Joint Frequency-domain Equalization and Antenna Diversity Combining for
Orthogonal Multicode DS-CDMA Signal Transmissions in A
Frequency-selective Fading Channel
Takeshi ITAGAKI*1 and Fumiyuki ADACHI*2
Dept. of Electrical and Communication Engineering, Graduate School of Engineering, Tohoku
University
05 Aza-Aoba, Aramaki, Aoba-ku, Sendai, 980-8579 Japan
E-mail: *1 itagaki@mobile.ecei.tohoku.ac.jp, *2 adachi@ecei.tohoku.ac.jp
Abstract
Orthogonal multicode direct sequence code divisionmultiple access (DS-CDMA) has a flexibility in offeringvariousdata
rate
services.
However,inafrequency-selective fading channel, the bit error rate (BER)
performance is severely degraded since othogonality amongspreading codes is partially lost. In this paper, we apply
frequency-domain equalization and antenna diversity
combining, used in multi-carrier CDMA (MC-CDMA), to
DS-CDMA in order to restore the code othogonality whileachieving frequency and antenna diversity effect. It is foundby computer simulations that the joint use of
frequency-domain equalization and antenna diversity
combining can significantly improve the BER performanceof orthogonal multicode DS-CDMA in a frequency-selective
fading channel.

Keywords
Frequency-domain
equalization,antennadiversity,
multicode DS-CDMA, frequency-selective fading

1. Introduction

2. Joint Frequency-domain Equalization andAntenna Diversity Reception
Transmission system model of orthogonal multicode
DS-CDMA using frequency-domain equalization andantenna diversity combining is illustrated in Fig. 1.
{c0 ( k )}{d i }

Insertion
of GI

…
…

…
S/P

{cC −1 ( k )}{c scr ( k )}Scramble-sequence

Walsh-Hadamard
sequence(a) Transmitter
From other
wm(0) branches

AWGN
MF

㰿
{cC∗ −1 (k )}㰿(b) ReceiverFig. 1 Transmission system model

P/S
…

…

…

wm(SF-1)

P/S
…
IFFT

…

Removalof GI{c0∗ (k )}∗{c scr
(k )}

FFT
…
S/P

Orthogonal multicode DS-CDMA is flexible inoffering various data rate services by changing the numberof parallel orthogonal spreading codes [1]. In a DS-CDMA
mobile communications system, orthogonal multicode
DS-CDMA is used for the downlink; each spreading code is
assigned to a different user. In DS-CDMA, the multipath
fading channel [2] is resolved by a bank of correlators(known as the rake fingers) into many distinct paths forcoherent rake combining. Coherent rake combining can
exploit the channel frequency-selectivity to improve the
BER performance through path diversity effect (similareffect to antenna diversity) [3]. However, as the number ofresolvable paths increases, the receiver complexity
increases due to increasing number of rake fingers.
Furthermore, increased inter-path interference (IPI)
resulting from time asynchronism of different paths offsets
the performance improvement by rake combining.
Recently, multi-carrier CDMA (MC-CDMA) has been
attracting much attention for broadband wireless multiple
access [4]. In MC-CDMA, many orthogonal subcarriers areused and the data symbol to be transmitted is spread over
several subcarriers using frequency-domain orthogonal
spreading code. At an MC-CDMA receiver, thefrequency-domain equalization is applied to the received
signal for restoring the code orthogonality. Throughfrequency-domain equalization and despreading, frequency
diversity effect is attained, resulting in significantly
improved BER performance of MC-CDMA compared tothat of DS-CDMA with coherent rake combining [5].
Quite recentry, application of frequency-domain
equalization to single-carrier (SC) transmission has been
attracting attention [6]. It is pointed out in [7] that thefrequency-domain equalization can be applied to orthogonal
multicode DS-CDMA for improving its BER performance ina severe frequency-selective channel. In this paper, joint
frequency-domain equalization and antenna diversity
combining is considered for the reception of the orthogonal
multicode
DS-CDMAsignalsinaseverefrequency-selective channel. Section 2 presents weights forjoint frequency-domain equalization and antenna diversity
combining. Various frequency-domain equalization schemes
using maximal ratio combining (MRC), equal gain
combining (EGC), orthogonal restoration combining (ORC)
and minimum mean square error combining (MMSEC) [8]are considered. In Sect. 3, the achievable BER
performances are evaluated by computer simulation. Someconclusions and future work are offered in Sect. 4.

{dˆi }

2.1. Insertion of Guard IntervalFrequency-domain Equalization

(GI)

for

Fast Fourier transform (FFT) is applied for decomposing
the received DS-CDMA signal into the SF subcarriercomponents (the term “subcarrier” is used althoughsubcarriers are not used for data modulation) for
frequency-domain equalization and antenna diversity
combining. At the receiver, the received DS-CDMA signal
with spreading factor SF on each diversity antenna issampled at the chip rate. The FFT windowsize is equal to
the spreading factor SF. For applying FFT, the received
signal waveform must be treated as a periodic function intime with the repetition period of SF chips. Rememberingthat the wireless channel consists of many time delayed
paths [2], cyclic extension of the transmitting waveform
needs to be applied; a copy of the last part of N g chips in
the transmitting signal waveform is inserted into thebeginning of the signal waveform as a guard interval (GI)as in MC-CDMA (see Fig. 2). The GI length T g needs to belonger than the largest time delay difference in the channel.
Letting T c be the chip period, T g is T g =T c N g and theeffective symbol length is T s =T c SF. Then, GI-inserted
DS-CDMA signaling period is T=T s +T g . Hence, the data
symbol rate decreases by a factor of (1+ T g /T s ) or (1+N g /SF) times compared to the no GI insertion system (whichuses rake combining instead of frequency-domain
equalization) and also power penalty of (1+T g / T s ) or (1+N g /SF) is produced.
After joint one-tap frequency-domain equalization andantenna diversity combining, the time-domain signal
waveform is obtained by applying inverse FFT (IFFT) fordata demodulation. Since the IFFT operation is the linear
combination of all SF subcarrier components, thewell-known frequency diversity effect can be attained in afrequency-selective fading channel.

signal waveform s(k) can be expressed using the equivalent
baseband representation as
C −1s (k ) = 2 Ec / Tc

i i

scr

i =0

where E c represents the chip energy. The GI-inserted signal
waveform can be expressed as~s (k ) = s(k mod SF ) , k = − N g ~ SF − 1 ,(2)which is transmitted over a frequency-selective fading
channel and received by M antennas at the receiver.
The channel is assumed to be composed of L distinct
propagation paths with different time delays. The complex
path gain and time delay of the lth path corresponding to
the mth antenna are respectively denoted by ξm,l and τl . The
signal waveform received on the mth antenna at time k may
be expressed as
L −1

τl

¦ξ ~s (k − T ) + η (k ) , (3)rm (k ) =

m, l

c

l =0

m

where { η m (k)} is the zero-mean complex Gaussian processwith variance 2N 0 /T c due to the additive white Gaussian
noise (AWGN) having the one-sided power spectrum
density N 0 . Block fading, where the path gains stay constantover one signaling period, is assumed. After removal of GI,
the received signal is decomposed into SF subcarriercomponents {Rm (n); n = 0 ~ SF − 1} by applying SF-point
FFT. R m (n) is given by
SF −1

Rm (n) =k¦ r (k ) exp(− j 2πn SF ) , (4)m

k =0= H m (n) S (n) + η~m (n)InsertionGI

Spread signal

Ng

SF chips

where H m (n) and S(n) are the Fourier transforms of thechannel impulse response and the transmitted multicode
DS-CDMA signal waveform, respectively. In Eq.(4),η~m (n) represents the noise component at the nthsubcarrier frequency. R m (n) is multiplied by the weightw m (n) for joint frequency-domain equalization and antenna
diversity combining to obtain

chips
-Ng

¦ d c (k )c (k ), k = 0 ~ SF − 1 , (1)

SF

0

time
k

Fig. 2 Insertion of GI

2.2. Signal representation
For simplicity, we consider the 0th signaling interval,i.e., a time interval of k=-N g ~SF-1. At the transmitter,binary data sequence is transformed into quadrature phaseshift keying (QPSK) symbol sequence {d i ; i=0~C-1}, eachsymbol is then spread using orthogonal spreading codes{ci (k ) = ±1; i = 0 ~ C − 1, k = 0 ~ SF − 1}ofspreadingfactor SF. The resultant orthogonal C parallel chipsequences are added (i.e., code multiplexed) and thenmultipliedbyascramblesequence{c scr ( k ) = ±1; k = ⋅⋅,−1,0,1,⋅⋅} for making the transmitting
signal noise-like.
Throughout the paper, the chip-spaced discrete time
representation is used. The generated multicode DS-CDMA~
R ( n) =M −1¦ R (n)w (n) . (5)
m

m

m =0

The time-domain signal waveform
SF-point IFFT is given by1~r (k ) =
SF

SF −1

~

{~r (k )} obtained byn¦ R (n) exp( j 2πk SF ) . (6)n=0Parallel despreading and descrambling operation isr (k )} to obtain the soft decision sample
performed on {~sequence {dˆ i } :
1
dˆi =
SF

SF −1

¦ r~(k )c (k )c (k ) , i = 0 ~ C − 1
k =0

∗
i

∗
scr

(7)

for succeeding data demodulation.

Propagation channel model

2.3. Weights
for
joint
frequency-domain
equalization
andantennadiversity
combining

No. of FFT samples
No. of antennas
Frequency-domain
equalization
Channel estimation∗­H m( n), MRC°∗
° H m ( n), EGC°
° H m ( n)°∗
° H m (n) , ORC° M −1wm (n) = ®
,(8)2
H m ( n)
°
° m=0
°
∗
Hm
( n)
°
, MMSEC
−1
1
−
M
°
§
·
E
2
c
°
¸
H m (n) + ¨¨ C¸°
© N0 ¹¯ m=0¦¦where E c /N 0 represents the average received chipenergy-to-AWGN power spectrum density ratio. The nth~subcarrier component R (n) of Eq. (5) after joint
frequency-domain equalization and antenna diversity
combining is now rewritten as
M −1§ M −1·~R ( n) = ¨wm (n) H m (n) ¸ S (n) +wm (n)η~m (n)
¨
¸
, (9)
m =0
© m=0
¹
~
= H (n) S (n) + η~′(n)¦¦where the first term is the signal component and the second
the noise component. ORC can completely restore thefrequency nonselective channel (thus, orthogonality of
spreading codes can be restored) but produces noise
enhancement. Restoration of frequency non-selectivity andnoise enhancement have a trade-off relationship. MMSEC
cannot completely restore the frequency non-selectivity but
minimizes the equalization error on each subcarrier.

3. Computer Simulation
Simulation condition is shown in Table 1. A very slow
L-path frequency-selective block Rayleigh fading channel
having uniform power delay profile is assumed. The timedelay τl of the lth path is assumed to be τl =lT c .
Table 1 Simulation condition
Data modulation
QPSK
Multicode
Spreading
BPSK
spreadingmodulation
Spreading factor SF=256
No.
parallel C=1~256
codes
Scramble code
M-sequence with a periodof 4095 chips
Guard interval
T g =32T c

Ideal

3.1. Frequency-domain ORC, EGC, MRC, and
MMSEC equalizations
The
average
BER
performanceswithfrequency-domain ORC, EGC, MRC, and MMSEC
equalizations are plotted as a function of the average
received signal energy per bit-to-AWGN power spectrum
density ratio E b /N 0 in Fig. 3 for various values of C in thecase of L=8 and M=1. E b/N 0 is defined as
E b/N 0 =(SF+N g )E c /N 0 . The MRC and EGC can achieve thefrequency diversity effect and there is no noise
enhancement. Hence, they provide very good BER
performances. However, this is only true for the single code
case (C=1). For the multicode case, the BER performancesof EGC and MRC degrade and the BER floors are seen due
to enhanced frequency-selectivity of the channel (the MRC
performance is much worse than the EGC performance). On
the other hand, the BER performance of ORC is insensitive
to the value of C because of perfect restoration of
frequency-nonselectivity, but its BER performance is evenworse than those of EGC and MRC for low
code-multiplexing order. This suggests that MMSEC can
only be used for frequency-domain equalization of
DS-CDMA signals. It can be clearly seen in Fig. 3(b) that
the MMSEC always provides the best BER performance
among the four equalization schemes. Hence, in thefollowing simulation, only MMSEC is considered.
1.E+00
L =8, M =1, C =1

ORC
EGC
MRC
MMSEC

1.E-01

Average BER

In this paper, a heuristic approach is taken following
the frequency-domain equalization used in MC-CDMA.
Various frequency-domain equalization schemes using MRC,
EGC, ORC and MMSEC are considered to compare theachievable BER performance in a frequency-selective
fading channel. Joint frequency-domain equalization anddiversity combining weight described in [9] can be used:

Block
Rayleighfadingchannel with L=1~32 paths
256 (=SF)
M=1~4
ORC, EGC, MRC, MMSEC

1.E-02

1.E-03

1.E-04

1.E-05
0

5

10

15

20

Average received E b /N 0 [dB](a) C=12530

1.E+00

1.E+00

M =1
L =8

L =8, M =1, C =256

1.E-01

Average BER

1.E-01

Average BER

Ideal Rake

1.E-02

1.E-03
ORC
EGC
MRC
MMSEC

1.E-04

1.E-050

510

1.E-02AWGN

1.E-03

䂾䇭C =16
㬍䇭C =32
䂔䇭C =64
䂺䇭C =128
䂦䇭C =256 MMSEC

1.E-04

1.E-05
15

20

25

0

30

20

25

35
C =256C =130

䂺 M =1
䂔 M =2
䂦 M =425
for BER=10-4

Performancecomparisonoffrequency-domain
MMSEC equalization and rake combining for multicode
case is illustrated in Fig. 4. As C increases, the BER
performance with rake combining significantly degrades
due to increasing inter-code interference (ICI) resulting
from IPI and hence BER floors appear. However, the BER
performance with frequency-domain MMSEC equalization
provides much better BER performance due to thefrequency diversity effect and no BER floor is present at
the cost of slightly reduced data rate and power penalty.
So far, we have assumed L=8. How the number of
propagation paths impacts the required E b /N 0 for BER=10 -4
with frequency-domain MMSEC equalization is plotted inFig. 5 for M=1, 2, and 4. It is clearly seen that as L
increases, the channel frequency-selectivity becomesstronger and increased frequency diversity effect can beobtained, thereby improving the BER performance withfrequency-domain MMSEC equalization as in the case of
MC-CDMA. The frequency diversity gain is defined here as
the reduced value in dB of the required E b /N 0 compared to
the L=1 case. The frequency diversity gain of as large as
about 25 䋨 19 䋩 dB is obtained for C=1 䋨 256 䋩 when L=32 and
M=1, respectively. Although the frequency diversity gain
becomes smaller as M increases, a gain of 4.7(3.5) dB canstill be achieved for C=1(256) when M=4.

15

Fig. 4 Performance comparison between frequency-domain
MMSEC equalization and rake combining for
multicode case. L=8.

Required average received E b / N 0 [dB]3.2. Performancecomparisonoffrequency-domain MMSEC equalization andrake combining

10

Average received E b /N 0 [dB]

Average received E b /N 0 [dB](b) C=256
Fig. 3 Performance comparison of frequency-domain ORC,
EGC, MRC, and MMSEC equalizations for L=8 and
M=1.

5

20
15
M =1
10

M =2
5

M =4

0
0

10

20

30

40

Number L of paths

Fig. 5 Impact of number L of paths.

3.3. Performance comparison of DS-CDMA and
MC-CDMA
It is interesting to compare the DS-CDMAperformance with MC-CDMA. Figure 6 illustrates the BER
performances with DS-CDMA and MC-CDMA both using
frequency-domain MMSEC equalization for the sametransmission condition, i.e., the same data rate and the same
spreading factor (spreading bandwidth), for an L=8-path
frequency-selective channel and no antenna diversity (M=1).
Also plotted is the result of OFDM using 256 subcarriers
and the same data rate. It is clearly seen that both
DS-CDMA and MC-CDMA with frequency-domain

MMSEC equalization can achieve almost the identical BER
performance because the same frequency diversity effect isobtained.
Figure 7 shows how antenna diversity improves the
BER performance for C=1 and 256. As the number M ofantennas increases, the BER performances of both
DS-CDMA and MC-CDMA with frequency-domain
MMSEC equalization consistently improves. Again there isno performance difference between DS-CDMA and
MC-CDMA. When M=4, the BER=10 -4 can be achieved at
the average E b/N 0 of as small as 5dB even for C=256.
1.E+00

M =1
L =8

DS-CDMA with MMSEC
MC-CDMA with MMSEC䋫䇭C =1
䂾䇭C =16
㬍䇭C =32
䂔䇭C =64
䂺䇭C =128
䂦䇭C =256

Average BER

1.E-01

1.E-02

1.E-03AWGN

4. Conclusion
In this paper, joint use of frequency-domain
equalization and antenna diversity combining wasconsidered for improving the DS-CDMA signaltransmission performance in a frequency selective fading
channel and the achievable BER performance was evaluatedby computer simulation. In the case of rake combining, as
the number C of codes increases, the BER performance withrake combining significantly degrades and BER floors
appear due to increasing IPI. However, the BER
performance with frequency-domain MMSEC equalization
provides much better BER performance due to thefrequency diversity effect and produces no BER floors at
the cost of slightly reduced data rate and power penalty. It
was shown that both DS-CDMA and MC-CDMA withfrequency-domain MMSEC equalization can achieve almost
identical BER performance for any number of diversity
antennas.
In this paper, ideal channel estimation was assumed.
Pilot assisted channel estimation can be applied to estimate
the channel gains. Pilot sequence design and the evaluationof the BER performance using a practical channel
estimation method is left for an interesting future study.

OFDM

References[1]

1.E-04[2]

1.E-05
0

5

10

15

20

25

[3]

Average received E b /N 0 [dB]Fig. 6 Performance comparison of DS-CDMA and
MC-CDMA both using frequency-domain MMSEC
equalization for no antenna diversity (M=1). L=8.

[4]
[5]

1.E+00L =8

DS-CDMA with MMSEC
MC-CDMA with MMSEC

Average BER

1.E-01[6]

䂹䇭C =256
䂺䇭C =1[7]

1.E-02

1.E-03

M =1

[8]

M =4
M =2

1.E-04[9]
1.E-05
0

5

10

15

20

Average received E b /N 0 per antenna[dB]

Fig. 7 Performance comparison of DS-CDMA and
MC-CDMA both using frequency-domain MMSEC
equalization when antenna diversity is used. L=8.

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