Multiuser System Model

Here we employ the multiuser, chip-rate sampled, CDMA ISI channel model described in detail in Wang and Poor [11], extended to time-varying channels. Index the users as $k=1,2, \cdots , K$ and chips per symbol as $j=1,2, \cdots , N$. The users' signaling waveforms upon which the user's bits are impressed can be written as

$$s_k(t)= ~ \sum_{j=1}^{N}c_k(j)\Psi(t-jT_c)$$ (1)

where N is the processing gain, c_k(j) is the pseudorandom code assigned to each user and $\Psi$ is the normalized chip waveform of duration T_c=T/N. At the receiver, assume chip-rate sampling over symbol times $i=1,2, \cdots , n$. The observed data is:

$$\sum_{k=1}^K {\bf a}_i^T (k) {\bf h}_i (j,k) ~ + ~ n_i (j) ~ ;$$ r_i (j) = (2)

$$~ ~ ~ ~ ~ ~ i=1,2, \cdots , n ~ ; ~ ~ j=1,2, \cdots , N ,$$

where the n_i (j) are zero-mean Gaussian noise samples uncorrelated across both i and j with variance $\sigma^2$, ${\bf h}_i (j,k)$ is the $M \times 1$ dimensional coefficient vector for the time-varying FIR channel for the k^th user and j^th chip position at symbol time i, M-1 is the memory depth (in symbols) of the channels, and ${\bf a}_i (k)$ is the vector of the k^th user symbols in the FIR channel at symbol time i. The ${\bf h}_i (j,k)$ are unknown (and time-varying at the symbol rate) and the problem is to optimally estimate the K users' symbols up to time n. Defining ${\bf a}_i ~ = [ {\bf a}_i^T (1), {\bf a}_i^T (2), \cdots , {\bf a}_i^T (K) ]^T$and ${\bf h}_i (j) ~ = [ {\bf h}_i^T (j,1), {\bf h}_i^T (j,2), \cdots , {\bf h}_i^T (j,K) ]^T$, we have

$$r_i (j) ~ = ~ {\bf a}_i^T {\bf h}_i (j) ~ + ~ n_i (j) ~ ~ .$$ (3)

The observation over the symbol interval i is

$${\bf r}_i ~ = ~ [ r_i (1) , ~ r_i (2) , \cdots , r_i (N) ]^T ~ ~ ,$$ (4)

and the observation up to symbol time n is

$${\bf r}^n ~ = ~ [ {\bf r}_1^T , {\bf r}_2^T , \cdots , {\bf r}_n^T ]^T ~ .$$ (5)

Let ${\bf A}^n$ and ${\bf H}^n$ denote the matrices of, respectively, all users' symbols and all time-varying FIR channel coefficients up to time n. (The constructions of ${\bf A}^n$ and ${\bf H}^n$ are not important in this discussion.) The joint probability density function of ${\bf r}^n$, conditioned on ${\bf A}^n$ and ${\bf H}^n$, is

$$f( {\bf r}^n \vert {\bf H}^n , {\bf A}^n ) ~ = \frac{1}{( \pi... ...ert r_i (j) - {\bf a}_i^T {\bf h}_i (j) \vert^2 / \sigma^2 } .$$ (6)

Define the $MKN \times 1$ dimensional vector of all channel coefficients at symbol time i as

$${\bf h}_i ~ = ~ [ {\bf h}_i^T (1) , {\bf h}_i^T (2) , \cdots , {\bf h}_i^T (N) ]^T$$ (7)

and consider the $N \times MKN$ dimensional symbol matrix at symbol time i

$${\bf A}_i ~ = \left[ \begin{array}{cccccc} {\bf a}_i^T & {\bf... ...{\bf0}_{1 \times MK} & . & . & {\bf a}_i^T \end{array} \right]$$ (8)

where ${\bf0}_{1 \times MK}$ is the $1 \times MK$ row vector of zeros. With these we have that

$$f( {\bf r}^n \vert {\bf H}^n , {\bf A}^n ) ~ = ~ \frac{1}{( \... ...bf r}_i - {\bf A}_i {\bf h}_i \vert\vert^2 / \sigma^2 } ~ ~ .$$ (9)

For the ML estimate of ${\bf A}^n$ we determine the ${\bf A}^n$ which maximizes

$$f( {\bf r}^n \vert {\bf A}^n ) ~ = ~ \int_{{\bf H}^n} f( {\bf... ...t {\bf H}^n , {\bf A}^n ) ~ f( {\bf H}^n ) ~ d {\bf H}^n ~ ~ .$$ (10)