IEICE TRANS. COMMUN., VOL.E92–B, NO.1 JANUARY 2009219

PAPER

Channel Capacity of MC-CDMA and Impact of Residual ICI

Koichi ADACHI†a), Student Member, Fumiyuki ADACHI††, and Masao NAKAGAWA†, Fellows

SUMMARY Orthogonal frequency division multiplexing (OFDM),which uses a number of narrowband orthogonal sub-carriers, is a promis-ing transmission technique. Also multi-carrier code division multi-access(MC-CDMA), which combines OFDM and frequency-domain spreading,has been attracting much attention as a future broadband wireless access. Itwas shown that MC-CDMA has lower channel capacity than OFDM, due tointer-code interference (ICI) resulting from orthogonality distortion causedby frequency-selective fading. Recently, many ICI cancellers have beenproposed to mitigate the effect of ICI. In this paper, we derive a channelcapacity expression for MC-CDMA assuming perfect ICI cancellation tak-ing into account both frequency diversity gain and space diversity gain andcompare it to that of OFDM. Furthermore, we derive a channel capacityexpression for the case of imperfect ICI cancellation to discuss the impactof the residual ICI.key words: MC-CDMA, channel capacity, ICI cancellation, theoreticalstudy

1. Introduction

For future wireless communication systems, the develop-ment of a broadband wireless access technique is required.In a broadband channel, the frequency-selectivity of thechannel gets stronger and the bit error rate (BER) per-formance of single-carrier transmission is significantly de-graded owing to the severe inter-path interference (IPI). Re-cently, multi-carrier transmission technique, which uses anumber of narrowband orthogonal sub-carriers, e.g., suchas orthogonal frequency division multiplexing (OFDM) andmulti-carrier code division multi-access (MC-CDMA), hasbeen attracting much attention [1].

In MC-CDMA, each data symbol to be transmitted isspread over a number of orthogonal sub-carriers by an or-thogonal spreading code. Simple one-tap frequency do-main equalization (FDE) per sub-carrier can be used to com-pensate the amplitude and phase distortions of each sub-carrier. The transmission rate can be increased by usingcode-multiplexing. However, the transmission performanceof MC-CDMA degrades compared to that of OFDM sincethe inter-code interference (ICI) remains after FDE (this ICIis called the residual ICI) [2]. FDE can also be applied to

Manuscript received January 15, 2008.Manuscript revised July 6, 2008.†The authors are with School of Science for Open and Envi-

ronmental Systems, Graduate School of Science and Technology,Keio University, Yokohama-shi, 223-8522 Japan.††The author is with the Department of Electrical and Com-

munication Engineering, Graduate School of Engineering, TohokuUniversity, Sendai-shi, 980-8579 Japan.

a) E-mail: kouichi@nkgw.ics.keio.ac.jpDOI: 10.1587/transcom.E92.B.219

improve the BER performance of single-carrier transmis-sion and single-carrier direct-sequence code division mul-tiple access (DS-CDMA) [3], [4]; however, the presence ofresidual ICI limits the BER performance improvement. Re-cently, several ICI cancellation techniques have been pro-posed for both MC- and DS-CDMA to reduce the residualICI and its effectiveness was confirmed by computer simu-lation [5]–[7].

In [8], the impact of the channel parameters onthe MC-CDMA channel capacity was studied assumingNakagami-m fading. The capacity comparison of MC-CDMA with minimum mean square error (MMSE)-FDE,DS-CDMA with MMSE-FDE, and OFDM was theoreticallystudied in [9]. The channel capacity of MC-CDMA withmatched filter was evaluated in [10]. However, to the bestof authors’ knowledge, the theoretically achievable upper-bound of MC-CDMA with ICI cancellation has not beenstudied.

In this paper, we derive a channel capacity expressionfor MC-CDMA assuming perfect ICI cancellation takinginto account both the frequency diversity and space diver-sity gain and discuss the impact of the channel parameterssuch as the number of propagation paths and power delayprofile. The channel capacity of MC-CDMA assuming per-fect ICI cancellation is compared to that of OFDM derivedin [11]. Furthermore, in this paper, we discuss the impact ofthe residual ICI and show that MC-CDMA provides largerchannel capacity than OFDM when the residual ICI is suffi-ciently suppressed.

The rest of the paper is organized as follows. In Sect. 2,the system model is presented and the channel capacities ofMC-CDMA assuming perfect ICI cancellation and assum-ing imperfect ICI cancellation are derived. The numericalcomputation results are given in Sect. 3. Section 4 concludesthe paper.

2. Transmission System Model

2.1 Overall System Model

The transmission system model is shown in Fig. 1. Inthis paper, the sample-spaced discrete time representationis used. At the transmitter, the binary information se-quence is first channel coded, data modulated, and thenserial-to-parallel (S/P) converted into U parallel sequences{du (n) ; n = 0 ∼ �Nc/S F� − 1}, u = 0 ∼ U − 1, whereU is called the code multiplexing order and �x� denotes

Copyright c© 2009 The Institute of Electronics, Information and Communication Engineers

220IEICE TRANS. COMMUN., VOL.E92–B, NO.1 JANUARY 2009

Fig. 1 Transmission system model.

the largest integer smaller than or equal to x. Each datasymbol sequence is spread by orthogonal spreading code{coc,u (k) ; k = 0 ∼ S F − 1} and then code multiplexed,where S F is called the spreading factor. The code multi-plexed chip sequence is multiplied by a common scramblesequence {cscr (k) ; k = 0 ∼ Nc−1} to make the resulting mul-ticode signal white Gaussian noise like and then interleavedin the frequency-domain using an S F × �Nc/S F� chip-block interleaver. It has been shown [12] that if the scram-ble sequence is not used, the transmission performance de-grades because of the periodic property of auto-correlationand cross-correlation functions of the orthogonal spreadingcodes. Nc-point inverse fast Fourier transform (IFFT) is ap-plied to a block of code multiplexed chip sequence of Nc

chips to generate an MC-CDMA signal.Without loss of generality, we consider the transmis-

sion of the multi-code MC-CDMA signal s (t) during t =0 ∼ Nc − 1. s (t) can be expressed using the equivalent low-pass representation as

s (t) =Nc−1∑k=0

S (k) exp

(j2π

tNc

k

), (1)

with

S (k) =U−1∑u=0

S u (k) cscr (k)

=√

2PU−1∑u=0

d (�k/S F�) coc,u (k mod S F) cscr (k) ,

(2)

where P = Ec/ (S F · Tc) is the transmit power per spread-ing code with Ec being the signal energy per FFT sampleand Tc being the FFT sample length. After the insertion ofcyclic prefix (CP) into the guard interval (GI), the multicodeMC-CDMA signal, {s (t mod Nc) ; t = −Ng ∼ Nc − 1}, istransmitted, where Ng is the GI length.

The channel is assumed to be an L-path frequency-selective fading channel with each path being subjected toindependent fading. The impulse response of the channelassociated with the nr-th receive antenna can be expressedas

hnr (τ) =L−1∑l=0

hnr ,lδ (τ − τl) , (3)

where hnr ,l is the path gain with E[∑L−1

l=0

∣∣∣hnr ,l

∣∣∣2] = 1 (E [.]

denotes the ensemble average operation and δ (.) is the deltafunction) and τl is the time delay of the l-th path betweenthe transmit antenna and the nr-th receive antenna.

After removing the GI, the received signal {rnr (t) ; t =0 ∼ Nc − 1} on the nr-th antenna can be expressed as

rnr (t) =L−1∑l=0

hnr ,l s ((t − τl) mod Nc) + nnr (t) , (4)

where nnr (t) is a zero mean additive white Gaussian noise(AWGN) process having variance 2σ2 = 2N0/Tc with N0

being the single-sided power spectrum density. The receivedsignal is decomposed into Nc frequency components by ap-plying Nc-point FFT. The k-th sub-carrier component of thereceived signal, Rnr (k), is expressed as

Rnr (k) =1

Nc

Nc−1∑t=0

rnr (t) exp

(− j2π

kNc

t

)

=1

Nc

Nc−1∑t=0

⎛⎜⎜⎜⎜⎜⎜⎝L−1∑l=0

hnr ,l s (t − τl) + nnr (t)

⎞⎟⎟⎟⎟⎟⎟⎠ exp

(− j2π

kNc

t

)

= Hnr (k) S (k) + Πnr (k) , (5)

where Hnr (k) and Πnr (k) are the Fourier transforms ofthe channel impulse response and the AWGN, respectively.They are given as

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

Hnr (k) =L−1∑l=0

hnr ,l exp

(− j2π

kNcτl

)

Πnr (k) =1

Nc

Nc−1∑t=0

nnr (t) exp

(− j2π

kNc

t

) , (6)

Letting cu (k) = coc,u (k) cscr (k), the k-th sub-carriercomponent can be rewritten as

Rnr (k) = Hnr (k)

⎛⎜⎜⎜⎜⎜⎜⎝√

2PU−1∑u=0

du

(⌊k

S F

⌋)cu (k)

⎞⎟⎟⎟⎟⎟⎟⎠ + Πnr (k) .

(7)

Equation (7) can be further rewritten as

ADACHI et al.: CHANNEL CAPACITY OF MC-CDMA AND IMPACT OF RESIDUAL ICI221

Rnr (k) =√

2PHnr (k)cu(k)du (�k/S F�)+ μnr (k) + Πnr (k), (8)

where μnr (k) is the ICI component given by

μnr (k) = Hnr (k)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝√

2PU−1∑u′=0�u

du′

(⌊k

S F

⌋)cu′ (k)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . (9)

2.2 Channel Capacity Expression Assuming Perfect ICICancellation

The received signal can be expressed using the matrix formas

R (k) = H (k) cu (k) du (�k/S F�) + μ (k) +Π (k)

=√

2P

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝H0 (k)...

HNr−1 (k)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ cu (k) du

(⌊k

S F

⌋)

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝μ0 (k)...

μNr−1 (k)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Π0 (k)...

ΠNr−1 (k)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , (10)

where H (k) is the channel gain column vector with the sizeof Nr × 1. R (k) is extended to the equivalent received signalvector R (n) with the size of (Nr · S F) × 1 for the n-th datasymbol. R (n) can be represented as [13]

R (n) =(R0 (nS F) , · · · ,RNr−1 ((n + 1) S F − 1)

)T

=√

2P

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

H0 (nS F) cu (0)...

H0 ((n + 1) S F − 1)×cu (S F − 1)

...HNr−1 (nS F) cu (0)

...HNr−1 ((n + 1) S F − 1)×cu (S F − 1)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

du (n)

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

μ0 (nS F)...

μ0

((n + 1) S F−1

)

...μNr−1 (nS F)

...

μNr−1

((n + 1) S F−1

)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Π0 (nS F)...

Π0

((n + 1) S F−1

)

...ΠNr−1 (nS F)

...

ΠNr−1

((n + 1) S F−1

)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(11)

By defining the diagonal spreading code sequence with thesize of (Nr · S F) × (Nr · S F) as

cu=diag(cu(0), · · · , cu(S F−1), · · · , cu(0), · · · , cu(S F−1)),

(12)

R (n) can be expressed as

R (n) = cu

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

H0 (nS F)...

H0 ((n + 1) S F − 1)...

HNr−1 (nS F)...

HNr−1 ((n + 1) S F − 1)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

√2Pdu (n)

+ μ (n) + Π (n)

= cuH (n)√

2Pdu (n) + μ (n) + Π (n) . (13)

We use the Gaussian approximation of ICI. UsingEq. (13), the channel capacity, C (n), for the n-th parallelchannel is given by [14]

C (n) = log2det As · det Ar

det Au, (14)

where det As, det Ar and det Au are given by⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

det As = det E[(√

2Pdu (n)) (√

2Pdu (n))∗]

det Ar = det E[R (n) RH (n)

]det Au = det E

[u (n) uH (n)

] , (15)

and

u (n) =(√

2Pdu (n) , RT (n))T. (16)

Since we are assuming independent fading and E[|Hnr (k) |2

]=

1, we have⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

E[du (n) d∗u (n)

]= 1

E[μ (n) μH (n)

]=

(U − 1) · 2Ec

S F · Tc· I(Nr·S F)

E[Π (n) Π

H(n)]=

2N0

Nc · Tc· I(Nr ·S F)

, (17)

where IM is the M × M identity matrix. Thus, Eq. (15) be-comes⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

det As = det(E[(√

2Pdu (n)) (√

2Pdu (n))∗])

=2Ec

Tc · S Fdet Ar = det

(E[R (n) RH (n)

])

= det

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝2Ec

Tc · S F· cuH (n) HH (n) cH

u

+

((U − 1) · 2Ec

Tc · S F+

2N0

Nc · Tc

)· I(Nr ·S F)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠det Au = det

(E[u (n) uH (n)

])

= det As · det

(((U − 1) · 2Ec

Tc · S F+

2N0

Nc · Tc

)· I(Nr ·S F)

)

.

(18)

Substitution of Eq. (18) into Eq. (14) gives

C (n) = log2 det(I(Nr ·S F) + Γ · cuH (n) HH (n) cH

u

). (19)

222IEICE TRANS. COMMUN., VOL.E92–B, NO.1 JANUARY 2009

Since det(Im×m + Xm×nYH

m×n

)= det

(In×n + YH

m×nXm×n

)[15],

C (n) can be rewritten as

C (n) = log2

(1 + Γ · HH (n) H (n)

)

= log2

⎛⎜⎜⎜⎜⎜⎜⎝1 + Γ ·Nr−1∑nr=0

(n+1)S F−1∑k=nS F

|Hnr (k) |2⎞⎟⎟⎟⎟⎟⎟⎠ , (20)

where Γ is the received signal-to-interference plus noisepower ratio (SINR) and given by

Γ =(1/S F) (Es/N0)

(U − 1) (1/S F) (Es/N0) + 1. (21)

Assuming perfect ICI cancellation, the received SINRreduces to

Γ =1

S F· Es

N0. (22)

The channel capacity of MC-CDMA with spreading factorS F and code multiplexing order U is given by

CMC =UNc

� NcS F �−1∑n=0

C (n)

=UNc

� NcS F �−1∑n=0

log2

⎛⎜⎜⎜⎜⎜⎜⎝1 + 1S F

Es

N0

Nr−1∑nr=0

(n+1)S F−1∑k=nS F

|Hnr (k) |2⎞⎟⎟⎟⎟⎟⎟⎠ .

(23)

On the other hand, the channel capacity of OFDM is givenby [11]

COFDM =1

Nc

Nc−1∑n=0

C (n)

=1

Nc

Nc−1∑n=0

log2

⎛⎜⎜⎜⎜⎜⎜⎝1 + Es

N0

Nr−1∑nr=0

|Hnr (n) |2⎞⎟⎟⎟⎟⎟⎟⎠ . (24)

2.3 Channel Capacity Expression Assuming Imperfect ICICancellation

First, we derive the conditional received SINR. The ICI re-mains (i.e., residual ICI) since the equivalent channel gain isdifferent from its average over all sub-carriers. The residualICI can be expressed as [6], [7]

Mnr (k) =

(Hnr (k) − A

Nr

)S (k) , (25)

where Hnr (k) = wnr (k) Hnr (k) (wnr (k) is the FDE weightfor the k-th sub-carrier at the nr-th receive antenna and isgiven in Eq. (28)) and A is the average equivalent channelgain given by

A =1

S F

Nr−1∑nr=0

(n+1)S F−1∑k=nS F

Hnr (k) . (26)

The analysis of channel capacity taking into account the

residual ICI is quite difficult if not impossible. In this pa-per, we introduce a heuristic approach; the degree of ICIcancellation is represented by the parameter α, α = 0 repre-sents perfect ICI cancellation. The received signal after ICIcancellation is expressed as

R (k) =Nr−1∑nr=0

(wnr (k) Rnr (k) − (1 − α) · Mnr (k)

)

= AS (k)+α

⎧⎪⎪⎨⎪⎪⎩Nr−1∑nr=0

(Hnr (k) − A

Nr

)⎫⎪⎪⎬⎪⎪⎭ S (k) +Nr−1∑nr=0

Πnr (k).

(27)

There is a tradeoff relationship between orthogonalityrestoration and noise enhancement. Although perfect or-thogonality can be restored by zero-forcing (ZF)-FDE, thenoise enhancement results in the performance degradation.MMSE-FDE can restore the orthogonality to some degreewhile suppressing the noise enhancement. However, theresidual ICI is present after MMSE-FDE. Thus, in this pa-per, assuming MMSE-FDE, the cancellation of the residualICI is considered. The MMSE equalization weight, takinginto account the receive antenna diversity and the residualICI, can be obtained as [6]

wnr (k) =H∗nr

(k)

α2 ·Nr−1∑nr=0

|Hnr (k) |2 +(

US F· Es

N0

)−1. (28)

The despreading operation is carried out to obtain thedecision variable as

du (n) =1

S F

(n+1)S F−1∑k=nS F

R (k) c∗u (k)

=1

S F

(n+1)S F−1∑k=nS F

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

AS (k)

+α

⎛⎜⎜⎜⎜⎜⎜⎝Nr−1∑nr=0

(Hnr (k) − A

Nr

)⎞⎟⎟⎟⎟⎟⎟⎠ S (k)

+

Nr−1∑nr=0

Πnr (k)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠c∗u(k)

=√

2PAdu (n) + μICI + μnoise, (29)

where the 1st term represents the desired signal component,the 2nd term the residual ICI, and the 3rd term the noisecomponent. μICI and μnoise are given as⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

μICI

= α

U−1∑u′=0�u

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝√

2Pdu′ (n)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1

S F

(n+1)S F−1∑k=nS F

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Nr−1∑nr=0

Hnr (k)

−A

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠×cu′ (k) c∗u (k)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠μnoise =

1S F

(n+1)S F−1∑k=nS F

Nr−1∑nr=0

Πnr (k) c∗u (k)

.

(30)

ADACHI et al.: CHANNEL CAPACITY OF MC-CDMA AND IMPACT OF RESIDUAL ICI223

The sub-carrier spacing is 1/Ts (where Ts = Tc · Nc

is the effective OFDM symbol length). The instantaneousreceived SINR is the function of the received Es/N0 and theset of instantaneous channel gains, {Hnr (k) ; k = 0 ∼ Nc −1}. The averaging operation over S F sub-carriers is carriedout in the despreading process. �Nc/S F� data symbols aretransmitted during one MC-CDMA symbol interval. Here,the instantaneous received SINR of the n-th symbol (n =0 ∼ �Nc/S F� − 1) is denoted as γn (Es/N0, {H (k)}). SinceμICI is the sum of many components, the sum of μICI andμnoise can be treated as a new zero mean complex valuedGaussian variable μ according to the central limit theorem.The variance of μ is given by

2σ2μ = E

[|μ|2]= 2σ2

ICI + 2σ2noise

= E[|μICI |2

]+ E[|μnoise|2

], (31)

Since the transmit signal is uncorrelated to each other andspreading sequence is orthogonal to each other, i.e.,⎧⎪⎪⎪⎨⎪⎪⎪⎩

E[S u′ (n) S ∗u′′ (n)

]=

2Ec

S F · Tcδ(u′ − u′′

)E[cu′ (k) c∗u (k) c∗u′′

(k′)

cu(k′)]= δ(k − k′

)δ(u′ − u′′

) ,(32)

the variance of μICI and μnoise can be given by (see Ap-pendix)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2σ2ICI =

α2

S F· 2Ec · (U − 1)

S F · Tc

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1

S F

(n+1)S F−1∑k=nS F

∣∣∣∣∣∣∣Nr−1∑nr=0

Hnr (k)

∣∣∣∣∣∣∣2

− |A|2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠2σ2

noise =1

S F2· 2N0

Nc · Tc

(n+1)S F−1∑k=nS F

Nr−1∑nr=0

∣∣∣wnr (k)∣∣∣2.

(33)

The conditional received SINR, γn (Es/N0, {H (k)}), is givenby

γn

(Es

N0, {H (k)}

)=

∣∣∣∣∣∣√

Ec

S F · TcAdu (n)

∣∣∣∣∣∣2

σ2μ

=

Es

N0

∣∣∣∣∣∣∣1

S F

(n+1)S F−1∑k=nS F

Nr−1∑nr=0

Hnr (k)

∣∣∣∣∣∣∣2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

α2 · Es

N0· (U − 1)

S F·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1S F

(n+1)S F−1∑k=nS F

∣∣∣∣∣∣∣Nr−1∑nr=0

Hnr (k)

∣∣∣∣∣∣∣2

−∣∣∣∣∣∣∣

1S F

(n+1)S F−1∑k=nS F

Nr−1∑nr=0

Hnr (k)

∣∣∣∣∣∣∣2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+

1S F

(n+1)S F−1∑k=nS F

Nr−1∑nr=0

∣∣∣wnr (k)∣∣∣2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(34)

Since the signal bandwidth is 1/Tc, the normalized channel

capacity (bps/Hz) can be expressed as

CMC =UNc

�Nc/S F�−1∑n=0

log2

(1 + γn

(Es

N0, {H (k)}

)). (35)

By substituting Eq. (34) into Eq. (35), the channel capacityof MC-CDMA assuming imperfect ICI cancellation can beobtained.

3. Numerical Results

The condition for numerical evaluation is summarized in Ta-ble 1. The number of sub-carriers is set to Nc = 256. Thespreading factor S F is varied from 1 to 256 and the full codemultiplexing is assumed, i.e., U = S F. The channel is as-sumed to be an L-path block Rayleigh fading channel withexponential power delay profile with the decay factor γ dB.The time delay of the l-th path is τl = l FFT samples. Thereceive antenna diversity is used with the number of receiveantennas of Nr = 1 ∼ 6. The performance loss owing tothe insertion of GI is not taken into account in the paper. Toobtain the diversity gain, a block interleaver with the size ofS F × �Nc/S F� is used except for Fig. 2 and Fig. 3(a).

3.1 Perfect ICI Cancellation Case

The channel capacity of MC-CDMA assuming perfect ICIcancellation, which is given by Eq. (23), is compared to thatof OFDM.

The channel capacity of MC-CDMA is plotted in Fig. 2as a function of the spreading factor S F for both cases of in-terleaving and no interleaving. It can be seen from the figurethat as spreading factor S F increases, the channel capacityincreases owing to the increased frequency diversity gain ifno frequency-domain block interleaving is used. However,if the interleaving is applied, even when the spreading factoris small (such as S F = 16), a sufficient frequency diversitygain can be obtained. It should be worth noticing that whenreceive antenna diversity is used (e.g., Nr = 4), the channel

Table 1 Simulation condition.

MC-CDMA OFDMNumber of

Nc = 256sub-carriers

Spreading factor S F = 1 ∼ 256����

Code multiplex U = S F����

ChannelBlock interleaver

����interleaver

FadingBlock Rayleigh

fadingChannel model Number of paths L = 1 ∼ 16

Decay factor γ = 0 ∼ 6 dBMaximum Doppler

fD � 0Hzfrequency

Number ofNr = 1 ∼ 6receive antennas

224IEICE TRANS. COMMUN., VOL.E92–B, NO.1 JANUARY 2009

Fig. 2 Impacts of S F and interleaver on the channel capacity ofMC-CDMA.

capacity is almost insensitive to S F except for the case ofS F = 1. This is because a sufficient space diversity gain isobtained and therefore, an additional gain due to frequencydiversity is small. The channel capacity of MC-CDMAdepends on the power gain due to spreading as shown inEq. (23). This means that the variability of the power gainaffects the channel capacity. The probability density func-

tion (pdf) of power gain (1/S F)∑Nr−1

nr=0

∑(n+1)S F−1k=nS F

∣∣∣Hnr (k)∣∣∣2

is plotted in Fig. 3. First, we consider the case of no an-tenna diversity. The variability of power gain can signifi-cantly be reduced by increasing S F when no interleaving isused. However, when interleaving is used, almost the samepower gain variability is seen irrespective of S F (except forS F = 1). This suggests that the achievable channel capac-ity increases as S F, but almost the same capacity can beobtained with interleaving. Next, we consider the case ofantenna diversity. Similar trends to the no antenna diversitycase can be observed; but the average power gain is muchhigher than no diversity case and therefore, much larger ca-pacity can be achieved with antenna diversity. These obser-vations support the results shown in Fig. 2.

The channel capacity of MC-CDMA and that ofOFDM are plotted in Fig. 4 with the number of receive an-tennas, Nr, as a parameter. The difference between the chan-nel capacity of MC-CDMA and that of OFDM becomessmaller owing to the increased space diversity gain as Nr

increases.The impact of channel parameters (the number of

multi-paths and decay factor) is shown in Fig. 5. The chan-nel capacity of OFDM is insensitive to the channel param-eters and is only sensitive to the received signal power asdiscussed in [11]. On the other hand, the channel capacityof MC-CDMA increases owing to the increased frequencydiversity gain as the number L of paths increases and/or thedecay factor γ reduces. The channel capacity of MC-CDMA

Fig. 3 Average equivalent channel gain.

Fig. 4 Impact of antenna diversity.

is almost insensitive to the channel parameters when Nr = 4.

ADACHI et al.: CHANNEL CAPACITY OF MC-CDMA AND IMPACT OF RESIDUAL ICI225

Fig. 5 Impact of channel parameters.

3.2 Imperfect ICI Cancellation Case

The channel capacity of MC-CDMA assuming imperfectICI cancellation, which is given by Eq. (35), is comparedto that of OFDM.

The channel capacity of MC-CDMA is plotted in Fig. 6as a function of the average received Es/N0 per receive an-tenna with the spreading factor S F and the ICI cancellationfactor α as parameters. As can be seen from the figure, thechannel capacity of MC-CDMA with α = 1 (without ICIcancellation) is always smaller than that of OFDM irrespec-tive of S F. On the other hand, the channel capacity of MC-CDMA with α = 0.1 is larger than that of OFDM for theaverage received Es/N0 < 20 dB, but smaller than that ofOFDM for the average received Es/N0 > 20 dB.

The impact of α on the channel capacity of MC-

Fig. 6 Impact of ICI cancellation factor α.

Fig. 7 Impact of ICI cancellation factor when receive antenna diversityis used.

CDMA is plotted in Fig. 7 with the number Nr of receiveantenna as a parameter for the average received Es/N0 =

0 dB and 10 dB. The difference between the channelca-pacity of MC-CDMA with α = 1 and that of OFDMgets smaller as Nr increases. This can be explained be-low. As discussed in Sect. 2.3, the residual ICI is presentsince the equivalent channel gain after diversity combin-ing∑Nr−1

nr=0 Hnr (k) =∑Nr−1

nr=0 wnr (k) Hnr (k) after FDE differsfrom the average equivalent channel gain A. However, thevariability of

∑Nr−1nr=0 Hnr (k) gets smaller and approaches the

value of A given by Eq. (26) as Nr increases as shown inFig. 8(a). As a consequence, the residual ICI reduces asshown in Fig. 8(b), resulting in the larger channel capac-ity. The channel capacity of MC-CDMA increases as α gets

226IEICE TRANS. COMMUN., VOL.E92–B, NO.1 JANUARY 2009

Fig. 8 Equivalent channel gain after FDE and ICI power.

smaller and almost the same channel capacity as OFDM canbe achieved if α = 0.4 ∼ 0.5. Here, only receive antennadiversity was considered. However, space-time transmit di-versity (STTD) [16] and site diveristy [17] can also be usedto mitigate the adverse effect of residual ICI.

4. Conclusion

In the previous literature [8]–[10], it has been shown thatthe channel capacity of MC-CDMA is worse than that ofOFDM. However, the performance degradation of MC-CDMA compared to OFDM is owing to the presence ofresidual ICI. In this paper, we derived a channel capacityexpression for MC-CDMA with ICI cancellation taking intoaccount both the frequency diversity gain and the space di-versity gain. Furthermore, a channel capacity expressionfor MC-CDMA assuming imperfect ICI cancellation wasderived and the impact of residual ICI on the channel ca-pacity was discussed. The numerical computation resultsshowed that if the residual ICI can be reduced to about 40%∼ 50%, MC-CDMA provides the channel capacity largerthan OFDM.

In this paper, we took a heuristic approach and intro-duced the ICI cancellation factor to discuss the impact ofresidual ICI on the channel capacity of MC-CDMA. Nextstep is to find the value of α for a practical ICI cancella-tion technique. To what extent the practical ICI cancellationtechnique can mitigate the residual ICI is left as an interest-ing future study topic.

Acknowledgment

This work is supported by Research Fellowship of the Japan

Society for the Promotion of Science for Young Scientists.

References

[1] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEECommun., Mag., vol.35, no.12, pp.126–133, Dec. 1997.

[2] F. Adachi, D. Garg, S. Takaoka, and K. Takeda, “Broadband CDMAtechniques,” IEEE Wireless Commun. Mag., vol.12, no.2, pp.8–18,April 2005.

[3] D. Falconer, S.L. Ariyavisitakul, A. Benyamin-Seeyarand, and B.Eidson, “Frequency domain equalization for single-carrier broad-band wireless systems,” IEEE Commun. Mag., vol.40, no.4, pp.58–66, April 2002.

[4] F. Adachi, T. Sao, and T. Itagaki, “Performance of multicode DS-CDMA using frequency domain equalization in frequency selec-tive fading channel,” Electron. Lett., vol.39, no.2, pp.239–241, Jan.2003.

[5] R. Dinis, P. Silva, and A. Gusmao, “An iterative frequency-domaindecision-feedback receiver for MC-CDMA schemes,” Proc. IEEEVTC’05-spring, pp.271–275, May-June 2005.

[6] K. Takeda, K. Ishihara, and F. Adachi, “Frequency-domain ICI can-cellation with MMSE equalization for DS-CDMA downlink,” IEICETrans. Commun., vol.E89-B, no.12, pp.3335–3343, Dec. 2006.

[7] K. Ishihara, K. Takeda, and F. Adachi, “Iterative frequency-domainsoft interference cancellation for multicode DS- and MC-CDMAtransmission and performance comparison,” IEICE Trans. Com-mun., vol.E89-B, no.12, pp.3344–3355, Dec. 2006.

[8] Z. Kang and K. Yao, “On the capacity and asymptotic performanceof MC-CDMA system over Nakagami-m fading channels,” Proc.IEEE VTC’04-spring, vol.2, pp.1153–1157, May 2004.

[9] M. Ma, Y. Yang, H. Cheng, and B. Jiao, “A capacity comparison be-tween MC-CDMA and CP-CDMA,” Proc. IEEE VTC’06-fall, pp.1–4, Sept. 2006.

[10] M. Debbah, “Capacity of a downlink MC-CDMA multi-cell net-work,” IEEE International Conference on Acoustics, Speech, andSignal Processing (ICASSP), vol.4, pp.761–764, May 2004.

[11] H. Bolcskei, D. Gesbert, and A.J. Paulraj, “On the capacity ofOFDM-based spatial multiplexing systems,” IEEE Trans. Commun.,vol.50, no.2, pp.225–234, Feb. 2002.

[12] D. Garg, High Speed Packet Access Techniques for BroadbandCDMA Mobile Radio, Doctoral Thesis, Tohoku University, March2005.

[13] K. Adachi, F. Adachi, and M. Nakagawa, “A study on channel ca-pacity of MC-CDMA,” IEICE Technical Report, RCS2007-91, Oct.2007.

[14] G.J. Foschini and M. Gans, “On limits of wireless communicationin a fading environment when using multiple antennas,” Wirel. Pers.Commun., vol.6, no.3, pp.311–335, March 1998.

[15] J.N. Franklin, Matrix theory, pp.80–81, Prentice-Hall, 1968.[16] S.M. Alamouti, “A simple transmit diversity technique for wire-

less communications,” IEEE J. Sel. Areas Commun., vol.6, no.8,pp.1481–1458, Oct. 1998.

[17] T. Inoue, S. Takaoka, and F. Adachi, “Frequency-domain equaliza-tion for MC-CDMA downlink site diversity and performance eval-uation,” IEICE Trans. Commun., vol.E88-B, no.1, pp.84–92, Jan.2005.

Appendix: Derivation of 2σ2ICI

and 2σ2noise

.

The variance of residual ICI, μICI, is given by

ADACHI et al.: CHANNEL CAPACITY OF MC-CDMA AND IMPACT OF RESIDUAL ICI227

2σ2ICI = E

[|μICI |2

]

= E

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∣∣∣∣∣∣∣∣∣∣α

U−1∑u′=0�u

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝S u′ (n)

S F

(n+1)S F−1∑k=nS F

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎜⎜⎜⎝Nr−1∑nr=0

Hnr (k) − A

⎞⎟⎟⎟⎟⎟⎟⎠×cu′ (k) c∗u (k)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

∣∣∣∣∣∣∣∣∣∣

2⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ .

(A· 1)

Since the transmitted signals are mutually uncorrelated andthe spreading sequences are orthogonal to each other as⎧⎪⎪⎪⎨⎪⎪⎪⎩

E[S u′ (n) S ∗u′′ (n)

]=

2Ec

S F · Tcδ(u′ − u′′

)E[cu′ (k) c∗u (k) c∗u′′

(k′)

cu(k′)] , (A· 2)

2σ2ICI becomes

2σ2ICI

=

(α

S F

)2· 2Ec · (U − 1)

S F · Tc

(n+1)S F−1∑k=nS F

∣∣∣∣∣∣∣Nr−1∑nr=0

Hnr (k) − A

∣∣∣∣∣∣∣2

=α2

S F· 2Ec · (U − 1)

S F · Tc

⎛⎜⎜⎜⎜⎜⎜⎜⎝ 1S F

(n+1)S F−1∑k=nS F

∣∣∣∣∣∣∣Nr−1∑nr=0

Hnr (k)

∣∣∣∣∣∣∣2

− |A|2⎞⎟⎟⎟⎟⎟⎟⎟⎠ .

(A· 3)

Since the noise components of different subcarriers areuncorrelated to each other, the variance of μnoise is given by

2σ2noise = E

[|μnoise|2

]

= E

⎡⎢⎢⎢⎢⎢⎢⎢⎣∣∣∣∣∣∣∣

1S F

(n+1)S F−1∑k=nS F

Nr−1∑nr=0

Πnr (k) c∗u (k)

∣∣∣∣∣∣∣2⎤⎥⎥⎥⎥⎥⎥⎥⎦

=1

S F2

(n+1)S F−1∑k=nS F

(n+1)S F−1∑k′=nS F

Nr−1∑nr=0

Nr−1∑n′r=0

E

[wnr (k)Πnr (k) c∗u (k)×w∗n′r (k′)Π∗n′r (k′) cu (k′)

]

=1

S F2· 2N0

Nc · Tc

(n+1)S F−1∑k=nS F

Nr−1∑nr=0

∣∣∣wnr (k)∣∣∣2 .

(A· 4)

Koichi Adachi received the B.E. and M.E.degrees in Information and Science Technologyfrom Keio University, Japan, in 2005 and 2007,respectively. Currently, he is a Japan Society forthe Promotion of Science (JSPS) research fel-low, and is studying toward his Ph.D. degree atSchool of Science for Open and EnvironmentalSystems, Keio University. His research interestsinclude MIMO and channel estimation. He wasthe recipient of 2007 RCS (Radio Communica-tion Systems) Active Research Award for active

participation and the same award for presenting excellent papers.

Fumiyuki Adachi received the B.S. andDr.Eng. Degrees in electrical engineering fromTohoku University, Sendai, Japan, in 1973 and1984, respectively. In April 1973, he joinedthe Electrical Communications Laboratories ofNippon Telegraph & Telephone Corporation(now NTT) and conducted various types of re-search related to digital cellular mobile commu-nications. From July 1992 to December 1999,he was with NTT Mobile Communications Net-work, Inc. (now NTT DoCoMo, Inc.), where he

led a research group on wideband/broadband CDMA wireless access forIMT-2000 and beyond. Since January 2000, he has been with Tohoku Uni-versity, Sendai, Japan, where he is a Professor of Electrical and Communi-cation Engineering at the Graduate School of Engineering. His researchinterests are in CDMA wireless access techniques, equalization, trans-mit/receive antenna diversity, MIMO, adaptive transmission, and channelcoding, with particular application to broadband wireless communicationssystems. From October 1984 to September 1985, he was a United KingdomSERC Visiting Research Fellow in the Department of Electrical Engineer-ing and Electronics at Liverpool University. He received the IEICE FellowAward in 2007 and was a co-recipient of the IEICE Transactions best paperof the year award 1996 and again 1998 and also a recipient of Achievementaward 2003. He is an IEEE Fellow and was a co-recipient of the IEEEVehicular Technology Transactions best paper of the year award 1980 andagain 1990 and also a recipient of Avant Grade award 2000. He was arecipient of Thomson Scientific Research Front Award 2004.

Masao Nakagawa was born in Tokyo,Japan in 1946. He received the B.E., M.E., andPh.D. degrees from Keio University, Yokohama,Japan, all in electrical engineering, in 1969,1971, and 1974 respectively. Since 1973, hehas been with the Department of the ElectricalEngineering, Keio University, where he is nowProfessor. His research interests are in CDMA,Consumer Communications, Mobile Commu-nications, ITS (Intelligent Transport Systems).Wireless Home Networks, and Visible Light

Communications. He received 1989 IEEE Consumer Electronics SocietyPaper Award, 1999-Fall Best Paper Award in IEEE VTC, IEICE Achieve-ment Award in 2000, IEICE Fellow Award in 2001, Ericsson Telecommuni-cations Award in 2004, YRP (Yokosuka Research Park) technical Award in2005, and IEEE Fellow Award in 2006. He is a co-author of “TDD-CDMAfor Wireless Communications” published by Artech House in which themain ideas of TDD-CDMA were given by him. He was the executive com-mittee chairman of International Symposium on Spread Spectrum Tech-niques and Applications in 1992 and the technical program committeechairman of ISITA (International Symposium on Information Theory andits Applications) in 1994. He is an editor of Wireless Personal Communi-cations and was a guest editor of the special issues on “CDMA NetworksI, II, III, and IV” published in IEEE JSAC in 1994 (I and II) and 1996 (IIIand IV).