Comp041122.qxd


The Journal of Engineering Research  Vol. 5, No.1 (2008)  1-6

1.  Introduction

Given the constraints of the high cost and limited avail-
ability of radio spectrum, efficient spectrum usage is a
cornerstone in the economic success of third generation
cellular systems. One novel approach to using available
resources more efficiently is the combination of existing
radio systems into coordinated, hybrid systems, in order to
combine the strength and capabilities of the individual
systems. The combination of systems adds a degree of
freedom in the sense that at any time the most spectrally
efficient transport system can be chosen depending for
example on quality-of-service requirements and traffic
characteristics.       

Most of the third generation systems use CDMA as the
multiple access method. Furthermore some systems use
TDMA or FDMA or both as the multiple access method.
CDMA systems use spreading methods to spread a narrow
band signal into a wideband signal.  Since the spreading
process does not change the power of the signal, the
power spectral density of the spread signal is lower than
the  power  density  of  the  narrowband  signal.  Figure 1
shows the principle of the spreading and de-spreading
process. The narrow band useful signal is spread and the
resulting  signal  is  transmitted over a  wideband channel 
_________________________________________
*Corresponding author’s e-mail: abbosh@itte.uq.edu.au

and de-spread in the receiver. As can be seen in Fig. 1, the
de-spreading process in the CDMA receiver operates as a
spreading process for a narrowband interfering signals. In
terms of coexistence of TDMA/FDMA and CDMA sys-
tems, the narrow band TDMA (or FDMA) channels act as
narrowband interferer for the CDMA system if they exist
in the same frequency band as the CDMA carrier. From
the TDMA (or FDMA) systems point of view, the CDMA
system acts like a noise-like interferer with a low power
spectral density. The receiver filter in the TDMA (or
FDMA) receiver will make sure that the noise-like inter-
fering signal will only be received in the narrow band-
width of the TDMA system. Thus, only a small part of the
interference power originating in the CDMA system will
be received, see Fig. 2.

The need for future multiple access schemes employ-
ing an overlay is stressed in  many papers (Schilling,
Lomp and Garodnick, 1993; Grieco and Schilling, 1994;
Himimy and Gupta, 1996; Widdowson, 1997;  Koorevaar
and Ruprecht, 1999). In all these papers simulation algo-
rithms were used to study the effects of overlaying differ-
ent schemes on the performance of the hybrid systems.  In
addition to simulation, some authors tried to study the
overlay process from a theoretical point of view (Singh,
1997;  Latchman and Chitamu, 2003). The inaccuracy of
the model used due to oversimplification of the assump-
tions  leads to incorrect expectations.         

Theoretical Investigation into Spectral Coexistence of CDMA
and TDMA Systems  

A.M.  Abbosh*1 and  M. A. Shehathah2

1School of ITEE, The University of Queensland, Brisbane Q4072, Australia
2Department of Electronics Engineering, University of Mosul, Iraq

Received 22 November 2004; accepted 11 October 2006

Abstract: The scarcity of available radio spectrum presently limits the extension of modern multimedia sys-
tems. This paper presents a theoretical investigation into the possibility of using a frequency overlay of a nar-
rowband Code Division Multiple Access (CDMA) System and a Time Division Multiple Access (TDMA)
System to provide a greater spectral efficiency. This paper shows that under certain conditions the two systems
can operate in the same frequency band and in the same area with a considerable improvement in the overall
capacity of the whole system.

Keywords: Capacity, CDMA, Hybrid system, TDMA

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The Journal of Engineering Research  Vol. 5, No.1 (2008)  1-6

In this paper, a Fourier transform is used to investigate
the benefits of spectral coexistence of CDMA and TDMA
systems. Also the model used in this paper considers the
case of complete overlapping between the two systems to
explore the worst case condition. Effects of the voice
activity ratio, the TDMA duty cycle and the code used in
CDMA system besides the lack of synchronization
between CDMA and TDMA are included in the analysis.

2.  Analysis

The analysis  considers the mutual effect of CDMA
and TDMA systems operating in the same area and using
the same frequency band.  The overall capacity of the
hybrid system is investigated to see whether the perform-

ance is degraded or enhanced. In the following analysis
the capacity of the system is assumed to be the total num-
ber of users working at the same time with a signal to
noise ratio above the threshold level.

2.1 Effect  of  the  CDMA Signal  on  the TDMA
Receiver

Assume a CDMA signal of the form:

(1)

where  Pc is the CDMA received power at the base sta-
tion, m(t) is the message signal,  c(t) is the chip sequence,
wc  is the radian carrier frequency.

f 

Spread Data  

f 

  Spread Data  

Interfering  
Signal  

× ×    Channel 

Code Code 

m(t) 

f 

Interfering 
signal 

f 

Message 
signal 

Message 
signal 

Spread 
Interfer ence  

f 

Fig.1: E ffect of TDMA signal on CDMA Receiver  
Figure 1.  Effect of the TDMA signal on the CMDA receiver

   Channel  

f 

CDMA 
interfering 

f 

  

BPF 

f 
CDMA signal  

TDMA 
signal 

f 

  

Figure 2.  Effect of the CMDA signal on the TDMA receiver

 )twcos()t(m)t(cP2)t(r cc=



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The Journal of Engineering Research  Vol. 5, No.1 (2008)  1-6

Let us assume that: Tb is the bit duration, Tc is the chip
duration, k1 = Tb / Tc is the number of chips per bit, C(f)
is the Fourier transform (F.T.) of c(t), M(f) is F.T. of  m(t)
and   R(f) is F.T. of r(t). As k1 >>1 according to definition
of CDMA then  M(f) appears as a Dirac delta function to
the relatively wideband  C(f) spectrum. For a single bit of
duration Tb then R(f) is: 

(2)

where CTb(f) is the F.T of the k-chips in a single bit dura-
tion Tb which is given by:

(3)

where ui = ±1 and sinc (x) = sin (µx) / µx.
The amount of the CDMA interference energy experi-

enced in a bandwidth equal to B by a TDMA signal work-
ing at carrier frequency  ft , is given as:

(4)

Substituting from (2) into (4);

(5)

Since f varies from ft - B/2 to ft + B/2 (as maximum),
and assuming ft = fc  (full overlaying of CDMA & TDMA);
hence:  f - ft = (fc ± B/2) - ft = ± B/2.   As ± B/2 is very
small relative to fc, we  can assume that sin c (± B/2 fc) =
1. Hence (5) can be written as:

(6)

Therefore, the interference power from the CDMA source
in that small bandwidth (B ) can be expressed as: 

(7)

Substituting (6) into (7);

(8)

Assuming that the result of the double integral of (8)
is:

(9)

where g1 is  a  constant that depends on  the bandwidth
(B ) and code type and length. The values of  g1 has been
calculated numerically using MATLAB program and
Walsh code. Depending on the parameter values shown in
Table 1, the value of g1 is shown to be equal to 0.0021.

The signal to noise ratio at the TDMA receiver (St ) can
be expressed as:

(10)

2.2  Effect of the TDMA Signal on the CDMA
Receiver

Let the TDMA signal be represented by: 

(11)

where b(t) is the information signal (message).  At the
CDMA receiver, the TDMA signal appears as:

(x) = g(t) c(t)                                                        (12)

Substituting  (11) into (12);

(13)

Let Td be the bit duration for the TDMA signal, kd =
integer (Td / Tc ) = number of chips per bit.

For a single bit of duration , Td, F.T of x(t) is:

(14)

CTd (f) is F.T of k2 chips in a single bit duration which
is given by:

(15)

The amount of the TDMA energy (Et) actually lying
within the passband of the CDMA receiver working at car-
rier frequency  fc can be written as in (6):

(16)

The value of the TDMA power interfered with the
CDMA signal is: 

 
)]ff(C)ff(C[

2
P

)f(R cTbcTb
c −++=

 
∑
=

−−=
1

1

)12()sinc()(
k

i

Tcifj
iccTb eufTTfC

π

df)f(R)f(E
f

2/Btf

2
i ∫

−
=

 
df

2f

2
B

tf

1k

1i
cT)1i2)(cff(jeiu)

cf
cffinc(scT2

cP2)f(iE ∫
−

∑
=

−−−−= π

 
dfeu

f

P
)f(E

2f

2
B

tf

1k

1i
cT)1i2)(cff(j

i2
c

c
i ∫

−

∑
=

−−−= π

 
df)f(EP

2
B

tf

2
B

tf
ici ∫

+

−

=

dfdfeu
f
P

P
2f

2
B

tf

1k

1i
cT)1i2)(cff(j

i
2
B

tf

2
B

tf
2

c

c
ci ∫

−

∑
=

−−−
∫

+

−

= π

 c1ci PgP =

 
oc11

t
t NPvgN

P
S

+
=

      Where tP  is TDMA received power, 1N =number 
of CDMA users, v = voice activity ratio ( ),5.0≤  oN = 
Thermal noise = KTB ; where K = Boltzman n’s 
constant = 1.38*10 -23 Joules/Kelvin, T  is the 
temperature in Kelvin.  

 )twcos()t(bP2)t(g tt=

 )twcos()t(c)t(bP2)t(x tt=

 
)]ff(C)ff(C[

2
P

)f(X tTdtTd
t −++=

 
∑
=

−−=
2

1

)12()(sinc)(
k

i

Tcifj
iccTd eufTTfC

π

 
dfeu

f

P
)f(E

2f

2
B

cf

2k

1i
cT)1i2)(tff(j

i2
c

t
i ∫

−

∑
=

−−−= π



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The Journal of Engineering Research  Vol. 5, No.1 (2008)  1-6

(17)

As before, the integral above can be assumed to be
equal to:

(18)

The value of g2 can be calculated for all the possible
Walsh codes. Figure 3 shows this result. It is to be con-
cluded here that due to lack of synchronization CDMA
and  TDMA systems, as they work separately, then all pos-
sible values of g2 are shown in Fig.  3 can be expected. To
analyze the worst case condition we shall take g2
=0.00386, see Fig. 3.

The signal to noise ratio at the CDMA receiver ( Sc )
after using (18) can be expressed as:

(19)

3. Capacity of the Hybrid System

In order to see if the spectral coexistence of CDMA and
TDMA systems is useful and possible we have to investi-
gate the variation of the system capacity
(CDMA+TDMA) as a function of number of CDMA
users and TDMA users for different CDMA received
power (or different TDMA received power), this study is
developed for constant values of Sc  and St .

Solving (10) for Pt :

(20)

Substituting in (19) we get:

(21)
Solving for N2 , we get:

(22)

Adding N1 to both sides of the above equation we get
the overall capacity NT :

(23)

Solving (21) for N1 :

(24)

By adding N2 to both sides of (24):

(25)
4.  Results of Analysis

Depending on the results of the theoretical analysis dis-
cussed in the last section, variations of the overall capaci-
ty in the hybrid system with different parameters is inves-
tigated for the worst case (that is, maximum overlapping;
fc = ft ). Values of the needed parameters are selected as
shown in Table (1) to comply with practical systems.
Figure 4 shows variations of the overall capacity with
number of CDMA users (N1) for different values of the
received CDMA power (Pc).  It is shown that for a con-
stant value of Pc, the overall capacity increases with
increasing  N1 till a certain limit after which any increase
in    N1 causes a sudden drop in the overall capacity. This
is because increasing the number of users  causes degra-

 
df)f(EP

2
B

cf

2
B

cf
iti ∫

+

−

=

Parameter  Value Parameter  Value  

bTB /1=  19200 Hz  cT  1/1.2288*10
6 s 

tS  
8  cS  0.04 

Table 1.  Values of different parameters used in the
paper

t2ti PgP =

Figure 3.  Variation of g2 with Walsh code index and
possible    synchronizations.    Maximum
value  for g2 (the worst case scenario) is
shown

 
ot22c1

c
c NPvgNvP)1N(

P
S

++−
=

θ

 otc11tt NSPvgNSP +=

 
oot1c1t22c1

c
c N)NSvgPNS(vgNvP)1N(

P
S

+++−
=

θ

 
)NSvgPNS(vgS
)NSvP)1N(S(P

N
ot1c1t2c

occ1cc
2 +

+−−
=

θ

1
ot1c1t2c

occ1cc
T N)NSvgPNS(vgS

)NSvP)1N(S(P
N +

+
+−−

=
θ

)]vgPSvgNSvPS[
]vPSNSNSvgNS[P

N
1ct22ccc

ccoccott22cc
1 θ

θ
+

−+−
=

2
1ct22ccc

ccocot22cc
T N)]vgPSvgNSvPS[

]vPSNSNSvgNS[P
N +

+
−+−

=
θ

θ



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The Journal of Engineering Research  Vol. 5, No.1 (2008)  1-6

dation in the signal to noise ratio.  If the amount of
increase in the number of users cause this  to be lower than
the threshold then all users will be assumed in the off state
and hence a sudden decrease in the capacity occurs.  This
suggests that it is of great importance to choose carefully
the number of CDMA users in the hybrid system. The case
is different when changing the number of TDMA users.
Figure 5 shows variations of the overall capacity with the
number of TDMA users (N2) for different values of Pc.  It
is obvious that the overall capacity increases with increas-
ing N2.

Depending on the above results, it is possible to say that
under certain limits on values of N1, N2, Pc and Pt , the
hybrid system (CDMA+TDMA) can have a total capacity
which is higher than the CDMA (or TDMA) system work-
ing alone by about 100%.  This conclusion can be
explained with the help of  Figs. 4 and 5. Maximum
capacity with a TDMA system working alone, ie.  N1 = 0,
or with a CDMA system working alone, ie. N2 = 0, is 48
users for the parameter values assumed in this paper.
While with the hybrid system maximum capacity can
reach up to 96 users. This means that the spectral coexis-

No. of CMDAmusers (N1)

O
ve

ra
ll 

ca
pa

ci
ty

Figure 4.  Variation of overall capacity with N1 for the different values of Pc

O
ve

ra
ll 

ca
pa

ci
ty

No. of TDMA users (N2)

Figure 5.  Variation of overall capacity with N2 for different values of Pc



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The Journal of Engineering Research  Vol. 5, No.1 (2008)  1-6

tence of CDMA and TDMA systems is possible and has
real benefits under certain conditions.

5. Conclusions

In this paper, theoretical investigations on the spectral
coexistence of narrow band CDMA and TDMA systems
has been presented. Results of analysis have proved the
possibility of the spectral coexistence of the two systems
under certain conditions which shows  that the available
frequency band can be used in an efficient way. It has been
shown that an increase of about 100% in the overall
capacity can be obtained for a certain range of values for
the number of CDMA users, number of TDMA users,
received CDMA power, and received TDMA power.

References

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Koorevaar, P. and  Ruprecht, J., 1999,  "Frequency
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Sinha, R., 1997,  "Spread Spectrum Interference Issues in
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