Multi-Carrier CDMA Systems Using Bridge Functions Slimane Ben Slimane Department of Signals, Sensors and Systems Royal Institute of Technology, Sweden Tel. +46 8 790 9353, Fax +46 8 790 9370 e-mail:
[email protected]
Abdullatif Glass Dept. of Communication and Electronic Engineering Royal Melbourne Institute of Technology University Tel: +61 3 9925 2976, Fax: +61 3 9662 1060 e-mail:
[email protected]
Abstract— Multi-Carrier CDMA systems appear to be a good candidate for high speed wireless data transmission. These systems avoid time domain equalization and have the capability of achieving high diversity gains over frequency selective fading channels. Orthogonal Walsh-Hadamard spreading waveforms have been commonly considered for these systems. However, reliable system performance over frequency selective fading channels requires complicated detector structures with a complexity that increases with the length of the spreading code. In this paper a multi-carrier CDMA system using the Bridge functions as spreading waveforms is considered and its performance over frequency selective fading channels is evaluated. The Bridge functions are three valued orthogonal functions which take the values +1, 0, and −1. Such a structure adds flexibility to the multi-carrier CDMA where the number of zeros in the spreading code can be used to decide on the frequency separation between the modulated subcarriers and thus avoiding the need for frequency interleaving. Another benefit is the possibility to use more powerful detectors at the receiver. Infact, this combined system can be seen as a generalization to multi-carrier CDMA where the Bridge functions play the role of bridging between the regular OFDM with no spreading and multi-carrier CDMA with WalshHadamard functions.
I. Introduction The continuous growth in traffic volume and emergence of new services such as wireless multimedia applications have begun to change the infrastructure of wireless telecommunication networks. The high capacity required to support such applications requires the use of the spectrum as efficiently as possible and also provides the flexibility in sharing the spectrum resources by multi-user. Spread spectrum systems using Code-Division Multiple Access (CDMA) have been widely used in wireless telecommunication networks. These include satellite, cellular, and the new generation personal and universal mobile telecommunication systems. There are several codes that have been considered for spectrum spreading which are based on orthogonal functions. Pseudorandom (PN) sequences, Gold and Kasami, Walsh-Hadamard and other spreading codes are widely used in the lit-
erature. Orthogonal frequency-division multiplexing (OFDM) based on the previous spreading technology has recently witnessed an extensive development for deployment in wireless telecommunication systems that includes but not limited to cellular systems and wireless LANs. In these systems multi-carrier CDMA is used to multiplex information signals from different users and then transmitted using a single carrier. The spreading strategy of multi-carrier CDMA is done in the frequency domain where the different chips of the same symbol are transmitted in parallel using OFDM modulation and orthogonal spreading codes are generally used. Since in typical mobile radio application using spread spectrum techniques the transmission channel is frequency-selective, multi-carrier CDMA can take full advantage of such channel behaviour and a diversity gain can be achieved. However, the maximum achievable diversity gain for typical mobile radio environments is limited by the transmission bandwidth and the transmission channel. Thus, it might not be necessary to use very longue codes and spread the multi-carrier signals over the whole bandwidth. Instead, one can use properly designed codes that can provide the same diversity gain and allow more powerful multi-user detectors such as maximum likelihood. In this paper, a novel proposal for spectrum spreading code is developed for the OFDM multi- carrier CDMA system. The Bridge functions have been considered as new spreading orthogonal functions. These functions are three valued functions taking the values −1, 0, and +1 and mathematically derived from block pulses and Walsh functions. The combination of Walsh functions and pulse codes represents a new bridging code between them and thus called Bridge functions. The symmetric copying of Walsh functions and the shift mode of block pulses are combined to construct Bridge functions [1], [2]. These generated functions have two extreme cases, when the three valued functions reduce to two valued functions: one of them is the Walsh functions and the other is the block pulse code. Bridge code offers more functions in each set compared with Walsh-Hadamard code and thus more users can be accommodated. A synchronous multi-carrier CDMA system is considered in which different data symbols are multiplexed on a single carrier using the above Bridge
functions as spectrum spreading code. The use of Bridge function in this structure adds flexibility to the multi-carrier CDMA system where the number of zeros in the spreading code can be used to decide on the frequency separation between the modulated sub-carriers and thus avoiding the need for frequency interleaving. The use of Bridge function in OFDM multi-carrier system can further be seen as a generalization to multi-carrier CDMA where the Bridge functions play the role of bridging between the regular OFDM with no spreading and multi-carrier CDMA with Walsh-Hadamard functions. The proposed multi-carrier OFDM system is analyzed mathematically and its transmission model performance over frequency selective fading channels is evaluated. The analysis shows that with the exploitation of the Bridge code properties inter-code interference (ICI) can be reduced. The limit of ICI is governed by the number of participating users from fullload capacity. Since in each Bridge set more functions can be generated compared with Walsh-Hadamard set, this leads to a significant reduction in ICI in comparison to Walsh-Hadamard code case. The analysis is supported by an example that emphasises on the use of Bridge functions and their relation to WalshHadamard functions in multi-carrier CDMA system. It is also shown in the analysis that recursive procedure can be adopted for the generation of Bridge functions similar to the well-known Hadamard matrices generation procedure. In this special case the shifting parameter of Bridge functions is set to zero. An alternative approach to multi-carrier CDMA system with the use of Bridge functions as quadrature spreading codes is also investigated. This approach is needed to overcome the situation where, depending on the load of the system, some sub-carrier frequencies might not be used depending on users spreading code. In this operation mode, each user can take the advantage of both frequency diversity and load variability. Furthermore, with this structure the orthogonality of the system is still preserved as well as the spectral efficiency. II. Multi-Carrier CDMA Transmission Model In this paper we consider a synchronous multicarrier CDMA system where the information from the different users are multiplexed at the transmitter and then transmitted using a single carrier frequency [3], [4]. The transmitter block diagram of the system is shown in Figure 1. Each user transmits N chips of data symbol over N different carriers, where N is the total number of chips per data symbol. When all chips are non-zero N becomes the Processing Gain (PG). As we are dealing with a synchronous system the frequency separation between adjacent subcarriers is taken as 1 ∆f = Ts
where Ts is the multi-carrier symbol duration. Here, we have assumed that the number of subcarriers is equal to the number of chips and is defined as N . Over one symbol interval, the multi-carrier CDMA transmitted signal with a full load takes the following form ("N −1 # ) X m x(t) = Re sm ej2π Ts t ej(2πfc t+θ) (1) m=0
where sm =
NX u −1 n=0
cn,m an , m = 0, 1, · · · , N − 1,
(2)
is the transmitted signal at subcarrier m, Nu is the total of active users, cn = [c0,n , c1,n , · · · , cN −1,n ]
T
is the spreading sequence of user n which corresponds to the nth column of the N × N matrix c0,0 ··· c0,N −1 .. .. .. C= (3) , . . . cN −1,0
· · · cN −1,N −1
an is the transmitted symbol of user n with E{|an |2 } = Es , and Es is the user symbol energy. In Figure 1, the spreading waveform is a periodic function with period T and is defined as N −1 1 X ci,n p(t − nTc ), cn (t) = √ N i=0
0 ≤ t ≤ Ts = N Tc ;
and p(t) =
√1 , Tc
0,
0 ≤ t ≤ Tc elsewhere
(4)
In a matrix form, the transmitted signals of the different subcarriers can be written as follows: s = [s0 , s1 , · · · , cN −1 ]T = C a. where [·]T refers to the transpose matrix and a is a column vector containing the information symbols from the Nu different users. user n binary data
baseband modulator
an
S/P
OFDM
1:N
modulator
x(t)
cn (t)
other users
Fig. 1. Transmitter block of a multi-carrier system using Bridge functions as spreading waveforms.
A. Channel Model The mobile radio channel is modeled by a tapped delay line with equivalent lowpass impulse response h(t) =
P −1 X i=0
αi δ(t − τi )
(5)
where αi is a complex Gaussian random variable with zero-mean and average power pi representing the tap weight of path i, τi is its relative time delay, and P is the total number of resolvable paths. We assume a normalized fading channel, i.e., P −1 X
pi = 1.
i=0
In the expression of h(t) we have assumed a slowly varying fading channel with a Doppler spread much smaller than the symbol duration Ts . The equivalent lowpass of the multi-carrier CDMA received signal takes the following form r(t) =
P −1 X l=0
αl e−j2πτi fc xl (t − τl ) + z(t),
(6)
where fc is the carrier frequency, xl (t) is the equivalent lowpass of the transmitted signal, and z(t) a complex Gaussian random process representing the thermal noise. The received signal is then correlated at each subcarrier frequency and sampled at the symbol rate (or equivalently an N point FFT receiver). Assuming a guard time interval longer than the maximum delay spread of the channel the correlator output sample of subcarrier k is given by rk = hk e
jφk
sk + z k
(7)
where hk e
jφk
=
P −1 X
k
αi e−j2πτi (fc + T )
(8)
i=0
is a complex Gaussian random variable representing the channel attenuation factor at subcarrier k. Considering the output samples of the N correlators as a vector of length N we get r = =
jφ0
jφN −1
It is obvious from the above expressions that if the delay spread is much smaller than the symbol duration of the multi-carrier CDMA signal than the fading amplitudes of the adjacent subcarriers will be very correlated. In this situation it is necessary to introduce some kind of frequency interleaving to reduce this correlation and increase the frequency diversity gain. However, when the number of subcarriers is chosen equal to the total number of chips per symbol such interleaving will not provide any extra diversity gain. It might therefore be better to reduce the number of chips per symbol. This will increase the frequency separation between adjacent chips and allows the use of more powerful detectors at the receiver [3]. Another possible solution that help in increasing the frequency separation between adjacent chips is to use some spreading codes than are able to automatically control this frequency separation and may avoid the need for frequency interleaving. The following sections will show that the Bridge functions [1], which are also orthogonal waveforms, have very nice properties that can be exploited for multi-carrier systems in frequency selective fading channels. III. Multi-Carrier CDMA using the Bridge Functions The Bridge functions are three valued functions taking the values -1, 0, and +1. These functions were introduced by Zhihua and Qishan [1] and are derived from a combination of the block pulses and the Walsh-Hadamard functions. Just like Walsh-Hadamard functions, the Bridge functions are orthogonal functions and can be represented by and N × N square matrix b0,0 b0,1 ··· b0,N −1 b1,0 b1,1 ··· b1,N −1 BN (j) = (10) .. .. .. . . . bN −1,0
(r0 , r1 , · · · , rN −1 ) HCa + z jφ1
of the channel. For a slowly varying channel, the frequency correlation function can be expressed as a function of the subcarrier separations. For a uniform delay profile of width Tm = τmax − τmin this correlation can be written as follows: o n Tm ρ(k) = E hn hn+k ej(φn+k −φn ) = sinc πk . Ts
(9)
is a diwhere H = h0 e , h1 e , · · · , hN −1 e agonal matrix representing the fading coefficients of the different subcarriers, a is the vector containg the symbols from the different users during the kth symbol interval, and z is the Gaussian noise vector. Assuming a fading channel with only diffuse paths, the fading channel coefficients hi ejφi are correlated complex Gaussian processes with zero-mean and unit 2 variance (i.e., E{|hl | } = 1). The correlation between adjacent channels is dependent on the delay spread
bN −1,1
· · · bN −1,N −1
where
p
bi,k = −1, 0, or +1
N = 2 , and j (0 ≤ j ≤ p) is an integer controlling the number of zeros in the Bridge sequence. For each value of j a new set of orthogonal Bridge functions is obtained. Therefore, for a given code length N = 2p one can generate p + 1 different sets of N orthogonal spreading codes. Among these sets we find two extreme cases. From one extreme we find the set of the Walsh-Hadamard spreading codes which is obtained when j = 0 BN (0) = HN ,
where in this case the three-valued function is reduced to a two-valued function (−1, +1), and on the other extreme the set of block pulses which is obtained when j = p, i.e., BN (p) = IN (the N × N identity matrix), which is ofcourse a two-valued function taking the values 0 and +1. It was shown in [1] that the Bridge functions can be generated using copying with (even or odd) symmetry. This method can be used to generate any particular code within the set independently from the other codes. Only two parameters are needed for generating such codes: The shifting parameter j and the order of the code (A detailed description of this method is given in [1]). It is well known that the Hadamard matrices can be generated using a recursive procedure. In this paper we consider a recursive procedure for generating the Bridge functions similar to that used to generate the Hadamard matrices. As each set of Bridge functions is defined by a particular shifting parameter j (0 ≤ j ≤ p), we define the dimension L with L = 2j . The Bridge function BN (j) is then obtained using the following recursive procedure: BL (j) = IL , B2i−1 L (j) B 2i−1 L (j) ,(11) B2i L (j) = B2i−1 L (j) −B 2i−1 L (j) where IL is the L × L identity matrix, B stands for the reverse sequence B, −B is the reverse sequence B with sign reversed, and i is any nonnegative integer. Notice that the recursive procedure for generating the Hadamard matrices is just a special case of the above procedure and is obtained when j = 0. This is natural since as mentioned earlier Hadamard functions belong to the family of the Bridge functions. Using Bridge functions as spreading waveforms for multi-carrier transmission systems each user is assigned a spreading waveform defined by a given column of the Bridge matrix BN (j). That is, N −1 1 X cn (t) = p bi,n p(t − iTc ), Nj i=0
0 ≤ t ≤ Ts = N Tc ,
n = 0, 1, · · · , N − 1,
where Nj represents the number of non-zero elements in the selected column of the Bridge matrix. Example 1: As an illustrative example let us consider the Bridge functions B8 (j). In this case we have p = 3 which gives a total of four different orthogonal sets. Following the recursive procedure given in (11) the matrix representation of these sets is obtained B8 (0) =
+1 +1 +1 +1 +1 +1 +1 +1
+1 −1 +1 −1 +1 −1 +1 −1
+1 −1 −1 +1 +1 −1 −1 +1
+1 +1 −1 −1 +1 +1 −1 −1
+1 +1 −1 −1 −1 −1 +1 +1
+1 −1 −1 +1 −1 +1 +1 −1
+1 −1 +1 −1 −1 +1 −1 +1
+1 +1 +1 +1 −1 −1 −1 −1
+1 0 +1 0 +1 0 +1 0
0 +1 0 +1 0 +1 0 +1
0 +1 0 −1 0 +1 0 −1
+1 0 −1 0 +1 0 −1 0
+1 0 −1 0 −1 0 −1 0
0 +1 0 −1 0 −1 0 +1
0 +1 0 +1 0 −1 0 −1
+1 0 +1 0 −1 0 −1 0
+1 0 0 0 +1 0 0 0
0 +1 0 0 0 +1 0 0
0 0 +1 0 0 0 +1 0
0 0 0 +1 0 0 0 +1
0 0 0 +1 0 0 0 −1
0 0 +1 0 0 0 −1 0
0 +1 0 0 0 −1 0 0
+1 0 0 0 −1 0 0 0
+1 0 0 0 0 0 0 0
0 +1 0 0 0 0 0 0
0 0 +1 0 0 0 0 0
0 0 0 +1 0 0 0 0
0 0 0 0 +1 0 0 0
0 0 0 0 0 +1 0 0
0 0 0 0 0 0 +1 0
0 0 0 0 0 0 0 +1
B8 (1) =
B8 (2) =
B8 (3) =
,
It is observed that the Hadamard matrix is given by BN (0) which corresponds to the spreading codes that have been widely studied and used for multi-carrier CDMA systems. The BN (p) matrix (the 4th in the above example) is the identity matrix which corresponds to the case of no spreading or basically the regular OFDM where each user is assigned one subcarrier. The other matrices are possible alternatives of other spreading codes that can be used for multicarrier CDMA systems. This means that by choosing a particular set of spreading codes one can decide on the level of signal spreading over the different subcarriers. With this degree of freedom one can make a compromise between frequency diversity gain and receiver complexity. Also if the maximum delay spread of the channel is very small compared to the symbol duration, it might not be necessary to spread the signal over the whole frequency spectrum. Notice that the non-zero values of each spreading code (column of the matrix) are equally spaced which ensures a good signal spreading over the total signal bandwidth. To illustrate the suitability of the Bridge functions we consider a multi-carrier CDMA with coherent BPSK and the Bridge functions B8 (j), j = 0, · · · , 3 as defined above. We look at the system performance when the multi-carrier CDMA signal is transmitted over a two path fading channel with delays τ0 = 0, τ1 = 0.10Ts, and equal power split. Figure 2 shows the average bit error probability for the single user case when maximum ratio combining is used at the receiver. We notice that changing the set of spreading codes does not affect the bit error probability. Thus, one can select the set of spreading codes that reduces the receiver complexity and preserves the achievable diversity gain. Figure 3 illustrates the performance of a given user for a full system over the
same two path channel. It is observed here also that changing the spreading codes has little effect on the system performance.
Average Bit Error Probability
BF (4) 16 BF (3) 16 BF (2) 16 BF16(1) BF (0)
−1
10
16
−2
10
MRC Detector
(cI2n , cQ 2n ) = (b2n , b2n+1 )
−3
10
and for user 2n+1 the same spreading codes but with a 90o phase rotation
−4
10
(cI2n+1 , cQ 2n+1 ) = (b2n+1 , b2n ).
−5
10
cn (t) = cIn (t) + jcQ n (t) These two codes can be chosen such that the inphase and the quadrature components of the user transmitted signal are always disjoint. That is, they do not share the same the subcarriers. For instance let us consider the example discussed in the previous section and let us assume that the Bridge function B8 (1) is used as spreading codes. We assign for user 2n the two codes
p
MC−CDMA using Bridge Functions with N=2 =16
0
10
the quadrature component of the user signal) to each user. Thus, the spreading code of user n becomes
0
5
10 15 20 25 Signla−to−Noise Raio, (dB)
30
35
Fig. 2. Average bit error probability of multi-carrier CDMA using Bridge functions over a two path fading channel for the single user case.
In this way each user signal can take advantage of both frequency diversity and load variability. Furthermore, with this structure the orthogonality of the system is still preserved as well as the spectral efficiency. V. Conclusions
p
MC−CDMA using Bridge Functions with N=2 =16
0
Average Bit Error Probability
10
BF (4) 16 BF (3) 16 BF (2) 16 BF16(1) BF (0)
−1
10
16
−2
10
MMSE Detector −3
10
−4
10
−5
10
0
5
10 15 20 25 Signla−to−Noise Raio, (dB)
30
35
Fig. 3. Average bit error probability of multi-carrier CDMA using Bridge functions over a two path fading channel for the full system case.
IV. Multi-Carrier CDMA with Quadrature Spreading Bridge Functions We have seen in the previous section that the use of Bridge functions as spreading codes for multicarrier CDMA gave a flexibility in the degree of signal spreading over the available broad band. One can also notice that depending on the spreading code assigned to each user, some users do not share the same subcarrier frequencies. This indicates that depending on the load of the system some subcarriers might not be used. One possible solution to take advantage of this situation is to assign quadrature spreading (two codes (cIn , cQ n ), one for the inphase component and one for
This paper reported a novel approach to multiple access technique using multi-carrier CDMA systems. The good properties of the Bridge functions as a spreading code is exploited to achieve multiplexing of several users data on a single carrier. This new orthogonal set is shown to be as a generalization to the Walsh-Hadamard orthogonal set and also provides an even signal spreading over the whole allocated bandwidth. Analysis of the new proposed multi-carrier CDMA system revealed that more users can be accommodated and this leads to increase spectrum efficiency. Furthermore, it offers more flexibility in system structure and also avoids the need for frequency interleaving. Finally, it offers a degree of freedom in trading off frequency diversity gain and receiver complexity. References [1] L. Zhihua and Z. Qishan, “Introduction to bridge functions,” IEEE Trans. Elect. Compatibility, vol. EMC-25, No. 4, pp. 459-464, November 1983. [2] I. Haidar and R. L. Brewter, “Code-division multiplex technique using bridge functions,” IEE Proceedings, vol. 132, No. 6, pp. 480-484, October 1985. [3] K. Fazal, S. Kaiser, and M. Schnell, “A flexible and high performance mobile communications system based on orthogonal multi-carrier SSMA,” Wireless Personal Communications 2, pp. 121-144, 1995. [4] N. Yee, J.-P. Linnartz, and G. Fetteweis, “Multicarrier CDMA in indoor wireless networks,” IEICE Trans. Commun., Japan, vol. E77-B, pp. 900-904, July 1994.
