Here (5.27) follows from the Definition (3.1); in (5.29), based on Proposition 3.3, we drop the condition E = 1 and incorporate into A; finally, we use (5.25) and (5.26) and rearrange it to get (5.30).
Note that log(1 − eν) + ν + )ν− log ν tends to 0 as ν goes to 0, and since ν is arbitrary, so we get the following upper bound:
χMAC≤ lim and the proof of Theorem 4.1 is concluded.
5.2 Derivation of Theorem 4.2 and Corollary 4.3
The proof consists of two parts. The first part is given already from (4.5), (4.6), and (4.8).
There it is shown that
1≤i≤mmax FχMISO,iG = max is a lower bound to χMAC. Note that this lower bound can be achieved by using an input that satisfies the peak-power constraint (2.9).
The second part will be to prove that max1≤i≤mFχMISO,iG is also an upper bound to χMAC. We will prove this under the assumption of an power-sharing average-power con-straint (2.11). Since the peak-power concon-straint (2.9) and the average-power concon-straint (2.10) are more stringent than (2.11), the result follows.
The proof of this upper bound relies strongly on Proposition 3.5. Note that the supre-mum in (4.10) over all joint distributions such that at least one user’s input distribution escapes to infinity.
5.2 Derivation of Theorem 4.2 and Corollary 4.3 Chapter 5
Continue with (5.33), where A is defined in (4.11), which is the set of joint input distri-butions such that X are independent and the input distribution of at least one user escapes to infinity when the available power E tends to infinity,
In the following we will focus on finding an upper bound on (5.33). First we assume x1 escapes to infinity, i.e.
E↑∞limQE
where in the first inequality (5.37), we define A1 as the set of all input distributions of the first user that escape to infinity, and take the supremum over all QX1 ∈ A1. The last inequality (5.38) then follows from splitting the inner integration into two parts and from the fact that the supremum of a sum is always upper-bounded by the sum of the suprema.
To simplify our life, we define:
I1 ! sup
5.2 Derivation of Theorem 4.2 and Corollary 4.3 Chapter 5 such that equation (5.38) becomes
E↑∞lim sup
5.2 Derivation of Theorem 4.2 and Corollary 4.3 Chapter 5
Here in equation (5.42) follows by Cauchy-Schwarz inequality; since
!xi!2
!x1!2+ · · · + !xm!2 ≤ 1, ∀i, (5.51) and by the inequality of arithmetic and geometric means,
!xi! !xj!
!x1!2+ · · · + !xm!2 ≤ !xi! !xj!
!xi!2+ !xj!2 ≤ 1
2,∀i, j, (5.52) equation (5.43) holds; in (5.44), −h(·) can be upper-bounded by a finite number η because of the regular fading assumption (2.5); and we can take constants out from the integration in (5.45); (5.46) follows by taking the supremum into the first integral which can only enlarge the expression; in (5.47) we exchange limit and integration, which is allowed by the Dominated Convergence Theorem in [7], we are allowed to swap limit and integration,
5.2 Derivation of Theorem 4.2 and Corollary 4.3 Chapter 5
because both E0!Hi!21
and η are finite; finally, (5.49) follows because QX1 escapes to infinity.
where we keep the term |HT1x1|2 unchanged, and bound the others by the Cauchy-Schwarz inequality. (5.57), we use the notation from Section 1.2; (5.58) follows because for every ε > 0 we can choose a, which is the arbitrary fixed number defined in equation (5.36) big enough such that the inequality holds. That is only hold if we get the continuity of h(HTx) in x for all
5.2 Derivation of Theorem 4.2 and Corollary 4.3 Chapter 5
xare finite and larger than 0, and the detailed proof of continuity is provided in Appendix A.
Continuing with (5.53),
I2 ≤ sup
5.2 Derivation of Theorem 4.2 and Corollary 4.3 Chapter 5 get the equation (5.62); the subsequent inequality (5.63) follows by taking the supremum into the first integral, which can only enlarge the expression; same as (5.47), since
sup
by the DCT in [7], it is allowed to exchange limit and integration because it can be upper bounded by a finite value in equation (5.64).
Continuing with (5.64), we get
I2 ≤ sup
5.2 Derivation of Theorem 4.2 and Corollary 4.3 Chapter 5
out from the integration because they are constant for x1; (5.69) holds since limE1↑∞supQX1∈A1Pr[!X1! ≥ !xi!] = 1; similar to (5.66) and (5.69), equation (5.70) and equation (5.71) follows from taking the constant out from the integration and the whole remaining integration is exactly 1; finally, in (5.72), for any ε we can choose a big enough, such that the inequality holds, the detailed proof is provided in Appendix B.
Finally, plugging (5.49), (5.72) into (5.33) and (5.41) and note that ε is arbitrary, we now have
Recall that we let X1 escapes to infinity in the beginning of the derivation. We can
5.2 Derivation of Theorem 4.2 and Corollary 4.3 Chapter 5
change this assumption to any Xi ,∀i = 1, 2, . . . , m, so we get χMAC= max
1≤i≤m
I sup
'ˆxi'=1
@
log π + E2
log !HTixˆi!23
− h.HTixˆi
/ AJ
(5.74)
= max
1≤i≤mFχMISO,iG. (5.75)
If we consider the case of SISO MAC, i.e., each user has just one antenna, we get the following result:
χMAC= max
1≤i≤m
Klog π + E0log |Hi|21 − h(Hi)L
(5.76)
= max
1≤i≤mFχSISO,iG. (5.77)
From the first part and the second part of proofs in Chapter 5.2, the whole proof of Theo-rem 4.3 and TheoTheo-rem 4.2 is concluded.
Discussion and Conclusion Chapter 6
Chapter 6
Discussion and Conclusion
In this thesis, the fading number of the multiple-access general fading channel is provided where each user is allowed to have more than one antenna. The result indicates that the MAC fading number is exactly equivalent to the single-user MISO fading number. In the special case that each user has only one antenna, the fading number is 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 the best channel situation 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.
Actually, we got the similar results in [8] and [10], which are the SISO Rician fading MAC without memory and with memory respectively, and now we extend to the general fading (they are allowed to be dependent between each other) and each user has more than one antenna, i.e., memoryless MISO general fading MAC.
In the analysis of the channel we have allowed for many different types of power con-straints. We grouped them into three categories: an individual peak-power constraint for each user, an individual average-power constraint for each user, and a combined power-sharing average-power constraint among all users. The power-power-sharing constraint does not make sense in a practical setup as it requires the users to share a commom battery, while their signals still are restricted to be independent. However, the inclusion of this case helps with the analysis. Moreover, it turns out that the pessimistic results described above even hold if we allow for such power sharing.
Recall that it is shown in [7, 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 [7, Proposition 19] if at least one user uses circularly symmetric input, then the MAC fading number is the same as the single user MISO fading number. From the results in this thesis, we learn that the capacity-achieving input distribution reduces the MAC to a single-user
Chapter 6
channel.
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 a theoretical result when SNR tends to infinity; in a practical situation, it is not necessary to reduce a multiple-access channel to a single-user channel for designing a system.
Actually, we also get the asymptotic capacity region of the multiple-access general fading channel at the same time. Because we need to reduce the multiple-user channel to the best single-user channel if we want to achieve the fading number, the asymptotic capacity is only one point in that region, not a line unless there are more than one user with best channels.
Figure (6.5) shows the asymptotic capacity region in the case of m = 2. We can see all the possible rate pairs, and the asymptotic sum-rate capacity is a point in the figure, the value is just equivalent to the capacity of the better channel. Figure (6.6) shows the special case that both of the users have equivalent channel capacity, and we can get the asymptotic sum-rate capacity by time-sharing, which is the reason why the asymptotic sum-rate capacity is a line in such a special case.
R(1) R(2)
C(1) C(2)
Asymptotic sum-rate capacity
Figure 6.5: The capacity region of memoryless general fading MAC (2-user case).
In the case of m users with m > 2, as we mention in Section 3.1, the m-user capacity region is given by the convex closure of all rate m-tuples.
Possible future work for the multiple-access general fading channel might be as follows:
• Considering the case with memory.
• Considering the case with side-information.
• Loosening the restriction that the receiver has only one antenna
• Deriving the nonasymptotic capacity.
The first two point might be easier. We need to modify the channel model, considering the time index and the effects of feedback, to get the better capacity since both of memory
Chapter 6
R(1) R(2)
C(1) C(2)
Asymptotic sum-rate capacity
Figure 6.6: The capacity region of a special case: 2-user case with C(1)= C(2). and side-information are helpful to our analysis. The difficulty in the third point is that we need to consider the more troublesome MIMO case, not only SISO and MISO cases.
Finally, the last point is the hardest task since we do not know much about the o(1) term in equation (3.17). All the asymptotic tricks is not suited to this problem anymore, i.e., we need to restart from the upper and lower bounds to the nonasymptotic capacity of the fading channel.
Derivation of Equation (5.58) Appendix A
Appendix A
Derivation of Equation (5.58)
Equation (5.58) holds only if we can prove the continuity of h(HTx) in x for all x are finite and larger than 0. Here note that we prove for a general case with H is a nT× nR fading matrix.
Let the sequence xn converge to x, !x! > 0. It then follows that the sequence Hxn
converges weakly to Hx. To simplify the notation, we let W = Hx, such that the sequence Wn converges weakly to W, i.e., Wn ⇒ W. Let the law of W be denoted by QW, the law of Wn by QWn, the law of a zero-mean Gaussian random variable NC.0, E0WW†1/
of convariance E0WW†1 by QW,G, and the law of a zero-mean Gaussian random variable NC
P 0, E
2
WnW†n
3Q
of convariance E2
WnW†n
3
by QWn,G. Then we obtain from the lower semi-continuity of relative entropy in [2] that
lim
n↑∞
D(QWn!QWn,G) ≥ D(QW!QW,G). (A.1)
But
D(QWn!QWn,G) = log(πe )ν + log det E2
WnW†n3
− h(Wn), (A.2) and
D(QW!QW,G) = log(πe )ν + log det E2
WW†3
− h(W), (A.3)
Moreover, since E0WW†1 and E2
WnW†n3
is continuous, and determinants are polynomials of the corresponding matrix, we get log det E2
WnW†n3
−→ log det E0WW†1.
So we obtain
n↑∞limh(Wn) ≤ h(W), (A.4)
i.e.,
n↑∞limh(Hxn) ≤ h(Hx). (A.5)
Appendix A
It therefore remains to prove the reverse inequality lim
n↑∞
h(Hxn) ≥ h(Hx). (A.6)
Let ˜Z ∼ NC(0, InR) be independent of H. By choosing σ small enough, the following inequality holds:
h(H + βσ ˜Z) − h(H) < ), (A.7)
where β is an arbitrary number and is finite and larger than 0.
It now follows from (A.7) that for any xn, !xn! > 0,
h(Hxn+ σ ˜Z) − h(Hxn) = I(Hxn+ σ ˜Z; ˜Z) (A.8)
= I D
H xn
!xn! + σ
!xn!Z; ˜˜ Z E
(A.9)
= I DP
H+ σ
!xn!Z˜Q xn
!xn!; ˜Z xn
!xn! E
(A.10)
≤ I D
H+ σ
!xn!Z; ˜˜ Z E
(A.11)
< ), (A.12)
where the first inequality follows by the data processing theorem, because Z˜ xn
!xn! "−− ˜Z"−− H + σ
!xn!Z˜ "−−P
H+ σ
!xn!Z˜ Q xn
!xn! (A.13)
forms a Markov chain, and where the last inequality follows from (A.7) with β = 'x1
n'. By scaling properties of differential entropy, we are allowd to only consider the case that xhas unit length !x! = 1, i.e. x = ˆx. It now follows from (A.12) that for !xn! > 0,
h(Hxn) > h(Hxn+ σ ˜Z) − ) (A.14)
= h(Hˆx+ H(xn− ˆx) + σ ˜Z) − ) (A.15)
≥ h(Hˆx+ H(xn− ˆx) + σ ˜Z6
6H(xn− ˆx) + σ ˜Z) − ) (A.16)
= h(Hˆx6
6H(xn− ˆx) + σ ˜Z) − ) (A.17)
= h(Hˆx) − I(Hˆx; H(xn− ˆx) + σ ˜Z) − ), (A.18) where the second inequality follows because conditioning cannot increase differential entropy.
Expanding the mutual information term we obtain:
I(Hˆx; H(xn− ˆx) + σ ˜Z) = h(H(xn− ˆx) + σ ˜Z) − h(H(xn− ˆx) + σ ˜Z6
6Hx)ˆ (A.19)
≤ nRlog
;E0!H!2F1
nR !xn− ˆx!2+ σ2
<
− nRlog σ2. (A.20) Here the inequality can be derived as follows. Firstly, note that since ˜Z is Gaussian and independent of H(xn− ˆx), we have
h(H(xn− ˆx) + σ ˜Z6
6Hˆx) ≥ h(H(xn− ˆx) + σ ˜Z6
6Hˆx, H(xn− ˆx)) (A.21)
= h(σ ˜Z) (A.22)
= nRlog(πe σ2). (A.23)
Appendix A
Secondly, note that because among all random vectors of a given expected squared norm, differential entropy is maximized by the vector whose components are IID Gaussian. Hence, we get
h(H(xn− ˆx) + σ ˜Z) ≤ nRlog
πe E2
!H(xn− ˆx) + σ ˜Z!23 nR
(A.24)
= nRlog
;πe.E0!H(xn− ˆx)!21 + nRσ2/ nR
<
(A.25)
≤ nRlog
; πe
;E0!H!2F1
nR !xn− ˆx!2+ σ2
<<
. (A.26)
Inequalities (A.18) and (A.20) combine to prove that
n↑∞limh(Hxn) ≥ h(Hx) − ), (A.27) and since ) > 0 is arbitrary, (A.6) is proven, which combines with (A.5) to prove the continuity of h(HTx) in x for all x with !x! > 0.
Derivation of Equation (5.72) Appendix B
Appendix B
Derivation of Equation (5.72)
First, let an be a monotonical increasing unbounded sequence with n ↑ ∞, and define
Plugging (B.1), (B.2) into (B.3), we get
βn= sup
Appendix B
where in (B.10) follows from DCT.
Since limn↑∞an tends to infinity, this means that
E so equation (5.72) holds.
Bibliography
[1] C. E. Shannon, “A mathematical theory of communication,” Bell System Technical Journal, vol. 27, pp. 379–423 and 623–656, July and October 1948.
[2] A. Lapidoth and S. M. Moser, “Capacity bounds via duality with applications to multiple-antenna systems on flat fading channels,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2426–2467, October 2003.
[3] S. M. Moser, “Duality-based bounds on channel capacity,” Ph.D. dissertation, ETH Zurich, October 2004, Diss. ETH No. 15769. [Online]. Available: http://moser.cm.nct u.edu.tw/publications.html
[4] ——, “The fading number of memoryless multiple-input multiple-output fading chan-nels,” IEEE Transactions on Information Theory, vol. 53, no. 7, pp. 2652–2666, July 2007.
[5] V. Sethuraman, L. Wang, B. Hajek, and A. Lapidoth, “Low-snr capacity of noncoherent fading channels,” IEEE Transactions on Information Theory, vol. 55, no. 4, pp. 1555–
1574, April 2009.
[6] G.-R. Lin and S. M. Moser, “The fading number of a multiple-access Rician fading channel,” March 2009, submitted to IEEE Information Theory Workshop (ITW).
[7] S. M. Moser, “Capacity analysis of multiple-access OFDM channels,” Department of Communication Engineering, National Chiao Tung University (NCTU), Zhudong, Taiwan, final report of project 4G Wireless Access Technology (G1-95003), from January 1, to December 31, 2006, funded by Industrial Technology Research Institute (ITRI). [Online]. Available: http://moser.cm.nctu.edu.tw/publications.html
[8] I. Csisz´ar and J. K¨orner, Information Theory: Coding Theorems for Discrete Memory-less Systems. Budapest: Academic Press, 1981.
[9] S. M. Moser, Advanced Topics in Information Theory (Lecture Notes), version 1, spring semester 2012, Information Theory Lab, Department of Electrical &
Computer Engineering, National Chiao Tung University (NCTU), March 2012.
[Online]. Available: http://moser.cm.nctu.edu.tw/scripts.html
Bibliography
[10] Y.-H. Chou, “The fading number of multiple-access Rician fading channel with memory,” Master’s thesis, Information Theory Lab, Department of Electrical &
Computer Engineering, National Chiao Tung University (NCTU), Hsinchu, Taiwan, August 2012, supervised by Prof. Dr. Stefan M. Moser. [Online]. Available: http://mo ser.cm.nctu.edu.tw/publications.html