where (6.65) follows from (6.31). Note that (6.31) also holds for asymmetric |d1| and |d2|.
We have shown that (6.31) is true by assuming R1 escapes to infinity in Section 6.2.1; for R2 escaping to infinity the proof is identical so we omit it.
From (6.58), (6.63) and (6.66) we obtain the following:
E↑∞lim sup
therefore our result follows for the asymmetric case.
6.3 Derivation of Theorem 5.4
In this section, we step further to general m-user SISO MAC. Our goal is to derive upper and lower bounds to dMAC. From the same argument in Section 4.1 we know that
χMAC≥ max
and hence we get
dMAC≥ max{|d1|, . . . , |dm|}. (6.69) Once the upper bound to dMACcan be shown to be equivalent to the lower bound in (6.69), we complete the proof of the result. Note that the lower bound (6.69) can be achieved by using an input that satisfies the peak-power constraint, while we will derive the upper bound under the average-power constraint.
Using the upper bound in (4.13) for the channel model (2.11), we can get after some steps the bound which with the right choice of the free parameters α, β, and ν leads to the following propo-sition.
Proposition 6.1. For the Rician fading MAC (2.9), an upper bound of the sum-rate fading number under the average-power constraint (2.8) is given as follows:
χMAC≤ lim
Here we defineA to be the set of joint input distributions such that all users are independent and at least one user’s input distribution escapes to infinity when the available powerE tends to infinity.
6.3 Derivation of Theorem 5.4 Chapter 6
Proof. A proof is provided in Appendix A.
Since ξ 7→ log(ξ) − Ei(−ξ) is monotonically increasing, the problem of deriving an upper bound to dMAC can be transformed to finding an upper bound of the expression:
E· |d1X1+· · · + dmXm|2
|X1|2+· · · + |Xm|2
¸
. (6.72)
Note that (6.72) is equivalent to
E· |d1|2|X1|2+· · · + |dm|2|Xm|2 can upper-bound it as follows:
sup
Here in (6.76) we define
D˜ = diag¡
In (6.77) we use Rayleigh-Ritz Theorem as in Section 6.2.2, and (6.78) follows because the maximum eigenvalue of ˜Dis |d1|2.
6.3 Derivation of Theorem 5.4 Chapter 6 Assuming that QR1 escapes to infinity, we can separate (6.82) into two kinds of products as follows Firstly, we look at (6.83) and note that
E↑∞lim sup
where in the first inequality we upper-bound by dropping terms in the denominator, and the last equality follows from (6.31).
In (6.84) we upper-bound by dropping terms in the denominator as follows:
E· 2|di||dj|RiRj Hence, the problem lies in to show that
E↑∞lim sup
To that goal again we let
E£
Assume if E ↑ ∞ then E1↑ ∞ without loss of generality. Moreover, note that 2rirj
r12+ ri2+ rj2 ≤ 2ri2
r21+ 2r2i ≤ 1 (6.90)
6.3 Derivation of Theorem 5.4 Chapter 6 an arbitrary choice of a > 0, we have
E↑∞lim sup
Here in (6.91) we define A1 as the set of all input distributions such that the first user escapes to infinity, and take the supremum over all joint distributions of QRi and QRj. In the subsequent inequality we apply (6.90) to replace rj by ri. In the last inequality we split the integration into two parts as in Section 6.2.1.
6.3 Derivation of Theorem 5.4 Chapter 6
For the first term in (6.93), we have
Elim1↑∞ sup
Here (6.95) follows from the fact that QR1 escapes to infinity and equivalent derivation as in Section 6.2.1.
As for the second term in (6.93), we have sup r1. The last inequality follows because a can be chosen arbitrarily.
We have shown that if QR1 escapes to infinity, then (6.85) and (6.87) hold. As for other users’ distribution escaping to infinity, we can easily reformulate (6.82) and follow the same steps to obtain
Therefore, the result follows from (6.78) and (6.99).
Other Observations Chapter 7
Chapter 7
Other Observations
This chapter contains some observations not related to the proof of the main results but still interesting. In Section 7.1 we review what a scale family is, and provide a proposition of previous results. In Section 7.2 we give an observation on the power usage of a capacity-achieving distribution in the multiple-access channel.
7.1 Generalization of Scale Family
The definition of a scale family is given in [3]: a scale family of input distributions{Qβ} is generated by a random vector with a given distribution Q1 that is then multiplied by the factor β > 0. Note that the Gaussian input signal is a scale family. In [3, Theorem 6.11]
it was shown that in the MIMO fading channel a scale family is sub-optimal in the sense that the mutual information is bounded in the availableE. As to the multiple-access fading channel, we have a proposition for the special case (2.14).
Proposition 7.1. Consider the channel given in (2.14). Assume that E£
|X1|2+|X2|2¤
= E£
kXk2¤
= 1. Then
E↑∞limsup
E>0
I³√ EX;√
EHTX+ Z´
<∞. (7.1)
Proof. Expanding the mutual information we get I³√
EX;√
EHTX+ Z´
≤ I³√ EX;√
EHTX´
(7.2)
= h(HTX)− h(HTX| X) (7.3)
= h(HTX)− E£
log πekXk2¤
(7.4)
= h(HTX)− E£
logkXk2¤
− log πe (7.5)
= h µ
HT X kXk
¶
− log πe (7.6)
≤ log πeVar³ HTXˆ´
− log πe (7.7)
= log Var³ HTXˆ´
. (7.8)
7.2 Observation on Power Usage Chapter 7
Here the first inequality follows from data processing inequality, and (7.4) follows because h(HTX| X = x) = log πeVar(H1x1+ H2x2) (7.9)
= log πe¡
|x1|2+|x2|2¢
(7.10)
= log πekxk2 (7.11)
then we take the expection over X; (7.6) follows from the scale property of the differen-tial entropy; in (7.7) we upper-bound the differendifferen-tial entropy by the Gaussian differendifferen-tial entropy.
Continuing on (7.8), we look at Var³ HTXˆ´
which is finite and therefore completes the proof.
Thus we learn that any scale family including Gaussian input is sub-optimal for this special two-user SISO MAC. Note that in this proposition the noise is assumed Gaussian, while in [3, Theorem 6.11] the noise can be any additive noise.
7.2 Observation on Power Usage
Consider the two-user multiple-access fading channel given in (2.14). If the input vector uses full available average power, i.e.,
E£ kXk2¤
=E, (7.18)
then one can define a new input vector as X˜ = 1
√EX. (7.19)
In this case note that
Eh°
7.2 Observation on Power Usage Chapter 7
Therefore X is a scale family of ˜X. This input vector X cannot achieve the asymptotic capacity whenE tends to infinity.
It is an unexpected observation that in order to achieve the asymptotic capacity, the input cannot use the full available average power. However, we also know from escaping to infinity that in order to achieve the asymptotic capacity, the cost function kXk2 should take values that also tend to infinity withE. We conclude that the capacity-achieving input cannot have an average power with linear growth rate as the available powerE, but should have an average power that goes to infinity withE with a slower growth rate than the linear growth rate.
The following example shows this behavior. Consider one of the capacity-achieving distributions of the single-user SISO fading channel
log|X|2∼ U([log log E, log E]) . (7.21) Note that this distribution also achieves the MAC fading number if we only allow the users with the best channel to transmit using this distribution. The average power of this input distribution can be computed as follows: first let
Y = log|X| ∼ Uµ· 1
2log logE,1 2logE
¸¶
. (7.22)
By changing the variable
|X| = eY, (7.23)
we have
f|X|(x) =
2
x·log E−log log E1 x∈h√
logE,√ Ei
,
0 otherwise. (7.24)
After a few steps of integration, the average power can be obtained:
E£
|X|2¤
= E − log E
logE − log log E. (7.25)
We can observe that this distribution does not use full power E but its average power also tends to infinity withE, which fits the previous discussion.
Discussion and Conclusion Chapter 8
Chapter 8
Discussion and Conclusion
In this thesis, the fading number of the multiple-access fading channel is provided in the two-user SISO and the m-user SISO case. The results of this study indicate that the MAC fading number is exactly equivalent to the single-user SISO fading number. In order to be able to achieve the fading number, we need to reduce the multiple-user channel to a single-user channel. This single user must have a maximum line-of-sight component and use a input distribution that escapes to infinity.
A possible reason for this rather pessimistic result might be that cooperation among users is not allowed. Therefore, the best strategy in the single-user MISO fading channel—
beam-forming among antennas on the transmitter side—can not be implemented. The users interfere with each other and this causes the degression in performance, i.e., without coop-eration between the users, signals transmitted from other users can only be interferences.
Recall that it is shown in [6, Lemma 6] that a capacity-achieving input distribution can be assumed to be circularly symmetric in the single-user fading channel. Also note that in [6, Proposition 19] if at least one user uses circularly symmetric input, then the MAC fading number is the same as the SISO fading number. From the results in this thesis, we learn that the capacity-achieving input distribution reduces the MAC to a single-user channel.
Hence one can assume the input distribution to be circularly symmetric, which exactly fits the two previous results.
The result shown in this thesis using the noncoherent capacity approach is obviously far below that of assuming the perfectly known channel state. Since the users on the transmitter side have no knowledge of the channel state, some techniques such as successive interference canceling cannot be utilized. However, real systems operate at low SNR. This is theoretical result when SNR tends to infinity; in practical situation, it is not necessary to reduce a multiple-access channel to a single-user channel for designing a system.
Possible future works for the multiple-access fading channel might be as follows:
• Generalizing to the MIMO case: the users and the receiver use multiple antennas. A possible approach could be first to consider the MISO case.
• Considering the case with memory.
Appendix
• Considering the case with side-information.
• Loosening the restriction of Rician fading and considering a general fading process.
• Deriving the nonasymptotic capacity. This is related to the upper and lower bounds to the nonasymptotic capacity of the fading channel.
Derivation of Proposition 6.1 Appendix A
Appendix A
Derivation of Proposition 6.1
To derive the upper bound in Proposition 6.1, we follow steps in [6, Section 4.2]. From Lemma 4.1 we have
I(X; Y )≤ −h(Y |X) + log π + α log β + log Γ
Appendix A
Here the first inequality follows from Lemma 4.1; in the subsequent equality we assume 0 < α < 1 such that 1− α > 0 and define
in the subsequent equality we use the fact that given X = x the channel output is Gaussian distributed; in the subsequent equality we evaluate the expected logarithm of a noncentral chi-square random variable as derived in [12], [2, Lemma 10.1], [3, Lemma A.6]; and the last equality follows from simple algebraic rearrangements.
Next we bound the following expressions:
E£
Here, (A.11) follows from dropping some nonnegative terms; (A.12) follows because log ξ− Ei(−ξ) ≥ −γ where γ ≈ 0.57 denotes Euler’s constant; and to derive (A.16) we used the Cauchy-Schwarz inequality and the fact that the input needs to satisfy the average-power constraint.
Appendix A
which follows from the monotonicity of ξ7→ log ξ − Ei(−ξ).
Together with (A.5) we then get
I(X; Y )≤ −1 + E Here the last inequality follows from Jensen’s inequality, the fact that log ξ− Ei(−ξ) is concave.
We will now make the following choices of the free parameters α and β:
α, α(E) = ν
for some constant ν ≥ 0, which leads to the following asymptotic behavior:
E↑∞lim
(Compare with [2, Appendix VII], [3, Sec. B.5.9].)
Hence, we have derived the following upper bound on the fading number of a Rician MAC:
Appendix A
+ 1 β
³¡1 +kdk2¢
E + σ2´ + ν
β − log µ
1 + log µ
1 + E σ2
¶¶ )
(A.28)
≤ lim
E↑∞sup
QX
( log
à E
" ¯
¯dTX¯
¯2
°°X°
°2
# !
− Ei Ã
−E
" ¯
¯dTX¯
¯2
°°X°
°2
#!
− 1 )
+ ǫν+ ν + log¡
1− e−ν¢
− log ν, (A.29)
where the supremum is taken over distributions QX that satisfy the average-power con-straint.
By letting ν tend to zero which makes sure that ǫν → 0 as can be seen from (A.6) and using Proposition 5.1 we complete the proof.
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