Channel equalization

The goal of equalization is to reduce the inter symbol interference (ISI) in order to estimate the transmitted bit. Indeed, since the GMSK main pulse duration is four times the bit period, inter symbol interference is unavoidable.

At the reception, the received signal is the convolution of the transmitted signals with a cascade of lters which are:

the modulation pulse shape a(t),

the transmission lter Fe(t) that ts the signal to the desired bandwidth,

the lter C(t) that corresponds to the propagation over the air,

The reception lterFr(t) that translates the signal to its base-band version.

The channel impulse response is dened as the cascade of these lters. The transmission scheme can then be written in the following way:

8 function, and b(t) an additive complex noise that is usually assumed to be zero mean and uncorrelated with s(t):

E[s(t)b()] = 0

h(t) is the channel impulse response. For convenience, h(t) is assumed to have a nite duration of p bit periods. p is the memory of the channel. One consequence of Shannon's work is the sampling theorem which states that it is possible to reconstruct any nite bandwidth signal from its samples at a rate that is greater or equal to twice its highest frequency.

In the GSM case, the bandwidth of each carrier is 200 kHz, so that the complex envelope is limited to 100 kHz. The bit rate, that is approximately equal to 271 kHz is then enough to satisfy the Shannon's criterion. From now on, all signals are sampled at the bit rate. The sampled signal is:

y(jT) = ^pX;1

k=0h(kT)dj^;k+b(jT) pj N (2.15)

19 2.4. CHANNEL EQUALIZATION The last equation can then be written in a more compact formula using ma-trices: The noise samples are assumed to be uncorrelated between each other, i.e.

E[

BB

] =²

I

.

One way of estimating the bits is the maximum likelihood (ML) detector. This detector is obtained by maximizing the likelihood of the observation with respect to the parameter to be estimated or equivalently by maximizing the log-likelihood of the observation:

^= argmaxlnp(y^j)

In our case,is a vector that contains the channel impulse response coecients and the unknown bits. In the context of Gaussian noise, the log-likelihood can be written in the following way:

lnp(

Y

^j

H

;

d

^{) =;1}₂^k

Y

^;

D

⁽

d

⁾

H

^{k2 (2.18)}

Several approaches are possible to solve the equation. The channel impulse response is unknown. It is possible to estimate it directly without any prior knowledge of the bits: this is called blind equalization. We will focus on the GSM case where a training sequence consisting of known bits is available in all bursts.

We will solve the system (2.18) in two steps: in the rst step, we assume that

d

refers to the training sequence and estimate the channel impulse response accordingly. In the second step,

d

refers to the unknown bits and the channel impulse response estimated from the rst step is used.

CHAPTER 2. WIRELESS COMMUNICATIONS 20 It should be noted that this scenario is not optimal since the estimations of the unknown bits and the channel impulse response are not performed jointly.

However, the low gain in performances given by the joint estimation does not justify its huge complexity increase.

We suppose that the channel impulse response is constant within one burst.

This approximation is usually veried at normal mobile speed. The estimation of the channel impulse response is straightforward using the training sequence:

H

^ =

D

^y(

d

)

Y

(2.19)

where

D

^y = (

D

D

)^;1

D

refers to the Moore-Penrose pseudo-inverse of

D

. Note that the matrix

D

^y(

d

) can be precomputed and stored as it only depends on the training sequence.

The error covariance of this estimator is:

E^h

H

^ ^;

H

H

^{^ ;}

H

ⁱ=²[

D

(

d

)

D

(

d

)]^;1

The training sequences are chosen for their good auto and inter correlation properties, i.e. they are almost uncorrelated with theirs own shifted versions:

D

⁽

d

⁾

D

⁽

d

^)m

I

This shows that the variance error is reduced by a factor m=q^;p+ 1.

_2r = ²

In the case we have some a priori information on the parameter to be es-m timated, the Maximum A Posteriori (MAP) estimator can be used successfully to minimize the covariance error E^h

H

^ ^;

H

H

^{^ ;}

H

ⁱ. Supposing that the channel impulse response is Gaussian zero mean with covariance matrix

K

h = E[

HH

], the MAP estimator is (see [7] for example):

H

^ ^{= h}

D

⁽

d

⁾

D

⁽

d

^{) +2}

K

^;h¹

i

;1

D

⁽

d

⁾

Y

=

K

h

D

(

d

)^h

D

(

d

)

K

h

D

(

d

) +²

I

^i;1

Y

(2.20) The second equation is obtained using the inverse matrix lemma. It avoids the inversion of the covariance matrix

K

h. Note that equation (2.19) is a special case of this equation when no information a priori is available, i.e.

K

h =¹.

If the noise is not white, its covariance

K

b = E[

BB

] can be incorporated:

H

^ ^{= h}

D

⁽

d

⁾

K

^;b¹

D

⁽

d

^{) +}

K

^;h¹

i

;1

D

⁽

d

⁾

K

^;b¹

Y

=

K

h

D

⁽

d

^)[

D

⁽

d

⁾

K

h

D

⁽

d

^{) +}

K

b]^;1

Y

^(2.21)

21 2.4. CHANNEL EQUALIZATION Once the channel impulse response is estimated, we estimate the unknown bits by minimizing:

lnp(

Y

^j

d

^)k

Y

^;

D

⁽

d

^{) ^}

H

^k2

where

d

represents in this case the unknown bits. The well known Viterbi algorithm [8] is usually used here to recursively estimate the unknown bits. This algorithm can also provide soft condence values for each bit enabling soft de-coding.

CHAPTER 2. WIRELESS COMMUNICATIONS 22

Chapter 3

Technologies for location

This chapter presents a description of possible technologies that might be used for locating a mobile handset in a radio cellular network. It explains why the best technology in the actual context is based on time of arrival.

In general, the idea is to gather as many measurements as possible and to exploit a large number of relevant observations. Each observation contributes to add some more information and enhances therefore the estimation accuracy. The set of relevant measurements can be obtained at the handset from signals coming from dierent base stations or can be measured from the signal coming from the mobile and impinging on several base stations. Accumulating both measurements (i.e. on downlink and uplink channels) enhances the nal estimation accuracy.

3.1 Distance estimation from signal strength

This is the most intuitive technique to locate a handset. The mobile measures the signal strength from several base stations. A good candidate is the BCCH channel since it is transmitted at full power. The mobile can compute the distance that separates it from the base station by means of an appropriate path loss empirical model. In general, the path loss is proportional to the distance between the mobile and the base station :

Pr

Pt =cd^; (3.1)

Two distances are required for locating a handset by a simple circular trilateration¹. This methodology has been analyzed in [9]. The main advantage of this method is that it can easily be implemented in GSM². Indeed, the mobile sends a

mea-1The ambiguity can be solved by selecting the intersection that is inside the area of interest.

2Some operators have already implemented this technology for some commercial applications such as home zone billing.

23

CHAPTER 3. TECHNOLOGIES FOR LOCATION 24 surements report every 0.48 s that includes the downlink signal levels of the best six base stations. The signal power level is expressed on a discrete scale running from 0 to 63.

The main drawback of this method is its low precision due mainly to shad-owing. However it may be assisted by a prediction map in order to correct the estimation resulting from the trilateration alone. Such prediction maps are avail-able in the database of an operator. They are generally obtained by simulation.

Another way of assisting the method is to provide collected eld measurements as training data; this is subject of next section.

Dans le document Localisation d'un mobile dans un reseau GSM (Page 35-41)