On The Effect of Self-Interference in Gaussian Two-Way ... - IEEE Xplore

2013 IEEE International Symposium on Information Theory

On The Effect of Self-Interference in Gaussian Two-Way Channels With Erased Outputs Seyed Ershad Banijamali

Kamyar Moshksar

Amir K. Khandani

Dep. of Electrical & Computer Eng. University of Waterloo Waterloo, ON, Canada Email: [email protected]

Dep. of Electrical & Computer Eng. University of Waterloo Waterloo, ON, Canada Email: [email protected]

Dep. of Electrical & Computer Eng. University of Waterloo Waterloo, ON, Canada Email: [email protected]

Abstract—It is well-known that the so-called Shannon Achievable Region (SAR) in a collocated two-user Gaussian TwoWay Channel (GTWC) does not depend on the self-interference that is due to the leakage of the signal transmitted by each user at its own receiver. This is simply because each user can completely remove its self-interference. In this paper, we study a class of GTWCs where each user is unable to cancel the self interference due to random erasures at its receiver. The mixture of the intended signal for each user and its self-interference is erased independently from transmission slot to transmission slot. It is assumed that both users adopt PAM constellations for transmission purposes. Due to the fact that both users are unaware of the erasure pattern, the noise plus interference at each user is mixed Gaussian. To analyze this setup, a sequence of upper and lower bounds are developed on the differential entropy of a general mixed Gaussian random variable where it is shown that the upper and lower bounds meet as the sequence index increases. Utilizing such bounds, it is shown that the achievable rate for each user is monotonically increasing in terms of the level of selfinterference and eventually saturates as self-interference grows to infinity. This saturation effect is justified analytically by showing that as self-interference increases, each user is enabled to extract the erasure pattern at its receiver. Treating the erasure pattern as side information, both users are able to cancel self-interference and decode the useful information at higher transmission rates.

Notation- Before proceeding, let us provide a list of notations adopted in this paper. For any x ∈ [0, 1], we define x  1 − x. Random variables are shown in bold, e.g., x with realization x. Sets are shown by calligraphic letters such as X . Probability Density Function (PDF) of a random variable x is shown by px (·). The probability of an event E and the expectation of a random variable x are denoted by P(E) and E[x], respectively. The entropy of a discrete random variable x is shown by H(x) and the differential entropy of a continuous random variable x is denoted by h(x). The mutual information between x and y is denoted by I(x; y). A Gaussian random variable with mean μ and variance σ 2 is shown by N(μ, σ 2 ). The PDF of such a random variable is shown by ϕ(·; μ, σ 2 ). The so-called Q-function is defined by  ∞ u2 Q(x)  √12π x e− 2 du . A Bernoulli random variable with parameter p ∈ (0, 1) is denoted by Ber(p). The acronym i.i.d. stands for independent and identically distributed and O(·) is

978-1-4799-0446-4/13/$31.00 ©2013 IEEE

Fig. 1.

Gaussian Two-Way Channel (GTWC)

the so-called big-O notation. I. I NTRODUCTION A. Prior Art and Motivation The collocated two-way channel introduced by Shannon [1] is the first proposed scenario in which two users attempt to transmit and receive information through a common channel simultaneously. Each user is represented by a node that can transmit and receive at the same time. The Gaussian Two-Way Channel (GTWC) shown in Fig. 1 is a particular example of such a scenario where each user attempts at transmitting information to the other user. A feature of this channel is that the signals transmitted by node 1 (node 2) to node 2 (node 1) also leak into the receiver of node 1 (node 2). This phenomenon is referred to as self-interference. Denoting the signal received at node k by y k , y 1 = h1,1 x1 + h2,1 x2 + z 1 y 2 = h1,2 x1 + h2,2 x2 + z 2

,

(1)

where hk,l is the channel coefficient from transmitter k to receiver l, xk is the signal transmitted by user k and z k is the additive ambient noise at node k. It is assumed that z 1 and z 2 are i.i.d. N(0, 1) random variables and xk satisfies the power constraint E[x2k ] ≤ Pk . Moreover, the channel coefficients are known to both nodes. The codebook of user k consists of 2nRk  codewords of length n where each codeword represent a message intended

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2013 IEEE International Symposium on Information Theory

for the other user. As n grows, Rk represents the transmission rate for user k. The so-called Shannon Achievable Region (SAR) [1] for a general memoryless two-way channel is the set of all (R1 , R2 ) that satisfy R1 ≤ I(x1 ; y 2 |x2 ) R2 ≤ I(x2 ; y 1 |x1 )

,

(2)

where x1 and x2 are independent random variables. By (2), SAR for GTWC does not depend on the coefficients h1,1 and h2,2 that represent self-interference. This is simply because users are able to cancel self-interference completely. In fact, I(x1 ; y 2 |x2 ) = I(x1 ; y 2 − h2,2 x2 |x2 ) = I(x1 ; h1,2 x1 + z 2 ) (3) and similarly, I(x2 ; y 1 |x1 ) = I(x2 ; h2,1 x2 + z 1 ).

(4)

As such, the channel from each transmitter to its intended receiver converts to an additive white Gaussian noise channel. This implies that random Gaussian codewords achieve the largest value for I(x1 ; y 2 |x2 ) and I(x2 ; y 1 |x1 ). In fact, selecting xi as N(0, Pi ), SAR is given by   R1 ≤ 12 log 1 + h21,2 P1 (5)  .  R2 ≤ 12 log 1 + h22,1 P2 It is shown in [2] that (5) is indeed the capacity region of a two-user GTWC. In the following, we consider a modification of GTWC motivated by the physics of the underlying system. This modification results in a nonlinear channel in the sense that each node can not simply remove self-interference by subtraction. Hence, in contrast to (3) and (4), I(x1 ; y 2 |x2 ) and I(x2 ; y 1 |x1 ) depend on x2 and x1 , respectively. Assuming user i can adjust hi,i in the interval [αi , ∞) for some αi > 0, our goal is to design hi,i in oder to achieve the largest SAR. B. Contribution We study a GTWC with discrete inputs and erased outputs as shown in Fig. 2. The random codebooks of both users are generated over the 2-PAM constellation. The mixture of signals received by each node is randomly erased from transmission slot to transmission slot before being added to the ambient noise. The point to point Gaussian erasure channel was first introduced in [3] to model channels with impulsive noise on top of Gaussian additive noise at the receiver side. In contrast to [3] where the erasure pattern is known to the receiver a priori, in this paper both nodes are unaware of the erasure pattern due to the random nature of erasures. In return, this makes each receiver be unaware of the presence of self-interference in the received signal. Therefore, subtracting self-interference does not necessarily remove self-interference. It is notable that the received signal by each receiver is a mixed Gaussian random variable and hence, there is no closed formula for the upper bounds in (2). In section III, we develop

Fig. 2.

GTWC with discrete inputs and erased outputs

a sequence of upper and lower bound on the differential entropy of a mixed Gaussian random variable. The bounds have the property that as the sequence index increases, the upper bounds decrease and the lower bounds increase and they meet as the sequence index tends to infinity. To our knowledge, this is the first time that arbitrarily tight bounds are developed on the differential entropy of a mixed Gaussian random variable. In section III, using the proposed bounds and through an example, we show that I(x1 ; y 2 |x2 ) and I(x2 ; y 1 |x1 ) increase and eventually saturate as h2,2 and h1,1 increase, respectively. In fact, it is verified1 that each node can use self-interference as a training tool in order to understand the erasure pattern. If self-interference is sufficiently large and the received signal is small in magnitude, the node understands that erasure has occurred. On the other hand, if the received signal appears to be at the same order of magnitude as self-interference, the node understands that erasure has not occurred and simply cancels self-interference. II. T IGHT BOUNDS ON THE DIFFERENTIAL ENTROPY OF MIXED G AUSSIAN RANDOM VARIABLES The following is the main result of this section: Theorem 1: Let x be a random variable with PDF px (x) = M (x−xm )2 − √pm 2P , where x1 < · · · < xM and (pm )M m=1 m=1 2πP e is a discrete probability sequence. Define x2 xm m , m = 1, · · · , M, (6) am = pm e− 2P , bm = P am + c+ , d = bm − bM , m = 1, · · · , M − 1, (7) m = aM m am − , d = b1 − bm , m = 2, · · · , M, (8) c− m = a1 m   M −1  d+ mx c+ (9) , x ∈ R+ , η + (x) = ln 1 + me  −

η (x) = ln 1 +

m=1 M −1 

 d− mx c− me

, x ∈ R+ .

(10)

m=1 1 + Let x+ ∗ ≥ 0 be the unique root of η (x) = ln 2 for aM < 2 + − and x∗ = 0 otherwise. Also, let x∗ ≥ 0 be the unique root of η − (x) = ln 2 for a1 < 12 and x− ∗ = 0 otherwise. Then for any N1lb , N1ub , N2 ≥ 0,

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β + α2N1lb +1,N2 ≤ h(x) ≤ β + α2N1ub ,N2 , 1 See

Proposition 1.

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2013 IEEE International Symposium on Information Theory

αN,N 





log e

s∈{+,−}

N  (−1)n n n=1

s∈{+,−} n=0

β



n!

j1 ,··· ,jM −1 ≥0 j1 +···+jM −1 =n

m=1

jm !

M −1

(csm )jm

m=1

  s  s xn+1 − xm xn − xm √ √ pm η s (xsn ) Q −Q . P P m=1

(12)

    M M M    log e xm xm 1 2 log(2πP ) + P+ pm xm − log aM pm Q − √ pm Q √ − log a1 2 2P P P m=1 m=1 m=1    

  M M   xm P − x2m P − x2m xm e 2P + xm Q − √ e 2P . (13) pm pm x m Q √ − log e bM − log e b1 − 2π 2π P P m=1 m=1

θ± (j1 , · · · , jM −1 ) 

M 

as e

1 2P

M −1 2 jm d± (±bs + m=1 m)

 Q

s=1

x+ n 

OUTPUTS

Let us consider a two-way channel where the received signals at nodes 1 and 2 are given by y 1 = e1 (h1,1 x1 + h2,1 x2 ) + z 1 . (16) y 2 = e2 (h1,2 x1 + h2,2 x2 ) + z 2 √ √ Here, xk ∈ Xk  {− Pk , P k } and P(Xk = x) = 12 , x ∈ Xk . Also, z 1 and z 2 are independent N(0, 1) random variables and e1 and e2 are independent Ber(e1 ) and Ber(e2 ) random variables, respectively, for known e1 , e2 ∈ (0, 1) at both nodes. We emphasize that e1 and e2 are unknown to both ends. The gain hk,k can be set at any value in [αk , ∞) by user k. Let us define R1∗  I(x2 ; y 1 |x1 ), R2∗  I(x1 ; y 2 |x2 ).

(17)

In this case, SAR is given by R1 ≤ suph2,2 ≥α2 R1∗ .

R1∗  h(y 1 |x1 ) − h(y 1 |x1 , x2 ),



 x± ∗ −P

±bs +

M −1 

 j m d± m

.

(18)

(19)

(14)

m=1

(15)

where h(y 1 |x1 ) =

1  h(e1 (h1,1 x1 + h2,1 x2 ) + z 1 ) 2

(20)

x1 ∈X1

and h(y 1 |x1 , x2 ) =

III. GTWC WITH DISCRETE INPUTS AND ERASED

R2 ≤ suph1,1 ≥α1 R2∗

1 √ P

nx+ nx− ∗ ∗ , x−  , n = 0, · · · , N2 . n N2 N2

where αN,N  and β are given in (12) and (13), respectively. Moreover, the error terms α2N1ub ,N2 + β − h(x) and     α2N1lb +1,N2 + β − h(x) scale like O N1ub + O N12 and 1     O N1lb + O N12 , respectively. 1 Proof: See appendix A.

We have

M −1

θs (j1 , · · · , jM −1 )

 M −1  N 



− log e



1 4

 x1 ∈X1 ,x2 ∈X2

h(e1 (h1,1 x1 + h2,1 x2 ) + z 1 ).

(21) To calculate h(e1 (h1,1 x1 + h2,1 x2 ) + z 1 ) in (20), note that e1 (h1,1 x1 +h2,1 x2 )+z 1 is a mixed Gaussian random variable where all its Gaussian components have unit variance and √ √ means 0, h1,1 x1 − h2,1 P2 and h1,1 x1 + h2,1 P2 with corresponding probabilities 1 − e1 , e21 and e21 . Similarly, in computing h(e1 (h1,1 x1 + h2,1 x2 ) + z 1 ) in (21), note that e1 (h1,1 x1 +h2,1 x2 )+z 1 is a mixed Gaussian random variable where both its Gaussian components have unit variance and means 0, h1,1 x1 + h2,1 x2 with corresponding probabilities 1 − e1 and e1 . As such, Theorem 1 can be utilized to derive arbitrarily tight upper and lower bounds on R1∗ . For example, let us consider a scenario where α1 = α2 = 1, h1,2 = h2,1 = 1 and P1 = P2 = 2dB. Let us focus on a particular setting where h1,1 = h2,2 = a and a ≥ 1. Fig. 3 presents plots of lower and upper bounds on the sum rate R1∗ +R2∗ in terms of a in a scenario where (e1 , e2 ) = (0.4, 0.2). Setting N (lb) = 5 and N (ub) = 6 guarantees a uniform distance of less than 0.01 between the upper and lower bounds on R1∗ + R2∗ . Moreover, it is seen that R1∗ + R2∗ is an increasing function of a and eventually saturates as a grows to infinity. This example motivates us to study the behaviour of R1∗ + R2∗ in the large self-interference regime where h1,1 and

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Then for any x1 ∈ X1 ,  h(u1 (x1 )) = −

0.5

Sum Rate (bits/sec/hz)

0.45

∞ −∞



0.4

=−

Upp er Bound N (ub) = 1

0.35



Lower Bound N (l b) = 0

0.3

−

Upp er Bound N (ub) = 2



Lower Bound N (l b) = 1

0.25

−

Upp er Bound N (ub) = 6

0.2

2

3 4 5 a; (h 1 , 1 = h 2 , 2 = a)

6

−∞ ∞ −∞



Lower Bound N (l b) = 5

1

∞

=−

7

pu1 (x1 ) (u) log pu1 (x1 ) (u)du

∞ −∞

ε(u, h1,1 )du

  e1 pu1 (x1 )|e1 (u|0) log(e1 pu1 (x1 )|e1 (u|0)) du   e1 pu1 (x1 )|e1 (u|1) log(e1 pu1 (x1 )|e1 (u|1)) du

∞ −∞

ε(u, h1,1 )du + hb (e1 ) + e1 h(u1 (x1 )|e1 = 0) +e1 h(u1 (x1 )|e1 = 1)

Fig. 3. Plots of lower and upper bounds on the sum rate R1∗ + R2∗ in a scenario where h1,2 = h2,1 = 1 and P1 = P2 = 2dB. It is seen that the bounds meet as the indices N1lb and N1ub increase.

 =−

∞ −∞

ε(u, h1,1 )du + hb (e1 ) + e1 h(z 1 )

+e1 h(h2,1 x2 + z 1 ), h2,2 tend to infinity. Intuitively, if h1,1 is sufficiently large, user 1 is capable of recognizing the realization of e1 . As such, the achievable information rate by user 1 saturates on I(x2 ; y 1 |x1 , e1 ) which is larger than I(x2 ; y 1 |x1 ).2 In fact, we have the following Proposition: Proposition 1: lim I(x2 ; y 1 |x1 ) = I(x2 ; y 1 |x1 , e1 ).

h1,1 →∞

(29)

where the last step is by the fact that h(u1 (x1 )|e1 = 0) = h(z 1 ) and h(u1 (x1 )|e1 = 1) = h(h1,1 x1 + h1,2 x2 + z 1 ) = h(h1,2 x2 +z 1 ) as additive constants do not change differential entropy. Invoking dominated convergence [4], it is shown in ∞ appendix B that limh1,1 →∞ −∞ ε(u, h1,1 )du = 0. Hence, we conclude that

(22)

lim h(u1 (x1 )) = hb (e1 ) + e1 h(z 1 ) + e1 h(h2,1 x2 + z 1 ),

h1,1 →∞

Proof: let us define u1 (x1 )  e1 (h1,1 x1 + h2,1 x2 ) + z 1 , x1 ∈ X1

(23)

(30) for any x1 ∈ X1 . Following similar lines that led us to (30), lim h(v 1 (x1 , x2 )) = hb (e1 ) + h(z 1 ),

and

h1,1 →∞

v 1 (x1 , x2 )  e1 (h1,1 x1 + h2,1 x2 ) + z 1 , xk ∈ Xk , k = 1, 2. (24) Then  1 1  h(u1 (x1 )) − h(v 1 (x1 , x2 )). R1∗ = 2 4 x1 ∈X1 x1 ∈X1 ,x2 ∈X2 (25) Note that pu1 (x1 ) (u) = e1 pu1 (x1 )|e1 (u|0) + e1 pu1 (x1 )|e1 (u|1),

(26)

where pu1 (x1 )|e1 (u|0) = ϕ(u; 0, 1) pu1 (x1 )|e1 (u|1) =

1 2



lim R1∗

h1,1 →∞

(27)

ε(u, h1,1 )  pu1 (x1 ) (u) log pu1 (x1 ) (u) −e1 pu1 (x1 )|e1 (u|0) log(e1 pu1 (x1 )|e1 (u|0)) −e1 pu1 (x1 )|e1 (u|1) log(e1 pu1 (x1 )|e1 (u|1)).

=

e1 h(h2,1 x2 + z 1 ) + e1 h(z 1 ) − h(z 1 )

= =

e1 (h(h2,1 x2 + z 1 ) − h(z 1 )) I(x2 ; y 1 |x1 , e1 ).

(32)

Similarly, one can see that limh2,2 →∞ R2∗ = I(x1 ; y 2 |x2 , e2 ). This verifies our earlier claim that both R1∗ and R2∗ saturate as self-interference increases and saturation rates correspond to having full side information on erasure patterns at the nodes. A PPENDIX A

Let us define the two-variable function ε(u, h1,1 ) by

(28)

that I(x2 ; y 1 |x1 , e1 ) = H(x2 ) − H(x2 |y 1 , x1 , e1 ) which is larger than H(x2 ) − H(x2 |y 1 , x1 ) = I(x2 ; y 1 |x1 ) due to the fact that conditioning reduces differential entropy. 2 Note

for any xk ∈ Xk , k = 1, 2. By (25), (30) and (31),

. x2 ∈X2 ϕ(u; h1,1 x1 + h2,1 x2 , 1)

(31)

M One can write p(·) as p(x) = g(x; P ) m=1 am ebm x , x2 1 where g(x; P ) = √2πP e− 2P . It is easy to see that  E[X 2 ] 1 +Pr{X > 0} ln aM p(x) ln p(x)dx = − ln(2πP )− 2 2P + Pr{X < 0} ln a1 + bM E[X1(X > 0)] + b1 E[X1(X < 0)]  ∞  ∞ p(x)η + (x)dx + p(−x)η − (x)dx. (33) +

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Straightforward calculations show that all term except the last two terms in (33) represent logβ e where β is given in (13). ∞ Next, let us consider the term 0 p(x)η + (x)dx. We write  ∞  x+  ∞ ∗ p(x)η + (x)dx = p(x)η + (x)dx+ p(x)η + (x)dx. x+ ∗

0

0

(34) We treat the integrals in (34)  separately. M −1 + d+ m x < 1. Applying 1- For x > x+ ∗ , we have m=1 cm e 3 Leibniz test for alternating series and for any N1lb ≥ 0, 2N1lb +1 (−1)n−1 M −1 + d+ x n m η + (x) ≤ and the difn=1 m=1 cm e n ference between the right and left side in this inequality is less  2(N1lb +1) M −1 + d+ 1 mx c e ≤ N1lb . than or equal to 2(N lb m m=1 1 +1) n 1  ∞ M −1 + d+ mx One may calculate the terms x+ p(x) c e dx m=1 m n ∗ + M −1 + dm x for any n ≥ 1 by expanding according m=1 cm e to the multinomial identity. This results in appearance of the term involving θs (j1 , · · · , jM −1 ) in (12). 2- Note that η + (·) is a decreasing function on [0, ∞). Let us show that η + (·) is also convex on [0, ∞). One can write   M −1 M −1   d2 + + d+ x + 2 d+ m mx 1+ cm e η (x) = c+ m (dm ) e 2 dx m=1 m=1  M −1  M −1   + + + 2 dm x + d+ x + cm (dm ) e cm e m m=1

−

M −1 

.

(35)

m=1

Using

x+

Cauchy-Schwartz

inequality and noting 2 that  M −1 + + d+ x m > 0, the term = m=1 cm dm e    2 1 1 + 1 1 + M −1 + 2 + 2 dm x 2 2 dm x is less (c+ m) e m=1 (cm ) dm e     + M −1 + M −1 + 2 d+ x + d mx . m than or equal to c (d ) e c e m=1 m m m=1 m

c+ m

∞ n−1 a alternating series n where an > 0 converges if n=1 (−1) is decreasing and lim a = 0. Moreover, any N ∈ N, a n n→∞ n 2N  2N +1 for n−1 ∞ n−1 a ≤ n−1 a ≤ an . n n n=1 (−1) n=1 (−1) n=1 (−1) 3 An

η + (y )

A PPENDIX B For notational simplicity, let us define f (u) = e1 pu1 (x1 )|e1 (u|0) and g(u, h1,1 ) = e1 pu1 (x1 )|e1 (u|1). Note that by (27), f (u) does not depend on h1,1 . Then ε(u, h1,1 ) = (f (u) + g(u, h1,1 )) ln(f (u) + g(u, h1,1 )) log e (36) −f (u) ln(f (u)) − g(u, h1,1 ) ln(g(u, h1,1 )). Since the function x ln(x) is smooth for x > 0 with derivative 1 + ln(x), the mean value theorem implies there exists ξ (depending on u and h1,1 ) such that g(u, h1,1 ) < ξ < f (u) + g(u, h1,1 )

(37)

and (f (u) + g(u, h1,1 )) ln(f (u) + g(u, h1,1 ))

d2 + dx2 η (x)

Applying this in (35) yields > 0 for any real x, i.e., η + (·) is convex. We partition the interval [0, x+ ∗ ] into N2 nx+ + ∗ subintervals with end-points xn = N2 for n = 0, · · · , N2 . N2 −1 + + + Define η(x)  η (xn+1 )1(x+ n ≤ x < xn+1 ) Nn=0 −1 2 + + + and η(x)  ≤ x < x+ n+1 ). n=0 η (xn )1(xn + ≤ η + (x) ≤ for x ∈ [0, x∗ ]. Then η(x)  x+ + + ∗ η(x), x ∈ [0, x∗ ] and we get 0 p(x)η (x)dx ≤  x+ N2 −1 + +  x+ n+1 ∗ p(x)η(x)dx = p(x)dx. n=0 η (xn ) x+ 0 n +   x+ x∗ + ∗ ≥ p(x)η(x)dx = Similarly, 0 p(x)η (x)dx 0  x+  x+ N2 −1 n+1 n+1 + p(x)dx. The term x+ p(x)dx can n=0 η(xn+1 ) x+ n n    M x+ −xm x+ −x √ be expressed as m=1 pm Q n√P m − Q n+1 . P  x+ ∗ p(x)η(x)dx − To bound the difference 0 +  x∗ + p(x)η (x)dx, note that η(x) − η + (x) ≤ 0

d

d − dx η(yn )(x+ − x+ = − ∗ dxN2 n and n) n+1  x+ x+ n+1 + + ∗ p(zn ) p(x)dx = p(zn )(xn+1 − xn ) = . N2 x+ n +  N2 −1 + + x n+1 + + Therefore, p(x)dx n=0 (η (xn ) − η (xn+1 )) x+ n + 2 d N2 −1 (x∗ ) (− dx η(yn ))p(zn ) . Since p(·) is equal to n=0 N22 d + η (·) are continuous on [0, x+ and − dx ∗ ], there are d + η (yn ) ≤ k1 constants k1 , k2 ∈ R+ such that − dx and p(zn ) ≤ k2 for all values of n. This yields  x+ N2 −1 + + k k (x+ )2 n+1 + + p(x)dx ≤ 1 2N2 ∗ n=0 (η (xn ) − η (xn+1 )) x+ n which approaches 0 as N2 tends to infinity.

m=1

2 + d+ mx c+ m dm e

+ η + (xn ) − η + (xn+1 ) for x ∈ [x+ n , xn+1 ]. Hence, +   x+ x ∗ p(x)η(x)dx − 0 ∗ p(x)η + (x)dx ≤ 0 ≤ 0  x+ N2 −1 + n+1 + p(x)dx. The upper n=0 (η (xn ) − η (xn+1 )) x+ n  x+ N2 −1 + + n+1 + + p(x)dx tends bound n=0 (η (xn ) − η (xn+1 )) x+ n to 0 as N2 grows to infinity. To see this note that by + the Mean Value Theorem, there are yn ∈ (x+ n , xn+1 ) + + + + + + and zn ∈ (xn , xn+1 ) such that η (xn ) − η (xn+1 ) =

−g(u, h1,1 ) ln(g(u, h1,1 )) = (1 + ln(ξ))f (u). Note that f (u) ≤

√e1 2π

(38)

√e1 . Using 2π ε(u,h ) ≤ (38), log 1,1 e

and g(u, h1,1 ) ≤

these bounds together with (36), (37) and   1 1 + 12 ln 2π f (u) − f (u) ln f (u). The right side of this inequality does not depend on h1,1 and it is an integrable function as f (u) represents a Gaussian pulse. Moreover, limh1,1 →∞ ε(u, h1,1 ) = 0 for any u ∈ R. Then one can use dominated convergence to exchange ∞ ε(u, h1,1 )du = the limit and integral, i.e., lim h →∞ 1,1 −∞ ∞ lim ε(u, h )du = 0 as desired. h1,1 →∞ 1,1 −∞ R EFERENCES [1] C. E. Shannon, ”Two-way communication channels”, in Proc. 4th Berkeley Symp. Math. Satist. Probab., vol. 1, 1961, pp. 611-644. [2] T. S. Han, ”A general coding theorem for the two-way channel”, IEEE Trans. on Inf. Theory, vol. IT-30, pp. 35-44, Jan. 1984. [3] A. Tulino, S. Verdu, G. Caire, S. Shamai ”The Gaussian Erasuare Channel,” ISIT 2007. IEEE International Symposium on Information Theory, pp.1721 - 1725, 24-29 June 2007. [4] R. M. Dudley, “Real analysis and probability”, Cambridge University Press, 2005.

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