# definition

definition

IE341 Handout #1

Prof. Pietro Belotti

Base Station Location

We are given a set ofncandidate locations for antennas S and a set of m terminals T .

The problem we have to solve is that of finding a subset of S that covers all terminals.

In class, we have modeled it as follows:

min ∑ j∈S c ∑ jy j

j∈S x ij = 1 ∀i ∈ I

x ij ≤ y j ∀i ∈ I,j ∈ S

p ig ij

η ∑ j+ h∈I\{i} p hg hj

≥ γx ij ∀i ∈ I,j ∈ S

x ij ∈ {0,1} ∀i ∈ I,j ∈ S

y j ∈ {0,1} ∀j ∈ S

0 ≤ p i ≤ P max ∀i ∈ I

Constraint #3, on the Signal-to-Interference Ratio (SIR), refers to the quality of signal

at a certain antennaj and is related to the fact that this is an uplink problem — we only

care for the communication from terminals to antennas. The constraint is nonlinear, and

we have seen that it is equivalent to the linear constraint

p i g ij ≥ γ⎝η j + ∑

h∈I\{i}

p h g ih

⎠−M(1−x ij )

∀i ∈ I,j ∈ S

However, this constraint and its nonlinear version are incorrect. Consider a problem

with one antenna and two terminals. Inevitably, both terminals are associated with the

only antenna. That implies I = {1,2}, S = {1}, and x 11 = x 21 = 1, as there is only

one antenna. Now, there are two SIR constraints:

p 1 g 11

p 2 g 21 +η 1

≥ γ

and

p 2 g 21

p 1 g 11 +η 1

≥ γ

but it’s clear that we can’t satisfy both constraints. In fact, γ > 1 as it is a desired

signal-to-noise ratio, and if one constraint is satisfied then the other cannot be satisfied.

This is in general true for the above model.

UMTS networks use transmission technology that allow for filtering interference

by using orthogonality of signals, which is defined in terms of the range of frequencies

that each signal use: an antenna is able to discern a signal originating at terminali ′ from

one that originated at terminali ′′ , or at least it can exclude most of the incoming signal

of i ′′ . In other words, the SIR constraint is actually “helped” by a filter that attenuates

the power of the interfering signals by an orthogonality factor u.

Therefore, the SIR constraint becomes:

p i g ij

η j +u ∑ h∈I\{i} p hg hj

≥ γx ij

and can be linearized as in the previous model, as u is a constant (an AMPL param).

IE341 Handout #1

Prof. Pietro Belotti

Antenna Configuration Optimization

The second wireless network design problem we have looked at is on the optimal configuration

of wireless antennas. As the input of such a problem is usually the result

of a BS location problem, the antennas are known in advance and their configuration

needs be determined. Also, we are interested in the downlink communication, i.e., from

antenna to terminal.

From an Optimization standpoint, the configuration simply determines a set of values

for the gain (or attenuation) g ij . Let us assume that there is a set F = {1,2... ,k}

possible configuration, and the gain from antenna j to terminal i in configuration f is

g ijf . The minimum SIR at terminal i is denoted γ i . The model is as follows

min ∑ j∈J P j

∑ j∈J x ij = 1 ∀i ∈ I

f∈F j

z jf = 1 ∀j ∈ J

P j = ∑ i∈I p ij

∀j ∈ J

P j ≤ P max

∀j ∈ J

p ij

∑f∈F j

g ijf z jf

η i+u ∑ k∈J\{j} P k

∑f∈F k

g ikf z kf

≥ γ i x ij ∀i ∈ I,∀j ∈ J

x ij ∈ {0,1}

∀i ∈ I,j ∈ J

z jf ∈ {0,1}

∀j ∈ J,f ∈ F j

p ij ≥ 0 ∀i ∈ I,j ∈ J.

The SIR constraint looks like the one for the BS location problem, and can be further

improved as follows:

p ij

∑f∈F j

g ijf z jf

η i +u ∑ k∈J\{j} p ≥ γ i x ij ∀i ∈ I,∀j ∈ J

ik

∑f∈F k

g ikf z kf

Linearizing it requires a little more effort, as now we have a nonlinear termp ij g ijf z jf .

It’s nonlinear as p ij and z jf are variables. Since z jf is binary, p ij g ijf z jf can be either

p ij g ijf or zero, and we summarize it with a new variable, r ijf , and two constraints

expressing these two values according to the value of z jf :

r ijf ≤ p ij g ijf +M(1−z jf )

r ijf ≥ p ij g ijf −M(1−z jf )

r ijf ≤ P max z jf

If we define the above three constraints for each i,j,f we get a new, “more linear”

constraint:

f∈F j

r ijf

η i +u ∑ k∈J\{j}

∑f∈F k

r ikf

≥ γ i x ij ∀i ∈ I,∀j ∈ J

which is easy to linearize as it looks as the SIR constraint in the BS location problem:

⎛ ⎞

r ijf ≥ γ i

⎝η i +u ∑ ∑

r ikf

⎠−M(1−x ij ) ∀i ∈ I,∀j ∈ J.

f∈F j f∈F k

k∈J\{j}

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