Third Term

Recalling the definition of E_k in (6.25), we lower-bound the fourth term on the RHS of (6.24) as follows:

6.2 Detailed Derivations for Three Terms in (6.24) Chapter 6 Here, (6.90) follows because the first term and last term in (6.89) are equal or greater than zero and H_b(E_k|Y^k−1_k−κ, G_k= 0) ≤ H_b(E_k) = H_b(β_k); in (6.92), we add {Θ_k}, which is IID

∼ U ((−π, π]) and independent of Yk. Because {Θ_k} is uniformly distributed, it destroys the phase of {Hk} and let {Hke^iΘk} becomes circularly symmetric. (6.94) follows because we drop e^Θk on both side of mutual information.

By Appendix C.1, we have β_kI,

so we further bound (6.96) as follows:

I,

-Chapter 6 Proof of Theorem 5.4

Here, in (6.100), we drop {Z^k−1_k−κ}, so the mutual information becomes smaller.

By Appendix C.2, we have βkI,%

and we further bound (6.101) as follows:

I,

6.2 Detailed Derivations for Three Terms in (6.24) Chapter 6

Here, (6.105) follows from taking the magnitude from H_k|Xk|e^iΘk; (6.106) follows be-cause we drop some terms in mutual information; (6.107) follows from the definition of differential entropy for unit vectors (see Section 3.1.2); (6.110) follows because dropping conditioning increases entropy.

Chapter 7

Discussion and Conclusion

In this thesis, we have shown that the asymptotic capacity of general regular SIMO fading channels with memory remains unchanged even if one allows causal noiseless feedback.

This once again shows the extremely unattractive behavior of regular fading channels at high SNR: besides the double-logarithmic growth [8] and the very poor performance in a multiple-user setup (where the maximum sum-rate only can be achieved if all users apart from one always remain switched off [16]), we now see that any type of feedback does not increase capacity in spite of memory in the channel.

Possible future works for the general regular fading channels with memory and feedback might include the following:

• Considering the case with multiple-input single-output, i.e., having several mobile phones (each having one antenna) communicating with one base station (having only one antenna). The difficulties for this case lies in the fact that now we not only need to optimize the phase and magnitude of the inputs, but also the direction of them.

• Considering the case with multiple-input multiple-output.

• The situation where both transmitter and receiver have access to causal partial side-information S_k about the fading, where by partial we mean that

n→∞lim 1 nI!

Sⁿ₁; Hⁿ₁"

<∞. (7.1)

Appendix A

Upper Bound (6.46)

In this appendix, we derive the following upper bound:

I!

We bound the first term as follows:

I! where in (A.7), C_IID(·) denotes the capacity without feedback or memory for a given power.

Because C_IID(·) is nondecreasing, and under the condition that Ek = 0, i.e., |Xk| ≤ ξmin,

Appendix A Upper Bound (6.46)

Here, (A.9) follows from conditioning that reduces entropy; and (A.10) follows because conditional on H^k−1₁ , H_k is independent of 7

. We first have the following inequality: where both h⁺(·) and h⁻(·) are nonnegative (see Section 3.1.1), we further bound the first term in (A.11) as follows:

−1 Here, (A.14) follows from Lemma A.12 in [10, Appendix A.4.2]; and (A.15) follows because

E#

Appendix A

Choosing ξ = 1, we then get from (A.15) and (A.11) E#

log %H_k%²)

)E_k= 0, G_k = 0$

≥ h! H_k)

)E_k= 0, G_k = 0"

−n_R+ 1

e − nRlog⁺ .πe

n_R E#

%H0%²)

)G₀= 0$ 1 − β_k

/

− ∆(nR,1)(A.18)

We put this back into (A.10), and get I!

X_k,H^k−1₁ ; Y_k)

)E_k= 0, G_k= 0"

− hλ! ˆH_ke^iΘk)

) ˆH_le^iΘl, E_k= 0, G_k= 0"

−nRE#

log %Hk%²)

)Ek= 0, Gk = 0$

≤ C_IID(ξ_min)

)G_k= 0) + h! Hk

))E_k= 0, Gk= 0"

− hλ! ˆHke^iΘk)

)H^k−1₁ , G_k= 0"

− h(Hk

))H^k−1₁ , G_k = 0) − n_Rh(H_k)

)E_k= 0, G_k= 0) + n_R(n_R+ 1) e + n²_Rlog⁺

.πe n_R

E#

%H0%²|G0 = 0$ 1 − βk

/

+ n_R∆(n_R,1) (A.19)

≤ CIID(ξ_min)

)G₀ = 0) − (n_R− 1)h(Hk

))E_k= 0, G_k= 0) − h_λ! ˆH₀e^iΘ0)

)H⁻¹_−∞, G₀= 0"

− h(Hk

))H^k−1₁ , G_k = 0) +n_R(n_R+ 1)

e + n²_Rlog⁺ .πe

n_R E#

%H₀%²|G₀ = 0$ 1 − β_k

/

+ n_R∆(n_R,1) (A.20)

≤ CIID(ξ_min)

)G₀ = 0) − (n_R− 1)h(H0|H⁻¹_−∞, G₀= 0) − h_λ! ˆH₀e^iΘ0)

)H⁻¹_−∞, G₀ = 0"

− h(H0

))H⁻¹_1−k, G₀= 0) + n_R(n_R+ 1)

e + n²_Rlog⁺ .πe

n_R E#

%H₀%²|G₀ = 0$ 1 − β_k

/

+ n_R∆(n_R,1) (A.21)

where (A.20) follows because we shift the time index in hλ(·) by k using the stationarity of {H_k}, and add more terms to it; we also shift Gk to G₀ in C_IID(·) since C_IID(·) is IID.

(A.21) follows because {Hk} is a stationary process, h(Hk|H^k−1₁ ) is nonincreasing in k, and therefore, we have

h(Hk|Ek= 0, Gk= 0) ≥ h(Hk|H^k−1₁ , E_k= 0, Gk = 0) (A.22)

= h(Hk|H^k−1₁ , G_k = 0) (A.23)

≥ h(Hk|H^k−1_−∞, G_k = 0) (A.24)

= h(H₀|H⁻¹_−∞, G₀ = 0). (A.25)

Appendix B

Upper Bound (6.81)

In this appendix, we derive the following upper bound:

I!

X_k; %H_k%|Xk|e^iΘk)

)E_k= 1, G_k= 0"

+ I,

X_k; ˆH_ke^i(Φk+Θk) ))

)%H^k%|Xk|, e^iΘk, E_k= 1, G_k= 0

-(B.1)

≤ − log 2 − h(Hk

))X_k, E_k = 1, G_k= 0) + (2n_R− 1)E[log %Hk%|Ek= 1, G_k = 0]

− E[log %Hk%|Ek = 1, G_k= 0] + µ log η + log Γ 5

µ,ν η

6

+ (1 − µ)E#

log %Hk%²)

)E_k = 1, Gk= 0$

− µE#

log |Xk|²)

)E_k= 1, Gk= 0$ + )ν,k

+1 ηE#

%Hk%²|Xk|²)

)E_k= 1, G_k = 0$ + ν

η + h_λ, Hˆ_ke^iΘk

))

)E^k= 1, G_k= 0

-, (B.2) using a similar approach as in [9, Appendix D].

First, we apply Lemma 11 in [9] to the first term in (B.1), i.e., we choose S = X_k and T = %H_k%|Xk|e^iΘk. Note that we need to condition everything on the events E_k= 1 and G_k= 0.

I!

X_k; %H_k%|Xk|e^iΘk)

)E_k= 1, G_k = 0"

≤ −h!

%Hk%|Xk|e^iΘk)

)X_k, E_k= 1, G_k= 0"

+ log π + µ log η + log Γ 5

µ,ν η

6

+ (1 − µ)E#

log(%Hk%²|Xk|²+ ν))

)E_k = 1, Gk = 0$ + 1

ηE#

%Hk%²|Xk|²)

)E_k= 1, G_k= 0$ +ν

η (B.3)

where µ, η > 0, and ν ≥ 0 can be chosen freely. Note that from a conditional version of Lemma 2 in [9] with m = 1 follows that

h!

%Hk%|Xk|e^iΘk)

)X_k= x_k, E_k= 1, G_k = 0"

= h! Θk

))Xk= xk, Ek= 1, Gk= 0"

+ h!

%Hk%|Xk|)

)e^iΘk, Xk = xk, Ek= 1, Gk= 0"

+ E#

log %Hk%|Xk|)

)X_k = xk, E_k= 1, Gk= 0$

(B.4)

= log 2π + h(%Hk%|Xk|)

)X_k= xk, E_k = 1, Gk = 0) + E#

log %H_k%|Xk|)

)X_k = x_k, E_k= 1, G_k= 0$

, (B.5)

Appendix B

where we have used that Θ_k is independent of all other random quantities and uniformly distributed on the unit circle. Taking the expectation over Xkconditional on Ek= 1, Gk= 0 and noting that by the law of total expectation

E_Xk where (B.8) follows from the scaling property of entropy with a real argument.

We choose 0 < µ < 1 (recall that µ is a free parameter!) such that 1 − µ > 0. Then

Appendix B Upper Bound (6.81) (B.16) into (B.3) yields

I!

Next, we continue with the second term in (B.1):

I!

Hence, using (B.21) and (B.18) we get the following upper bound for (B.1):

I!

Appendix B

+ (1 − µ)E#

log %H_k%²)

)E_k = 1, G_k = 0$

− µE#

log |X_k|²)

)E_k= 1$ + )_ν,k +1

ηE#

%Hk%²|Xk|²)

)E_k= 1$ +ν

η + h_λ! ˆH_ke^iΘk)

)E_k= 1, G_k= 0"

(B.23) Here, (B.23) follows from a conditional version of Lemma 2 in [9] similar to (B.4)–(B.9) which allows us to combine the second and the last term in (B.22).

Appendix C

Upper Bounds (6.97) and (6.102)

In this appendix, we will find bounds (6.97) and (6.102).

C.1 δ₁(κ, ξ_min)

We first derive upper bound (6.97):

β_kI,

H_k|Xk|e^iΘk+ Z_k; Z^k−1_k−κ) ))

%H_l|Xl|e^iΘl+ Z_l&k−1

l=k−κ, E_k= 1, G_k= 0

-≤ δ₁(κ, ξ_min) + H_b(βk). (C.1)

We start as follows:

β_kI,

Hk|Xk|e^iΘk+ Zk; Z^k−1_k−κ) ))

%Hl|Xl|e^iΘl+ Zl

&k−1

l=k−κ, E_k = 1, Gk= 0

-= βkh, Z^k−1_k−κ)

))

%Hl|Xl|e^iΘl+ Zl

&k−1

l=k−κ, E_k= 1, Gk= 0

-− βkh, Z^k−1_k−κ)

))

%Hl|Xl|e^iΘl+ Zl

&k

l=k−κ, E_k= 1, Gk = 0

-(C.2)

≤ βkh, Z^k−1_k−κ)

))E^k= 1, Gk= 0

-− βkh, Z^k−1_k−κ)

))

%H_l|Xl|e^iΘl+ Z_l&k

l=k−κ,|Xl|^k_l=k−κ,Z_k, E_k = 1, G_k= 0

-(C.3)

= β_kh, Z^k−1_k−κ)

))E^k= 1, G_k= 0

-− βkh .

Z^k−1_k−κ )) )) ) :

H_le^iΘl+ Z_l

|Xl|

;k−1 l=k−κ

,H_ke^iΘk,|Xl|^k_l=k−κ, E_k= 1, G_k= 0 /

. (C.4) Here (C.3) follows from conditioning that reduces entropy. The reason why we do not drop Ek is because we have βk in front of the mutual information, if we drop Ek now, we will not be able to get rid of β_k later. In (C.4) we drop Z_k since {Z_k} are IID. In order to get rid of the dependence on input, we take an infimum:

β_kh, Z^k−1_k−κ)

))E^k= 1, Gk= 0

-− βkh .

Z^k−1_k−κ )) )) )

:

H_le^iΘl+ Z_l

|Xl|

;k−1 l=k−κ

,H_ke^iΘk,|Xl|^k_l=k−κ, E_k= 1, G_k= 0 /

C.1 δ1(κ, ξmin) Appendix C thus the smaller the entropy of Z^k−1_k−κ would be. From this stage, the dependence on input inside mutual information is gone (except Ek), but we still have βk in front of mutual information, therefore we add 1 − β_k to get rid of β_k as follows:

-Appendix C Upper Bounds (6.97) and (6.102) {Zk} are stationary processes.

If ξmin goes to infinity, h57

Next, we derive upper bound (6.102):

β_kI,%

H_l|Xl|e^iΘl&k−1

l=k−κ; Z_k)

)) H^k|Xk|e^iΘk+ Z_k, E_k= 1, G_k= 0

-≤ δ2(κ, ξmin) + H_b(βk). (C.14)

The derivation is similar to (C.2)–(C.13).

β_kI,% Here (C.16) follows from conditioning that reduces entropy and for the same reason as in Section C.1, we keep E_k = 1. In order to get rid of the dependence on input, we take an infimum:

C.2 δ2(κ, ξmin) Appendix C

Here, (C.19) follows because the smaller γ_k is, the more Z_k can reflect in H_ke^iΘk + ^Z_γ^k

k, thus the smaller the entropy of Zk would be. Next, we want to get rid of βk as follows:

β_kI {Zk} are independent of each other; (C.26) follows because {Hk} and {Zk} are stationary processes.

-converges to h,%

Hle^iΘl&−1

Appendix D

Causal Interpretations for Independence

In Figures D.3–D.5 we prove the following independence claims used in (6.21), (6.73), and (6.84).

• (M, Y^k−1₁ ) ⊥⊥ Yk when conditioned on (X_k,H^k−1₁ );

• Xk ⊥⊥ Yk when conditioned on (HkXk);

• H^k−1₁ ⊥⊥ Yk when conditioned on (Xk,Hk).

X₁ X₂ X₃ X_k−1 X_k

F₁ F₂ F₃ F_k−1 Fk

Z₁ Z₂ Z₃ Z_k−1 Z_k

Y₁ Y₂ Y₃ Y_k−1 Yk

H₁ H₂ H₃ H_k−1 H_k

M

Figure D.3: The relevant subgraph of V showing the independence of (M, Y₁^k−1) and Yk

when conditioned on (X_k,H^k−1₁ ).

Appendix D

X₁ X₂ X₃ X_k−1 X_k

F₁ F₂ F₃ F_k−1 F_k

Z₁ Z₂ Z₃ Z_k−1 Zk

Y₁ Y₂ Y₃ Y_k−1 Y_k

H₁ H₂ H₃ H_k−1 Hk

M

Figure D.4: The relevant subgraph of V showing the independence of X_k and Y_k when conditioned on (HkX_k).

X₁ X₂ X₃ X_k−1 Xk

F₁ F₂ F₃ F_k−1 Fk

Z1 Z2 Z3 Z_k−1 Zk

Y₁ Y₂ Y₃ Y_k−1 Yk

H1 H2 H3 H_k−1 Hk

M

Figure D.5: The relevant subgraph of V showing the independence of H^k−1₁ and Yk when conditioned on (X_k,H_k).

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