First- and Second-Moment Constrained Gaussian Channels
aa r X i v : . [ c s . I T ] F e b First- and Second-Moment Constrained GaussianChannels
Shuai Ma ∗†‡ and Mich`ele Wigger ∗∗ LTCI, Telecom Paris, IP Paris, 91120 Palaiseau, France † National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China ‡ School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116,[email protected], [email protected]
Abstract —This paper studies the channel capacity of intensity-modulation direct-detection (IM/DD) visible light communication(VLC) systems under both optical and electrical power con-straints. Specifically, it derives the asymptotic capacities in thehigh and low signal-to-noise ratio (SNR) regimes under peak,first-moment, and second-moment constraints. The results showthat first- and second-moment constraints are never simultane-ously active in the asymptotic low-SNR regime, and only in fewcases in the asymptotic high-SNR regime. Moreover, the second-moment constraint is more stringent in the asymptotic low-SNRregime than in the high-SNR regime.
I. I
NTRODUCTION
The ever-increasing number of wireless devices and high-speed communication requirements cause a spectrum scarcityof conventional radio-frequencies (RF). A promising solu-tion is visible light communication (VLC) with its abundantunlicensed spectrum [1], [2]. In particular, when utilizingthe simple and practical intensity modulation–direct detection(IM/DD) technology, transmitters directly modulate informa-tion onto the real, non-negative optical intensity of the VLCsignals (in contrast to RF signals which modulate the complexfield) and receivers apply photodetectors to measure incomingoptical intensities. For eye safety reasons and hardware limi-tations, both the maximum and average optical intensities ofVLC transmit signals typically have to be restricted. Sincethese apply directly to the intensities, they impose both peakand first-moment constraints on the transmit signals. Addi-tional second-moment constraints are imposed by limitationsof the electronic circuits that control the transmit signal, suchas the boundedness of the linear amplification regime andelectric power consumption [3]–[8].A close-form expression for the capacity of such IM/DDsystems is still unknown, even when some of the first orsecond-moment constraints are relaxed. However, bounds andasymptotic results in the high- and low signal-to-noise ratio(SNR) regimes are known under certain relaxations. For exam-ple, various upper and lower bounds on the capacity, as well asits exact high- and low-SNR asymptotics, have been derivedunder only a first-moment constraint without a second-momentconstraint [9]–[15].In this work, we derive the exact expressions for theasymptotic high- and low-SNR capacities under peak, first-moment, and second-moment constraints. Our results show that in the asymptotic low-SNR regime, only one of thetwo moment constraints is stringent. Specifically, the second-moment constraint is active if the peak-constraint A timesthe first-moment-constraint α A exceeds this second-momentconstraint α A , and otherwise the first-moment constraint isactive. This can be seen as a consequence of the optimality ofon-off keying in the asymptotic low-SNR regime. Our resultsfurther show that for most constraint-parameters ( α , α ) alsoin the high-SNR regime, only one of the moment-constraints isactive. Interestingly, the second-moment constraint is inactiveover a larger region of ( α , α ) -pairs in the high-SNR regimethan in the low-SNR regime, and the first-moment constraintover a smaller region. An additional second-moment constraintis thus more restrictive in the low-SNR regime than in thehigh-SNR regime. In the asymptotic high-SNR regime, wefurther observe a small region of ( α , α ) -pairs where bothmoment-constraints are simultaneously active and limit theasymptotic capacity.II. C HANNEL M ODEL
Consider a typical VLC communication link, where thetransmitter is equipped with a single LED or laser and thereceiver with a single photodetector. The photodetector mea-sures the incoming light intensity, which can be modeled as Y = x + Z, (1)where x denotes the input signal produced by the transmitter’sLED or laser, and Z is standard additive white Gaussian noiseindependent of x . Note that, in contrast to the input x , theoutput Y can be negative.Inputs x are subject to both a peak and an average opticalpower (average-intensity) constraints: X ∈ [0 , A ] , (2) E [ X ] ≤ α A , (3)for some fixed parameters A > and α ∈ (0 , . Theseconstraints come from (eye- and skin-) safety reasons, andfrom limitations (caused by non-linearities) on the opticaloperating regimes of LEDs and lasers.Due to battery limitations on the attached RF circuit andpower amplifier limitations, the second moment of the transmit signal also has to be restricted: E (cid:2) X (cid:3) ≤ α A . (4)We denote the capacity of the channel (1) with allowedpeak power A , maximum average power α A , and maximumsecond moment α A by C ( α , α , A ) . It is given by [16] C ( α , α , A ) = sup P X I ( X ; Y ) , (5)where the supremum is over input laws P X satisfying (2)–(4).Notice that, for any random variable X ∈ [0 , A ] , we have E (cid:2) X (cid:3) ≤ E [ X ] A and of course E [ X ] ≤ p E [ X ] . Therefore,whenever α < α , the second moment constraint (4) is inac-tive in view of the first moment-constraint (3), and whenever √ α < α , the first moment constraint (3) is inactive inview of the second moment-constraint (4). Moreover, for any α ≥ / , the first-moment constraint (3) is not active, and for α ≥ / , the second-moment constraint (4). In fact, by thesymmetry of the Gaussian density, for any input X , we have I ( X ; Y ) = I ( X ′ ; Y ) for the derived input X ′ = A − X , whichhas smaller first and second moments than X if E [ X ] ≥ / A : E [ X ′ ] = A − E [ X ] ≤ / A ≤ E [ X ] , (6)and E (cid:2) X ′ (cid:3) = A − E [ X ] A + E (cid:2) X (cid:3) ≤ E (cid:2) X (cid:3) . (7)We can thus limit the optimization in (5) to random variables X with first moments not exceeding / A and, by E (cid:2) X (cid:3) ≤ E [ X ] A , with second moments not exceeding / A .As a consequence: C ( α , α , A ) = C (1 , α , A ) , ∀ α ≥ max {√ α , / } , (8)and C ( α , α , A ) = C ( α , , A ) , ∀ α ≥ max { α , / } . (9)In the remainder of the paper, we present bounds on thecapacities, and establish the exact asymptotic results in thehigh and low SNR regimes, respectively.The following functions will be used throughout the paper.For i = 0 , , , , , define: ζ i ( λ , λ ) := Z y i e − λ y − λ y d y. (10)III. T HE ASYMPTOTIC HIGH -SNR
CAPACITY
Consider first the asymptotic high-SNR regime, where α , α are fixed and A grows without bound. Theorem 1:
Depending on the parameters α , α > , theasymptotic high-SNR capacity satisfies one of the followinglimiting behaviours.1) If α ≥ and α ≥ , then both the first- and second-moment constraints are inactive and lim A →∞ (cid:18) C ( α , α , A ) − log A √ πeσ (cid:19) = 0 . (11) 2) If < α < / is such that the unique solution λ ∗ tothe equation (in λ ) λ − e − λ − e − λ = α (12a)satisfies λ ∗ ) − e − λ ∗ (cid:16) λ ∗ (cid:17) − e − λ ∗ < α , (12b)then only the first moment constraint is active and lim A →∞ (cid:18) C ( α , α , A ) − log A √ πeσ (cid:19) = log ζ ( λ ∗ ,
0) + λ ∗ α . (12c)3) If < α < / is such that the unique solution λ ∗ tothe equation (in λ ) p πλ (cid:0) (2 λ ) − − α (cid:1)(cid:20) − Q (cid:16)p λ (cid:17)(cid:21) = e − λ , (13a)satisfies p πλ ∗ α (cid:20) − Q (cid:16)p λ ∗ (cid:17)(cid:21) > − e − λ ∗ (13b)then only the second moment constraint is active and lim A →∞ (cid:18) C ( α , α , A ) − log A √ πeσ (cid:19) = log ζ (0 , λ ∗ ) + λ ∗ α . (13c)4) Else, both moment constraints are active and lim A →∞ (cid:18) C ( α , α , A ) − log A √ πeσ (cid:19) = log ζ ( λ ∗ , λ ∗ ) + λ ∗ α + λ ∗ α , (14)for λ ∗ , λ ∗ > the unique solution to the equations p πλ e λ λ (cid:18) α + λ λ (cid:19)(cid:20) Q (cid:18) λ √ λ (cid:19) − Q (cid:18) λ + 2 λ √ λ (cid:19)(cid:21) = 1 − e − ( λ + λ ) (15a)and r πλ e λ λ (cid:18) α − λ − λ λ (cid:19)(cid:20) Q (cid:18) λ √ λ (cid:19) − Q (cid:18) λ + 2 λ √ λ (cid:19)(cid:21) = 12 λ e − ( λ + λ ) (cid:18) λ λ − (cid:19) − λ λ . (15b)IV. T HE ASYMPTOTIC LOW -SNR
CAPACITY
Consider now the asymptotic low-SNR regime, where α , α are again kept fixed and A → . Proposition 2:
Given parameters α , α > , lim A ↓ C ( α , α , A ) A = max T ∈ [0 , E [ T ] ≤ α E [ T ] ≤ α Var [ T ] . (16) Proof:
The achievability follows directly from Prelov’sand Verd´u’s classical result on the mutual information of peak- constrained channels [17, Corollary 2]. The converse followsby the well-known Gaussian max-entropy bound: C ( α , α , A ) ≤ max 12 log (cid:18) Var [ X ] σ (cid:19) , where the maximization is over random variables X ∈ [0 , satisfying (2)–(4). Defining T := X/ A and using that lim t ↓ bt ) t = b , for any constant b > , establishes thedesired asymptotic converse bound. Lemma 3:
The maximization in Proposition 2 is attained bya binary random variable T ∈ { , A } : max T ∈ [0 , E [ T ] ≤ α E [ T ] ≤ α Var ( T ) = max T ∈{ , A } : E [ T ] ≤ α E [ T ] ≤ α Var ( T ) (17) Proof:
Fix T satisfying the conditions in the minimizationand construct a new random variable T ′ ∈ { , A } with p A :=Pr[ T ′ = A ] = E [ T ] A and Pr[ T ′ = 0] = 1 − p A . Notice that E (cid:2) ( T ′ ) (cid:3) = p A A = E (cid:2) T (cid:3) and E [ T ′ ] = p A A = E (cid:2) T (cid:3) A ≤ E [ T ] · AA = E [ T ] . (18)The new random variable T ′ thus also satisfies the conditionsin the maximization, and moreover it has larger objectivefunction (variance) than T because Var [ T ′ ] = E (cid:2) ( T ′ ) (cid:3) − ( E [ T ′ ]) ≥ E (cid:2) ( T ) (cid:3) − ( E [ T ]) = Var [ T ] .Combining Proposition 2 with Lemma 3 establishes thedesired low-SNR asymptotics. Theorem 4:
For any parameters α , α > : lim A ↓ C ( α , α , A ) A = p ∗ (1 − p ∗ ) , (19)where p ∗ := min { α , α , / } . Proof:
By Lemma 3: max T ∈ [0 , E [ T ] ≤ α E [ T ] ≤ α Var ( T ) = max p A ∈ [0 , p A ≤ α p A ≤ α p A (1 − p A ) . (20)Since the function t t (1 − t ) is continuous and monoton-ically increasing over [0 , / but monotonically decreasingover [1 / , , the maximum value is obtained for p A =min { α , α , / } . Plugging this into Proposition 2 establishesthe desired result.V. D ISCUSSION OF A SYMPTOTIC R ESULTS
Figure 1 illustrates the regions of ( α , α ) -pairs where boththe first and the second-moment constraints, i.e., (3) and (4),are active. At any SNR values, the first-moment constraint (3)is not active on the right of the the blue dash-dotted line, see(8), and the second-moment constraint (3) is not active abovethe red dash-dotted line, see (9).In the asymptotic low-SNR regime, only one of the twoconstraints is active, unless α = α < / in which caseboth constraints are active, or α , α ≥ / in which case noconstraint is active. Otherwise, the first-moment constraint isactive when α < min { α , / } , i.e., above the dashdotted . . . . . . . . . . α α Fig. 1: The figure illustrates the regions where the two moment constraints(3) and (4) limit the (asymptotic) capacity. blue line, and the second-moment constraint is active when α < min { α , / } , i.e., below the dashdotted blue line.This contrasts the high-SNR regime where both constraintsare inactive for α ≥ / and α ≥ / . Generally, the first-moment constraint (3) is inactive for all ( α , α ) -pairs on theright of the red solid line shown in Figure 1. The second-moment constraint (4) is inactive for all pairs lying above theblue solid line. Both the first and second-moment constraintsare thus simultaneously active only in the small white regionthat lies in between the solid blue and red lines.Overall, it can be noted that the second-moment constraintis more stringent in the low-SNR regime than in the high-SNR regime, where it is inactive for more ( α , α ) -pairs.Surprisingly, we observe that in the asymptotic regimes bothconstraints are simultaneously active in very few cases.VI. P ROOF OF T HEOREM A. Lower Bound
We first lower-bound the capacity with some simpleentropy-manipulations and by using the entropy-maximizinginput-density f ∗ X ( x ) over [0 , A ] . Under constraints (2)–(4), f ∗ X ( x ) has the form: f ∗ X ( x ) = ( A ζ ( λ , λ )) − · e − λ A x − λ A x , x ∈ [0 , A ] , (21)where the parameters λ , λ have to be chosen to satisfy Z A f ∗ X ( x ) · x d x ≤ α A , (22a) Z A f ∗ X ( x ) · x d x ≤ α A . (22b)Given the form in (21), through a simple variable substitu-tion y = x A , one can prove that (22) are equivalent to ζ ( λ , λ ) ζ ( λ , λ ) ≤ α , (23a) ζ ( λ , λ ) ζ ( λ , λ ) ≤ α , (23b) where recall that the functions ζ i , for i = 0 , , . . . , , aredefined in (10). Then, C ( α , α , A ) ≥ I f ∗ X ( X ; Y ) (24) = h f ∗ X ( Y ) − h ( Z ) ≥ h f ∗ X ( Y | Z ) − h ( Z ) (25) = h f ∗ X ( X ) − h ( Z ) (26) = E f ∗ X [ − log f ∗ X ( X )] −
12 log(2 πeσ ) (27) = log( A · ζ ( λ , λ )) + λ A E f ∗ X [ X ] + λ A E f ∗ X (cid:2) X (cid:3) −
12 log(2 πeσ ) (28) = log (cid:18) A · ζ ( λ , λ ) √ πeσ (cid:19) + λ α + λ α , (29)where all ( λ , λ ) satisfying (23) yield valid lower bounds. B. Upper bound
We turn to the duality-based upper bound with the choiceof output density f Y ( y ) = τ · f (1) Y ( y ) + (1 − τ ) · f (2) Y ( y ) , (30)where τ ∈ (0 , is a parameter that we specify later on; f (1) Y ( y ) is a probability density function over the interval I :=[0 , A ] of the form f (1) Y ( y ) = 1 A · ζ ( λ , λ ) e − λ A y − λ A y · { y ∈ I} , (31)where λ , λ ≥ are free parameters, over which we willoptimize in a latter stage; and f (2) Y ( y ) is a probability densityfunction over the rest of the real line I c := R \I : f (2) Y ( y ) = √ πσ e − y σ if y < , √ πσ e − ( y − A )22 σ if y > A . (32)For the choice in (30), the duality-based upper bound yields C ( α , α , A ) (33) ≤ E f ∗ Y [ − log f Y ( Y )] −
12 log(2 πeσ ) (34) ≤ E f ∗ Y h − log (cid:16) τ f (1) Y ( Y ) (cid:17)(cid:12)(cid:12)(cid:12) Y ∈ I i · P f ∗ Y ( I )+ E f ∗ Y h − log (cid:16) (1 − τ ) f (2) Y ( Y ) (cid:17)(cid:12)(cid:12)(cid:12) Y ∈ I c i · P f ∗ Y ( I c ) −
12 log(2 πeσ ) (35) = log A · ζ ( λ , λ ) τ · P f ∗ Y ( I )+ E f ∗ Y h − log (cid:16) τ · f (2) Y ( Y ) (cid:17)(cid:12)(cid:12)(cid:12) Y ∈ I c i · P f ∗ Y ( I c )+ (cid:18) λ A E f ∗ Y [ Y | Y ∈ I ] + λ A E f ∗ Y (cid:2) Y | Y ∈ I (cid:3)(cid:19) · P f ∗ Y ( I ) −
12 log(2 πeσ ) . (36)Following similar steps as, e.g., in [18, Eq. (209)–(226)], weobtain the following lemmas. Lemma 5:
For the Gaussian-tail distribution defined in (32): E f ∗ Y h − log (cid:16) f (2) Y ( Y ) (cid:17)(cid:12)(cid:12)(cid:12) Y ∈ I c i ≤ log √ πeσ . (37) Proof:
We have Z −∞ √ π e − ( y − x )22 (cid:18) log √ π + y (cid:19) d y = log √ π Q ( x ) + 12 x Q ( x ) + 12 Q ( x ) − x √ π e − x (38) ≤ (cid:18) log √ π + 12 (cid:19) Q ( x ) (39)ans similarly Z ∞ A √ π e − ( y − x )22 (cid:18) log √ π + ( y − A ) (cid:19) d y ≤ (cid:18) log √ π + 12 (cid:19) Q ( A − x ) . (40)Therefore, E f ∗ Y h − log (cid:16) f (2) Y ( Y ) (cid:17)(cid:12)(cid:12)(cid:12) Y ∈ I c i · P f ∗ Y ( I c )= − Z I c f ∗ Y ( y ) log (cid:16) f (2) Y ( Y ) (cid:17) d y (41) = − Z I c Z A f ∗ X ( x ) 1 √ π e − ( y − x )22 d x · log (cid:16) f (2) Y ( Y ) (cid:17) d y (42) = − E f ∗ X (cid:20)Z I c √ π e − ( y − X )22 log (cid:16) f (2) Y ( Y ) (cid:17) d y (cid:21) (43) = (cid:18) log √ πe + 12 (cid:19) E f ∗ X [ Q ( X ) + Q ( A − X )] . (44)Since P f ∗ Y ( I c ) = E f ∗ X (cid:20)Z I c √ π e − ( y − X )22 d y (cid:21) (45) = E f ∗ X [ Q ( X ) + Q ( A − X )] (46)we obtain the desired result. Lemma 6:
For the distribution in (31): E f ∗ Y [ Y | Y ∈ I ] · P f ∗ Y ( I ) ≤ E f ∗ Y [ Y ] + 1 √ π (cid:16) − e − A (cid:17) (47) = E f ∗ X [ X ] + (cid:18) − √ π e − A (cid:19) (48)and E f ∗ Y (cid:2) Y | Y ∈ I (cid:3) · P f ∗ Y ( I ) ≤ E f ∗ Y (cid:2) Y (cid:3) = E f ∗ X (cid:2) X (cid:3) + σ . (49) Proof:
The inequality in (49) follows simply because Y ≥ with probability . The inequality in (47) is provedas follows: E f ∗ Y [ Y | Y ∈ I ] · P f ∗ Y ( I )= Z A f ∗ Y ( y ) · y d y (50) = Z A Z A f ∗ X ( x ) 1 √ π e − ( y − x )22 d x · y d y (51) = E f ∗ X "Z A √ π e − ( y − X )22 y d y (52) = E f ∗ X [ X (1 − Q ( X ) − Q ( A − X ))]+ E f ∗ X (cid:20)(cid:18) √ π e − X − √ π e − ( A − X )22 (cid:19)(cid:21) (53) < E f ∗ X [ X ] + 1 √ π · (cid:16) − e − A (cid:17) , (54)where the last inequality holds because the Q ( · ) -function ispositive and the exponential function monotonically increas-ing.We continue with our upper bound. By plugging theselemmas into (36) and choosing τ = A · ζ ( λ , λ ) A · ζ ( λ , λ ) + √ πeσ , (55)we obtain: C ( α , α , A ) ≤ log A · ζ ( λ , λ ) τ · P f ∗ Y ( I ) + log √ πeσ − τ · P f ∗ Y ( I c )+ λ A E f ∗ X [ X ] + λ A (cid:0) E f ∗ X (cid:2) X (cid:3) + σ (cid:1) + λ A (cid:18) − √ π e − A (cid:19) −
12 log(2 πeσ ) (56) = log (cid:16) A · ζ ( λ , λ ) + √ πeσ (cid:17) · P f ∗ Y ( I )+ log (cid:16) A · ζ ( λ , λ ) + √ πeσ (cid:17) · P f ∗ Y ( I c )+ λ A E f ∗ X [ X ] + λ A (cid:0) E f ∗ X (cid:2) X (cid:3) + σ (cid:1) + λ A (cid:18) − √ π e − A (cid:19) −
12 log(2 πeσ ) (57) ≤ log (cid:16) A · ζ ( λ , λ ) + √ πeσ (cid:17) + λ α + λ α + λ σ A + λ A (cid:18) − √ π e − A (cid:19) −
12 log(2 πeσ ) (58) = log (cid:18) A · ζ ( λ , λ )2 πeσ (cid:19) + λ α + λ α + λ σ A + λ A (cid:18) − √ π e − A (cid:19) . (59)We can conclude that for any choice of λ , λ ≥ : lim A →∞ (cid:18) C ( α , α , A ) − log A √ πeσ (cid:19) ≤ log ζ ( λ , λ ) + λ α + λ α . (60) C. Distinction of the Four Cases
We now show that the case distinction proposed in thetheorem partitions the set of all ( α , α ) -parameters andthat the described choice of λ ∗ , λ ∗ -parameters exists in eachsubset. More specifically, we show that the proposed casedistinction coincides with the case distinction that arises whenminimizing the right-hand side of (60), i.e., the function Γ( λ , λ ) := log ζ ( λ , λ ) + λ α + λ α , (61) over the choices λ , λ > , and we show that the λ ∗ , λ ∗ values given in the theorem are the minimizers of this function.Consider the partial derivatives of this function: ∂ Γ ∂λ = − ζ ( λ , λ ) ζ ( λ , λ ) + α (62a)and ∂ Γ ∂λ = − ζ ( λ , λ ) ζ ( λ , λ ) + α , (62b)as well as its Hessian matrix H Γ( λ , λ ) := ∂ Γ( λ ,λ ) ∂λ ∂ Γ( λ ,λ ) ∂λ ∂λ ∂ Γ( λ ,λ ) ∂λ ∂λ ∂ Γ( λ ,λ ) ∂λ (63) = (cid:18) ζ ( λ , λ ) − ζ ( λ , λ ) c ( λ , λ ) c ( λ , λ ) ζ ( λ , λ ) − ζ ( λ , λ ) (cid:19) , (64)where c ( λ , λ ) := ζ ( λ , λ ) − ζ ( λ , λ ) · ζ ( λ , λ ) . (65)Since for any pair ( λ , λ ) the Hessian H Γ( λ , λ ) is a two-by-two matrix with positive trace and determinant, all itseigenvalues are positive, and the Hessian itself is positivedefinite for all ( λ , λ ) . As a consequence, the function Γ( λ , λ ) is jointly strictly convex in both arguments and theminimizer ( λ ∗ , λ ∗ ) of Γ( λ , λ ) , for λ , λ ≥ is accordinglyobtained as follows, depending on the values of α and α :1) If both partial derivatives of Γ at the origin are strictlypositive, i.e., − ζ (0 , ζ (0 ,
0) + α = −
12 + α > , (66) − ζ (0 , ζ (0 ,
0) + α = −
13 + α > , (67)then λ ∗ = λ ∗ = 0 .2) If for some λ ′ > the partial derivatives of Γ satisfy − ζ ( λ ′ , ζ ( λ ′ ,
0) + α = 0 , (68) − ζ ( λ ′ , ζ ( λ ′ ,
0) + α > , (69)then λ ∗ = λ ′ and λ ∗ = 0 .3) If for some λ ′ > the partial derivatives of Γ satisfy − ζ (0 , λ ′ ) ζ (0 , λ ′ ) + α > (70) − ζ (0 , λ ′ ) ζ (0 , λ ′ ) + α = 0 , (71)then λ ∗ = 0 and λ ∗ = λ ′ .4) If for some λ ′ , λ ′ > the partial derivatives of Γ at ( λ ′ , λ ′ ) are both zero, i.e., − ζ ( λ ′ , λ ′ ) ζ ( λ ′ , λ ′ ) + α = 0 (72) − ζ ( λ ′ , λ ′ ) ζ ( λ ′ , λ ′ ) + α = 0 , (73) then λ ∗ = λ ′ and λ ∗ = λ ′ .Since the strictly convex function Γ( λ , λ ) has exactly oneminimizing pair, combined with continuity considerations, thisconcludes the proof of the theorem.VII. A CKNOWLEDGEMENT
The authors thank Lina Mroueh for interesting discussions.R
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