here, we can deal with MUI as an interference noisecoming from the channel, then the spreading code with longerlength has better capability of MUI suppression.

Then, we utilize the initial estimate d(u) or the decisionfeedback result d(u) to yield s(u). As the channel estimateh(u) is obtained as well, we can use (3) to calculate the desiredsignal for mth user in the following:

v(m) = r −P∑

u=1u�=m

h(u) ⊗ s(u)

= h(m) ⊗ s(m) + g(m) + z (13)

where g(m) denotes the residual MUI for mth user and

g(m) =P∑

u=1u�=m

h(u) ⊗ s(u) −P∑

u=1u�=m

h(u) ⊗ s(u) (14)

Performing despreading for the signal after removing MUI,from (13) we can express the despreaded signal as follows:

y(m)n =

1Lc

(n+1)Lc−1∑t=nLc

v(m)t c

(m)t

= h(m)n ⊗ d(m)

n + w(m)n (15)

where the noisy term w(m)n is

w(m)n =

1Lc

(n+1)Lc−1∑t=nLc

g(m)t c

(m)t + w(m)

n (h, c, d) + zn. (16)

After DFT, we have

Y(m)k = H

(m)k D

(m)k + W

(m)k (17)

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FFT IFFT

IFFT C P

FBg

( )mnd

( )ˆ mnd

( ) ( ) ( ), , 1 , 2* m m mk FF k FF k FFG G G

( )0, 1mFFG

( )1, 1

mN FFG

( )0, 2mFFG

( )1, 2mN FFG

( )mny

( )mkY

Fig. 2: Structure of the Decision Feedback SC FDE Receiver

where W(m)k = DFTN{w(m)

n }N−1n=0 . Obviously, the residual

MUI effect g(m) can be reduced by the spreading operation.Hence, the noise variance in (17) is far less than that in (11).From (17), we have the MMSE equalizer whose coefficientscan be computed by

G(m)2,k =

H(m)∗k∣∣∣H(m)

k

∣∣∣2 + σ2w

σ2d

(18)

and σ2w = E[W (m)H

k W(m)k ].

C. SC-FDE with Decision Feedback

Since the noise W(m)k in (17) involves the term

W(m)k (h, c, d). As explained in section III. A and (10), M

(m)n−l

can be very small based on choosing proper c(m)t and a

sufficiently large length of Lc. However, if those conditionsare not perfect such that M

(m)n−l cannot be ignored, the effect

of W(m)k (h, c, d) brings ISI. In this case the DFE can be

developed to get rid of ISI. A decision feedback SC-FDEalgorithm can be found in [2] [3], which is shown in Fig.2. In this figure, the MMSE solution G

(m)k,FF is the coefficient

of the feedforward filter (FF) in the frequency domain, whichprimarily eliminates the effect of channel response. From (17),the FFT output signal first passes through G

(m)k,FF and then

is transformed into the time-domain signal after IFFT. Thedecision feedback equalized signal d

(m)n is obtained by

d(m)n =

1N

N−1∑k=0

(G(m)k,FF Y

(m)k )ej 2π

N kn −NFB∑l=1

g(m)l,FB d

(m)n−l (19)

where d(m)n is the decision output signal, g

(m)l,FB is the time-

domain feedback filter coefficient, and NFB is the tap numberof the feedback filter, which value is chosen based on ensuringthat the ISI from previous symbols is canceled. Hence, notingthat from (17), the ISI effect is related to w

(m)k (h, c, d), we

can choose NFB = Lh in this case. By MMSE criterion, theFF coefficients can be obtained by

G(m)k,FF =

⎛⎜⎝ H

(m)∗k∣∣∣H(m)

k

∣∣∣2 +σ2w

σ2d

⎞⎟⎠ × (1 + G

(m)k,FB). (20)

TABLE I: Complexity Comparison of the FDE

Proposed Conventionalmultiplication multiplication

FFT N2 log2(N) NLc

2 log2(NLc)IFFT N

2 log2(N) NLc

2 log2(NLc)Equalizer N NLc

total N [log2(N) + 1] NLc[log2(NLc) + 1]

We also note that the signal d(m)n in (19) requires previous

feedback output signals d(m)n−l with l = 1, 2, · · · , NFB . The

first term in (20) is G(m)k,FF1 which is the same result obtained

in (18) and is used to calculate the last NFB symbols at theinitial NFB symbols. The second term is G

(m)k,FF2 in which

G(m)k,FB =

NF B∑l=1

g(m)l,FBe−j 2π

N kl. The FB coefficients g(m)l,FB can

be referenced to the work [2].

IV. COMPLEXITY COMPARISON

The proposed scheme moves the code despreader to the fontof the frequency domain equalizer in order to reduce the costof equalization. The comparison of computational complexityis listed in Table I. Since the main cost of complexity ismultipliers, we list the required number of multiplication. Thesimilar result can be observed for the number of additionoperation. Here, we consider the MUI cancellation at chip rate.The complexity due to the MUI canceler is therefor the samefor the both schemes. The main difference is on the operationof FFT and IFFT. The FDE in the conventional scheme workswith the FFT size of Lc times more than that used in theproposed scheme. For an example of numerical comparison,assuming N = 64 and Lc = 31, the required numberof complex multiplication for the FDE in the conventionalscheme is about 23808 while that in the proposed scheme isonly 448. The conventional scheme leads to a ratio of about53 more than the proposed scheme. As the number of NLc

increases, the proposed scheme saves more computational loadthan the conventional scheme.

V. SIMULATION RESULTS

In our simulation, the transmitted data are randomly gen-erated through Gray-coded 16-QAM modulation and N = 64symbols are used for a block at symbol rate. The length ofCP is chosen as 1

4N and the tap number of the multipathchannel is set to be 8 which is the effective channel delayspread at symbol rate. The spreading codes are pseudo random(PN) sequences with two different length 15 and 31 forthe purpose of comparison. To compare the MUI removalefficiency, two and four users with independent multipathchannels are simulated. For each user, we assume two initialuncorrelated symbol blocks, i.e., 128 pilots are transmittedfor LSE channel estimation. We regularly insert block symbolpilots for noise power estimation.

In Figs. 3 and 4 we compare the conventional CP-CDMAand the proposed CP-CDMA schemes with Lc = 15 andLc = 31 for the case of two users. The efficiency of the

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

100

CP−CDMA (two users, 15 chips)

SNR(dB)

BE

R

proposed(w/o DFE,MUIC)proposed(+DFE)convent.(w/o DFE,MUIC)convent.(+DFE)proposed(+MUIC)convent.(+MUIC)proposed(+DFE,+MUIC)convent.(+DFE,+MUIC)Ideal(one user, no MUI)

Fig. 3: Comparison of BER performance with Lc=15 for twousers.

0 5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

CP−CDMA (two users, 31 chips)

SNR(dB)

BE

R

proposed(w/o DFE,MUIC)proposed(+DFE)convent.(w/o DFE,MUIC)convent.(+DFE)proposed(+MUIC)convent.(+MUIC)proposed(+DFE,+MUIC)convent.(+DFE,+MUIC)Ideal(one user, no MUI)

Fig. 4: Comparison of BER performance with Lc=31 for twousers.

DFE and MUI cancellation (MUIC) functions is also shown.The proposed scheme without (w/o) using MUIC results in theworst performance since the move of the code despreader leadsto severe MUI. In contrast, the conventional scheme is betterthan the proposed when the DFE and MUIC are not utilized.As the proposed scheme utilizes the MUIC function, the BERperformance shows dramatic improvement. The performanceimproved by the DFE is not as significant as that obtainedby MUIC because the effect due to W

(m)k (h, c, d) is far less

than MUI. When the DFE and MUIC functions are bothused, the proposed scheme achieves rather close result to theconventional scheme. The same comparison for the case offour users is shown in Fig. 5. In this case, the proposed schemeonly shows a slightly worse performance than the conventionaleven though the MUI effect becomes more serious. However,the proposed scheme gains great reduction of complexity.

VI. CONCLUSION

We study a new CP-CDMA receiver with proposed DFE andMMSE MUI cancellation algorithm. The merit of the new CP-CDMA receiver is that it can be performed with a smaller FFTsize than the conventional CP-CDMA receiver. To preserve

5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

100

CP−CDMA (four users, 31 chips)

SNR(dB)

BE

R

proposed(w/o DFE,MUIC)proposed(+DFE)convent.(w/o DFE,MUIC)convent.(+DFE)proposed(+MUIC)convent.(+MUIC)proposed(+DFE,+MUIC)convent.(+DFE,+MUIC)Ideal(one user, no MUI)

Fig. 5: Comparison of BER performance with Lc=15 for fourusers.

this property, the proposed algorithm can work in a multipathchannel of long delay spread as a typical OFDM systemcan do. Besides, the proposed algorithm achieves almostsimilar performance to the conventional CP-CDMA receiver.Although the proposed algorithm has shown its effectivenessfor CP-CDMA on dealing with the severe multipath channelas OFDM typically can combat, the MUI free reception in thiscase is still under work.

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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 IEEE ICC 2011 proceedings