Discussion on Power Constraints



 X₁

... Xm





, (2.3)

the subvectors X_i∈ Cⁿⁱ denoting the input vectors of each user are statistically independent

Xi ⊥⊥ Xj, ∀ i "= j. (2.4)

We assume that the random variable Z is a spatially white, zero-mean, circularly symmetric Gaussian noise, i.e., Z ∼ NC.0, σ²/ for some σ² >0.

As for the fading vector H, the distribution is general, with the assumption of finite power

E0!H!²1 < ∞, (2.5)

and finite differential entropy

h(H) > −∞, (2.6)

the last assumption is usually denoted as regular fading.

2.2 A Simple Special Case of the Channel Model

For simplicity, we sometimes assume that each user and the receiver use only one antenna, i.e., n1 = n2 = · · · = nm = 1, such that nT= m. This reduces (2.2) to the multiple-access SISO case. Note that

Y = H^Tx+ Z, (2.7)

where x ∈ C^m denotes the input vector, with n_T components xi in x are independent of each other, but the components of H can be dependent, see Figure (2.2).

2.3 Discussion on Power Constraints

In the given setup we can consider several possible constraints on the power. We use E to denote the maximum allowed total instantaneous power in the peak-power constraint, and to denote the allowed total average power in the average-power constraint. For both cases we get

snr! E

σ². (2.8)

Note that the total power still must be split and distributed among all users. In our channel model, we consider three different scenarios:

2.3 Discussion on Power Constraints Chapter 2

Figure 2.1: The memoryless multiple-access channel with multiple antennas at transmitter side and one antenna at receiver side.

Figure 2.2: The memoryless multiple-access channel with only one antenna at transmitter and receiver sides.

2.3 Discussion on Power Constraints Chapter 2

• Peak-Power Constraint: At every time-step, every user i is allowed to use a power of at most ^κ_mⁱE:

Pr2

!Xi!² > κi

mE3

= 0, (2.9)

for some fixed numbers κi>0.

• Average-Power Constraint: Averaged over the length of a codeword, every user i is allowed to use a power of at most ^κ_mⁱE:

E0!X_i!²1 ≤ κi

mE, (2.10)

for some fixed numbers κi>0.

• Power-Sharing Average-Power Constraint: Averaged over the length of a code-word all users together are allowed to use a power of at most ¯κE:

E 4 _m

'

i=1

!Xi!² 5

≤ ¯κE, (2.11)

for some fixed numbers ¯κ >0.

Note that if κ_i = 1 for all i, we have the special case where all users have an equal power available. Also note that in (2.9) and (2.10) we have normalized the power to the number of user m. This might be strange from an engineering point of view; however, in regard of our freedom to choose κ_i, it is irrelevant, and it simplifies our analysis because we can esaily connect the power-sharing average-power constraint with the average of the constants {κi}^m_i=1, i.e., if we define

¯ κ! 1

m

m

'

i=1

κ_i, (2.12)

then the three constraints are in order of strictness: the peak-power constraint is the most stringent of the three constraints in the sense of that if (2.9) is satisfied for all i = 1, . . . , m, then the other two constraints are also satisfied; and the average-power constraint is the second most strigent in the sense that if (2.10) is satisfied for all i, then also the power-sharing average-power constraint (2.11) is satisfied. In the remainder of this thesis we will always assume that (2.12) holds.

For some comments about even more general types of power constraints, we refer to the discussion in Chapter 6.

It is worth mentioning that the slackest constraint, i.e., the power-sharing average-power constraint, implicitly allows a form of cooperation: while the users are still assumed to be statistically independent, we do allow cooperation concerning power distribution. This is not very realistic (it implies that our cellphones can share batteries), however, it helps the derivation and it will turn out that the asymptotic sum-rate capacity is unchanged irrespective of which constraint is assumed. Based on this, we can choose one of the power constraint arbitrarily in our derivation, but not all of them in the same time.

Mathematical Preliminaries & Previous Results Chapter 3

Chapter 3

Mathematical Preliminaries &

Previous Results

In this chapter we review some important concepts and some related previous results, includ-ing some known result of the Rician fadinclud-ing MAC, for which case, the exact fadinclud-ing number is already provided.

The channel model considered is (2.2). In Section 3.1 we review the channel capacity and make a further generalization to the maximum possible sum rate of multiple users. In Section 3.2 we introduce the fading number. In Section 3.3 we provide the concept of input distributions that escape to infinity and a lemma which shows that under some conditions the input distribution must escape to infinity. In Section 3.4 we extend the notion of escaping to infinity to multiple users. In Section 3.5 we review a known bound of the sum-rate capacity for our case. Finally, in Section 3.6 we get the exact value of the m-user SISO MAC Rician fading number. The concepts we use in this chapter are mainly based on [2], [3], [6], and [9].

3.1 The Channel Capacity

In this section we first review the definition of channel capacity provided by Shannon in [1]. Further we give the definition of the maximum possible sum rate of the multiple-access channel; it is basically identical to the channel capacity, but takes multiple users into consideration.

Recall that in a discrete memoryless channel (DMC), the channel capacity is defined as C! max

QX

I(X; Y ), (3.1)

where the maximization is taken over all possible input distributions QX(·). When the concept is generalized to the continuous case, i.e., the input and output take values in continuous alphabet, a power constraint must be taken into consideration: for the peak

3.1 The Channel Capacity Chapter 3

power constraint (2.9)

C! max

QX

Pr[^|X|2≤E]⁼¹

I(X; Y ), (3.2)

or for the average power constraint (2.10) C! max

QX

E[^X2]^≤E

I(X; Y ), (3.3)

where the maximization is taken over all the input distributions satisfying the constraint.

In the generalization to the memoryless multiple-user channel, we use C to denote the maximum possible sum rate. The (sum-rate) capacity of the channel (2.2) is given by

C= sup

Q_X

I(X; Y ), (3.4)

where the supremum is taken over the set of all probability distributions on X for which the m subvectors are independent and which satisfy the power constraint, i.e.,

Pr2

!Xi!² > κ_i mE3

= 0, (3.5)

for a peak-power constraint, or

E0!Xi!²1 ≤ κ_i

mE, (3.6)

for an average-power constraint.

The most general concept of capacity in a multiple-access scenario is the capacity region.

The capacity region of the multiple-access channel is defined to be the closure of the set of all achievable rate tuples. An example of a 2-user capacity region is provided in figure 3.3, it is a common region for 2-user MAC. Speaking precisely, we have given three fixed numbers:

I₁ ! I.X⁽¹⁾; Y6

6X⁽²⁾/, (3.7)

I₂ ! I.X⁽²⁾; Y6

6X⁽¹⁾/, (3.8)

I₃ ! I.X⁽¹⁾, X⁽²⁾; Y/. (3.9) These three numbers together with the constraints R⁽¹⁾≥ 0 and R⁽²⁾≥ 0 specify a pentagon of achievable rate pairs:

R⁽¹⁾ ≥ 0 R⁽²⁾ ≥ 0 R⁽¹⁾ ≤ I1 R⁽²⁾ ≤ I2

R⁽¹⁾+ R⁽²⁾ ≤ I₃











(3.10)