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2 2 2 2 2 im1 (k l ) = ik l ,im1 + qk l ,im1 , 2 = E ik l ,im1 + qk l ,im1
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and normalization A2 l 2 /2 A2 l /2. k k A similar equation can be obtained for the Q-channel too. It has been assumed that all interference per path components are independent. In what follows we will simplify the notation by dropping all indices im1 so that im1 (k l ) kl . With no power control (npc) kl will depend only on the channel characteristics. In partial power control (ppc) only the rst multipath component of the signal is measured and used in a power control (open or closed) loop. Full power control ( fpc) will normalize all components of the received signal and rake power control (rpc) will normalize only those components combined in the rake receiver. The (rpc) control seems to be more feasible because these components are already available. These concepts for ideal operation are de ned by the following equations npc kl = kl , f pc
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k k (19.12)
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where L 0 is the number of ngers in the rake receiver. The contemporary theory in this eld does not recognize these options which causes a lot of misunderstanding and misconceptions in the interpretation of the power control problem in the CDMA network. Although fpc is not practically feasible, the analysis including fpc should provide the reference results for the comparison with other, less ef cient options. Another problem in the interpretation of the results in the analysis of the power control imperfections is caused by the assumption that all users in the network have the same problem with power control. Hence, the imperfect power control is characterized with the same variance of the power control error. This is more than pessimistic assumption and yet it has been used very often in analyses published so far. The above discussion is based on the signal amplitude seen by the receiver (asr). System losses due to difference between the asr and maa will be discussed in the next section. If we now introduce matrix m with coef cients kl , k, l except for m1 = 0 and use notation 1 for vector of all ones, Equation (19.10) becomes m1 Ybm = (19.13) 1 m 1T + 0 Rb /S
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Compared with Equation (19.10), the index i is dropped in order to indicate that the same form of equation is valid for both, the I- and Q-channel de ned by Equation (19.8). 19.1.4 Multipath channel: pointer tracking error, rake receiver and interference canceling If L 0 - ngers rake receiver (L 0 L) with combiner coef cients wmr (r = 1, 2, . . . , L 0 ) and interference canceller are used, the signal-to-noise ratio will become Ybm = where
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L0 (L rm 0 ) f (m, , c, r)K + 0 Rb /S
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(19.14)
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2 wmr = wm wT ; m
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wm = (wm1 , wm2 , . . . , wmn )
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(19.15)
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is due to Gaussian noise processing in the rake receiver, and noise density 0 becomes due to additional signal processing. Also we have f (m, , c, r) = = 1 K
k=1 k=m
L 2 wmr kl (1 Ckl ) +
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L 2 wmr ml (1 Cml )
l=1 l=r
1 (19.16) wm 1 cmr 1T wT m K with cmr being a matrix of size K L with coef cients kl (1 Ckl ) except for mr (1 Cmr ) = 0 and Cml is ef ciency of the canceller. The parameter kl = E p( , ) { kl } is the average value of the interfering signal power kl coming from direction (azimuth, elevation) ( kl , kl ) = ( m + kl , m + kl ) with respect to the reference pointer p( m , m )of the useful signal. Formally this can be represented as kl =
, ,
kl ( , )PDF( kl , kl ) d d d( kl ) d( kl )
(19.16a)
were PDF( kl , kl ) is probability density function of the arguments. For the rst insight into the system performance a uniform distribution of kl and kl can be assumed along with a rectangular shape of T ( , ) and R( , ) in the range 0 + , 0 + , so that evaluation of Equation (19.16a) becomes trivial. For the amplitudes Akl seen by the (L receiver asr, the parameter rm 0 ) in Equation (19.14), called rake receiver ef ciency, is given as