On the Capacity of Free-Space Optical Intensity Channels

On the Capacit y of F ree-Space Optical In tensit y Channels Amos Lapidoth Stefan M. Moser Mic h ` ele A. Wigger ∗ No v em ber 3, 2018 Abstract New upper and lower bounds a re presented on the capacity of the free- space optical intensit y channel. This c hannel is c haracter ized b y inputs that are nonnegative (represen ting the transmitted optical in tensity) and b y outputs that are corrupted b y additive white Ga ussian noise (becaus e in free space the disturbances a rise fro m many indep endent so urces). Due to battery and safety reasons the inputs are simultaneously constrained in b o th their average a nd pea k pow er. F or a fix e d ratio of the av erage p ow er to the p eak p ow er the difference b etw een the uppe r and the lower bounds tends to zero as the av erage power tends to infinity , and the ra tio of the upp e r and low er bo unds tends to one as the average power tends to zero. The c a se where only a n av erage-p ow er constraint is imp osed on the input is treated separately . In this ca s e, the difference of the upp er and low er b ound tends to 0 as t he av erage pow er tends to infinity , and their ratio tends to a constant as the p ower tends to zero. 1 In tro duction W e consider a channel mod el for short-range optical comm unication in free space suc h as the inf r ared comm unication b et w een electronic handheld d evices. W e assume a c hannel mo del b ased on intensity mo dulation , wh ere the in put signal mo dulates the optical in tensit y of th e e mitted ligh t. Th us, the inp u t signal is proportional to the ligh t in tensit y and is therefore nonn egativ e . W e further assume that at the receiv er a front-e nd photod etecto r measures th e inciden t optica l in tensit y of the incoming ligh t and p ro duces an output signal w h ic h is prop ortional to th e detected intensit y . W e mod el the am bien t ligh t conditions by a Gaussian disturbance. Moreo v er, w e assume that the li ne-of-sigh t c omp onent is d ominan t and ignore any effects due to m ultiple-path prop agation lik e fading or inte r-symbol interference. 1 Optical comm unication is r estricted not only b y battery p ow er but also, for safet y reasons, by th e m axim um allo w ed p eak p o w er. W e therefore consider simultaneously ∗ A. Lapidoth and M. A. Wigger are with the Departmen t of Information T echnolog y and Electri- cal Engineering, Swiss F ed eral Institute of T echnology (ETH) in Zuric h, Switzerland; S . M. Moser is with the Department of Communication En gineering, National Chiao T un g Universit y (N CTU) in Hsinch u, T aiwan. The work of S . M. Moser was sup p orted in part by ETH und er TH-23 02-2 and in part by th e National Science Council, T aiw an, under NS C 96-2221-E-009-012-MY3. Presen ted in part at the Wintersc h o ol on Co ding and Information Theory 2005, Bratisla v a, Slov akia. 1 F or more details on th e channel mo del see Section 2. Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 1 t wo constrain ts: an av erage-p o wer constrain t E and a maxim um allo w ed p eak p ow er A . The s itu ation wh er e only a p eak-p o w er constraint is imp osed, corresp ond s to E = A . The case of only an av erage-p o wer constrain t is treated separately . The d escrib ed system is c alled the fr e e-sp ac e optic al intensity cha nnel and has previously b een studied in [1], [2], [3 ], [4], [5]. In [3] it has b een p ro ved that the capacit y-ac hieving probability measur e for this channel is discrete, and in [4], [5] upp er and lo wer b ound s on this c hannel’s capacit y hav e b een deriv ed. Related c h annel mo d els used to describ e optical comm u nication are the Poisson c hannel, see [6], [7], [8], [2] for the discrete-time c hannel and [9], [10], [11], [12], [13], [14], [15] for the cont inuous-time c hannel, and a v ariation of the free-space optical optical in tensit y wh ere the noise d ep ends on the in put [2, Chapter 4], [16]. In this w ork w e presen t new upp er and low er b ound s on the ca pacit y of the free-space optical intensit y channel and study the capacit y’s asymptotic b eha v ior at h igh a nd lo w p o wers. The m axim um gap b et wee n the u pp er an d lo wer boun ds nev er exceeds 1 nat w hen the ratio of the av erage-p o wer constrain t to the p eak-p ow er constrain t is larger than 0 . 03 or when only an av erage-p o wer constraint is imp osed. F or the case of a verag e-p o w er and p eak-p o we r constraint s, asymptotically when the a v ailable av erage and p eak p ow er tend to infinity with their ratio h eld fixed , the upp er and lo wer b ound coincide, i.e. , their d ifference tends to 0. When th e a v ailable a verag e and p eak p o wer tend to 0, with their ratio held fi xed, the ratio of the upp er and l o w er b ound tends to 1. F or the case o f o nly an a verag e-p o we r constraint the prop osed up p er and lo we r b ound coincide asym p totical ly for high p o we r, i.e . , their difference tends to 0 as th e p o w er tends to infinity . A t lo w pow er their ratio tends to 2 √ 2. The deriv ation of th e up p er b ounds is based on a general t ec h nique introdu ced in [17] using a du al expression of m u tual in formation. W e will not state it in its full generalit y but only in the fo rm needed in th is pap er. F or more details and f or a pro of see [17, Sec. V], [2 , Ch . 2]. Prop osition 1. Assume a memoryless channel with input alphab et X = R + 0 and output alphab et Y = R wher e c onditional on the input x ∈ X the distribution on the output Y is denote d by the pr ob ability me asur e W ( ·| x ) . 2 Then, for arbitr ary distribution R ( · ) over Y , the channel c ap acity und er a p e ak- p ower c onstr aint A and an aver age- p ower c onstr aint E is up p er-b ounde d by C ( A , E ) ≤ sup Q E Q  D  W ( ·| X )   R ( · )  , (1) wher e the supr emum is taken over al l pr ob ability laws Q on t he input X satisfying Q ( X > A ) = 0 and E Q [ X ] ≤ E . Her e, D ( ·k· ) stands for the r elative entr opy [18, Ch. 2]. Pr o of. See [17 , S ec. V].  There are t wo chall enges in using (1). The fir st is in finding a cle v er c hoice o f the la w R that will lead to a go o d upp er b ound. Th e second is in upp er-b ounding the 2 The p roposition requ ires certain measurability assumptions on th e law W ( ·|· ) which we omit for simplicity . Ho wev er, the channel law under consideration (see its density (9) ahead) satisfies these assumptions. See [17, S ec. V], [2, Ch. 2] for a description of the assumptions. Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 2 supremum on the righ t-hand side of (1). T o handle this sec ond c hallenge w e shall resort to some further b ounding, e.g. , Jens en’s in equalit y [18, Ch. 2.6]. T o deriv e th e lo we r b ounds we app ly t w o different tec hniques: one for the high- p o w er r egime, and the other for the lo w-p o wer regime. F or h igh p ow ers we use the en trop y p o wer inequalit y (see Lemma 16) and the theory of en tropy maximizing distributions [18, Ch. 11]. Asymptotically , the differences of these lo wer boun ds and so me of the upp er b ound s derive d using dualit y tend to 0 as the p o we r tends to in fi nit y , and thus the b ounds are t igh t at high pow er. A t lo w p ow ers w e low er- b ound capacit y considering binary input distributions; a c hoice wh ic h w as in spired b y [19] and [3]. In the cases inv olving a p eak-p ow er constrain t, the asymptotic b eha vior of the corresp ond ing mutual inform ation is s tu died usin g [20]. Wh en only an a v erage-p o wer constraint is imp osed, a low er b ound on the asymp totic b eha vior of the m u tual information is derived. In the cases inv olving a p eak-p o wer constraint the asymptotic expression of th e m utual information for bin ary in puts and some of the dualit y-based upp er b ound s are asymptotica lly tight at lo w p o we r, i.e. , their ratio tends to one as th e p o wer tends to 0. When on ly an a v erage-p o wer constraint is imp osed, th e d eriv ed low er b ound on the asymptotic expression of the m utual information is not tigh t w ith any o f the dualit y-based upp er b ounds. Indeed, th e ratio of the b est u pp er b ound with this low er b ound tends to 2 √ 2 when the a verage p o w er tends to 0. The rest of t he pap er is structured as follo w s . After some remarks ab out nota- tion at the end of th is sec tion, w e defin e the considered c hannel mo d el in d etail in the su b sequen t Section 2 . Section 3 conta ins some mathematical preliminaries. In Section 4 we state our main results, i.e. , the upp er and low er b ounds on c hannel capacit y an d th e asymptotic results. The detailed deriv ations of th e lo w er b ounds, the upp er b ound s, and the asym p totic results can b e foun d in Sections 5, 6, and 7 , resp ectiv ely . F or random qu an tities w e u se u pp ercase letters and for their realizations lo wer- case letters. Scalars are typica lly d enoted using Greek letters or lo w er-case Roman letters. A few exceptions are the follo wing symb ols: C stands fo r capacit y , D ( ·k· ) denotes the relativ e entrop y b et wee n t wo pr obabilit y measures, and I ( · ; · ) stands for the m utual information. Moreo v er, the capitals Q , W , and R denote probabilit y measures: • Q ( · ) denotes a generic probab ility measure on the channel input; • f or any input x ∈ X , W ( ·| x ) repr esents a probabilit y m easure on the c hannel output when the c hannel input is x ; • R ( · ) denotes a generic p r obabilit y measure on the c hannel outp u t. The exp ression I ( Q, W ) stands for the mutual information b et ween inpu t X and output Y of a c hannel with tr an s ition p robabilit y measure W when t he input h as distribution Q , i.e. , I ( Q, W ) , I ( X ; Y ) . (2) The symb ol E denotes a verag e p o wer and A stands for p eak p o w er. W e denote the mean- η , v ariance- σ 2 real Gaussian distribu tion by N R  η , σ 2  . All rates sp ecified in this pap er are in nats p er c hannel-use, and all logarithms are natural logarithms. Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 3 2 Channel Mo del 2.1 Ph ysical Description In free-space optical communicatio n the input signal u sually is transmitted by means of li gh t emitting dio des (LED) or laser diod es (LD). Con ve nti onal and most inex- p ensive dio des emit infrared light of wa v elength b et wee n 85 0 and 95 0 nanometers. F or such high frequencies, practical systems often apply int ensit y mo dulation where the trans mitter mo dulates th e optical intensit y of the emitted light , a nd hence the input signal is pr op ortional to the optical intensit y . Th e receiv er first measures the inciden t optical intensit y of the incoming ligh t by means of a f ron t-end ph otod etec- tor a nd pro duces an output signal wh ic h is p rop ortional t o the d etected intensit y . Based on this output signal the r eceiv er deco des the transmitted data. F or our mo del we neglect the impact of fading or inter-sym b ol interference due to m ultiple-path prop agation and assume that the direct line-of-sigh t path is d ominan t. In t he a bsence of a protectiv e medium li k e, e.g. , a fib er cable, the d omin an t noise source is assumed to b e strong ambien t ligh t. Even if optical filters are app lied to reduce the impact of this noise, it typicall y has m uc h larger p o we r than the actual signal and causes high-intensit y shot noise in the outp u t signal. In a first appro ximation this sh ot n oise can b e assu med to b e additiv e and indep endent of th e signal itself. The maxim u m allo w ed optica l p eak p ow er of the transmitted signal has to b e constrained, e.g. , to guaran tee eye safet y . Moreo ver, to increase batte ry life-time, w e also co nstrain the allo we d optical a v erage p ow er. Since the optical p ow er is prop ortional to the optical in tensit y , and the in p ut signal mo dulates the optical in tensit y , in the describ ed system the optica l p o wer is prop ortional to the inpu t signal. Th us, the constraints on the optical p eak and a ve rage p o wer ha v e to b e transformed into p eak and a v erage co nstraints on the inpu t signal (and not on its square as u s ual in radio comm unication). F or a more detailed description of th e free-space optical int ensit y c h annel see [21]. 2.2 Mathematical Channel Model W e w ill no w translate the ab o ve physical c hann el description int o a sim p lified time- discrete c hannel mo d el. The time- k c hannel ou tp ut ˜ Y k is giv en by ˜ Y k = x k + ˜ Z k . (3) Here, x k denotes the time- k c h annel inp ut and the rand om pro cess { ˜ Z k } mo dels the additiv e noise. As describ ed ab ov e this noise is mainly caused b y strong ambien t ligh t. W e therefore approximat e it as a constan t intensit y term η and some in tensity- fluctuations around η . Because these fluctuations are caused by man y ind ep enden t sources, we assume that they are indep end en t and id entical ly distrib uted (I ID) zero- mean Gauss ian with a giv en v ariance σ 2 > 0. I.e. , { ˜ Z k } ∼ I ID N R  η , σ 2  . (4) Since η is constan t, we ma y without loss of generalit y neglect it b ecause th e receiv er can alw a ys subtr act or add any constant signal. W e th en define a new c hannel output random v ariable Y k = x k + Z k , (5) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 4 where { Z k } ∼ I ID N R  0 , σ 2  . Notice that the new c hannel outp uts { Y k } represent the fluctuations of the electrical outp ut s ignal arou n d its w orking p oin t η . Our c hannel mo del is memoryless and therefore w e d rop the time-index k . The c h annel output Y is then give n by Y = x + Z , (6) where x den otes the channel inp ut that is prop ortional to the optical intensit y and therefore cannot b e negativ e, x ∈ R + 0 , (7) and where the additiv e noise is Z ∼ N R  0 , σ 2  . (8) Hence, th e conditional p robabilit y l a w W ( ·| x ) of the output Y giv en i npu t x ∈ R + 0 has densit y f Y | X ( y | x ) = 1 √ 2 π σ 2 e − ( y − x ) 2 2 σ 2 , x ∈ R + 0 , y ∈ R . (9) It is important to note that, unlike the input, th e output Y ma y b e negativ e since the noise in tro duced at the receiv er can b e negativ e. The restrictions on the optical p eak and a v erage p o w er are translated in to a p eak-p o w er and an av erage-p o wer constrain t on the inpu t, resp ectiv ely: Pr[ X > A ] = 0 , (10) E [ X ] ≤ E , (11) for some fixed parameters E , A > 0. No te that th e a v erage-p o wer constrain t is on the exp ectation of the c hann el input an d not on its square. W e denote the ratio b et w een a v erage p o w er and p eak p ow er by α , α , E A , (12) where 0 < α ≤ 1. Note that for α = 1 th e a v erage-p o wer constrain t is in activ e, in the sense that it has no influence on the capacit y and is automatically satisfied. This means that α = 1 corresp onds to the case with only a p eak-p ow er constrain t. Similarly , α ≪ 1 corresp onds to a d ominan t a verag e-p o w er constrain t and only a v ery wea k p eak-p o wer constraint. W e denote the capacit y of the described c h annel with p eak-p ow er constraint A and a verage- p ow er constraint E b y C ( A , E ). The capacit y is giv en by [22] C ( A , E ) = sup Q I ( Q, W ) (13) where the suprem um is ov er all la ws Q on X ≥ 0 satisfying Q ( X > A ) = 0 and E Q [ X ] ≤ E . When only an a verage- p ow er constrain t is imp osed, capacit y is denoted by C ( E ). It is giv en as i n (13) except that the sup rem um is tak en ov er a ll laws Q o n X ≥ 0 satisfying E Q [ X ] ≤ E . Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 5 3 The Q -F unction Definition 2. The Q -function is define d b y Q ( ξ ) , Z ∞ ξ 1 √ 2 π e − t 2 2 d t, ∀ ξ ∈ R . (14) Some of the prop erties of this fun ction are r ecalled in the follo wing lemma. Lemma 3. The Q -function satisfies Q ( − ξ ) + Q ( ξ ) = 1 , ∀ ξ ∈ R , (15) and Q ( 0) = 1 2 , (16) and is b ounde d by 1 √ 2 π ξ e − ξ 2 2  1 − 1 ξ 2  < Q ( ξ ) < 1 √ 2 π ξ e − ξ 2 2 , ξ > 0 , (17) and Q ( ξ ) ≤ 1 2 e − ξ 2 2 , ξ ≥ 0 . (18) Its first and se c ond derivatives ar e given by Q ′ ( ξ ) = − 1 √ 2 π e − ξ 2 2 , ∀ ξ ∈ R , (19) and Q ′′ ( ξ ) = ξ √ 2 π e − ξ 2 2 , ∀ ξ ∈ R . (20) Thus, Q ( · ) is monotonic al ly strictly de cr e asing for al l ξ ∈ R , strictly c onc ave over ( −∞ , 0) , and strictly c onvex over (0 , ∞ ) . Pr o of. The pro of of (15 ) and (16) follo ws b ecause 1 √ 2 π e − ξ 2 2 is s ymmetric around 0, and b ecause it equals the density of a standard Gaussian rand om v ariable and hence in tegrates to 1. F or a pro of of the b ou n ds (17) and (18) s ee [23, pp. 83–84 ].  Lemma 4. L et ξ 0 , γ ≥ 0 b e nonne gative c onstants, and let the function f ( · ) b e define d as f ( ξ ) , 1 − Q ( ξ 0 + ξ ) − Q ( ξ 0 + γ − ξ ) , ξ ∈ [0 , γ ] . (21) Then f ( · ) is strictly c onc ave over [0 , γ ] and symmetric ar ound ξ = γ 2 . F urthermor e, it is incr e asing over  0 , γ 2  , de cr e asing over  γ 2 , γ  , and takes on its maximum value at ξ = γ 2 . Pr o of. See App endix A.  Lemma 5. F or µ ≥ 1 √ e and ξ ≥ 0 : ξ Q ( ξ − µ ) ≤ µ. (22) Pr o of. See App endix B .  Lemma 6. F or µ, ξ ≥ 0 : 1 − Q ( ξ − µ ) ≤ 1 − Q ( − µ ) + ξ µ Q ( − µ ) . (23) Pr o of. See App endix C .  Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 6 4 Results The results of this p ap er are partially b ased on the results in [24] and [2 , Ch . 3]. Lemma 7. Given p e ak-p ower c onstr aint A and aver age-p ower c onstr aint E the c a- p acity C ( A , E ) of the f r e e- sp ac e optic al intensity channel (6) has a u ni q ue input distribution Q ∗ that achieves the su pr emum in (13) . Pr o of. See [3].  Using this lemma together with the symmetry of the channel la w ((6) a nd (8)) and the conca vit y of m utual information in the inp ut distrib ution, the f ollo wing lemma can b e prov ed. Lemma 8. If the al lowe d aver age p ower E is lar ger than half the al lowe d p e ak p ower A , then the optimal input distribution Q ∗ in (13) satisfies E Q ∗ [ X ] = 1 2 A . (24) Thus, C ( A , α A ) = C  A , A 2  , 1 2 < α ≤ 1 . (25) Pr o of. See App endix D .  T o state our results we distin gu ish b et wee n th r ee differen t cases: • C ase I: b oth an a v erage-p o wer and a p eak-p o w er constrain t are imp osed, with α ∈  0 , 1 2  ; • C ase I I : b oth an a v erage- and a p eak-p o w er constrain t are imp osed, w ith α ∈  1 2 , 1  ; • C ase I I I: only an a verag e-p o w er constraint is imp osed. W e present firm u pp er and lo we r b ounds on the c hann el capacit y in all three cases. In all three cases their gap tends to 0 as th e av ailable p ow er tend s to infinity , and th u s, we can deriv e the asymptotic capacit y at high p ow er. W e also pr esen t the asymptotics of capacit y at low p o wer. F or case I and I I we state them exactly , i. e . , w e present asymptotic u pp er and lo w er b oun ds w hose ratio tends to 1 as the p o wer tends to 0. F or case I I I we presen t asymptotic up p er and lo wer b ounds whose r atio tends to 2 √ 2 as the p ow er tends to 0. 4.1 Bounds on Channel Capacit y for Case I Theorem 9. If 0 < α < 1 2 , then C ( A , α A ) is lower-b ounde d by C ( A , α A ) ≥ 1 2 log 1 + A 2 e 2 αµ ∗ 2 π eσ 2  1 − e − µ ∗ µ ∗  2 ! , (26) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 7 and upp e r- b ounde d by e ach of the two b ounds C ( A , α A ) ≤ 1 2 log  1 + α (1 − α ) A 2 σ 2  , (27) C ( A , α A ) ≤  1 − Q  δ + α A σ  − Q  δ + (1 − α ) A σ  · log A σ · e µδ A − e − µ ( 1+ δ A ) √ 2 π µ  1 − 2 Q  δ σ  ! − 1 2 + Q  δ σ  + δ √ 2 π σ e − δ 2 2 σ 2 + σ A µ √ 2 π  e − δ 2 2 σ 2 − e − ( A + δ ) 2 2 σ 2  + µα 1 − 2 Q δ + A 2 σ !! . (28) Her e, µ > 0 and δ > 0 ar e fr e e p ar ameters, and µ ∗ is the unique solution to α = 1 µ ∗ − e − µ ∗ 1 − e − µ ∗ . (29) The existence and un iqueness of a solution to (29) is guarantee d b y the follo wing lemma. Lemma 10. L et ϕ b e a function fr om the p ositive r e als to the op en interval  0 , 1 2  ϕ : µ 7→ 1 µ − e − µ 1 − e − µ . (30) Then, ϕ is monotonic al ly strictly de cr e asing and bi je ctive with the fol lowing limiting b ehavior: lim µ ↑∞ ϕ ( µ ) = 0 , (31) lim µ ↓ 0 ϕ ( µ ) = 1 2 , (32) lim µ ↑∞ ( µ · ϕ ( µ )) = 1 . (33) Pr o of. See App endix E.  A sub optimal but useful choic e f or the free parameters in up p er b ound (28) is δ = σ log  1 + A σ  , (34) µ = µ ∗  1 − e − α δ 2 2 σ 2  , (35) where µ ∗ is the solution to (29 ). Figures 1 and 2 depict the b ounds of Theorem 9 for α = 0 . 1 and 0 . 4, where (28) is numerically minimized o v er δ, µ > 0. Theorem 11 . If α lies in  0 , 1 2  , then lim A ↑∞  C ( A , α A ) − log A σ  = − 1 2 log 2 π e − (1 − α ) µ ∗ − log(1 − αµ ∗ ) (36) and lim A ↓ 0 C ( A , α A ) A 2 /σ 2 = α (1 − α ) 2 . (37) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 8 −10 −5 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 P S f r a g r e p l a c e m e n t s A σ [dB] C [nats p er c hannel u se] low er boun d (26), α =0 . 1 upp er b ound (27) , α =0 . 1 upp er b ound (28) , α =0 . 1 −30 −25 −20 −15 −10 −5 0 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P S f r a g r e p l a c e m e n t s A σ [dB] C [nats p er c hannel u se] low er boun d (26), α =0 . 1 upp er b ound (27) , α =0 . 1 upp er b ound (28) , α =0 . 1 Figure 1: Bounds of T heorem 9 f or α = 0 . 1 when upp er b ound (28) is n umerically minimized ov er δ, µ > 0. Th e maximum gap b et we en upp er and lo wer b ound is 0.68 nats (for A σ ≈ 10 . 5 dB). Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 9 −10 −5 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 P S f r a g r e p l a c e m e n t s A σ [dB] C [nats p er c hannel u se] low er boun d (26), α =0 . 4 upp er b ound (27) , α =0 . 4 upp er b ound (28) , α =0 . 4 −30 −25 −20 −15 −10 −5 0 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P S f r a g r e p l a c e m e n t s A σ [dB] C [nats p er c hannel u se] low er boun d (26), α =0 . 4 upp er b ound (27) , α =0 . 4 upp er b ound (28) , α =0 . 4 Figure 2: Bounds of Theorem 9 for α = 0 . 4 with numerically optimized upp er b ound (28). The maxim um gap b et ween upp er and lo w er b oun d is 0.52 nats (for A σ ≈ 6 . 4 d B). Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 10 4.2 Bounds on Channel Capacit y for Case I I Theorem 12 . If α ∈  1 2 , 1  , then C ( A , α A ) is lower-b ounde d by C ( A , α A ) ≥ 1 2 log  1 + A 2 2 π eσ 2  , (38) and is upp er-b ounde d by e ach of the two b ounds C ( A , α A ) ≤ 1 2 log  1 + A 2 4 σ 2  , (39) C ( A , α A ) ≤ 1 − 2 Q δ + A 2 σ !! log A + 2 δ σ √ 2 π  1 − 2 Q  δ σ  − 1 2 + Q  δ σ  + δ √ 2 π σ e − δ 2 2 σ 2 , (40) wher e δ > 0 is a fr e e p ar ameter. A useful but sub optimal choic e f or δ is δ = σ log  1 + A σ  . (41) Figure 3 depicts the b ound s of Theorem 12, where u pp er b oun d (40) is numerically minimized o ver δ > 0. Theorem 13 . If α lies in  1 2 , 1  , then lim A ↑∞  C ( A , α A ) − log A σ  = − 1 2 log 2 π e (42) and lim A ↓ 0 C ( A , α A ) A 2 /σ 2 = 1 8 . (43) Note that ( 42) an d (43) exhibit the w ell-kno wn asymptotic b eha vior of the ca- pacit y of a Gaussian channel un der a p eak-p o wer constraint only [22]. Based on the right- hand sid es of (36) and (42) we d efine χ ( α ) , ( − 1 2 log 2 π e − (1 − α ) µ ∗ − log(1 − αµ ∗ ) , 0 < α < 1 2 , − 1 2 log 2 π e, 1 2 ≤ α ≤ 1 . (44) Th us for α ∈ (0 , 1), χ ( α ) represents t he second term in the high SNR a symptotic expansion of t he c h annel capacit y C ( A , α A ). It is d epicted in Figure 4. Note that when α tends to 0, th en χ ( α ) tends to −∞ . Th is can b e seen by rewriting χ ( α ) for α ∈  0 , 1 2  using (29 ) as χ ( α ) = − 1 2 log 2 π e − αµ ∗ − log µ ∗ 1 − e − µ ∗ , α ∈  0 , 1 2  , (45) and then noting that by Lemma 10 (in particular b y (31) and (33)) µ ∗ ↑ ∞ and αµ ∗ ↑ 1 w h en α ↓ 0. Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 11 −20 −15 −10 −5 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 P S f r a g r e p l a c e m e n t s A σ [dB] C [nats p er c hannel u se] low er boun d (38) upp er b ound (39) upp er b ound (40) −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 P S f r a g r e p l a c e m e n t s A σ [dB] C [nats p er c hannel u se] low er boun d (38) upp er b ound (39) upp er b ound (40) Figure 3: Bounds on capacit y for α ∈  1 2 , 1  according to Theorem 12, where upp er b ound (40) is numerically minimized o ver δ > 0. Th e maximum gap b et w een up p er and lo wer b ound is 0.50 nats (for A σ ≈ 6 . 4 d B). Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 P S f r a g r e p l a c e m e n t s α χ [nats p er channel use] Figure 4: The term χ ( α ) for α ∈ (0 , 1]. 4.3 Bounds on Channel Capacit y for Case I I I Theorem 14 . In the absenc e of a p e ak-p ower c onstr aint th e channel c ap acity C ( E ) is lower-b ounde d by C ( E ) ≥ 1 2 log  1 + E 2 e 2 π σ 2  , (46) and is upp er-b ounde d by e ach of the b ounds C ( E ) ≤ log  β e − δ 2 2 σ 2 + √ 2 π σ Q  δ σ  − log  √ 2 π σ  − δ E 2 σ 2 + δ 2 2 σ 2  1 − Q  δ σ  − E δ Q  δ σ  + 1 β  E + σ √ 2 π  , δ ≤ − σ √ e , (47 ) C ( E ) ≤ log  β e − δ 2 2 σ 2 + √ 2 π σ Q  δ σ  + 1 2 Q  δ σ  + δ 2 √ 2 π σ e − δ 2 2 σ 2 + δ 2 2 σ 2  1 − Q  δ + E σ  + 1 β  δ + E + σ √ 2 π e − δ 2 2 σ 2  − 1 2 log 2 π eσ 2 , δ ≥ 0 , (48) wher e β > 0 and δ ar e fr e e p ar ameters. Bound (47) only holds for δ ≤ − σ e − 1 2 , while b ound (48) only holds for δ ≥ 0 . A sub optimal but useful choic e f or the free parameters in b ou n d (47) is δ = − 2 σ r log σ E , for E σ ≤ e − 1 4 e ≈ − 0 . 4 dB , ( 49) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 13 β = 1 2  E + σ √ 2 π  + 1 2 s  E + σ √ 2 π  2 + 4  E + σ √ 2 π  √ 2 π σ e δ 2 2 σ 2 Q  δ σ  , (50) and for the free parameters in b ound (48) is δ = σ log  1 + E σ  , (51) β = 1 2  δ + E + σ √ 2 π e − δ 2 2 σ 2  + 1 2 s  δ + E + σ √ 2 π e − δ 2 2 σ 2  2 + 4  δ + E + σ √ 2 π e − δ 2 2 σ 2  √ 2 π σ e δ 2 2 σ 2 Q  δ σ  . (52 ) Figure 5 dep icts the b ounds of T heorem 14 when the upp er b ound s (47) and (48) are numerical ly minim ized ov er the allo wed v alues of β and δ . Theorem 15 . In the c ase of only an aver age-p ower c onstr aint, lim E ↑∞  C ( E ) − log E σ  = 1 2 log e 2 π (53) and lim E ↓ 0 C ( E ) E σ p log σ E ≤ 2 , (54) lim E ↓ 0 C ( E ) E σ p log σ E ≥ 1 √ 2 . (55) Note that the asymptotic up p er and lo we r b ound at lo w SNR do not coincide in the sense that their ratio equals 2 √ 2 ins tead of 1. 5 Deriv ation of the Firm Lo w er Bounds T o deriv e the lo wer b ounds in Section 4 we u se the en trop y p o wer inequalit y . Lemma 16 (Entrop y P o wer I nequalit y). If X and Z ar e indep endent r andom variables with densities, then e 2 h ( X + Z ) ≥ e 2 h ( X ) + e 2 h ( Z ) . ( 56) Pr o of. See [18 , T h eorem 17.7.3].  One can find a low er b ound on capacit y by dropping the maximization and c h o osing an arbitrary inpu t d istribution Q in (13 ). Ho wev er, in order to get a tigh t b ound , this c h oice of Q sh ould yield a m utual information that is reasonably close to capacit y . S u c h a c hoice is difficult to find and might mak e the ev aluation of I ( Q, W ) in tractable, b ecause already for relativ ely “easy” distribu tions Q the corresp ondin g m utual inform ation is d ifficult to compute. W e circum v en t these p roblems by u s ing Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 14 −30 −25 −20 −15 −10 −5 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 P S f r a g r e p l a c e m e n t s E σ [dB] C [nats p er c hannel u se] low er boun d (46) upp er b ound (47) upp er b ound (48) −40 −35 −30 −25 −20 −15 −10 −5 0 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 P S f r a g r e p l a c e m e n t s E σ [dB] C [nats p er c hannel u se] low er boun d (46) upp er b ound (47) upp er b ound (48) Figure 5: Bounds on capacit y ac cording to Th eorem 14 when up p er b oun ds (47) and (48) are n umerically minimized o ver allo wed v alues of β , δ . Th e maximum gap b et w een up p er and lo we r b oun d is 0.57 nats (for E σ ≈ 2 . 8 d B). Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 15 the ent ropy p o wer inequalit y (5 6). F or any p robabilit y distribution Q with d en s it y C ≥ I ( Q, W ) (57) = h ( Y ) − h ( Y | X ) (58) = h ( X + Z ) − h ( Z ) (59) ≥ 1 2 log  e 2 h ( X ) + e 2 h ( Z )  − h ( Z ) (60) = 1 2 log 1 + e 2 h ( X ) 2 π eσ 2 ! , (6 1) where (60) follo ws from Lemma 16. T o mak e this lo w er boun d as tight as p ossible w e w ill c ho ose a distribu tion Q that maximizes differen tial en trop y u nder the giv en constrain ts [18, Ch. 12]. 5.1 Lo wer Bound (26) of Theorem 9 The inp ut distribution Q 1 that maximizes different ial en trop y und er a nonnegativit y constrain t, a p eak constraint , and an av erage constrain t has the d ensit y [18 ] 1 A · µ ∗ 1 − e − µ ∗ e − µ ∗ x A , 0 ≤ x ≤ A , (62) where µ ∗ has to b e c hosen suc h th at the a v erage-p o wer c onstraint is satisfied, i.e. , µ ∗ is giv en as the solution to α = 1 µ ∗ − e − µ ∗ 1 − e − µ ∗ . (63) By Lemma 10 suc h a solution alw a ys exists for 0 < α < 1 2 and is u nique. T he b oun d (26) no w follo ws fr om (61) by compu ting h ( X ) u nder the probability la w Q 1 . 5.2 Lo wer Bound (38) of Theorem 12 The uniform distrib u tion o v er [0 , A ] maximizes differentia l en trop y u nder a nonneg- ativit y and a p eak constrain t [18 ]. W e c ho ose Q 2 to b e this un iform distrib ution and note that then E Q 2 [ X ] = A 2 , and hence Q 2 satisfies any a verag e-p o w er constraint larger than A 2 . Low er b ound (38) follo ws directly from (61) by compu ting h ( X ) under the la w Q 2 . Notice that the uniform distrib ution Q 2 represent s the limit of the input d istr i- bution Q 1 in S ection 5.1 when α ↑ 1 2 . Indeed, b y Le mma 1 0, and in p articular by the limit (32), lim α ↑ 1 2 µ ∗ = 0 , (64) and hence when α ↑ 1 2 the density of Q 1 con verges p oin twise to the uniform d ensit y o ver [0 , A ]. Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 16 5.3 Lo wer Bound (46) of Theorem 14 The distribu tion Q 3 that maximizes differenti al en trop y un der a nonn egativit y con- strain t and an exp ectation constraint is the exp onential density [18] 1 E e − x E , x ≥ 0 . (65) The b ound (46) follo w s fr om (61) by compu ting h ( X ) u nder the law Q 3 . Note that t he den s it y of Q 3 represent s the p oin twise limit of the densit y of Q 1 when α ↓ 0. Ind eed, by Lemma 10, and in particular b y (31) and (33), lim α ↓ 0 µ ∗ = ∞ , (66) lim α ↓ 0 ( αµ ∗ ) = 1 . (67) Also, using α = E A w e can rewrite the d ensit y of Q 1 , i.e. , (62), as αµ ∗ E · 1 1 − e − µ ∗ e − αµ ∗ E x , 0 ≤ x ≤ E α , (68) whic h by (66) and (67) tends to (65) wh en α ↓ 0. 6 Deriv ation of the Firm Upp er Bounds The deriv ation of the up p er b ou n ds in S ection 4 is based on Prop osition 1: C ≤ sup Q E Q  D  W ( ·| X )   R ( · )  (69) = sup Q E Q  − Z ∞ −∞ log d R ( y ) d W ( y | X )  − 1 2 log 2 π eσ 2 . (70) Hence, w e need to specify a distribution R , ev aluate the in tegral (70), and finally upp er-b ound the supr em u m in (70) . T o upp er-b ound the su prem um w e shall present upp er b ound s on E Q h − R ∞ −∞ log d R ( y ) d W ( y | X ) i whic h hold for arb itrary input laws Q satisfying the imp osed p ow er constraints. 6.1 Upp er Bound (27) of Theorem 9 T o d er ive th e first upp er b ound (27) w e c ho ose an output distribution R 1 corre- sp ondin g to a Gaussian r an d om v ariable of mean E and of v ariance ( σ 2 + ( A − E ) E ), i.e. , R 1 has densit y: f 1 ( y ) = 1 p 2 π ( σ 2 + E ( A − E )) e − ( y −E ) 2 2 σ 2 +2 E ( A −E ) , y ∈ R . (71) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 17 F or arbitrary la w Q satisfying E Q [ X ] ≤ E and Q ( X > A ) = 0 this yields E Q  − Z ∞ −∞ log f 1 ( y ) d W ( y | X )  = log q 2 π  σ 2 + E ( A − E )  + E Q  X 2 + σ 2 − 2 E X + E 2 2 σ 2 + 2 E ( A − E )  (72) ≤ log q 2 π  σ 2 + E ( A − E )  + E Q  ( A − 2 E ) X + σ 2 + E 2 2 σ 2 + 2 E ( A − E )  (73) ≤ log q 2 π  σ 2 + E ( A − E )  + ( A − 2 E ) E + σ 2 + E 2 2 σ 2 + 2 E ( A − E ) (74) = 1 2 log 2 π e  σ 2 + E ( A − E )  , (75) where the first inequalit y follo ws fr om X 2 ≤ A X du e to the p eak-p o we r constraint, and where the second inequ alit y follo ws fr om the a v erage-p ow er constrain t usin g that E A = α ≤ 1 2 , i. e. , A − 2 E ≥ 0. Since the r esu lting up p er b oun d in (75) do es not dep end on the inp ut law Q , (75) also u p p er-b oun ds the su p rem um in (70), and (27) is p ro ved. 6.2 Upp er Bound (28) of Theorem 9 T o deriv e (28) we choose the la w on the ou tp ut R 2 to ha v e d en sit y: f 2 ( y ) =                1 √ 2 π σ e − y 2 2 σ 2 , y < − δ, 1 A · µ ( 1 − 2 Q ( δ σ )) e µδ A − e − µ ( 1+ δ A ) e − µy A , − δ ≤ y ≤ A + δ , 1 √ 2 π σ e − ( y − A ) 2 2 σ 2 , y > A + δ , (76) where δ > 0 and µ > 0 are f ree parameters. This leads to the follo win g expression E Q  − Z ∞ −∞ log R ′ 2 ( y ) d W ( y | X )  = E Q  Z − δ −∞ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log  √ 2 π σ e y 2 2 σ 2  d y  | {z } c 1 − E Q " Z A + δ − δ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log e − µy A A µ  1 − 2 Q  δ σ  e µδ A − e − µ ( 1+ δ A ) ! d y # | {z } c 2 + E Q  Z ∞ A + δ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log  √ 2 π σ e ( y − A ) 2 2 σ 2  d y  | {z } c 3 . (77) W e in v estigate eac h t erm individu ally . W e start with c 1 : c 1 = E Q  Z − δ −∞ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log  √ 2 π σ e y 2 2 σ 2  d y  (78) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 18 = E Q  log  √ 2 π σ  · Q  δ + X σ  + 1 2 σ 2 Z − δ −∞ y 2 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 d y  (79) ≤ E Q  log  √ 2 π σ  · Q  δ + X σ  + 1 2 σ 2 Z − δ −∞ y 2 1 √ 2 π σ e − y 2 2 σ 2 d y (80) = E Q  log  √ 2 π σ  · Q  δ + X σ  + 1 2 Q  δ σ  + δ 2 √ 2 π σ e − δ 2 2 σ 2 , (81) where the inequalit y f ollo ws from the assum p tion δ > 0 that ensur es that ( y − x ) 2 ≥ y 2 for all x ≥ 0 an d y ≤ − δ . S imilarly w e get for c 3 : c 3 = E Q  Z ∞ A + δ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log  √ 2 π σ e ( y − A ) 2 2 σ 2  d y  (82) = E Q  log  √ 2 π σ  · Q  δ + A − X σ  + E Q  1 2 σ 2 Z ∞ A + δ ( y − A ) 2 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 d y  (83) ≤ E Q  log  √ 2 π σ  · Q  δ + A − X σ  + 1 2 σ 2 Z ∞ A + δ ( y − A ) 2 1 √ 2 π σ e − ( y − A ) 2 2 σ 2 d y (84) = E Q  log  √ 2 π σ  · Q  δ + A − X σ  + 1 2 Q  δ σ  + δ 2 √ 2 π σ e − δ 2 2 σ 2 . (85) Here, th e inequalit y follo ws b ecause ( y − x ) 2 ≥ ( y − A ) 2 for all x ≤ A and y ≥ A + δ . Finally , for c 2 w e ha v e c 2 = E Q " Z A + δ − δ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log e µδ A − e − µ ( 1+ δ A ) 1 − 2 Q  δ σ  A µ e µy A ! d y # (86) = E Q "  1 − Q  δ + X σ  − Q  δ + A − X σ  log e µδ A − e − µ ( 1+ δ A ) 1 − 2 Q  δ σ  A µ !# + E Q  1 − Q  δ + X σ  − Q  δ + A − X σ  µ A X  + E Q  µσ A √ 2 π  e − ( δ + X ) 2 2 σ 2 − e − ( A + δ − X ) 2 2 σ 2  . (87) Plugging c 1 , c 2 , and c 3 in to (77 ) and combining th is with (70) w e get the f ollo wing b ound : C ≤ Q  δ σ  + δ √ 2 π σ e − δ 2 2 σ 2 − 1 2 + E Q    1 − Q  δ + X σ  − Q  δ + A − X σ  log   A  e µδ A − e − µ ( 1+ δ A )  √ 2 π σ µ  1 − 2 Q  δ σ      + E Q  1 − Q  δ + X σ  − Q  δ + A − X σ  µ A X  + E Q  µσ A √ 2 π  e − ( δ + X ) 2 2 σ 2 − e − ( A + δ − X ) 2 2 σ 2  . ( 88) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 19 It is sho wn in App endix F that log   A  e µδ A − e − µ ( 1+ δ A )  √ 2 π σ µ  1 − 2 Q  δ σ    ≥ 0 (8 9) for an y v alues of A , σ, δ, µ > 0. Therefore w e ma y use Jensen’s inequalit y com bin ed with the conca vity sh o wn in Lemma 4 to conclude that E Q    1 − Q  δ + X σ  − Q  δ + A − X σ  log   A  e µδ A − e − µ ( 1+ δ A )  √ 2 π σ µ  1 − 2 Q  δ σ      ≤  1 − Q  δ + E Q [ X ] σ  − Q  δ + A − E Q [ X ] σ  log   A  e µδ A − e − µ ( 1+ δ A )  √ 2 π σ µ  1 − 2 Q  δ σ    (90) ≤  1 − Q  δ + α A σ  − Q  δ + (1 − α ) A σ  log   A  e µδ A − e − µ ( 1+ δ A )  √ 2 π σ µ  1 − 2 Q  δ σ    . (91) Here, the last inequalit y follo ws from th e av erage-p o wer constraint, the assu mption that α ≤ 1 2 , and from the fact shown in Lemma 4 that ξ 7→  1 − Q  δ + ξ σ  − Q  δ + A − ξ σ  is m onotonicall y increasing f or 0 ≤ ξ ≤ A 2 . Next, w e once more use Lemma 4 to upp er-b ound  1 − Q  δ + x σ  − Q  δ + A − x σ  b y its maximum v alue that is tak en on for x = A 2 : E Q  1 − Q  δ + X σ  − Q  δ + A − X σ  µ A X  ≤ E Q " 1 − Q δ + A 2 σ ! − Q δ + A − A 2 σ !! µ A X # (92) ≤ 1 − Q δ + A 2 σ ! − Q δ + A 2 σ !! µ A α A (93) = µα 1 − 2 Q δ + A 2 σ !! . (94) And finally , we u se the mo notonicit y of the exp onen tial fu nction and the fact that X ∈ [0 , A ] to show the follo wing: E Q  µσ A √ 2 π  e − ( δ + X ) 2 2 σ 2 − e − ( A + δ − X ) 2 2 σ 2  ≤ µσ A √ 2 π  e − δ 2 2 σ 2 − e − ( A + δ ) 2 2 σ 2  . (95) Finally , com bin ing (88) with (91), (94), and (95) yie lds the b ound on c hannel ca- pacit y giv en in (28). Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 20 6.3 Upp er Bound (39) of Theorem 12 T o deriv e b oun d (39) we c ho ose a Gaussian outpu t law R 3 with densit y f 3 ( y ) = 1 r 2 π  σ 2 + A 2 4  e − ( y − A 2 ) 2 2 σ 2 + A 2 2 . (96) This yields E Q  − Z ∞ −∞ log R ′ 3 ( y ) d W ( y | X )  = log s 2 π  σ 2 + A 2 4  + E Q " X 2 + σ 2 − A X + A 2 4 2 σ 2 + A 2 2 # (97) ≤ log s 2 π  σ 2 + A 2 4  + σ 2 + A 2 4 2 σ 2 + A 2 2 (98) = 1 2 log 2 π eσ 2  1 + A 2 4 σ 2  , (99) where the inequality follo ws b ecause X 2 ≤ A X due to the peak-p ow er constrain t. Com bined with (70) this yields the claimed resu lt. Note that the relation E Q [ X ] ≤ α A has not b een used. Therefore this b oun d is v alid for all α ∈ [0 , 1] and esp ecially for all α ∈  1 2 , 1  . 6.4 Upp er Bound (40) of Theorem 12 The deriv ation of this b oun d is similar to th e deriv ation of (28). W e choose an output distribu tion R 4 with dens it y: f 4 ( y ) =            1 √ 2 π σ e − y 2 2 σ 2 , y < − δ, 1 − 2 Q ( δ σ ) A +2 δ , − δ ≤ y ≤ A + δ , 1 √ 2 π σ e − ( y − A ) 2 2 σ 2 , y > A + δ , (100) where δ > 0 is a f r ee parameter. This leads to the follo wing expression: E Q  − Z ∞ −∞ log R ′ 4 ( y ) d W ( y | X )  = E Q  Z − δ −∞ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log  √ 2 π σ e y 2 2 σ 2  d y  | {z } c 1 + E Q " Z A + δ − δ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log A + 2 δ 1 − 2 Q  δ σ  d y # | {z } ˜ c 2 + E Q  Z ∞ A + δ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log  √ 2 π σ e ( y − A ) 2 2 σ 2  d y  | {z } c 3 . (101) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 21 W e hav e already upp er-b ound ed c 1 and c 3 in (81) and (85) (assuming that δ > 0). Similarly to c 2 , we compute ˜ c 2 as follo ws: ˜ c 2 = E Q " Z A + δ − δ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 log A + 2 δ 1 − 2 Q  δ σ  d y # (102) = E Q "  1 − Q  δ + X σ  − Q  δ + A − X σ  log A + 2 δ 1 − 2 Q  δ σ  # . (103 ) Plugging c 1 , ˜ c 2 , and c 3 in to (101) and com bining th is w ith (70 ) we get C ≤ E Q "  1 − Q  δ + X σ  − Q  δ + A − X σ  log A + 2 δ √ 2 π σ  1 − 2 Q  δ σ  # − 1 2 + Q  δ σ  + δ √ 2 π σ e − δ 2 2 σ 2 . (104) F or an y A > 0 and δ > 0 A + 2 δ √ 2 π σ  1 − 2 Q  δ σ  ≥ 2 δ σ √ 2 π  1 − 2 Q  δ σ  ≥ 1 , (1 05) where the fi rst inequalit y follo ws from d ropping A and the second inequ ality is pr o ven in App end ix F, Equation (273). Hence, log A + 2 δ √ 2 π σ  1 − 2 Q  δ σ  ≥ 0 , (106) and we can u se Lemma 4 to up p er-b oun d  1 − Q  δ + x σ  − Q  δ + A − x σ  b y its maxi- m um v alue that is take n on for x = A 2 : E Q "  1 − Q  δ + X σ  − Q  δ + A − X σ  log A + 2 δ √ 2 π σ  1 − 2 Q  δ σ  # ≤ 1 − Q δ + A 2 σ ! − Q δ + A − A 2 σ !! log A + 2 δ √ 2 π σ  1 − 2 Q  δ σ  (107) = 1 − 2 Q δ + A 2 σ !! log A + 2 δ √ 2 π σ  1 − 2 Q  δ σ  . (108) Again w e hav e n ot used the relation E Q [ X ] ≤ α A , and hence th e b ound is v alid for arbitrary α ∈ [0 , 1]. 6.5 Upp er Bound (47) of Theorem 14 One of the main c hallenges of d er ivin g the upp er b ounds of Th eorem 14 using dualit y is that without a p eak-p ow er constrain t the inpu t can b e arbitrarily large (alb eit with small p robabilit y). This mak es it m uc h harder to fi nd b ounds on expressions lik e E Q  X 2  . Still, we shall d eriv e up p er b ound (47) using duality . W e c h o ose a distribu tion R 5 with densit y f 5 ( y ) =          1 β e − δ 2 2 σ 2 + √ 2 π σ Q ( δ σ ) e − y 2 2 σ 2 , y < − δ, 1 β e − δ 2 2 σ 2 + √ 2 π σ Q ( δ σ ) e − δ 2 2 σ 2 e − y + δ β , y ≥ − δ , (109) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 22 where δ ∈ R and β > 0 are fr ee parameters. This leads to the follo wing expression: E Q  − Z ∞ −∞ log R ′ 5 ( y ) d W ( y | X )  = log  β e − δ 2 2 σ 2 + √ 2 π σ Q  δ σ  + E Q  1 2 σ 2 Z − δ −∞ y 2 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 d y  + E Q  δ 2 2 σ 2 Z ∞ − δ 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 d y  + E Q  1 β Z ∞ − δ ( y + δ ) 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 d y  (110) = log  β e − δ 2 2 σ 2 + √ 2 π σ Q  δ σ  + E Q  1 2 + X 2 2 σ 2  Q  δ + X σ  + δ − X 2 √ 2 π σ e − ( δ + X ) 2 2 σ 2  | {z } c 4 + E Q  δ 2 2 σ 2  1 − Q  δ + X σ  | {z } c 5 + E Q  δ + X β  1 − Q  δ + X σ  + σ √ 2 π β e − ( δ + X ) 2 2 σ 2  | {z } c 6 . (111) W e no w restrict the fr ee parameter δ to satisfy δ ≤ − σ √ e (112) and con tin ue as follo ws. F or arbitrary in put la w Q such that E Q [ X ] ≤ E : E Q  1 2 Q  δ + X σ  ≤ 1 2 ; (113) E Q  X 2 2 σ 2 Q  δ + X σ  = E Q  X 2 σ · X σ Q  X σ − − δ σ  (114) ≤ E Q  X 2 σ · − δ σ  (115) ≤ − δ E 2 σ 2 ; (11 6) E Q  δ − X 2 √ 2 π σ e − ( δ + X ) 2 2 σ 2  ≤ 0; (117) E Q  δ 2 2 σ 2  1 − Q  δ + X σ  = E Q  δ 2 2 σ 2  1 − Q  X σ − − δ σ  (118) ≤ E Q  δ 2 2 σ 2  1 − Q  δ σ  + X − δ Q  δ σ  (119) ≤ δ 2 2 σ 2  1 − Q  δ σ  − E δ Q  δ σ  ; (120) E Q  δ β  1 − Q  δ + X σ  ≤ 0; (121) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 23 E Q  X β  1 − Q  δ + X σ  ≤ E Q  X β  ≤ E β ; (122 ) E Q  σ √ 2 π β e − ( δ + X ) 2 2 σ 2  ≤ σ √ 2 π β . (123) Here, the first inequalit y (113) follo ws fr om Q ( ξ ) ≤ 1 for all ξ ∈ R ; (115) follo w s from Lemma 5 using assumption (112); in the su bsequent inequ alit y (11 6 ) we use the a verag e-p o w er constrain t together w ith (11 2 ); (117 ) follo ws from X ≥ 0 and δ < 0 (b y (112)); in (119) we use Lemma 6, and the su bsequent inequalit y (120) follo ws aga in from the p o wer constraint together with (1 12); (121 ) is due to (1 12); (122) follo ws b ecause Q ( ξ ) ≥ 0 for all ξ ∈ R ; and in th e last inequalit y (12 3 ) w e upp er-b ound e − ( δ + X ) 2 2 σ 2 b y 1. Com bining (113 ) –(123) w ith (111 ) and (70) yields the claimed r esult. 6.6 Upp er Bound (48) of Theorem 14 The b ound (48) follo w s from th e same c h oice ( 109) as we h av e u sed f or the b ound (47). Ho wev er, here we will restrict the free parameter δ to b e nonn egativ e: δ ≥ 0 . (124) W e can th en b ound c 4 as c 4 = E Q  1 2 σ 2 Z − δ −∞ y 2 1 √ 2 π σ e − ( y − X ) 2 2 σ 2 d y  (125) ≤ E Q  1 2 σ 2 Z − δ −∞ y 2 1 √ 2 π σ e − y 2 2 σ 2 d y  (126) = 1 2 Q  δ σ  + δ 2 √ 2 π σ e − δ 2 2 σ 2 , (127) where the inequ alit y follo ws from the assumption δ ≥ 0 and the nonnegativit y of X that ensure that ( δ + X ) 2 ≥ δ 2 . Moreo ve r, u sing the conca vity and monotonicit y of ξ 7→ (1 − Q ( ξ )) for ξ ≥ 0 (see Lemma 3) and Jens en ’s inequalit y , w e b oun d c 5 as follo ws : c 5 = E Q  δ 2 2 σ 2  1 − Q  δ + X σ  ≤ δ 2 2 σ 2  1 − Q  δ + E σ  , (12 8) and, using the nonnegativit y o f Q ( · ) and of X , we get c 6 = E Q  δ + X β  1 − Q  δ + X σ  + σ √ 2 π β e − ( δ + X ) 2 2 σ 2  (129) ≤ E Q  δ + X β + σ √ 2 π β e − δ 2 2 σ 2  (130) ≤ δ + E β + σ √ 2 π β e − δ 2 2 σ 2 . (131) Com bining (111 ), (127), (128), and (131) with (70 ) yields the claimed result. Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 24 7 Deriv ation of Asymptotic Results 7.1 High-SNR A symptotic Expression (36) in Theorem 11 T o deriv e (36) w e c ho ose th e free parameter δ of Theorem 9 as in (34), and set the parameter µ equal to µ ∗ , the solution to (29). Then, lim A ↑∞ δ A = 0 , (132) lim A ↑∞ δ = ∞ , (133) lim A ↑∞ δ e − δ 2 2 σ 2 = 0 , (134) and therefore from (28) follo w s th at for α ∈  0 , 1 2  : lim A ↑∞  C ( A , α A ) − log A σ  ≤ log 1 − e − µ ∗ √ 2 π µ ∗ − 1 2 + µ ∗ α. (13 5) On th e other hand, from (26) follo ws lim A ↑∞  C ( A , α A ) − log A σ  ≥ − 1 2 log 2 π e + αµ ∗ + log 1 − e − µ ∗ µ ∗ . (136) By Equiv alence (29) and basic arithmetic r eform ulations it can b e shown that the t wo b oun d s (135) and (136) coincide and equal the limit (36). 7.2 High-SNR A symptotic Expression (42) in Theorem 13 T o d eriv e (42) we use lo wer b ound (38) and up p er b oun d (40) for δ as in (41 ). By this c h oice of δ : lim A ↑∞ δ A = 0 , (137) lim A ↑∞ δ = ∞ , (138) lim A ↑∞ δ e − δ 2 2 σ 2 = 0 , (139) lim A ↑∞ Q  A + 2 δ 2 σ  log  A + 2 δ  = 0 , (140) and hence from upp er b ound (40) follo w s lim A ↑∞  C ( A , α A ) − log A σ  ≤ lim A ↑∞ ( log 1 + 2 δ A 1 − 2 Q  δ σ  − 2 Q  A + 2 δ 2 σ  log A + 2 δ σ √ 2 π  1 − Q  δ σ  − 1 2 log 2 π e + Q  δ σ  + δ √ 2 π σ e − δ 2 2 σ 2 ) (141) = − 1 2 log 2 π e. (142) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 25 On th e other hand, by lo wer b ou n d (38): lim A ↑∞  C ( A , α A ) − log A σ  ≥ lim A ↑∞ 1 2 log  σ 2 A 2 + 1 2 π e  = − 1 2 log 2 π e. (143) The t w o b ounds coincide and ther efore they pro v e (42) in Theorem 13. 7.3 High-SNR A symptotic Expression (53) in Theorem 15 T o deriv e (53) we use b ound (48) with the follo win g choic e of the fr ee p arameters β and δ : β , E , (144) δ , σ r log E σ , (145) for E ≥ σ . Then, lim E ↑∞ δ = ∞ , (146) lim E ↑∞ δ E = 0 , (147) lim E ↑∞ δ e − δ 2 2 σ 2 = 0 , (148) lim E ↑∞ e δ 2 2 σ 2 E = 0 . (149) Hence, w e get fr om (48) lim E ↑∞  C ( E ) − log E σ  ≤ lim E ↑∞    − δ 2 2 σ 2 + log   1 + √ 2 π σ e δ 2 2 σ 2 Q  δ σ  E   + δ 2 2 σ 2 + δ + E + σ √ 2 π e − δ 2 2 σ 2 E − 1 2 log 2 π e      (150) = 1 − 1 2 log 2 π e = 1 2 log e 2 π . (151) On th e other hand, we get fr om lo wer b ound (46) that lim E ↑∞  C ( E ) − log E σ  ≥ 1 2 log e 2 π . (152) These t wo b oun ds coincide and therefore p ro ve (53 ) in T h eorem 15. 7.4 Lo w-SNR Asymptotic Expression (37) in Theorem 11 In order to p ro ve the lo w -S NR asymptotic expression (3 7 ) in Th eorem 11, w e de- riv e an asymptotic low er b oun d that com bined with u pp er b ou n d (27) yields the desired result. T he low er b ound we p rop ose is based on Theorem 2 in [20]. F or the c h annel (6) under co nsideration, the tec hnical conditions A–F in [20] are fulfilled, Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 26 and Theorem 2 in [20] states that for p eak-constrained inputs | X | < A the m utual information satisfies I ( X ; Y ) = V a r ( X ) 2 σ 2 + o ( A 2 ) , (153) where o ( A 2 ) decreases faster to 0 th an A 2 , i.e. , lim A ↓ 0 o ( A 2 ) A 2 = 0 . (154) W e restrict atten tion to settings where 0 < A < 1. Then, the b inary inp ut X = ( 0 , with prob. 1 − α, A (1 − A ) , with prob. α, (155) is nonnegativ e and p eak-constrained, an d it satisfies th e a verag e-p o w er constrain t E [ X ] ≤ α A . Hence, by (153) and since for the c hoice of X in (155) V ar ( X ) = α (1 − α ) A 2 (1 − A ) 2 , I ( X ; Y ) = α (1 − α ) A 2 (1 − A ) 2 2 σ 2 + o ( A 2 ) , 0 < A < 1 , 0 < α < 1 2 , (156) and we obtain the follo wing asymptotic lo wer b ound on the capacit y for α ∈  0 , 1 2  : lim A ↓ 0 C ( A , α A ) A 2 /σ 2 ≥ α (1 − α ) 2 , 0 < α < 1 2 . (157) F ur thermore, by upp er b ound (27) and since log (1 + ξ ) ≤ ξ , for ξ ≥ 0, lim A ↓ 0 C ( A , α A ) A 2 /σ 2 ≤ α (1 − α ) 2 , 0 < α < 1 2 . (158) The lo w-S NR asymptotic expression (37) is then established by the last t wo inequ al- ities. 7.5 Lo w-SNR Asymptotic Expression (43) in Theorem 13 T o prov e the lo w-SNR asymptotic expression (43) we deriv e an asymp totic low er b ound which com bined with u pp er b oun d (39) yields the d esired result. The lo we r b ound w e prop ose is again b ased on T h eorem 2 in [20]. W e c h o ose a nonnegativ e and p eak-limited b in ary in put X which equ ip robably tak es on the v alues 0 an d A (1 − A ), for 0 < A < 1. Then, w e apply the s ame steps as in the previous section, and in analogy to (156) obtain I ( X ; Y ) = A 2 (1 − A ) 2 8 σ 2 + o ( A 2 ) , 0 < A < 1 , (159) and lim A ↓ 0 C ( A , α A ) A 2 /σ 2 ≥ 1 8 , 1 2 ≤ α ≤ 1 . (160) F ur thermore, by upp er b ound (39) and from log (1 + ξ ) ≤ ξ , for all ξ ≥ 0, lim A ↓ 0 C ( A , α A ) A 2 /σ 2 ≤ 1 8 , 1 2 ≤ α ≤ 1 . (161) The lo w -S NR asymptotic expr ession (43) n o w follo ws by the last t wo inequalities. Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 27 7.6 Lo w-SNR Asymptotic Expressions ( 54) in Theorem 15 The asymptotic u p p er b ound (54 ) fo llo w s fr om up p er b ound (47) . W e choose δ a s in (49), i.e. , δ , − 2 σ r log σ E , (162) and β , 1 E . (163) Then, from (47): C ( E ) E σ p log σ E ≤ log  1 E √ 2 π σ e − δ 2 2 σ 2 + Q  δ σ   E σ p log σ E + 2 σ p log σ E · E 2 σ 2 · E σ p log σ E + 2 log σ E E σ p log σ E 1 − Q  δ σ  | {z } = Q ( − δ σ ) + E 2 σ p log σ E Q  δ σ  ! + E  E + σ √ 2 π  E σ p log σ E (164) = log  E √ 2 π σ 3 + Q  δ σ   E σ p log σ E + 1 + 2 p log σ E E σ Q  − δ σ  + Q  δ σ  + σ  E + σ √ 2 π  p log σ E . (165) Next w e note that lim E ↓ 0 r log σ E = ∞ , (166) lim E ↓ 0 Q  δ σ  = 1 , (167) lim E ↓ 0 p log σ E Q  − δ σ  E σ = 0 , (168) and, using Q ( ξ ) ≤ 1 and log(1 + ξ ) ≤ ξ f or all ξ ≥ 0, that lim E ↓ 0 log  E √ 2 π σ 3 + Q  δ σ   E σ p log σ E ≤ lim E ↓ 0 log  E √ 2 π σ 3 + 1  E σ p log σ E (169) ≤ lim E ↓ 0 E √ 2 π σ 3 E σ p log σ E (170) = 0 . (171) T ogether with (165) this leads to lim E ↓ 0 C ( E ) E σ p log σ E ≤ 2 . (172) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 28 7.7 Lo w-SNR Asymptotic Expressions ( 55) in Theorem 15 W e sh all deriv e a new asymptotic lo w er b ound at low p o we rs whic h p r o ves (55). T he lo wer b ound is obtained b y lo w er-b ound ing the mutual in formation I ( Q, W ) for Q the probability measure with p robabilit y mass function q ( x ) = ( 1 − E x 1 , if x = 0 , E x 1 , if x = x 1 , (173) where for sufficientl y sm all E w e c ho ose x 1 , σ r c log σ E , (174) for some constan t c > 2. Note that x 1 ↑ ∞ as E ↓ 0. In the remaining of this section w e assume E σ ≤ 1 2 so that the pr obabilit y mass function in (173) is well -defined. The probabilit y d ensit y of the c hannel outp u t Y corresp ond ing to the input with probabilit y mass fun ction (173) is given by f Y ( y ) =  1 − E x 1  1 √ 2 π σ 2 e − y 2 2 σ 2 + E x 1 · 1 √ 2 π σ 2 e − ( y − x 1 ) 2 2 σ 2 . (175) In order to ev aluate th e m utual information I ( Q, W ) for the c h osen binary inp ut distribution we write it as (see [25]) I ( Q, W ) = Z D  W ( ·| x )   W ( ·| 0)  d Q ( x ) − D  R ( · )   W ( ·| 0)  . (176) W e can th en ev aluate the first term on the righ t-hand side as Z D  W ( ·| x )   W ( ·| 0)  d Q ( x ) = E x 1 D  W ( ·| x 1 )   W ( ·| 0)  (177) = E x 1 Z ∞ −∞ 1 √ 2 π σ 2 e − ( y − x 1 ) 2 2 σ 2 log   e − ( y − x 1 ) 2 2 σ 2 e − y 2 2 σ 2   d y (178) = E x 1 Z ∞ −∞ 1 √ 2 π σ 2 e − ( y − x 1 ) 2 2 σ 2  y x 1 σ 2 − x 2 1 2 σ 2  d y (179) = E x 1  x 2 1 σ 2 − x 2 1 2 σ 2  (180) = E x 1 2 σ 2 = √ c 2 · E σ r log σ E . (181) Ev aluating the second term is more difficult, and in fact w e only derive an upp er b ound on it wh ich exh ib its th e desired a symptotic b eha vior at lo w SNR. W e shall sho w that lim E ↓ 0 D  R ( · )   W ( ·| 0)  E σ p log σ E ≤ √ c 2 − 1 √ c , (182 ) from whic h follo w s by (176) and (181) lim E ↓ 0 I ( Q, W ) E σ p log σ E ≥ 1 √ c . (183) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 29 The desired asymp totic lo w er b ound in (55) then follo ws b ecause (183 ) holds for any c > 2. Th us, in th e remaining o f this sectio n w e wish to pr ov e (182 ). T o this end , we write D  R ( · )   W ( ·| 0)  = Z ∞ −∞ f Y ( y ) log     1 − E x 1  1 √ 2 π σ 2 e − y 2 2 σ 2 + E x 1 1 √ 2 π σ 2 e − ( y − x 1 ) 2 2 σ 2 1 √ 2 π σ 2 e − y 2 2 σ 2    d y (184) = Z x 1 2 −∞ f Y ( y ) log  1 − E x 1 + E x 1 e y x 1 σ 2 − x 2 1 2 σ 2  d y | {z } c 7 + Z x 1 2 + x 1 c x 1 2 f Y ( y ) log  1 − E x 1 + E x 1 e y x 1 σ 2 − x 2 1 2 σ 2  d y | {z } c 8 + Z ∞ x 1 2 + x 1 c f Y ( y ) log  1 − E x 1 + E x 1 e y x 1 σ 2 − x 2 1 2 σ 2  d y | {z } c 9 (185) and upp er-b ound c 7 , c 8 , and c 9 . W e start with up p er-b ound ing c 7 where y ≥ x 1 2 : c 7 ≤ Z x 1 2 −∞ f Y ( y ) log  1 − E x 1 + E x 1 e x 1 σ 2 x 1 2 − x 2 1 2 σ 2  d y (186) = Z ξ 1 −∞ f Y ( y ) log 1 d y = 0 , (187) and hence, lim E ↓ 0 c 7 E p log(1 / E ) ≤ 0 . (188) Next w e examine c 8 . Using E x 1 ≥ 0 and log(1 + ξ ) ≤ ξ for all ξ ≥ 0 we get log  1 − E x 1 + E x 1 e y x 1 σ 2 − x 2 1 2 σ 2  ≤ log  1 + E x 1 e y x 1 σ 2 − x 2 1 2 σ 2  ≤ E x 1 e y x 1 σ 2 − x 2 1 2 σ 2 , (18 9) and hence, c 8 ≤ Z x 1 2 + x 1 c x 1 2 f Y ( y ) E x 1 e y x 1 σ 2 − x 2 1 2 σ 2 d y (190) = Z x 1 2 + x 1 c x 1 2  1 − E x 1  E x 1 1 √ 2 π σ 2 e − y 2 2 σ 2 +  E x 1  2 1 √ 2 π σ 2 e − ( y − x 1 ) 2 2 σ 2 ! e y x 1 σ 2 − x 2 1 2 σ 2 d y (191) =  1 − E x 1  E x 1 Z x 1 2 + x 1 c x 1 2 1 √ 2 π σ 2 e − ( y − x 1 ) 2 2 σ 2 d y Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 30 +  E x 1  2 e x 2 1 σ 2 Z x 1 2 + x 1 c x 1 2 1 √ 2 π σ 2 e − ( y − 2 x 1 ) 2 2 σ 2 d y (192) =  1 − E x 1  E x 1  Q  x 1 2 σ − x 1 cσ  − Q  x 1 2 σ   +  E x 1  2 e x 2 1 σ 2 Q  3 x 1 2 σ − x 1 cσ  − Q  3 x 1 2 σ  ! (193) = 1 − E σ √ c p log σ E ! E σ √ c p log σ E Q  √ c 2 − 1 √ c  r log σ E  − Q  √ c 2 r log σ E  ! +  E σ  2 − c c log σ E Q  3 √ c 2 − 1 √ c  r log σ E  − Q  3 √ c 2 r log σ E  ! (194) where in the last equalit y w e used the defin ition of x 1 . W e analyz e the limiting b eha viors of the t wo su mmands separately . F or the fi rst term lim E ↓ 0  1 − E σ √ c √ log σ E  E σ √ c √ log σ E  Q  √ c 2 − 1 √ c  p log σ E  − Q  √ c 2 p log σ E  E σ p log σ E = 0 . (195) T o deal with the second term we fur ther upp er-b ound it using (17) and th e n onneg- ativit y of Q ( · ):  E σ  2 − c c log σ E Q  3 √ c 2 − 1 √ c  r log σ E  − Q  3 √ c 2 r log σ E  | {z } ≥ 0 ! <  E σ  2 − c c log σ E · 1 √ 2 π p log σ E  3 √ c 2 − 1 √ c   E σ  c 2 ( 3 2 − 1 c ) 2 (196) =  E σ  1 2 + 1 2 ( c 4 + 1 c )  log σ E  3 / 2 · 1 √ 2 π  3 c 3 / 2 2 − √ c  . (197) Note that whenever c > 2 then  c 4 + 1 c  > 1 and  3 c 3 / 2 2 − √ c  6 = 0, and th erefore lim E ↓ 0 c 8 E σ p log σ E ≤ 0 . (198) Finally , w e examine the limiting b eha vior of c 9 . T o this end we rewrite c 9 as c 9 = Z ∞ x 1 2 + x 1 c f Y ( y ) log  e y x 1 σ 2 − x 2 1 2 σ 2  d y | {z } c 9 , 1 + Z ∞ x 1 2 + x 1 c f Y ( y ) log  E x 1  d y | {z } c 9 , 2 + Z ∞ x 1 2 + x 1 c f Y ( y ) log   1 +  1 − E x 1  E x 1 e − y x 1 σ 2 + x 2 1 2 σ 2   d y | {z } c 9 , 3 . (199) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 31 W e separately examine c 9 , 1 , c 9 , 2 , and c 9 , 3 and start with c 9 , 1 : c 9 , 1 = Z ∞ x 1 2 + x 1 c  1 − E x 1  1 √ 2 π σ 2 e − y 2 2 σ 2 + E x 1 1 √ 2 π σ 2 e − ( y − x 1 ) 2 2 σ 2  ·  y x 1 σ 2 − x 2 1 2 σ 2  d y (200) =  1 − E x 1  Z ∞ x 1 2 + x 1 c 1 √ 2 π σ 2 e − y 2 2 σ 2  y x 1 σ 2 − x 2 1 2 σ 2  d y + E x 1 Z ∞ x 1 2 + x 1 c 1 √ 2 π σ 2 e − ( y − x 1 ) 2 2 σ 2  x 1 σ 2 ( y − x 1 ) + x 2 1 2 σ 2  d y (201) =  1 − E x 1  | {z } ≤ 1 x 1 σ 2 · σ √ 2 π e − x 2 1 ( 1 2 + 1 c ) 2 2 σ 2 −  1 − E x 1  | {z } ≥ 0 if E σ ≤ 1 2 x 2 1 2 σ 2 Q  x 1 2 σ + x 1 cσ  | {z } ≥ 0 + E x 1 x 1 σ 2 · σ √ 2 π e − x 2 1 2 σ 2 ( 1 c − 1 2 ) 2 + E x 1 2 σ 2 Q  x 1 cσ − x 1 2 σ  | {z } ≤ 1 (202) ≤ x 1 σ √ 2 π e − x 2 1 2 σ 2 ( 1 2 + 1 c ) 2 + E σ · 1 √ 2 π e − x 2 1 2 σ 2 ( 1 c − 1 2 ) 2 + E x 1 2 σ 2 (203) = √ c p log σ E √ 2 π  E σ  c 2 ( 1 2 + 1 c ) 2 + 1 √ 2 π  E σ  c 2 ( 1 c − 1 2 ) 2 +1 + E σ · √ c 2 r log σ E (204) where in the last step w e used the defi nition of x 1 (174). Again, since c > 2 we ha v e c 4 + 1 c > 1, and th erefore lim E ↓ 0 √ c √ log σ E √ 2 π  E σ  c 2 ( 1 2 + 1 c ) 2 E σ p log σ E = lim E ↓ 0 √ c √ 2 π  E σ  1 2 ( c 4 + 1 c ) − 1 2 = 0 . (205) Moreo ve r, lim E ↓ 0 1 √ 2 π  E σ  c 2 ( 1 c − 1 2 ) 2 +1 E σ p log σ E = lim E ↓ 0 1 √ 2 π p log σ E  E σ  c 2 ( 1 c − 1 2 ) 2 = 0 (206) and lim E ↓ 0 E σ · √ c 2 p log σ E E σ p log σ E = √ c 2 , (207) and w e conclude that lim E ↓ 0 c 9 , 1 E σ p log σ E ≤ √ c 2 . (208) Next w e analyze c 9 , 2 . Note that E x 1 ≤ 1, for E σ ≤ 1 2 , and hence log E x 1 ≤ 0. Therefore, c 9 , 2 =  1 − E x 1  | {z } ≥ 0 if E σ ≤ 1 2 Q  x 1 2 σ + x 1 cσ  | {z } ≥ 0 + E x 1 Q  x 1 cσ − x 1 2 σ  ! log  E x 1  (209) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 32 ≤ E x 1 Q  x 1 cσ − x 1 2 σ  log  E x 1  (210) = E σ √ c p log σ E Q  1 √ c − √ c 2  r log σ E  log E σ √ c p log σ E ! . (211) Since c > 2 the term Q  1 √ c − √ c 2  p log σ E  tends to 1 when E tends to 0, and therefore lim E ↓ 0 c 9 , 2 E σ p log σ E = lim E ↓ 0 − log σ E − 1 2 log c − 1 2 log log σ E √ c log σ E = − 1 √ c . (212) Finally , w e analyze c 9 , 3 . Using that x 1 E − 1 ≥ 0 if E σ ≤ 1 2 w e lo w er-b oun d y ≥ x 1 2 + x 1 c to get c 9 , 3 = Z ∞ x 1 2 + x 1 c f Y ( y ) log  1 +  x 1 E − 1  e − y x 1 σ 2 + x 2 1 2 σ 2  d y (213 ) ≤ Z ∞ x 1 2 + x 1 c f Y ( y ) log  1 +  x 1 E − 1  e − x 2 1 cσ 2  d y ( 214) =   1 − E x 1  | {z } ≤ 1 Q  x 1 2 σ + x 1 cσ  + E x 1 Q  x 1 cσ − x 1 2 σ   · log  1 +  x 1 E − 1  | {z } ≤ x 1 E e − x 2 1 cσ 2  (215) ≤  Q  x 1 2 σ + x 1 cσ  + E x 1 Q  x 1 cσ − x 1 2 σ   log  1 + x 1 E e − x 2 1 cσ 2  (216) ≤  1 2 e − x 2 1 2 σ 2 ( 1 2 + 1 c ) 2 + E x 1 · 1 2 e − x 2 1 2 σ 2 ( 1 c − 1 2 ) 2  ·  x 1 E e − x 2 1 cσ 2  (217) =   1 2  E σ  c 2 ( 1 2 + 1 c ) 2 + 1 2 p c log σ E  E σ  c 2 ( 1 c − 1 2 ) 2 +1   √ c r log σ E . (218) Here, (217) follo ws by (18) and by log (1 + ξ ) ≤ ξ , for ξ ≥ 0. Since for c > 2 we ha ve c 2  1 2 + 1 c  2 > 1 and  1 c − 1 2  2 > 0, we obtain the f ollo wing limiting b ehavi or: lim E ↓ 0 c 9 , 3 E σ p log σ E ≤ 0 . (219) By (208), (212), and (219) w e conclude that lim E ↓ 0 c 9 E σ p log σ E ≤ √ c 2 − 1 √ c (220) and hence, as w e ha ve set out to pro v e, b y com bining (188), (198), and (220) we obtain lim E ↓ 0 D ( f Y k f Y | X =0 ) E σ p log σ E ≤ √ c 2 − 1 √ c . (221) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 33 A Pro of of Lemma 4 W e start by sh o win g th at f ( · ) is strictly conca v e. S ince ξ 0 , γ ≥ 0, the functions ξ 7→ Q ( ξ 0 + ξ ) and ξ 7→ Q ( ξ 0 + γ − ξ ) are strictly con vex o ver [0 , γ ] by Lemma 3. Then, also the function ξ 7→ ( Q ( ξ 0 + ξ ) + Q ( ξ 0 + γ − ξ )) m ust b e strictly conv ex ov er [0 , γ ] and we conclude that the fu nction f : ξ 7→ (1 − Q ( ξ 0 + ξ ) − Q ( ξ 0 + γ − ξ )) is strictly conca ve. The symmetry of f ( · ) aroun d ξ = γ 2 can b e seen by noting that for all ξ ′ ∈  0 , γ 2  f  γ 2 + ξ ′  = 1 − Q  ξ 0 + γ 2 + ξ ′  − Q  ξ 0 + γ − γ 2 − ξ ′  (222) = 1 − Q  ξ 0 + γ 2 + ξ ′  − Q  ξ 0 + γ 2 − ξ ′  (223) is id en tical to f  γ 2 − ξ ′  = 1 − Q  ξ 0 + γ 2 − ξ ′  − Q  ξ 0 + γ − γ 2 + ξ ′  (224) = 1 − Q  ξ 0 + γ 2 − ξ ′  − Q  ξ 0 + γ 2 + ξ ′  . (225) Finally , that f ( · ) has its maximum at γ 2 and is monotonically strictly increasing ov er  0 , γ 2  follo ws by the symmetry and the strict conca vit y . B Pro of of Lemma 5 F or ξ ≤ µ we hav e ξ |{z} ≤ µ Q ( ξ − µ ) | {z } ≤ 1 ≤ µ, (226) b ecause ξ ≥ 0 and Q ( · ) is n onnegativ e. Let us th en assu me that ξ > µ and introdu ce a v ariable sub stitution y = ξ − µ . Then for y > 0, we get ξ Q ( ξ − µ ) = ( y + µ ) Q ( y ) (227 ) ≤ ( y + µ ) 1 2 e − y 2 2 (228) ≤ y 2 e − y 2 2 + µ 2 (229) ≤ 1 2 e − 1 2 + µ 2 (230) ≤ µ. (231 ) Here, the first inequalit y (228) follo w s from (18); in (2 29) we u pp er-b ound e − y 2 2 b y 1; then in (230) w e replace y e − y 2 2 b y its m axim um e − 1 2 ; and the final in equalit y (231) holds b ecause we ha v e assu med that µ ≥ e − 1 2 . C Pro of of Lemma 6 Define f 1 ( ξ ) , 1 − Q ( ξ − µ ) , (232) f 2 ( ξ ) , 1 − Q ( − µ ) + ξ µ Q ( − µ ) . (233) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 34 W e shall sho w that for all ξ ≥ 0 f 1 ( ξ ) ≤ f 2 ( ξ ) , (234) where µ is a nonnegativ e constant. Let us start with the case 0 ≤ ξ ≤ µ . F rom Lemma 3 it follo ws that f 1 ( · ) is strictly con v ex o ver [0 , µ ]. Moreo ver, note that f 1 (0) = f 2 (0) and that the slop e of f 1 ( · ) at ξ = 0 is smaller than the slop e of f 2 ( · ): ∂ ∂ ξ f 1 ( ξ )     ξ =0 = 1 √ 2 π e − ( ξ − µ ) 2 2     ξ =0 (235) = 1 √ 2 π e − µ 2 2 (236) = 1 µ · µ √ 2 π e − µ 2 2 (237) ≤ 1 µ · 1 √ 2 π e − 1 2 (238) < 1 µ · 1 2 (239) ≤ 1 µ  1 − Q ( µ )  (240) = 1 µ Q ( − µ ) (2 41) = ∂ ∂ ξ f 2 ( ξ ) . (242) Here, (238 ) follo ws from upp er-b oundin g µ e − µ 2 2 b y its maximum e − 1 2 ; the sub sequen t inequalit y (239 ) f rom the fact that 1 √ 2 π e < 1 2 ; and the su bsequent tw o steps ( 240) and (241) follo w from Lemma 3. Hence for small v alues of ξ , f 1 ( ξ ) ≤ f 2 ( ξ ). Sin ce f 1 is co nv ex, it can in tersect with the linear function f 2 at most one more time apart from the in tersection at ξ = 0. How ev er, for ξ = µ , f 1 ( µ ) = 1 2 , (243) f 2 ( µ ) = 1 , (244) i.e. , f 2 is still larger than f 1 , so no in tersection has tak en place in the inte rv al (0 , µ ]. F or ξ > µ , n o intersecti on can tak e p lace either b ecause by d efinition f 2 ( ξ ) > 1 > f 1 ( ξ ) , ξ > µ. (245) This completes the pr o of. D Pro of of Lemma 8 W e first state an auxiliary pr op osition wh ic h is used to pro v e th e lemma. Prop osition 17. L et the r andom variable X take value in t he interval [0 , A ] , and let Z ∼ N R  0 , σ 2  b e indep endent of X . Th en, ther e exists a r andom variable ˜ X taking value in [0 , A ] and indep endent o f Z that satisfies E h ˜ X i = 1 2 A (246) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 35 and I ( ˜ X ; ˜ X + Z ) ≥ I ( X ; X + Z ) . (247) Pr o of. Define ¯ X = A − X and note that I ( X ; X + Z ) = I ( X ; A − X − Z ) (24 8) = I ( X ; A − X + Z ) (24 9) = I ( A − X ; A − X + Z ) (250) = I ( ¯ X ; ¯ X + Z ) , (2 51) where (24 8 ) and (250 ) follo w b ecause I ( U ; V ) = I ( U ; g ( V )) whenev er g is one-to - one; w here (249) follo w s from the symmetry of the c en tered Gaussian; and where (251) follo ws from the defin ition of ¯ X . Let B b e a binary random v ariable that tak es on the v alues 0 an d 1 equiprobably and indep enden tly of the pair ( X , Z ). Define the r andom v ariable ˜ X equ al to X when B = 1 and equal to ¯ X w hen B = 0. W e sho w that ˜ X (w h ic h tak es v alue in [0 , A ]) satisfies b oth (2 46) and (247). Condition (246) follo ws by the total la w of exp ectation, by th e definition o f ˜ X , by the indep endence of B and ( X , ¯ X ), a nd b ecause E  ¯ X  = A − E [ X ]: E h ˜ X i = 1 2 E h ˜ X    B = 1 i + 1 2 E h ˜ X    B = 0 i = 1 2 E [ X ] + 1 2 E  ¯ X  = 1 2 A . (25 2)  Condition (247) follo ws b ecause conditioning redu ces differen tial entrop y , b ecause ˜ X is indep enden t of ( X , ¯ X , Z ), and b y (251): I ( ˜ X ; ˜ X + Z ) = h ( ˜ X + Z ) − h ( Z ) (253) ≥ h ( ˜ X + Z | B ) − h ( Z ) (254) = 1 2 h ( ˜ X + Z | B = 1) + 1 2 h ( ˜ X + Z | B = 0) − h ( Z ) (255) = 1 2  h ( X + Z ) − h ( Z )  + 1 2  h ( ¯ X + Z ) − h ( Z )  (256) = 1 2 I ( X + Z ; X ) + 1 2 I ( ¯ X + Z ; ¯ X ) (257) = I ( X ; X + Z ) . (258) With the aid of Prop osition 17 we n o w p ro ve Lemma 8: Pr o of (Pr o of of L e mma 8) . Let Q ∗ denote the capacit y-ac hieving input d istribution (whic h exists b y Le mma 7 ). Th en , by Pr op osition 17, th er e exists an inpu t distri- bution ˜ Q with a verage p ow er E ˜ Q [ X ] = 1 2 A (259) suc h that I ( ˜ Q, W ) ≥ I ( Q ∗ , W ) . (260) Whenev er α ≥ 1 2 , ˜ Q is a v alid inpu t distribu tion in the optimization in (13) and by (260) it ac hiev es capacit y . But b y the uniqueness of the capacit y-ac hieving input distribution (Lemma 7) the distribu tions Q ∗ and ˜ Q must coincide, and therefore by (259) E Q ∗ [ X ] = 1 2 A . (261)  Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 36 E Pro of of Lemma 10 W e start by pro ving that the function ϕ ( · ) is m onotonicall y strictly decreasing. This follo ws immediately by taking th e der iv ative of ϕ ( · ) ϕ ′ ( µ ) = − 1 µ 2 − e − µ (1 − e − µ ) 2 , (262) whic h is strictly negati v e for µ ∈ (0 , ∞ ). T hen, by the strict monotonicit y and b ecause ϕ ( · ) is con tin uous, ϕ ( · ) is bijectiv e. W e are left with p ro vin g the asymptotic results. Whereas the first limit (31) follo ws directly , the second limit (32 ) follo ws by rewriting the fun ction ϕ ( · ) as ϕ ( µ ) = 1 − (1 + µ ) e − µ µ (1 − e − µ ) , (263) and then applying t wo times de l’Hˆ opital’s rule. Finally , the third limit (33) is obtained by observing that µϕ ( µ ) = 1 − µe − µ 1 − e − µ (264) and that µe − µ tends to zero as µ tends to infinity . This concludes the p ro of of the lemma. F App endix for the P ro of of the Upp er B ound (28) It r emains to show that log   A  e µδ A − e − µ ( 1+ δ A )  √ 2 π σ µ  1 − 2 Q  δ σ    ≥ 0 , ∀ A , σ, δ, µ > 0 . (265 ) W e in v estigate the f ollo wing expr ession: A  e µδ A − e − µ ( 1+ δ A )  √ 2 π σ µ  1 − 2 Q  δ σ  ≥ A  e µδ A − e − µ δ A  √ 2 π σ µ  1 − 2 Q  δ σ  (266) = A µδ · 1 2  e µδ A − e − µδ A  √ 2 π ( 1 − 2 Q ( δ σ )) 2 δ σ (267) = A µδ · sinh  µδ A  √ 2 π ( 1 − 2 Q ( δ σ )) 2 δ σ , (268) where the inequalit y follo ws b ecause w e dr op a factor e − µ ≤ 1. No w we n ote that since sinh( · ) i s a conv ex function o ver [0 , ∞ ) for an y triple 0 ≤ ξ 0 < ξ 1 < ξ 2 < ∞ , sinh( ξ 1 ) lies b elo w the line segmen t conn ecting sin h( ξ 0 ) and sinh( ξ 2 ), i.e. , sinh( ξ 1 ) ≤ ξ 1 − ξ 0 ξ 2 − ξ 0 (sinh( ξ 2 ) − sinh( ξ 0 )) . (269) Lapidoth, Moser, Wi gger, Nov ember 3, 2018, submitted 37 By c ho osing ξ 0 = 0 and since sin h(0) = 0 we conclude that sinh( ξ 1 ) ξ 1 ≤ sinh( ξ 2 ) ξ 2 , ∀ 0 < ξ 1 < ξ 2 . (270) Hence the function ξ 7→ sinh( ξ ) ξ is monoto nically increasing o ver (0 , ∞ ) and has its infimum in the limit ξ ↓ 0, i.e. , sinh( ξ ) ξ ≥ lim ξ ′ ↓ 0 sinh( ξ ′ ) ξ ′ = 1 , ∀ ξ > 0 , (271) where the limit follo ws b y d e l’Hˆ opital’s rule. Similarly , sin ce the function ξ 7→ √ 2 π 2 (1 − 2 Q ( ξ )) is conca v e o ver [0 , ∞ ) and since √ 2 π 2 (1 − 2 Q (0)) = 0, √ 2 π 2 (1 − 2 Q ( ξ 1 )) ξ 1 ≥ √ 2 π 2 (1 − 2 Q ( ξ 2 )) ξ 2 , ∀ 0 < ξ 1 < ξ 2 . (272) Hence the fun ction ξ 7→ √ 2 π 2 (1 − 2 Q ( ξ )) ξ is monotonically decreasing on (0 , ∞ ) and has its supr em u m in the limit wh en ξ ↓ 0, i.e . , √ 2 π (1 − 2 Q ( ξ )) 2 ξ ≤ lim ξ ′ ↓ 0 √ 2 π (1 − 2 Q ( ξ ′ )) 2 ξ ′ = 1 , ∀ ξ > 0 , (273) where again the limit follo ws b y de l’Hˆ op ital’s rule. Th us, for any A , σ, µ, δ > 0: log   A  e µδ A − e − µ ( 1+ δ A )  √ 2 π σ µ  1 − 2 Q  δ σ    ≥ log    A µδ · sinh  µδ A  √ 2 π ( 1 − 2 Q ( δ σ )) 2 δ σ    (274) ≥ log     inf ξ > 0 n 1 ξ · sinh( ξ ) o sup ξ > 0 n √ 2 π (1 − 2 Q ( ξ )) 2 ξ o     (275) = log 1 1 = 0 . (276) Ac kn o wledgmen ts F ru itful d iscussions with Ding-Jie Lin are gratefully ac kn o wledged. References [1] S. Hranilo vic and F. R. Ksc hischang, “Capacit y b oun ds for p o wer- and band - limited o ptical intensit y c hannels c orrup ted by Gaussian n oise,” IE EE T r ans- actions on Inform ation The ory , v ol. 50, no. 5, pp . 784–795, May 2004. [2] S. M. 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