Model and Optimum Receiver

 

Consider the following model for the complex envelope of the received signals in a communication system with multiple antennas, slow and flat fading, and additive non-Gaussian noise and interference. Let denote the vectors in a J-ary signal constellation, where each signal vector has dimension . (The superscripts T, H, and * will denote the transpose, conjugate transpose, and complex conjugate operations, respectively.) The elements of may be taken as m time samples of the transmitted waveform that encodes signal j. If the system contains N receiving antennas, then the vector of observations for one received symbol at antenna k is modeled as

 

where is one of the transmitted signal vectors , is the complex fading coefficient at antenna k, and is the vector of additive noise samples. The observations at all N antennas can be combined into a single vector of the form

 

where is an vector of fading coefficients, denotes Kronecker product, and is formed by stacking the vectors in the same manner as is formed by stacking the vectors. The objective is to determine which signal was transmitted by processing the observations in (2).

We will assume that the observations from T known training symbols are available at the N antennas. The received training symbols will be denoted as follows. Define as the sample at time i of training symbol t, and . Then the model in (1) is extended to

 

to describe the observations from all T training symbols. In (3) represents the vector of noise samples observed during the reception of training symbol t . Equation (3) describes the observations that are available in the training set. Note that the signal vectors are known for each . Also note that the fading coefficients are modeled as being fixed over the entire training set, i.e. they do not vary with t.

Some observations and further development of the model are as follows.

1. The fading is flat or frequency non-selective, so it is modeled by a single complex coefficient that describes the amplitude scaling and phase shift experienced by the transmitted signal. The fading is slow, which means that the fading coefficient is constant for several symbols to allow estimation of the fading coefficients.
2. The vectors in (2) are realizations of complex-valued random vectors .
3. is a discrete random vector that takes the values with probabilities .
4. The fading coefficients, the components of , are modeled as independent and identically distributed (iid) random variables. This implies that the antennas are spaced sufficiently far apart so that the fading is independent from antenna to antenna. Although the probability density function (pdf) of affects the system performance, our processing is not based on any particular distribution for . Simulation examples we ran typically assumed Rayleigh fading.
5. The additive noise vector has components that are iid, with each component modeled as an L-term mixture of Gaussian pdfs. The pdf of is

    

   A straightforward extension of (4) allows the noise parameters to vary from antenna to antenna.

6. The model is for a single user scheme, with any multiuser interference considered part of the additive non-Gaussian noise.

The model in (4) includes cases that are similar to a truncated-sum version of Middleton's canonical class A noise model [4, 7, 8], which has received extensive study. The truncated version of Middleton's model with L=2 or 3 terms has been shown to be a good approximation for some cases of interest [11, 12], including highly impulsive noise. In order to model impulsive noise, a common convention for the noise pdf parameters [13] is to take L=2, and , so that large noise samples are produced every so often by the larger variance term.

If the fading coefficient vector and the noise pdf parameters are known exactly, then detection based on the model in (2) reduces to detecting one of J known signals in additive noise. The decision rule that maximizes the probability of correct decisions is the maximum a posteriori (MAP) detector, which chooses the signal with index j that maximizes the posterior probability . The MAP detector is equivalent to choosing the index that maximizes , which can also be determined by using likelihood ratio (LR) test statistics. An explicit form for this test is obtained easily using the model we have outlined. The resulting test is optimum with regard to minimizing the probability of incorrect decisions only when the fading coefficients and the noise pdf parameters in (4) are known. In practice, these quantities must be estimated from the data and updated over time as the communication environment changes. Our approach is to apply the MAP test just described with the Gaussian mixture pdf in (4), using the estimates for and in place of the true values for these parameters. A procedure for obtaining maximum likelihood estimates for these parameters using the EM algorithm is presented in the next section. The use of maximum likelihood parameter estimates in a MAP test is a common procedure, and the resulting detector is known as a generalized likelihood ratio test.

An alternative detector structure, the MD-MMSE detector, is based on minimizing the distance between each possible signal and a minimum mean-squared error (MMSE) estimate of the transmitted signal. We have proven that the MD-MMSE detector is equivalent to the MAP detector in many cases of interest, including equal-energy binary signaling cases. The MD-MMSE detector has the structure of a class of neural networks called Gaussian mixture basis function networks (GMBFNs) [14],[15],[16],[17].