2.2 Linear Receivers for Synchronous CDMA

2.2.1 Synchronous CDMA Signal Model

We start by considering the most basic multiple-access signal model: a baseband K - user time-invariant synchronous additive white Gaussian noise (AWGN) system, employing periodic (short) spreading sequences and operating with a coherent BPSK modulation format. (An approach to adaptive detection in (long) aperiodic code DS-SS systems is developed in [61].) As noted in Chapter 1, the continuous-time waveform received by a given user in such a system can be modeled as follows :

Equation 2.1

where M is the number of data symbols per user in the data frame of interest; T is the symbol interval; A _k , , and s _k ( t ) denote, respectively, the received complex amplitude, the transmitted symbol stream, and the normalized signaling waveform of the k th user; and n ( t ) is the baseband complex Gaussian ambient noise with independent real and imaginary components and with power spectral density s ² . It is assumed that for each user k , is a collection of independent equiprobable ±1 random variables , and the symbol streams of different users are independent. For the direct-sequence spread-spectrum format, each user's signaling waveform is of the form

Equation 2.2

where N is the processing gain, is a signature sequence of ±1's assigned to the k th user, and y ( ·) is a chip waveform of duration T _c = T/N and unit energy

At the receiver, the received signal r ( t ) is filtered by a chip-matched filter and then sampled at the chip rate. The sample corresponding to the j th chip of the i th symbol is thus given by

Equation 2.3

The resulting discrete-time signal corresponding to the i th symbol is then given by

Equation 2.4

Equation 2.5

with

where is a complex Gaussian random variable with independent real and imaginary components; and [Here N _c ( ·, ·) denotes a complex Gaussian distribution and I _N denotes an N x N identity matrix.] and .

Suppose that we are interested in demodulating the data bits of a particular user, say user 1, , based on the received waveforms . A linear receiver for this purpose can be described by a weight vector such that the desired user's data bits are demodulated according to

Equation 2.6

Equation 2.7

Note that the linear equalizers and multiuser detectors discussed in Chapter 1 can all be written in this form, as will be seen below. In case the complex amplitude A ₁ of the desired user is unknown, we can resort to differential detection. Define the differential bit as

Equation 2.8

Then using the linear detector output ^[1] (2.6), the following differential detection rule can be used:

^[1] For simplicity, we will use the term "detector" to refer to the overall detector (2.6) “(2.7) or (2.6) and (2.9), to the detection statistic (2.6), and to the detector's weight vector.

Equation 2.9

Substituting (2.4) into (2.6), the output of the linear receiver w ₁ can be written as

Equation 2.10

In (2.10), the first term on the right-hand side contains the useful signal of the desired user, the second term contains the signals from other undesired users ”the multiple-access interference (MAI), and the last term contains the ambient Gaussian noise. The simplest linear receiver is the conventional matched filter, where w ₁ = s ₁ . As noted in Chapter 1, such a matched-filter receiver is optimal only in a single-user channel (i.e., K = 1). In a multiuser channel (i.e., K > 1), this receiver may perform poorly since it makes no attempt to ameliorate the MAI, a limiting source of interference in multiple-access channels. Two popular forms of linear detectors that are capable of suppressing the MAI are the linear decorrelating detector and the linear minimum mean-square-error (MMSE) detector, which are discussed next .

2.2.2 Linear Decorrelating Detector

A linear decorrelating detector for user 1, , is such that when correlated with the received signal r [ i ], it results in zero MAI [i.e., the second term in (2.10) is zero]. In particular, the linear decorrelating detector d ₁ for user 1 satisfies

Equation 2.11

Equation 2.12

Denote by e _k a K -vector with all entries zeros except for the k th entry, which is 1. Assume that the user signature sequences are linearly independent [i.e., the matrix has full column rank, rank ( S ) = K ]. Let be the correlation matrix of the user signature sequences. Then R is invertible. The following result gives the expression for the linear decorrelating detector.

Proposition 2.1: The linear decorrelating detector for user 1 is given by

Equation 2.13

Proof: It is easily verified that

Equation 2.14

Therefore, (2.11) and (2.12) hold.

The output of the linear decorrelating detector is given by

Equation 2.15

with

Equation 2.16

where, by (2.13),

Equation 2.17

and where in (2.17), [ A ] _i,j denotes the ( i , j )th element of the matrix A . Note that by the Cauchy “Schwartz inequality, we have

Equation 2.18

Since s ₁ = 1 and , it then follows that d ₁ 1. Hence, by (2.16), we have Var{ v ₁ [ i ]} s ² (i.e., the linear decorrelating detector enhances the output noise level).

2.2.3 Linear MMSE Detector

While the linear decorrelating detector is designed to eliminate the MAI completely at the expense of enhancing the ambient noise, the linear MMSE detector, , is designed to minimize the total effect of the MAI and the ambient noise at the detector output. Specifically, the linear MMSE detector for user 1 is given by the solution to the following optimization problem:

Equation 2.19

Denote . The following result gives the expression for the linear MMSE detector.

Proposition 2.2: The linear MMSE detector for user 1 is given by

Equation 2.20

Proof: First note that any linear detector must lie in the column space of S [i.e., m ₁ range ( S )]. This is because any component outside this space does not affect the signal components of the detector output [i.e., the first and second terms of (2.10], and it merely increases the noise level [i.e., the third term of (2.10)]. Therefore, we can write m ₁ = Sx ₁ for some where

Equation 2.21

Hence (2.20) is obtained.

The output of the linear MMSE detector is given by

Equation 2.22

with

Equation 2.23

where, using (2.20), we have

Equation 2.24

Equation 2.25

Note that unlike the decorrelator output (2.15), the linear MMSE detector output (2.22) contains some residual MAI. However, we will in general have m ₁ < d ₁ , so that the effects of ambient noise are reduced by the linear MMSE detector.