10

Some advancements in spread spectrum systems development

10.1 Multiuser reception and suppressing MAI

In Section 4.1 we met two options for making decisions on the data in a K-user CDMA system. One of them runs the optimal (ML) procedure realized by the so-called multiuser receiver, while the other involves a single-user or conventional procedure. The conventional receiver treats MAI as no more than an additional random noise, fully ignoring the deterministic nature of signatures and correlations between them. On the other hand, multiuser algorithms utilize a priori knowledge about signature codes or, at least, their ensemble correlation properties. In this section we are going to discuss briefly the ideas underlying multiuser reception, starting with the simplest case of synchronous CDMA.

10.1.1 Optimal (ML) multiuser rule for synchronous CDMA

In order to make the discussion free of secondary details let us consider the plainest, yet general enough, model of K-user DS CDMA involving real signatures and BPSK data transmission. The model covers, among others, any system with BPSK signature and data modulation. As in Section 7.2, within this subsection we consider a fully synchronous case when both chips and borders of data symbols (bits) of all users are strictly aligned in time. This, along with the assumption of the independence of consecutive data bits of any user, permits limiting the observation interval to a single bit duration: T ¼ Tb. Then the group signal of K users:

K

X

Spread Spectrum and CDMA: Principles and Applications Valery P. Ipatov

2005 John Wiley & Sons, Ltd

b ¼ (b1, b2, . . . , bK ) is the vector of data bits of K users (bit pattern) and sk(t) is the kth user’s signature.

As was mentioned in Section 4.1, the globally optimal (ML) procedure involves

of b minimizing the Euclidean distance (or its square d (s, y)) between the observation y(t) and group signal (10.1). Calculating d2(s, y) in the same way as in (4.3) results in:

is, as usual, correlation of the observation y(t) with the kth signature, kl is the correlation coefficient of the kth and lth signatures, and the presence of amplitudes Ak allows use of a convenient normalization of signatures:

ZT

Ek ¼ kskk2¼ s2kðtÞ dt ¼ 1; k ¼ 1; 2; . . . K

0

Let us introduce two matrices: G ¼ diag(A1, A2, . . . , AK ), a diagonal K K matrix of users’ amplitudes, and C ¼ [ kl], k, l ¼ 1, 2, . . . , K, the correlation matrix of signatures. With designation z ¼ (z1, z2, . . . , zK ) for the vector of correlations (10.3), the squared distance (10.2) becomes (superscript T symbolizing vector–matrix transpose):

The first term in the right-hand side of (10.4) is fixed with a current observation y(t) and

When a CDMA system is not oversaturated (K N; see Section 7.2), all signatures are allowed to be orthogonal, so that kl ¼ kl and C ¼ IK , where IK is the Kth order identity

matrix. Then bGCGbT ¼ PK¼ A2 does not depend on the user’s bit pattern b. As was

k 1 k

noted in Section 4.1, multiuser detecting then degenerates into a conventional one, where

^

the sign of the correlation zk defines the estimate bk of the kth user’s bit bk. In the case of non-orthogonal signatures (e.g. oversaturated CDMA, K > N) the conventional receiver yields to the ML one, but the latter may appear prohibitively complex. Indeed, the data vector b is strictly restrained by the BPSK alphabet limitation bk ¼ 1, and no

procedure exists that is more computationally efficient than just trying all 2K possible bit patterns and comparing the results of their substitution to the right-hand side of (10.5). Therefore, ML multiuser detection according to (10.5) has exponential complexity versus the number of users (see numerical example in Section 4.1). On the other hand, the motivation to involve multiuser detection is often strongest when the number of users is so great that conventional detection fails due to the high level of MAI. This explains the interest in various quasi-optimal multiuser algorithms, some of which are surveyed in the rest of the section.

10.1.2 Decorrelating algorithm

Let us start with a conventional (i.e. correlation-based) receiver of user number one’s data. According to (10.1) the observation:

K

X

k¼2

where n1 ¼ R0T n(t)s1(t) dt is a noise sample at the first correlator output. The second term of (10.7) is MAI, and the question is whether it may be suppressed to zero by means of some linear transform of the input observation. Whatever this linear operation is, finally we should have some MAI-free substitute &1 of z1, i.e. the scalar, producing the decision on the first user’s current bit as:

Any linear operation transforming y(t) into a scalar may be described as correlation:

ZT

&1 ¼ yðtÞuðtÞ dt ð10:9Þ

0

differing from (10.3) by only a reference signal u(t). Therefore, we are going to suppress MAI declining matched reference s1(t) in favour of mismatched one u(t), i.e. at the cost of loss in SNR with respect to a thermal noise. We have already resorted to this trick when seeking zero-forcing filters to remove autocorrelation sidelobes (see Section 6.12). Using (10.6) in (10.9) replaces (10.7) by:

K

X

where ku is the correlation coefficient of the kth signature with u(t), normalization

of reference u(t) is the same as for signatures, and n01 ¼ RT n(t)u(t) dt is a noise contri-

0

bution in &1.

Let us put signatures and a reference signal u(t) in the form (2.50) typical of DS CDMA with real-valued signatures:

where ui, i ¼ 0, 1, . . . , N 1 is a real code sequence of the reference u(t). Using the vector notation of code sequences ak ¼ (ak, 0, ak, 1, . . . , ak, N 1), u ¼ (u0, u1, . . . , uN 1) (see Section 7.2) and setting with no lost of generality chip energy E0 ¼ 1, we come to equationskl ¼ (ak, al) ¼ akaTl , ku ¼ (ak, u) ¼ akuT . To remove the MAI term in (10.10) independently of amplitudes and bits of interfering users we need to fulfil K 1 conditions:ku ¼ 0, k 2. In other words, the reference code u should be a solution of the linear equation set akuT ¼ uaTk ¼ 0, k ¼ 2, 3, . . . K. Preserving the non-zero useful effect 1u has to be non-zero, hence u is just a properly scaled solution v of the equation:

When all signature vectors are linearly independent, the system (10.12) may have a set of solutions, but among all vectors v satisfying (10.12) we choose the one that is a linear combination of signatures, i.e. rows of AT : v ¼ xAT , where x is an unknown K-dimensional row vector. The reason for such a choice is that inclusion in v of any component orthogonal to the space of the signature vectors will only increase the norm of v, i.e. the noise component at the correlator output, with no increase of the useful first term in (10.10). With this substitution (10.12) becomes:

xAT A ¼ xC ¼ e1

Linear independence of signatures (columns of A) means rank K of the correlation

K K matrix C ¼ AT A, i.e. its invertibility and uniqueness of solution of the equation above: x ¼ e1C 1. Then:

is a desired solution of (10.12), whose scaling u ¼ v/kvk results in the normalized decorrelation reference u, so that uaT1 ¼ 1u. This normalization is practically unnecessary, having no effect on the sign of &1 in the decision rule (10.8).

Physically the reference vector (10.13) is just orthogonal to all signatures but the first, entirely eliminating MAI at the output of a correlator tuned to the first user’s signal. We would find the reference signal for the kth user’s receiver in the same way, replacing e1 in (10.13) by ek, whose unique non-zero component is the kth one.

The main drawback of the described decorrelating receiver is its working capacity with only linearly independent signatures. If this demand is not observed, any attempt to