IEICE TRANS. COMMUN., VOL.E86–B, NO.7 JULY 2003

2203

LETTERUplink Link Capacity of DS-CDMA Packet Mobile
Communications with Rake Combining andTransmit Power ControlZhiSen WANG† , Student Member, Eisuke KUDOH†a) ,and Fumiyuki ADACHI† , Regular Members

SUMMARY Without transmit power control (TPC) andRake combining, the uplink capacity of a direct sequence code
division multiple access (DS-CDMA) packet mobile communication system signiﬁcantly degrades due to the near-far problemand multipath fading. In this letter, assuming a single cell system with an interference-limited channel, the impact of the joint
use of Rake combining and TPC on the uplink capacity is evaluated by computer simulation. Slow TPC is found to give a link
capacity larger than fast TPC. This is because, with slow TPC,
the received signal power variations due to fading remain intact
and this results in a larger capture eﬀect.
key words: DS-CDMA, packet communication, transmit powercontrol, Rake combining, link capacity, capture eﬀectof authors’ knowledge, how the joint use of TPC andRake combining impacts the packet link capacity hasnot been fully understood.
The remainder of this letter is organized as follows.
Section 2 presents formulations of the throughput andoutage probability. Section 3 derives expressions for
signal-to-interference plus noise power ratio (SINR) assuming a single cell system with an interference-limited
channel. Section 4 evaluates the link capacity by computer simulation. Section 5 gives some conclusions.
2.

1.

Throughput and Outage Probability

Introduction

In a packet mobile communication system, a packet
with larger power can be received correctly even when
the packets collide. This is known as the capture effect [1], [4]. An interesting question is whether the capture eﬀect can be obtained when packet transmission iscombined with direct sequence code division multiple
access (DS-CDMA), in which multiple users transmit
their packets using the same carrier frequency. Packets
transmitted by diﬀerent users suﬀer from diﬀerent pathloss, shadowing loss and multipath fading and received
by a base station with signiﬁcantly diﬀerent power fromothers. This produces too large multi-access interference (MAI) and may oﬀset the capture eﬀect. Transmitpower control (TPC) and Rake combining can reducethe variations in the received signal power and then reduce the MAI. However, the capture eﬀect may be lostwhen TPC and Rake combining are jointly used. It isinteresting to see how the joint use of TPC and Rake
combining impacts the link capacity of a DS-CDMApacket mobile communication system. There are twotypes of TPC: fast TPC and slow TPC. In Ref. [2], theimpact of control delay in the fast TPC on the packet
link capacity is theoretically investigated. In Ref. [3],the link capacity with fast TPC but without Rake combining is theoretically analyzed. However, to the best
Manuscript received February 3, 2003.
The authors are with the Department of Electrical andCommunication Engineering, Graduate School of Engineering, Tohoku University, Sendai-shi, 980-8579 Japan.
a) E-mail: kudoh@mobile.ecei.tohoku.ac.jp†Outage occurs if the transmission quality drops below the required quality of services (QoS), i.e., packet
throughput and delay. The link capacity is deﬁned
as the maximum number of active users that satisﬁes
the allowable outage probability (Qallow ) and Qallow isdeﬁned as the maximum outage probability that satisﬁes the system requirement. In a packet communication system, automatic repeat request (ARQ) isused. Assuming inﬁnite number of retransmissions (inﬁnite delay is allowed before successful transmission ofa packet), the throughput S is given by
S = 1 − p (K, λ) ,(1)where p(K, λ) is the average packet error rate with K
and λ being the number of active users and the packet
occurrence rate, respectively. The outage occurs only if
throughput is less than the required value. The outage
probability Q is given by
Q = Prob[S < Sreq ],(2)where Sreq is the required throughput which is deﬁned
as the minimum throughput that satisﬁes the required
QoS.
As K increases, the MAI increases and therefore,
the outage probability increases. We assume that theoccurrence rate of original packets is the same for all active users and is denoted by λ0 . The packet occurrence
rate λ is (1 + packet retransmission rate) × λ0 . When
TPC is used, since packet errors occur equally likely for
all active users, λ is given by

IEICE TRANS. COMMUN., VOL.E86–B, NO.7 JULY 2003

2204

λ=λ0.
1 − p(K, λ)(3)

From Eqs. (1) and (3), we haveS = λ0 /λ.

(4)There exist K − 1 active interfering users. Assumingthat the original and retransmitted packets are randomly produced, p(K, λ) can be computed usingK−1
K −1p(K, λ) =pk ·λk (1 − λ)K−k−1 , (5)kk=0

where pk is the conditional packet errorrate when kK −1interfering packets are received andis thekbinomial coeﬃcient. Assuming a slotted packet transmission and a block fading (the fading stays almost
constant over a packet), pk may be given by
pk = 1 − [1 − pb (γk )]N ,(6)where pb (γk ) is the bit error rate (BER) with γk being the received SINR when k interfering packets arereceived and N is the number of bits in a packet. Assuming coherent BPSK data modulation and that thesum of colliding packets is approximated as a Gaussian
process, pb (γk ) is given by [5]γk1,(7)pb (γk ) = erfc
22√ ∞
2
where erfc(x) = (2/ π) x e−y dy is the complimentary error function. The SINR expressions to computethe BER are derived in Sect. 3.
Solving Eq. (3) theoretically is quite diﬃcult if not
impossible. We resort to an iterative computation. Letg(λ)=λ0 /[1 − p(K, λ)]. The iterative computation is asfollows.
Step 1: Set K=0.
Step 2: Increase the value of K by one and set λ = λ0
and compute g(λ).
Step 3: Set λ=g(λ) and compute g(λ).
Step 4: Repeat step 3 until the diﬀerence between previous and new values of λ is within a given
bound. In our computation, the bound is taken
to be 10−4 .
Step 5: Obtain the throughput S using Eq. (4).
Step 6: Repeat step 2 to step 5 until S < Sreq isreached. The maximum number of K which
satisﬁes S ≥ Sreq is the link capacity C.
If TPC is not used, packets transmitted from different users reach the base station with diﬀerent powers
due to diﬀerent path loss, shadowing loss and multipath
fading. Therefore, throughputs are diﬀerent for diﬀerent users and must be treated as random variables. No
analytical method is available to ﬁnd the throughput.
We resort to Monte Carlo simulation. The Monte Carlosimulation is performed as follows.

Step 1: Set K=0.
Step 2: Increase the value of K by one and set users’ locations and generate path loss, shadowing loss
and multipath fading associated with each user.
Step 3: Generate original packets with the occurrence
rate of λ0 .
Step 4: Generate packet errors according to the BER
computed using SINR γ associated with eachuser’s packet received at the base station(Eqs. (6), (7), and (10) are used).
Step 5: If packet error occurs in step 4, retransmit thepacket of that user.
Step 6: Repeat step 3 to step 5 to obtain the packet
error rate of each user.
Step 7: Change locations of all users and repeat step 3
to step 6 to compute the probability distribution of the throughput S. The probability ofS ≤ Sreq represents the outage probability Q.
Step 8: Repeat step 2 to step 7 until Q ≥ Qallow .
The maximum number of K that satisﬁes Q <Qallow is the link capacity C.
3.

SINR Expressions

We assume a single cell system with an interferencelimited channel and ideal TPC based on the received
signal power. A frequency selective block fading channel having L discrete paths is assumed. Below, expressions for γk to compute the packet error rate pk using
Eqs. (6) and (7) are derived for no TPC, fast TPC andslow TPC.
(a) No TPC
Assuming the interference-limited channel, the received
signal r(t) may be expressed, using equivalent baseband
representation, as [6]L−1K−12Ai βiξi,l di (t − τi,l )ci (t − τi,l ) ,r(t) =i=0

l=0(8)
where i denotes the user index, Ai is the average received power and βi (=0 or 1) represents the transmission state; “βi =1” represents transmission of a packet
and “βi =0” otherwise. di (t) and ci (t) are the datamodulated waveform and the spreading chip waveform,respectively. ξi,l and τi,l are the complex path gain
and the time delay of the lth path, respectively. Timedependency of the path gain is dropped for the sakeL−12of simplicity. It is assumed that E[ l=0 |ξi,l | ] = 1,where E[·] is the ensemble average operation. Ai isgiven by
Ai = Pi ri−α 10−ηi /10 ,(9)where Pi , ri , α, and ηi are the transmit power, thedistance from the base station normalized by the cell

LETTER

2205

radius, the path loss exponent, and the shadowing loss[dB], respectively. The received faded signal is resolved
into L path components by L matched ﬁlters (Rakeﬁngers) for coherent Rake combining based on maximal ratio combining (MRC) methods [5]. Ideal channel
estimation is assumed. Without loss of generality, the0th user is considered as the desired user.
In the lth Rake ﬁnger, the received signal r(t) ismultiplied by the delayed replica of the spreading waveform c0 (t) synchronized to the lth path time delay, andintegrated over one data symbol duration to get the lth
ﬁnger output. Then, the lth ﬁnger output is multipliedby the complex conjugate of ξi,l and coherently combined with other ﬁnger outputs to produce the Rake
combiner output. The received interference is the sumof self-interference (due to inter-path interference) andother-user interference and is approximated as a white
Gaussian process. The received SINR γ after Rake
combining is therefore, given by
γ=−η /10
P0 r0−α 10 kL−1

22|ξ0,l |l=02SF η0 L−1 L−1−α − 10r10P
 0 0
l=0 m=0.

∗ 2ξ0,m ξ0,l

L−1
 K−1
 L−1 ∗
ξi,m ξ0,l+βi Pi ri−α 10−ηi /10m=l2 

l=0 m=0i=1(10)(b) Fast TPC
The transmit power of each mobile station is controlled
so that the received instantaneous signal power is kept
at the prescribed target value for all users. With fast
TPC, the ith user’s transmit signal power Pi becomes
Pi =Ptarget,
L−1
2−α −ηi /10
ri 10
l=0 |ξi,l |(11)where Ptarget is the target received signal power. The
0th user’s SINR γ with fast TPC is given byγ=2SF
L−1
2
2

|ξ0,l |

 . (12)

l=0L−1
 L−1∗ 2
ξ0,m ξ0,l l=0 m=0m=l
L−1L−1∗ 2  ξξ
i,m 0,lL−1K−1  l=0 m=02|ξ0,l |βi +L−1
i=1l=02|ξ |i,ll=0(c) Slow TPC
The transmit power of each user is controlled so thatthe power variations due to path loss and shadowing
loss are completely removed but those due to fading
remain intact. The ith user’s average transmit power
Pi becomes
Pi =P̄target−α −ηi /10 ,ri 10(13)where P̄target is the target average signal power. The
0th user’s SINR γ with slow TPC, is given by2
L−1
2|ξ0,l |2SF
γ = L−1 L−1∗ 2
ξ0,m ξ0,l+l=0 m=0l=0K−1
βii=1m=l

L−1 L−1.
∗ 2
ξi,m ξ0,ll=0 m=0(14)
So far, SINR expressions without TPC, with fast
TPC and with slow TPC have been developed. γk in
Eq. (6) is given by Eqs. (10), (12) and (14) withk=K−1
βi ,(15)i=1i.e., when k interfering packets are received.
4.

Computer Simulation

The uplink capacity is evaluated by the Monte-Carlosimulation. Table 1 shows the simulation parameters.
Figure 1 plots the uplink capacity normalized by
the spreading factor SF as a function of the number Lof propagation paths. When L=1, since no Rake combining eﬀect is expected, too large MAI is produced.
This oﬀsets the capture eﬀect and none of the colliding
packets survives. However, as L increases, the packet
Table 1Simulation parameters.

IEICE TRANS. COMMUN., VOL.E86–B, NO.7 JULY 20032206

smaller than with TPC due to the near-far problem.
Figure 2 plots the normalized link capacity C/SF
as a function of spreading factor when L=2. Below
SF ≤ 4, only a single user can be accommodated, i.e.,
C/SF ≈ 0. However, as the value of SF increases beyond 4, the normalized link capacity starts to increase.
For SF > 8, the normalized link capacity with fast
TPC becomes almost constant, while that with slow
TPC still continues to increase owing to the captureeﬀect.
5.

Fig. 1

Normalized link capacity C/SF as a function of L.

Conclusions

This letter evaluated by computer simulation the impact of TPC and Rake combining on the uplink capacity in a DS-CDMA packet mobile communication system. It was found that the slow TPC provides larger
link capacity than the fast TPC owing to the captureeﬀect if more than one propagation path are present(L > 1). However, when only a single path is present(L=1), the slow TPC provides smaller capacity thanthe fast TPC because the large MAI oﬀsets the capture eﬀect. In this letter, the single cell system wasconsidered. An extension to multiple cell system is left
for further study.
References

Fig. 2 Normalized link capacity C/SF as a function ofspreading factor SF.

power variations reduce due to increased Rake combining eﬀect and hence the probability of packet survival increases. As L increases, the normalized capacity increases with the slow TPC and maximizes when
L=3. Beyond L=3, however, the normalized capacity
decreases and approaches that with fast TPC.
On the other hand, in the case of fast TPC, thenormalized link capacity is almost constant because all
packets are received with the same power. The reason
for slightly larger normalized link capacity for small
L is due to less interpath interference produced on thedesired packet. The link capacity without TPC is much

[1] I. Widipangestu, A.J.T. Jong, and R. Prasad, “Capture
probability and throughput analysis of slotted ALOHA
and unslotted np-ISMA in a Rician/Rayleigh environment,”
IEEE Trans. Veh. Technol., vol.43, no.3, pp.457–465, Aug.
1994.
[2] D.K. Kim and F. Adachi, “Performance analysis of closedloop link power control for packet transmission over DSCDMA in a multipath fading environment,” IEICE Trans.
Commun., vol.E84-B, no.9, pp.2635–2643, Sept. 2001.
[3] J.C. Arnbak and W. van Blitterswijk, “Capacity of a slotted
ALOHA in Rayleigh fading channels,” IEEE J. Sel. Areas
Commun., vol.SAC-5, no.2, pp.261–299, Feb. 1987.
[4] R. Prasad, “Performance analysis of mobile packet radio
network in real channels with inhibit multiple access,” IEE
Proc.-I, vol.138, no.5, pp.458–464, Oct. 1991.
[5] J.G. Proakis, Digital communications, 3rd ed., McGraw-Hill,
1995.
[6] F. Adachi, “Reverse link capacity of orthogonal multi-code
DS-CDMA with multiple connections,” IEICE Trans. Commun., vol.E85-B, no.11, pp.2520–2526, Nov. 2002.