Adaptive Modeling and Identification of Nonlinear MIMO Channels Using Neural Networks - Adaptive Signal Processing in Wireless Communications - page 136

nonlinearly transformed by a memoryless nonlinearity g i (.). The outputs of these non-
linearities are then linearly combined by an L × M matrix H = [ h ij ].
he j t h output of the MIMO channel can be expressed as

M

∑ 1

yn () = hngxnNn () (()) + () , (5.1)

j ji i i j

i =

where N j is a white noise.
The system input-output relationship can be expressed in a matrix form as

     

yn
yn

()
()
...
()

gxn (() )

(())
...
(())

Nn ()

1 11 1




 




 




 




 




 




 

gxn

+ NNn

2 ()
...
()

2 22

=× H . (5.2)

yn gxn Nn

L      

L MM

Matrix H is a propagation matrix [1] that may be time varying.
In our modeling approach, only the structure of the MIMO system is assumed known.
That is, we know that the MIMO system is composed of linear memoryless blocks and a
linear combining matrix, but we do not know what their values and behaviors are.

5.2.2 Neural Network Scheme

The neural network [2, 3, 5] used for modeling the nonlinear MIMO system is repre-
sented in Figure 5.2 . It is composed of M neural network blocks. Each block k has a
scalar input x k ( n ) ( k = 1, …, M ), N neurons, and a scalar output:

N

∑ 1

( ) =…

NN () n = c faxnb () + , k 1 , , M , (5.3)

k ki ki k ki

i =

where f is the NN activation function (a sigmoid transform). a ki , c ki , b ki represent, respec-
tively, the input weight, bias term, and output weight of the i t h neuron of the k th block.
he output NN k of the k th block is connected to the j t h output of the system through
weight w jk . he system j t h output is then expressed as

M

∑ 1

sn () = wNNn j (), = 1 ,, . L (5.4)

j jk k

k =

Weig hts w jk will be put in a matrix form: W = [ w jk ]; j = 1, …, L ; k = 1, …, M .