The user SINR is a very important indicator in this thesis. In this chapter we are going to demonstrate the user SINR and the associated Shannon capacity.

Figure 5.1. Transmission paths for a repeater installation.

Figure 5.1 shows the transmission paths from BS and repeater to user. Now we only
consider single cell. BS transmits signals to both repeaters and users; repeaters
amplify the signal and re-transmit to users. From the user‟s point of view the only
*possible paths it could receive are from BS and cell repeaters. P**BS** and P**REP* are
*transmitting power of BS and repeater. G**ij** is the path gain from j*^{th}* transmitter to i** ^{th}*
receiver. It can encompass path loss, shadowing, antenna gain, coding gain, and other
factors.

Assume each user is assigned different power. The total transmitted power (could be
*of BS or repeater) is P**total**. For i** ^{th}* user it is allotted

(2)
* is the power ratio of user i to total power which value is between 0 and 1. It is the *
weighting of how much power is allocated to a user. And . The rest of the
power _{ }

* is regarded as interference from transmitter to other users *

*except user i. The overhead channel power is*

_{ }.

User BS

Repeater

*P*_{BS}*P*_{REP}

*G*_{user_BS }*G**user_REP*

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The signal power may decay during transmission, but the ratio of every user‟s power stay unchanged even the signal goes through fading. It becomes

* * _{ } _{ }

* (3) *

When user receives signal the above part is the desired information and the other is
unwanted which has power
_{ } _{ } _{ } (4)

Assume there are users and repeaters with power in a cell and
signals from different paths can be processed by the user (upper bound). The SINR of
*user i is *

_{ } _{ } _{ } _{ }

_{ } _{ } _{ } _{ } _{ } _{ }

* is the orthogonality factor which ranges from 0 to 1. * is the thermal noise.

Both the numerator and denominator include signals from BS and repeaters.

Traditionally we do power allocation on each user according to its channel condition.

However in repeater embedded system, every user‟s power is determined by BS,
repeaters only amply signal. We can see from the above formula, * for j = 1,…, *
are the variables and they are affected by repeater gains. The power allocation is done
on repeaters not on users. Every transmission each repeater has to decide the gain
associated with each other.

Equation (5) is the single cell case. If we consider multi-cell, the only difference from equation (5) is the inter-cell interference. The formula becomes

_{ } _{ } _{ } _{ }

_{ } _{ } _{ } _{ } _{ } _{ } _{ }

* can come from other-cell BS and/or repeaters. *

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The capacity of a user can be calculated as

(7)

And the cell capacity is

(8)

So far we have proposed the user SINR and cell capacity. These are the measurements of the repeater buried system performance. The next thing we want to know is how much improvement can repeaters increase? This is done by solving the following optimization problem:

subject to

_{ }

The objective function is the overall system throughput. It is optimized over the set of all feasible powers . The transmit power of every repeater has to be a positive value.

The second set of constrains gives a limit on repeater‟s output power. The last set of constraints is the data rates demanded by existing system users.

We use cvx which is a modeling system for disciplined convex programming developed by Michael Grant and Stephen Boyd to solve this problem. However since the mapping from repeater transmitting power to SINR function is not convex, it is impossible to directly deal with it via cvx. Alternatively we take the approach to approximate the non-convex Shannon capacity equation into piecewise linear functions, which can be managed by cvx.

To do so, we first look at the characteristic of Shannon capacity:

= _{ } _{ } _{ } _{ } _{ } _{ } _{ }
_{ } _{ } _{ } _{ } _{ } _{ } _{ } * *

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As seen above, Shannon capacity is decomposed into and – functions, which takes the sum of linear variables as the input. Therefore, if we can approximate the and – into linear functions, Shannon capacity is also approximated into linear functions. Fortunately is a concave function, which can be easily approximated as the sum of piecewise affine functions:

(9)

*where n is an index variable and * is a positive value. Specifically, the parameters of
each line and the number of lines can be adjusted according to the required precision.

By using this approximation technique, we can represent part of Shannon functions; and are approximation parameters.

While can be directly approximated through simple intersection, –
cannot be done in such way, but can be approximated by getting the union region. To
*do so, we need to select a proper affine function depending on domain x. More *
specifically we will add the virtual infinite value to the other affine functions except
the proper affine function to safely ignore them. We take the selection technique again,
*and hence, an indicator variable v**in* is introduced to choose the proper affine function,
which gives the biggest value for a given input. Finally we obtain the inequality
*A constrain for v**in* is also required:

_{ } * *

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Finally we can define the optimization problem as follows:

re-write it more clearly, it would be like this:

*For k = 1,…,N *

18 performance is close to the optimization result.

With the knowledge of repeater the most apparent factors one might think of affecting repeater gains are the distance and user distribution. The former means the distance from repeater to users or repeater to the cell edge. With the existence of multi-cell interference the critical thing is to avoid repeater transmits severer interference which might damage the surrounding users and when the repeater is close to the cell edge the gain should be low for the same reason. The latter means to tune the repeater gains according to the user distribution.

We try to verify the above assumptions by observing the optimization result. The conclusion however does not conform the assumptions, namely it is not directly related to the distance and user distribution. The most critical factor is the SINR ratio of repeater to its posterior users. If the posterior users of a repeater are all good users (whose SINR are greater than their anterior repeater) the repeater should turn diminish its gain to avoid interfering the covered users. On the other hand if there are many bad users (whose SINR are smaller than their anterior repeater) the repeater should raise its gain in order to enhance the signal qualities of covered users.

There are three types of situations: 1). . All of the posterior users get improvements on SINR and hence the capacity. The repeater gain can be as large as possible to increase the user qualities without any damage. 2). . It is the most common situation in which the repeater is beneficial to some users but harmful to the others. High quality users are sacrificed to increase the SINR of low quality users. 3). . It is the worst case that the repeater installation has no benefit to the posterior users. In this case the repeater gain should be as small as possible or just turned off.

The concept of the proposed algorithm is based on the above criteria and progressively converge the repeater gains.