An Experimental Approach to CDMA and Interference Mitigation phần 4 pdf

68 Chapter 2

where U U mk km , , is the shorthand notation for cross-correlation coefficient

( )() m k c c R between the two spreading codes of channel m and channel k , ( )i A

is the amplitude of signal on channel i , ( )i

k d is the kth data symbol on chan￾nel i , and ( )i Qk is the ith noise component. The effect of MAI is apparent,

and it is also clear that it can be potentially destructive. Assume for instance

that U z 1,2 0 (codes 1 and 2 are not orthogonal) and that (2) (1) A A  : the

MAI term (2) (2) U1,2A dk in (2.150) may overwhelm the useful term (1) (1) A dk for

data detection of channel 1. This phenomenon is called the nearfar effect:

user 2 can be considered as located near the receiver in the BS, thus received

with a large amplitude, whilst user 1 (the one we intend to demodulate) is the

far user and is weaker than user 2. Generalizing to N users, equations

(2.150) can be easily cast into a simple matrix form. If we arrange the cross￾correlation coefficients Ui k, into the correlation matrix

1,2 1,

2,1 2,

,1 ,2

1

1

1

N

N

N N

§ · U U ¨ ¸ U U

U U © ¹

R

"

# %#

"

(2.151)

and if we also introduce the diagonal matrix of the user amplitudes

^ ` (1) (2) ( ) diag , ,..., N A AA A , we have

z ARd  Ȟ (2.152)

where the column vectors z , d , and Ȟ simply collect the respective sam￾ples of received signal, data, and noise. A simple multiuser detector is the

decorrelating detector that applies a linear transformation to vector z to

provide N ‘soft’ decision variables relevant to the N data bits to be esti￾mated. Collecting such N decision variables ( )i

k g into vector g , the linear

joint transformation on the matched filter output vector z is just

1 g Rz (2.153)

where 1 R is the decorrelating matrix, so that

    1 1 g R ARd   Ȟ Ad R Ȟ

     ^ ` 1 2

1 2 diag , ,..., N Ad Ad A d k k Nk  Ȟc (2.154)

We have thus

2. Basics of CDMA for Wireless Communications 69

     1 2 , ,..., T N

kk k ª º gg g ¬ ¼ g (2.155)

where the superscript T

denotes transposition, () () () () i ii i

k kk g Ad Qc , and ( )i

k Qc

is just a noise component. It is apparent that the MAI has been completely

cancelled (provided that the correlation matrix is invertible) or, in other

words, the different channels have been decorrelated. The drawback is an

effect of noise enhancement owed to the application of the decorrelating ma￾trix 1 R : the variance of the noise components in Ȟc is in general larger than

that the components in Ȟ . Therefore, the decorrelating detector works fine

only when the MAI is largely dominant over noise.

A different approach is pursued in the design of the Minimum Mean

Square Error (MMSE) multiuser detector: the linear transformation is now

with a generic N Nu matrix Z whose components are such that the MMSE

between the soft output decision variables in g and the vector of the data

symbols is minimized

g Zz , with Z such that ^ `

2

E min g d  , (2.156)

where E{ } denotes statistical expectation. Solving for Z we have

  1 2 2   ZR A VQ , (2.157)

with 2 VQ indicating the variance of the noise components in (2.150). The

MMSE detector tries to optimize the linear transformation both with respect

to MAI and to noise. If noise is negligible with respect to MAI, matrix

(2.157) collapses into the decorrelating matrix 1 R . Vice versa, if the MAI is

negligible, the matrix Z is diagonal and collapses just into a set of scaling

factors on the matched filters output that do not affect data decisions at all

(and in fact in the absence of MAI, the outputs of the matched filters are the

optimum decision variables without any need of further processing).

From this short discussion about MUD it is clear that in general such

techniques are quite challenging to implement, either because they require

non-negligible processing power (for instance, to invert the decorrelating or

the MMSE matrices), and because they also call for a priori knowledge or

real time estimation of signal parameters, such as the correlation matrix. But

the potential performance gain of MUD had also an impact on the

standardization of 3G systems (UMTS in Europe), in that an option for short

codes in the downlink was introduced just to allow for the application of

such techniques in the BS [Ada98], [Dah98], [Oja98], [Pra98].