1 The Generalized CDMA 1.1 Introduction One of the basic concepts in communication is the idea of allowing several transmitters to send information simultaneously over a communication channel. This concept is described by the terms multiple access and multiplexing.Thetermmultiple access is used when the transmitting sources are not co-located, but operate autonomously as a multipoint-to-point network, while when the transmitting sources are co-located, as in a point-to-multipoint network, we use the term multiplexing. There are several techniques for providing multiple access and multiplexing, which belong to one of two basic categories: the orthogonal and the pseudo-orthogonal (PO) division multiple accesses. In orthogonal multiple access the communication channel is divided into sub-channels or user channels which are mutually orthogonal, i.e. are not interfering with each other. In pseudo-orthogonal multiple access, on the other hand, there is interference between user channels since they are not perfectly orthogonal to each other. The traditional Time Division and Frequency Division Multiple Access methods (TDMA and FDMA), as well as the Orthogonal Code Division Multiple Access (O-CDMA), are orthogonal multiple accesses, while the conventional asynchronous CDMA is a pseudo-orthogonal multiple access. Orthogonal division multiple access is achieved by assigning an orthogonal code or sequence to each accessing user (orthogonal code-sequences are presented in Chapter 2). Orthogonal sequences provide complete isolation between user channels. However, they require synchronization so that all transmissions arrive at the receiver at a given reference time (global synchronization). Pseudo-orthogonal multiple accesses, such as the asynchronous CDMA, are implemented with pseudo-random noise codes or sequences (PN-sequences) which suppress the other user interference only by the so- called spreading factor or processing gain. The pseudo-orthogonal approach, however, does not require global synchronization. The capacity (i.e. the maximum number of accessing users) of an orthogonal multiple access is fixed, and is equal to the length or the size of the orthogonal code, which is also equal to the spreading factor. In pseudo-orthogonal multiple access, on the other hand, the capacity is not fixed but is limited by the interference between users. Such a system is said to have a ‘soft’ capacity limit, since excess users may be allowed access at the expense of increased interference to all users. In general, the capacity in Pseudo- Orthogonal (PO) or Asynchronous (A) CDMA is less than the spreading factor. In order to enhance capacity, PO-CDMA sytems utilize multiple access interference CDMA: Access and Switching: For Terrestrial and Satellite Networks Diakoumis Gerakoulis, Evaggelos Geraniotis Copyright © 2001 John Wiley & Sons Ltd ISBNs: 0-471-49184-5 (Hardback); 0-470-84169-9 (Electronic) 2 CDMA: ACCESS AND SWITCHING cancellation techniques known as multiuser detectors (see Chapter 10). Such techniques are implemented at the receiver and they attempt to achieve (in the best case) what orthogonal codes provide at the transmitter in an orthogonal multiple access system, i.e. to eliminate the other user interference. Each of these two approaches is more efficient if it is used in the appropriate application. For example, Orthogonal CDMA (O-CDMA) can be used more efficiently in fixed service or low mobility wireless applications where synchronization is easier to achieve. Also, the O-CDMA is preferable in the forward wireless link (base-to- mobile), since no synchronization is required in this case. Asynchronous CDMA, on the other hand, is more appropriate in the reverse link (mobile-to-base) high mobility environment. The use of different access methods, however, led to the development of incompatible technologies and communication standards. In this chapter we attempt to provide an approach for unifying the multiple access communications. This approach is based on a user encoding process which is applied in order to integrate different access methods. Based on the proposed point of view, we represent a transmitter by a symbol encoder, and a user encoder, as illustrated in Figure 1.1. The symbol encoding provides channel encoding and symbol keying, while the user encoding provides the system and the user access into the communication link. The user encoding, in particular, is defined as the process in which a code sequence is used for both (1) to ‘spread’ the operating domain (i.e. time or spectrum), and (2) to identify each particular user in that domain. In this process the operation of spreading is required in order to create a ‘space’ in the channel which will contain all accessing or multiplexed users. The encoded signal will then depend upon: (1) The type of code sequence used. That is, the code sequences may be mutually orthogonal or pseudo-orthogonal, real or complex. (2) The type of spreading. Spreading may take place either in the frequency domain, called spread-spectrum, or in the time domain, called spread-time. (3) The pulse-shape of the data symbol. The pulse-shape, for example, may be time-limited or bandwidth-limited. : : Symbol Encoder User Encoder Symbol Decoder User Decoder User Code User Code SYNC Data Data Figure 1.1 The multiuser data communications process. GENERALIZED CDMA 3 TDMA G-TDMA DS-CDMA G-PDMA G-CDMA FDMA G-FDMA FH-CDMA Figure 1.2 The G-CDMA as the super-set of the multiple access methods. Each set of parameters (1), (2) and (3) defines a multiple access method or a type of user encoder. The combination of these parameters, (1), (2) and (3), will then create a large set of multiple accesses in which the conventional methods are only special cases, as illustrated in Figure 1.2. This super-set multiple access method is called Generalized CDMA (G-CDMA). Using this approach, in addition to the conventional methods, new multiple access methods have been created, such as the Generalized-TDMA and the Generalized-FDMA. Our purpose in this chapter, however, is not to examine and compare the performance of the new access methods, but to use them for demonstrating the continuum of the user encoding process. In the next section we present user encoding by real sequences, with spread- spectrum or spread-time, having synchronous or asynchronous access. We have reviewed the conventional asynchronous CDMA and have derived the traditional time division multiple access from the orthogonal spread-time CDMA. In Section 1.3 we present user encoding by complex sequences, with spread-spectrum or spread-phase, having synchronous or asynchronous access. In this case we have defined the generalized Frequency Division Multiple Access (FDMA) as a complex CDMA scheme, and from it we have derived the traditional FDMA and the frequency hopping CDMA. We have also presented a spread-phase CDMA and a Phase Division Multiple Access (PDMA) scheme. In Section 1.4 we present composite multiple access methods such as the spread-spectrum and spread-time multiple access using the method of extended orthogonal sequences presented in Chapter 2. This work was originally presented in reference [1]. 1.2 User Encoding by Real Sequences Let us now consider user encoding by sequences which are real numbers. First we assume the case of square pulse (time-limited) waveforms and binary (±1) sequences. 4 CDMA: ACCESS AND SWITCHING In particular, let a signal d i (t) of a data sequence of K symbols of user i, d i (t)= K−1  k=0 d i,k p T d (t −kT d )wherep T (t)=  1for0≤ t<T 0 otherwise Also, let the code-sequence c i (t) assigned to user i be given by c i (t)= L−1  l=0 c i,l p T c (t −lT c )1≤ i ≤ M where L is the length of the sequence, M is the number of sequences, T d is the duration of the data symbol and T c is the duration of the code symbol, and R d =1/T d is the data rate and R c =1/T c is the code rate. The encoded signal of user i is then s i (t)=d i (t) c i (t). The symbol  indicates the operation of user encoding, and is specified in each case we examine. As a result of encoding, s i (t) may be a spread-spectrum or a spread-time signal. Hence, we may distinguish the cases of spread-spectrum and spread-time described in the following subsections. 1.2.1 Spread-Spectrum In the case of spread-spectrum, the length of the data symbol is N times longer than the length of the encoding symbol T c . Hence, we define the ratio N SS ≡ T d T c = R c R d = N to be the spreading factor,whereN is an integer N>1, and T d = NT c .Therateof s i (t)isthenR c >R d , which means that the required bandwidth has to be spread to accommodate the rate R c = NR d . The encoded symbol or the spread time-pulse is called a chip. Considering a spread-spectrum process, we may again distiguish two cases. In the first case, spreading is achieved with orthogonal squences, and such a system is called USER ENCODER c i R c R d d i 1 : i : N s i = d i c i R c r c i = N d i . USER DECODER ∑ j s j r = c i . c i1 c i2 c i3 c i4 c i1 c i2 c i3 c i4 T d T c d i =1 d = i −1 N=4 Figure 1.3 The Spreading Process in Orthogonal CDMA. GENERALIZED CDMA 5 − 1 − G d (f) G ss (f) T d 1 T d 1 T c 1 T c 0 Spread Bandwidth Data Bw Figure 1.4 The power spectrum of data and spread signal. orthogonal or synchronous CDMA. In the second case, spreading is achieved with Pseudo-random Noise (PN) sequences. Then we have the conventional asynchronous CDMA, also called direct sequence CDMA (DS/CDMA). The Orthogonal CDMA Orthogonal CDMA (O-CDMA) is based on binary orthogonal sequences of length N. That is, the spreading factor is equal to the sequence length, which is also equal to the number of sequences. Hence, M = N = L.Letd i be a data symbol of user i, and c i ≡ [c 1i ,c 2i , , c Ni ]bethei th orthogonal code vector (sequence), i =1, , N; d i,j ,c ij ∈{−1, +1}. The encoded data vector of user i, s i is defined as follows: s i ≡ d i c i ≡ [d i c 0,i ,d i c 1,i , , d i c N−1,i ]. Assuming K consecutive data symbols, the transmitted signal of the O-CDMA is described by the equation s i (t)= K−1  k=0 d k,i c i (t −kT d )wherec i (t)= N−1  l=0 c l,i p T c (t −lT c )1≤ i ≤ N The transmitted signal s i (t) has a rate R c =1/T c = N/T d = NR d ,sinceT d = NT c . This means that the required bandwidth of the transmitted signal is N times wider than the bandwidth of the data d i (t), (spread-spectrum). Hence, the spreading factor is N ss = R c R d = T d T c = N>1. The spreading process is illustrated in Figure 1.3. Assuming that each chip is a square time pulse with duration T c , the spectrum of the 6 CDMA: ACCESS AND SWITCHING spreaded signal is (see Figure 1.4) G ss (f)=T c  sin πfT c πfT c  2 That is, the chip pulse is time-limited but spectrally unlimited. Therefore, a band- limiting filter (LPF) has to be used to limit the bandwidth in this case. Now, we assume that all N users accessing the system are synchronized to a reference time so that chips and symbols from all users are aligned at the receiver. Also, omitting the thermal noise and the impact of the band-limiting filter, the received signal at the input of the decoder is given by r(t)= K  j=1 s j (t)= K  j=1 N−1  l=0 d k,j c l,j p T c (t −lT c ) After the A/D converter the received signal may be represented by r = N  j=1 s j = N  j=1 d j c j The decoding process consists of taking the inner product between vectors r and c i . That is, r ·c i = N  i=1 N  j=1 d j c j · c i = Ld i since c j · c i = N  k=1 c kj c ik =  Lifi= j 0 if i = j The Asynchronous DS/CDMA In the asynchronous DS/CDMA we use Pseudo-random Noise (PN) sequences with length L,whereL ≥ N (T d = NT c ). PN-sequences are defined in Chapter 2 and are represented here by a continuous time function c i (t)=  L−1 l=0 c l,i p T c (t − lT c ), where p T c (t) is a square time-pulse with length T c ,andc l,i ∈{−1, +1}. The continuous time autocorrelation function R i (τ)ofc i (t) is then defined by R i (τ)= 1 L  L 0 c i (t)c i (t + τ)dt R i (τ) has been evaluated and is equal to R i (τ)=  l q(τ −lLT c ) where q(τ )=  1 − |τ| LT c (1 + 1 L )for|τ |≤T c − 1 L for T c ≤|τ|≤LT c /2 GENERALIZED CDMA 7 )(R i τ = = τ 0 L 1 − c T c T − 1 A. )f(S c f 0 c LT 1 c T 1 c T 1 − B. Figure 1.5 The power spectrum of the data and the spread signal. R i (τ) is shown Figure 1.5-A. The power spectral density S c (f)ofc i (t) is then the Fourier transform of R i (τ), and is given by S c (f)= L +1 L 2  sin πfT c πfT c    ∞  n=−∞,n=0 δ(f −n/LT c )   + 1 L 2 δ(f) Since R i (τ) is a periodic function with period L, S c (f) is a line spectrum. As L increases the spectral lines get closer together. S c (f) is shown in Figure 1.5-B. Now let c i (t) be assigned to the i th user. Also, let a sequence of K data symbols d i (t)= K−1  k=0 d k,i p T d (t −kT d ) where d k,i ∈{−1, +1}. The encoded signal of user i is then s i (t)=d i (t)c i (t)= K−1  k=0 d k,i c i (t −kT d )= K−1  k=0 N−1  l=0 d k,i c l,i p T c (t −lT c ) The signal s i (t) is transmitted at a carrier frequency f o (f o  1/T c ), which is s i (t) √ 2P i cos(2πf o t + θ i ), where P i is the power of the transmitted signal of user i. 8 CDMA: ACCESS AND SWITCHING Assuming M transmitting users, and omitting the thermal noise component, the received signal is given by r(t)= M  j=1 √ 2Pd j (t −τ j )c j (t −τ j )cos(2πf o t + φ j ) Since all users are transmitting asynchronously, the time delays (τ j ,forj =1, 2, , M) are different from each other. Also, φ j = θ j −2πτ j . Without loss of generality, we may assume θ i =0andτ i = 0, since we are only concerned with the relative phase shifts modulo 2π and time delays modulo T d . Then, 0 ≤ τ j <T d and 0 ≤ θ j < 2π for j = i. We have also assumed that each signal presents the same power P to the receiver. This assumption is satisfied with a power control mechanism. The transmitted signal s i (t), is recovered by correlating the received signal r(t) with the locally generating signal c i (t)cos2πf o t of user i, over the period of the symbol k =0: Z i =  T d 0 r(t)c i (t)cos2πf o tdt =  P/2    d i,0 T d + M  j=1(j=i) [d j,−1 R j,i (τ k )+d j,0 R  j,i (τ k )] cos φ j    The first term in the above expression d i,0 T d is the desired signal of user i, while the summation term represents the interference from all other users j,touseri. The interference is expressed in terms of the continuous-time partial cross-correlation functions R j,i and R  j,i , defined by R j,i (τ)=  τ 0 c j (t −τ)c i (t)dt and R  j,i (τ)=  T d τ c j (t −τ)c i (t)dt In order to evaluate the interference term we consider the phase shifts, time delays and data symbols as mutually independent random variables. The interference term in the above equation of Z i is random and may be treated as noise. Now, to evaluate the variance of Z i , we assume, without loss of generality, that φ i =0,τ i =0and d i,0 = 1. Then, Var{Z i } = P 4T d M  j=1  τ 0 [R 2 j,i (τ)+R  j,i 2 (τ)]dτ = P 4T d M  j=1 N−1  l=0  (l+1)T c lT c [R 2 j,i (τ)+R  j,i 2 (τ)]dτ for 0 ≤ lT c ≤ τ ≤ (l+1)T c ≤ T d . The expected values have been computed with respect to the mutually independent random variables φ j ,τ j ,d j,−1 and d j,0 for 1 ≤ j ≤ M and j = i. We have assumed that φ j is uniformly distributed on the interval [0,π]and τ j is uniformly distributed on the interval [0,T d ]forj = i. Also, the data symbols d j,k are assumed to take values +1 and −1 with equal probability. GENERALIZED CDMA 9 The Var{Z i } has been evaluated approximately in [2], and is found to be Var{Z i }≈PT 2 d (M −1)/6N The Signal-to-Interference Ratio (SIR) is defined as the ratio of the desired signal  P/2 T d divided by the rms value of the interference,  Var{Z i }.Thenwehave, SIR i ≡  P/2 T d  Var(Z i ) =  P/2 T d  PT 2 d (M −1)/6N ≈  3N M −1 where N is the spreading factor and M is the number of accessing users. 1.2.2 Spread-Time As in the case of spread-spectrum, spreading in time creates the ‘space’ in which multiple users may access the communication medium. In Spread-Time (ST) each encoding symbol may span one or more data symbols and each data symbol is repeated on every encoding symbol for the length of the sequence. Orthogonal Spread-Time CDMA Let d i be the k th symbol of user i and c i an orthogonal code sequence given by the vector c i ≡ [c 1i ,c 2i , , c Ni ]fori =1, , N where d i ,c ji ∈{−1, +1}. The encoded time-spread symbol is then given by the vector s i = d i c i =[d i c 1i ,d i c 2i , , d i c Ni ] (Since this is an orthogonal system N = M = L, L is the sequence length.) The transmitted signal s i (t) is then given by s i (t)=d i N−1  n=0 c ni P T d (t −nT d )for0≤ t ≤ NT d s i (t) has the same rate R d =1/T d as the data signal d i (t), while the rate of the code sequence is R c = R d /N . This means that the required bandwidth of the transmitted signal is the same as d i (t), while the required time for the transmission of its data symbols is N times longer (spread-time). Hence, given the length of the encoding symbol T c , and the length of the data symbol, T d , we define the ST-Spreading Factor to be the ratio N ST = T c T d = R d R c = N>1 At the receiving end the signal is given by r(t)= N  j=1 d j N−1  n=0 c ni P T d (t −nT d ) 10 CDMA: ACCESS AND SWITCHING In the above equation we have assumed that the symbols from all transmitting users are synchronized at the input of the receiver. We have also assumed that all arriving signals present equal power to the receiver. Also, the thermal noise component has been omitted and the impact of band-limiting filter has been ignored. After the A/D converter the received signal can be represented by the vector r = N  j=1 s j = N  j=1 [d j c 1j ,d j c 2j , , d j c Nj ] The transmitted symbol d i will then be recovered by taking the inner product of the vector r with the corresponding code vector c i of user i r ·c i = N  j=1 s j · c i = N  j=1 d j c j · c i = Nd i since c j · c i = N  k=1 c kj c ki =  N if i = j 0ifi = j Now, let us consider having a sequence of K data symbols of user i represented by the vector d i ≡ [d 1i ,d 2i , , d Ki ]. The encoded data vector of user i, s i , is then the r c i = N d i . USER ENCODER c i R c R d d i USER DECODER 1 : i : N s i = c i x d i R d ∑ j s j r = c i .       − 11 11 C = N=2, K=3 User 1 User 2 Spread-Time: NTc d 11 d 12 d 13 d 11 d 12 d 13 d 21 d 22 d 23 −−−d 21 d 22 d 23 T c T d Data Time Encoded user data: : User Code Vector, size N : User Data Vector, size K R c : Code Rate T c : Code Symbol Length c i d i R s : Symbol Rate T d : Data Symbol Length C : Orthogonal Code Matrix Figure 1.6 The Generalized Time Division Multiple Access (G-TDMA). [...]... PN-sequence (wi ∈ {+1, −1}), with L N 14 1.3.1 CDMA: ACCESS AND SWITCHING Spread-Spectrum In this section, as in that for the Spread-Spectrum (SS) CDMA with real encoding sequences, we examine the orthogonal and pseudo-orthogonal SS -CDMA, but with complex sequences Here, we also derive the conventional Frequency Division Multiple Access (FDMA) and the frequency hopping CDMA as special cases of a more general... all transmiting users However, the synchronization requirement in this case, unlike the spread-spectrum orthogonal CDMA, can be easily achieved since the length of the code symbol (or time slot) is N times longer than the data symbol Also, the ST Orthogonal CDMA, like the spread-spectrum DS /CDMA, requires power control The use of pseudo-random (PN) sequences with this type of spread-time accesses is also... Frequency-Hopping CDMA, where in,k indicates the frequency bin of the next hop for the (k + 1)th symbol The values of in,k are determined by an orthogonal or PN-code which is assigned to the nth user If the codes are orthogonal, consecutive symbols of different users may ‘hop’ simultaneously without ‘hitting’ the same bin A simple approach is to set in,k = 1 Asynchronous Complex CDMA In asynchronous complex CDMA. .. multiplexing of multiple symbol rates within the orthogonal CDMA channel Let us consider a basic symbol rate R and multiples of R, Rk = kR for k = 1, 2, 4, 8, 12, 16, 20, Also let Rc (Rk < Rc ) be the chip rate (spreading rate) of the O -CDMA Then we can use the method of extended sequences to accommodate multiple transmission rates in the O -CDMA network This method consists of concatenating two orthogonal... presenting a unified multiple access method called Generalized CDMA The GCDMA represents a super-set of multiple accesses created by the process of user encoding which is performed with different types of encoding sequences The choice of spread-spectrum with real encoding sequences provides the conventional (synchronous and asynchronous) CDMA The choice of spread-time with real encoding sequences creates... 2 We have also presented applications of composite access methods for wireless two-step access networks, and for multiplexing multiple symbol rates in orthogonal CDMA channels 28 CDMA: ACCESS AND SWITCHING References [1] D Gerakoulis ‘G -CDMA: a Unifying Approach to Multiple Access Communications’ AT&T Labs-Research Technical Memorardum HA1360000000306-01TM [2] M B Pursley ‘Performance Evaluation for... interfering with each other 26 CDMA: ACCESS AND SWITCHING R16 ( 32 ) h 1 R16 h R8 (2) W1 R8 Σ ( 32 ) 2 R16 h (2) W2 R4 (4) W1 : R4 Σ Σ R16 h (4) W4 R2 (8) W1 : R2 Σ ( 32 ) 3 Rc ( 32 ) 4 R16 h (8) W8 ( 32 ) 5 : : h R = 19.2 ks/s R ( 32 ) 32 = 16 ⋅ R = 8 ⋅ R2 = 4 ⋅ R 4 = 2 ⋅ R8 1 16 1 Rc = 32 ⋅ R = 9.8304 Mc/s 1 Figure 1.11 Example of multiplexing the rates, R2 , ,R16 , into a O -CDMA channel B: Multiplexing... distributed 18 CDMA: ACCESS AND SWITCHING over all bins, thus providing diversity in frequency selective fading channels On the other hand, FDMA does not require power control We also assume that all received signals are synchronized in both time and phase Therefore, this type of FDMA is a synchronous one and it does not require a ‘guard-band’ between the frequency bins If we assume asynchronous complex CDMA. .. and the sequence hj identifying the group j among the other groups of (k) (N ) transmission rates The rate of wi is Rk and the rate of hj is the O -CDMA chip rate Rc = KR The data of user (i, j) will then be recovered by despreading first the GENERALIZED CDMA (N ) sequence hj 27 (k) and then the sequence wi , then T Tk r( )hj ( ) cos(ωo )d Z= 0 wi (ν)dν = ±Ai,j kN Tc 0 where the symbol length Tk = 1/Rk... sequences are real is translated into Spread-Phase (SP) when the spreading sequences are complex As a result of spread-phase CDMA (following an equivalent approach as in the previous section), we derive the Phase Division Multiple Access (PDMA) methods Orthogonal Spread-Phase CDMA As in the case of spread-time with real sequences, the encoded signal in this case has the following form: −1 sn (t) = xn . communications process. GENERALIZED CDMA 3 TDMA G-TDMA DS -CDMA G-PDMA G -CDMA FDMA G-FDMA FH -CDMA Figure 1.2 The G -CDMA as the super-set of the multiple. conventional asynchronous CDMA, also called direct sequence CDMA (DS /CDMA) . The Orthogonal CDMA Orthogonal CDMA (O -CDMA) is based on binary orthogonal sequences of