On Secrecy Rate Analysis of MIMO Wiretap Channels Driven by Finite-Alphabet Input
SSUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 1
On Secrecy Rate Analysis of MIMO WiretapChannels Driven by Finite-Alphabet Input
Shafi Bashar,
Student Member, IEEE , Zhi Ding,
Fellow, IEEE , and ChengshanXiao
Fellow, IEEE
Abstract
This work investigates the effect of finite-alphabet input constraint on the secrecy rate of a multi-antenna wiretap channel. Existing works have characterized maximum achievable secrecy rate or secrecycapacity for single and multiple antenna systems based on Gaussian source signals and secrecy code.Despite the impracticality of Gaussian sources, the compact closed-form expression of mutual infor-mation between linear channel Gaussian input and corresponding output has led to broad use of theGaussian input assumption in physical secrecy analysis. For practical considerations, we study the effectof finite discrete-constellation on the achievable secrecy rate of multiple-antenna wire-tap channels.Our proposed precoding scheme converts the underlying multi-antenna system into a bank of parallelchannels. Based on this precoding strategy, we develop a decentralized power allocation algorithm basedon dual decomposition to maximize the achievable secrecy rate. In addition, we analyze the achievablesecrecy rate for finite-alphabet inputs in low and high SNR regions. Our results demonstrate substantialdifference in secrecy rate between systems given finite-alphabet inputs and systems with Gaussian inputs.
Index Terms
Wiretap channel, eavesdropping, information-theoretic security, secrecy rate, finite-alphabet input.
S. Bashar and Z. Ding are with the Department of Electrical and Computer Engineering, Univ. of California, Davis, CA,95616, USA. e-mail: { shafiab, ding } @ece.ucdavis.edu.C. Xiao is with the Department of Electrical and Computer Engineering, Missouri University of Science and Technology,Rolla, MO, 65409, USA. e-mail: [email protected] material is based upon work supported by the National Science Foundation under Grants No. 0520126. September 17, 2018 DRAFT a r X i v : . [ c s . I T ] J a n UBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 2
I. I
NTRODUCTION
Wireless communications, with increasing coverage and applications, are vulnerable to po-tential security compromises such as passive eavesdropping and active jamming. Traditionally,network planners have relegated system security considerations to higher network layers of theOSI protocol stack through authentication and cryptography. However, in recent years, there havebeen growing research interests in the security analysis of wireless systems from a physical layerand information theoretic perspective. In a wiretap channel environment originally introducedby Wyner [ ? ], a sender “Alice” wishes to transmit a secret message to the intended receiver“Bob” in the presence of a passive eavesdropper “Eve”. Wyner [ ? ] showed that when the Alice-to-Eve channel is degraded from the Alice-to-Bob channel, Alice can encode and send securemessages to the destination at a non-zero secrecy rate. In [ ? ], a generalization for the non-degraded broadcast channel is proposed, and in [ ? ], secrecy capacity of a Gaussian wiretapchannel is shown to be achievable by adopting a random Gaussian codebook. In [ ? ], the secrecycapacity of a multi-antenna Gaussian wiretap channel is shown to be achievable using a suitableinput covariance matrix and by encoding the message using a Gaussian random codebook.For both single- and multi-antenna Gaussian wiretap channels, the codebook that achievessecrecy capacity turns out to be Gaussian. However, such codebooks are not implementable inpractice. In real world systems, input codebook consists of finite set of equi-probable constellationpoints (e.g. M -QAM, M -PAM etc.). Therefore, in contrast to the Gaussian codebook, practicalwiretap codes must consist of finite-alphabet symbols. Because of this constraint, the achievablesecrecy rate for a finite-alphabet input scenario would differ from the secrecy rate achieved bya Gaussian codebook.A recent work [ ? ] considered the effect of M -PAM input on the secrecy rate of a Gaussianwiretap channel and provided the necessary condition for power allocation to maximize theachievable secrecy rate. In [ ? ], results were also extended to the case of parallel Gaussian wiretapchannels. In [ ? ], we have investigated the effect of finite-alphabet input on the ergodic secrecyrate of a multiple-input single-output and single-eavesdropper (MISOSE) system. To continueour progresses in this work, we investigate the effect of finite-alphabet input in a more generalsetting of a multiple-input multiple-output and multiple-eavesdropper (MIMOME) system. Thespecific contributions of the work are summarized below : September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 3 • In order to quantify the effect of finite-alphabet input on MIMOME systems, we propose theapplication of precoding matrix to transform the MIMOME channel into a bank of parallelchannels. • We propose a power allocation optimization framework based on decentralized dual decom-position technique to maximize the achievable secrecy rate of MIMOME systems with anarbitrary but known input distribution. • We provide secrecy rate analysis of MIMOME systems with finite-alphabet inputs at lowand high SNR regions. Our findings suggest that similar to the Gaussian wiretap channel,proper transmission power should be diverted at high signal-to-noise ratio (SNR) in caseof finite-alphabet input albeit with different effect.We organize the rest of the paper as follows. We begin with the system model in section II. Insection III we propose a linear precoding scheme that transforms the MIMOME wiretap channelinto a set of parallel channels. Based on this precoding scheme, we reformulate the secrecyrate problem for an arbitrary input distribution. In section IV, we develop a decentralized powerallocation algorithm based on dual decomposition that maximizes the achievable secrecy rate foran arbitrary distribution. In section V, we further consider the special case of Gaussian input andpresent a modified water-filling power allocation strategy by considering the secrecy constraint.We then extend the modified water-filling power allocation scheme to analyze the secrecy ratefor an arbitrary input distribution in both low and high SNR regions. In section VI we presentnumerical results before concluding with section VII.II. P
RELIMINARIES AND S YSTEM D ESCRIPTION
Throughout this work, we use notations tr ( . ) , det ( . ) and superscript {·} H , respectively, todenote the trace, the determinant, and the conjugate transpose of a matrix. A. System Model
We consider a MIMO (multiple-input multiple-output) wiretap system model in which thetransmitter (Alice), the intended receiver (Bob), and the passive eavesdropper (Eve), respectively,have m a (transmit), m b (receive), and m e (eavesdrop) antennas. Denote the received signals at September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 4
Bob and Eve as y b and y e , respectively. Their received signals are written as y b = H b x + n b (1) y e = H e x + n e (2)where H b ∈ C m b × m a and H e ∈ C m e × m a denote, respectively, the flat-fading MIMO channels,from Alice-to-Bob and from Alice-to-Eve. The noise n b ∈ C m b and n e ∈ C m e are zero-meanidentity matrix variance complex Gaussian random vectors independent of each other. The datasignal is x ∈ C m a transmitted by Alice in the form of x = Ws , in which W is a linearprecoding matrix. We denote s as a random vector with zero mean entries and identity correlationmatrix. We constrain the total transmission power by a peak level P T , i.e., tr { K x } ≤ P T , where K x = E { xx H } is the covariance matrix of the transmitted signal vector.The secrecy capacity of the above system is achievable by using a Gaussian random codebook[ ? ]. First, let A (cid:23) B denote that A − B is non-negative definite. The secrecy capacity is thesolution of the following optimization problemmaximize K x log det (cid:0) I + H b K x H Hb (cid:1) − log det (cid:0) I + H e K x H He (cid:1) subject to : K x (cid:23) , K x = K H x tr ( K x ) ≤ P T . (3)In order to realize an achievable secrecy rate for an arbitrary input signaling, we will generalizethe objective function. Instead of the optimal Gaussian signaling as used in the above optimizationproblem, we replace the objective function with the more general form of I ( x ; y b ) − I ( x ; y e ) ,where I ( x ; y ) represents the mutual information between the input and output vectors x and y .The optimization problem described in (3) is a non-convex optimization problem (exceptfor the special case of m b = m e = 1 [ ? ], [ ? ]). Even for simple cases, the objective functionpossesses a number of local maxima. Therefore, the optimum value of K x is in general notknown. In particular, for an arbitrary input signaling without any closed-form mutual information,solving the above optimization problem can be a difficult task. Thus, instead of solving theabove optimization problem, we consider a particular linear precoding scheme similar to the oneprovided in [ ? ]. The proposed linear precoding scheme first transforms the MIMOME channelinto a bank of parallel channels. This step allows us to gain a better understanding of theeffect of finite-alphabet input on the secrecy of the system. It enables us to gain better insights September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 5 into future system implementation. Such linear precoding scheme is justifiable from practicalsystem implementation perspective. In addition, at high SNR, such precoding scheme is knownto achieve the capacity of a MIMOME system [ ? ]. B. PreliminariesDefinition 1:
Similar to [ ? ], we define the following subspaces S b = null ( H b ) ⊥ ∩ null ( H e ) S b,e = null ( H b ) ⊥ ∩ null ( H e ) ⊥ S e = null ( H b ) ∩ null ( H e ) ⊥ S n = null ( H b ) ∩ null ( H e ) . (cid:3) In fact, subspace S b corresponds to the class of input with non-zero gain towards the directionof Bob only. Subspace S b,e corresponds to the class of input with non-zero gain in the directionof both Bob and Eve. S e corresponds to the class of input with non-zero gain in the directionof Eve only. Finally, S n is the subspace with non-zero gain in the direction occupied by neitherBob nor Eve. Define k = rank (cid:18)(cid:104) H Hb H He (cid:105) H (cid:19) and hence dim ( S n ) = m a − k . In addition,we define, r = dim ( S b ) and s = dim ( S r,e ) . Therefore, dim ( S e ) = k − r − s . Definition 2:
We recall the following definition of generalized singular value decomposition(GSVD) [ ? ], [ ? ] that we will use for our analysis. The GSVD of the pair ( H b , H e ) takes thefollowing form H b = Ψ b Σ b (cid:104) k m a − k Ω − (cid:105) Ψ Ha H e = Ψ e Σ e (cid:104) k m a − k Ω − (cid:105) Ψ Ha where Ψ a ∈ C m a × m a , Ψ b ∈ C m b × m b , and Ψ e ∈ C m e × m e are unitary matrices. Ω ∈ C k × k is anon-singular matrix. Σ b ∈ C m b × k and Σ e ∈ C m e × k have the following form September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 6 Σ b = k − r − s s rm b − r − s s b r Σ e = k − r − s s rk − r − s I 0 0 s e m e − k + r D b = diag ( { b , . . . , b s } ) and D e = diag ( { e , . . . , e s } ) are diagonal real matrices with dimension s × s . The diagonal entries of D b and D e are arranged in the following orders: < b ≤ . . . b s < > e ≥ . . . ≥ e s > and b i + e i = 1 , for i = 1 , . . . , s. (cid:3) III. S
ECRECY R ATE P ROBLEM F ORMULATION
A. Proposed Precoding Strategy
Given the above two definitions in hand, we propose the following precoding matrix W = Ψ a BP / (4)where, B is defined as follows B = k m a − kk Ω 0 m a − k (5) P = diag ( { p , . . . , p m a } ) is a diagonal power allocation matrix. The proposed precoding matrixin Eq. (4) is similar to the precoding strategy defined in [ ? ]. However, unlike the precoding September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 7 ˜ H b = k − r − s s r m a − km b − r − s s diag (cid:0)(cid:8) b √ p k − r − s +1 , . . . , b s √ p k − r (cid:9)(cid:1) k diag (cid:0)(cid:8) √ p k − r +1 , . . . , √ p k (cid:9)(cid:1) (10) ˜ H e = k − r − s s m a − k + rk − r − s diag (cid:0)(cid:8) √ p , . . . , √ p k − r − s (cid:9)(cid:1) s diag (cid:0)(cid:8) e √ p k − r − s +1 , . . . , e s √ p k − r (cid:9)(cid:1) m e − k + r (11)scheme in [ ? ], the diagonal elemens { p i } in the power allocaion matrix can have differentvalues. As will be evident in the subsequent sections, such differentiation is extremely importantdue to the finite nautre of the input constellations considered in this work.By using the above precoding strategy, the system equations of (1), (2) become y b = Ψ b Σ b (cid:104) k m a − k I 0 (cid:105) P / s + n b (6) y e = Ψ e Σ e (cid:104) k m a − k I 0 (cid:105) P / s + n e . (7)Pre-multiplying (6) and (7) with Ψ Hb and Ψ He , respectively, we have the following equivalentequations ˜ y b = ˜ H b s + ˜ n b (8) ˜ y e = ˜ H e s + ˜ n e , (9)where we use the notations ˜ y b = Ψ Hb y b , ˜ y e = Ψ He y e , ˜ n b = Ψ Hb n b and ˜ n e = Ψ He n e . The newequivalent channel matrices ˜ H b = Σ b (cid:104) I 0 (cid:105) P / and ˜ H e = Σ e (cid:104) I 0 (cid:105) P / are specified inEqs. (10) and (11).From Eqs. (10), (11), we observe that the new system equations (8), (9) in fact transformthe MIMOME system (1), (2) into a bank of parallel channels. Fig. 1 shows the resulting September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 8 parallel channel model. In this parallel channel model, input symbols s , . . . , s k − r − s are onlyobserved by Eve, symbols s k − r − s +1 , . . . , s k − r are received by both Bob and Eve, whereas symbols s k − r +1 , . . . , s k are only received by Bob. Finally, symbols s k +1 , . . . , s m a are lost by receivers ofboth Bob and Eve. B. Reformulation of Secrecy Rate Problem
In this section, we relax the secrecy capacity problem of Eq. (3) using the precoding matrixpresented in Sec. III-A. We generalize the problem into an achievable secrecy rate problem forarbitrary input distribution. To this end, we present the following proposition.
Proposition 1:
Define I ( γ ) = I (cid:0) s ; √ γ s + n (cid:1) , ν = k − r − s , p = { p , . . . , p m a } , { ω i } = diag (cid:0) Ω H Ω (cid:1) . When input s of a MIMOME system is a random vector with zero mean entriesand identity correlation matrix, by using the precoding matrix W defined in Eq. (4), we canachieve the following secrecy rate for an arbitrary distribution of s maximize p (cid:88) i : b i >e i (cid:20) I (cid:18) b i ω ν + i p ν + i (cid:19) − I (cid:18) e i ω ν + i p ν + i (cid:19)(cid:21) + k (cid:88) j = k − r +1 I (cid:18) ω j p j (cid:19) subject to : (cid:88) i : b i >e i p ν + i + k (cid:88) j = k − r +1 p j ≤ P T (12) Proof:
See Appendix A.We observe that, the application of the proposed precoding matrix can transform the MIMOMEproblem into a distributed secrecy rate problem for a bank of parallel channels. We note that,for a given alphabet set, in general the above optimization problem should be jointly optimizedover the input probability distribution and the power allocation. However, practical modulationconstellations are generally constrained to be equi-probable. Therefore, here we will considerequi-probable input alphabet and focus on power allocation optimization. Next we propose apower allocation algorithm to solve the above optimization problem.IV. P
OWER ALLOCATION ALGORITHM FOR ARBITRARY INPUT DISTRIBUTION
Even though the above optimization problem is convex for Gaussian input, for an arbitraryinput distribution, this is in general not the case. In addition, the lack of closed-form expressionfor mutual information makes the problem even more difficult to solve. In order to find an efficient
September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 9 sub-optimal solution, we will revert to the decomposition technique. We note that, without thesum power constraint, the optimization problem (12) can be decoupled into a number of parallelproblems each involving only one variable p i . However, the sum power constraint compels us tosolve a larger optimization problem jointly involving multiple variables. For problem such as (12)involving a complex constraint, a dual decomposition method [ ? ], [ ? ] based on the Lagrangianof the objective function enables us to readily decompose the problem into a number of parallelsub-problems each involving a single variable. These subproblems are linked through a masterproblem that updates the dual variable during each iterations of the subproblems. By introducinga dual variable µ , we can write the Lagrangian of the optimization problem (12) by relaxing thecoupling constraint as followsmaximize p (cid:88) i : b i >e i (cid:20) I (cid:18) b i ω ν + i p ν + i (cid:19) − I (cid:18) e i ω ν + i p ν + i (cid:19)(cid:21) + k (cid:88) j = k − r +1 I (cid:18) ω j p j (cid:19) − µ (cid:32) (cid:88) i : b i >e i p ν + i + k (cid:88) j = k − r +1 p j (cid:33) + µP T subject to : p i ≥ . (13)Note that, the above optimization problem is decoupled in terms of p i . We obtain the followingtwo subproblems. Subproblem 1 : for all i such that b i > e i maximize p i ≥ (cid:20) I (cid:18) b i ω ν + i p ν + i (cid:19) − I (cid:18) e i ω ν + i p ν + i (cid:19)(cid:21) − µp ν + i (14) Subproblem 2 : for all j = k − r + 1 , . . . , k maximize p i ≥ I (cid:18) ω j p j (cid:19) − µp j (15)Subproblems 1 and 2 are linked through the master problem which is the dual optimizationproblem of (12). The master problem updates the value of the dual variables µ . Let p ∗ i denotethe solution found from the subproblems. The master problem can be written as follows Master Problem : minimize µ ≥ (cid:88) i : b i >e i (cid:20) I (cid:18) b i ω ν + i p ∗ ν + i (cid:19) − I (cid:18) e i ω ν + i p ∗ ν + i (cid:19)(cid:21) + k (cid:88) j = k − r +1 I (cid:18) ω j p ∗ j (cid:19) − µ (cid:32) (cid:88) i : b i >e i p ∗ ν + i + k (cid:88) j = k − r +1 p ∗ j (cid:33) + µP T . (16) September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 10
Subproblem 2 is a convex problem and can be solve optimally. However, in general subproblem1 is not convex. Even though subproblems 1 and 2 only involve single variable, in most situations,a closed form or analytic expression for the mutual information is not known for an arbitraryinput distribution. In order to solve subproblems 1 and 2, we resort to a recent result on finite-alphabet research [ ? ], [ ? ] that relates the mutual information and the minimum mean squareerror (MMSE) at the receiver through d I ( ρ ) dρ = mmse ( ρ ) . (17)Note that the function “mmse” for different discrete constellations (e.g., M -PSK, M -QAM etc.,where M is the number of constellation points) has been given in [ ? ]. Using Eq. (17), theoptimum value of p i in subproblems 2 can be solved from the following equations ω j mmse (cid:18) ω j p ∗ j (cid:19) − µ = 0 , for j = k − r + 1 , . . . , k. (18)Eq. (18) can be further expressed as follows p ∗ j = ω j mmse − (min { , µ ω j } ) , for j = k − r + 1 , . . . , k (19)where we used the fact that mmse − (1) = 0 .For subproblem 1, we can derive the following necessary condition for optimality b i ω ν + i mmse (cid:18) b i ω ν + i p ∗ ν + i (cid:19) − e i ω ν + i mmse (cid:18) e i ω ν + i p ∗ ν + i (cid:19) − µ = 0 , for i : b i > e i (20)Next, we propose a sufficient condition for the optimality of the solution of Eq. (20). In thisregard, we define the following MMSE difference function f mmseD ( p, b i , e i , ω ν + i ) = b i ω ν + i mmse (cid:18) b i ω ν + i p ∗ ν + i (cid:19) − e i ω ν + i mmse (cid:18) e i ω ν + i p ∗ ν + i (cid:19) . (21)We note that, a similar condition has been proposed in [ ? ] [see Theorem 6 in [ ? ]] for a parallelGaussian wiretap channel with M -PAM inputs. Proposition 2:
If the MMSE difference function in Eq. (21) admits a unique zero p (cid:48) and isstrictly monotonically decreasing for ≤ p ≤ p (cid:48) , then the optimal solution (cid:8) p ∗ ν + i (cid:9) of Eq. (20)can be given as followswhen b i ≤ e i September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 11 p ∗ ν + i = 0 when b i > e i p ∗ ν + i = p (cid:48) if µ = 0 b i ω ν + i mmse (cid:18) b i ω ν + i p ∗ ν + i (cid:19) − e i ω ν + i mmse (cid:18) e i ω ν + i p ∗ ν + i (cid:19) = µ, < p < p (cid:48) if µ > . Proof:
See Appendix B.In following proposition, we describe a condition for the MMSE difference function to admita unique zero solution.
Proposition 3: If g ( ρ ) = ρ mmse ( ρ ) is a strictly uni-modal function and b i > e i , then theMMSE difference function f mmseD ( p, b i , e i , ω ν + i ) admits a unique zero solution for p > . Proof:
See Appendix C.In Figure 6, we present the plot of g ( ρ ) = ρ mmse ( ρ ) vs. ρ for BPSK, QPSK, -QAM, and -QAM input constellations. We observe that g ( ρ ) of all four constellations shows strictly uni-modality in the region of interest. Furthermore, in Figures 7, 8, 9, and 10, we illustrate graphicallythe MMSE difference function f mmseD ( p, a ) vs. power p for various values of a = e i (cid:30) b i . Here,we also assume ω ν + i = 1 without any loss of generality. We observe that, the MMSE differencefunction exhibits strictly monotonically decreasing behavior in the range ≤ p ≤ p (cid:48) for allfour constellations. Based on Propositions 2 and 3, and Figures 6-10, we see that the optimalitycriterion described in proposition 2 holds for common constellations of BPSK, QPSK, -QAM,and -QAM. Therefore, the solution of subproblem 1 in Eq. (19) will be unique for at leastthese four constellations.The combined algorithm to solve the optimization problem (12) is presented in Algorithm (1). Algorithm 1
Dual decomposition algorithm for (12)1) Initialize dual variable µ ≥ .2) Find solution of subproblem 1 and 2 by solving (20) and (19), respectively.3) Update dual variable as µ = (cid:34) µ + α (cid:32) (cid:88) i : b i >e i p ν + i + k (cid:88) j = k − r +1 p j − P T (cid:33)(cid:35) +
4) Go to Step 2 until stopping criterion is reached.We note that the master problem in Eq. (16) is differentiable with respect to the dual variable µ . Therefore, in Step 3 of the above algorithm, we use the gradient method to update the dual September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 12 variable. Here, the parameter α > denotes an appropriate stepsize, which can be either aconstant or time-varying. In our simulation, we choose a fixed stepsize α . If the stepsize issufficiently small, then the solution of the above algorithm will converge to the solution of theoptimal dual variable µ ∗ . A detail description on the choice of stepsize and stopping criterioncan be found in [ ? ], [ ? ] and the references therein. If the original optimization problem in Eq.(12) is convex, then the duality gap will be zero. Therefore, the solution of the dual problem inEq. (16) will also provide the optimal solution of the original problem. However, if the originalproblem is not convex, then there exists a positive duality gap and the solution of Eq. (16) willbe a suboptimal solution of the original problem.V. S ECRECY R ATE A NALYSIS OF
MIMOME S
YSTEM
A. Gaussian Input Case
In this section, we will present the power allocation problem for the special case of Gaussianinput distribution based on the framework considered above. Even though the result for Gaussianinput is well known [ ? ], [ ? ], [ ? ], results in this section will provide additional insight for thefinite-alphabet input scenarios to be considered later.For Gaussian input, the MMSE equation is simplymmse ( γ ) = 11 + γ . (22)Let us denote { p g i } as the optimum power allocation for Gaussian input. Based on Eq. (22),solutions of subproblem 1 can be found by solving the following equation b i e i b i − e i ω ν +1 (cid:0) p g ν +1 (cid:1) + 1 b i − e i p g ν +1 + (cid:20) ω ν +1 b i − e i − µ (cid:21) = 0 . (23)Thus, the optimal power allocation is p g ν + i = , if µ ≤ ω ν + i b i − e i (cid:115)(cid:18) ω ν + i b i e i (cid:19) + 4 ω ν + i b i e i ( b i − e i ) (cid:18) µ − ω ν + i b i − e i (cid:19) − ω ν + i b i e i , if µ > ω ν + i b i − e i . (24)Similarly, based on Eq. (22), solution of subproblem 2 can be found as p g i = , if µ ≤ ω i µ − ω i , if µ > ω i . (25) September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 13
We notice that, the solution of subproblem 2 presented in Eq. (25) can also be obtained fromEq. (23) for by replacing b i with and e i with . In the absence of eavesdropper, the solution ofthis simplified problem is the famous water-filling solution as given in Eq. (25), where the baselevel is ω i and water level is the inverse of the dual variable µ . When the security constraintis present, however, the problem takes on an interesting structure. In such case, we can stillconsider the water level as µ . However, we will use ω ν + i b i − e i as a base level to take into accountthe additional constraint due to secrecy. For the parallel channels j = k − r + 1 , . . . , k , withcomponents only towards Bob’s direction, the base level will still be ω j (since, b j − ν = 1 and e j − ν = 0 ). Similar to water-filling, we will allocate power only when the water level is abovethe base level. However, the power level in this case will not be the difference between the waterlevel and base level, i.e. µ − ω ν + i b i − e i . Instead, it will be a non-linear function of the differenceas given in Eq. (24). As a result, the achievable secrecy rate can be written as R g s = (cid:88) i : b i >e i log b i ω ν + i p g ν + i e i ω ν + i p g ν + i + k (cid:88) j = k − r +1 log (cid:18) ω j p g j (cid:19) . (26) B. Low SNR Approximation1) Second order optimal signaling:
For second order optimal signaling [ ? ], the first and thesecond order derivatives of the mutual information achieved at zero SNR matches with thoseachieved using Gaussian input. In general, quadratic symmetric signaling such as QPSK or anyother signaling distribution that can be written as a mixture of QPSK (i.e., M -QAM, for M ≥ )are second order optimal. For second order optimal signaling, the low SNR approximation ofMMSE (i.e., the first derivative of mutual information) is the same as that of Gaussian signaling.Hence, we can use the same water-filling power allocation solution as presented in Eq. (24),(25) in Section V-A.
2) Non-second order optimal signaling: -D signaling schemes such as BPSK and M -PAMare not second order optimal. A low SNR approximation of such signaling is given in [ ? ] asmmse ( ρ ) = 1 − ρ + o (cid:0) ρ (cid:1) . (27)Based on the above equation, low SNR power allocation (cid:8) p low i (cid:9) for non-second order optimalsignaling are given below in two cases: September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 14 p low ν + i = , when µ ≤ ω ν + i b i − e i ω ν + i µ (cid:18) µ − ω ν + i b i − e i (cid:19) when µ > ω ν + i b i − e i , { i : b i > e i } (28) p low i = , when µ ≤ ω i ω i µ (cid:18) µ − ω i (cid:19) , when µ > ω i , i = k − r + 1 , . . . , k. (29)The achievable secrecy rate at low SNR can be approximated as follows R low s = (cid:88) i : b i >e i (cid:0) b i − e i (cid:1) (cid:16) p low ν + i − (cid:0) p low ν + i (cid:1) (cid:17) + k (cid:88) j = k − r +1 (cid:16) p low j − (cid:0) p low j (cid:1) (cid:17) . (30)We notice that, similar to the case of Gaussian and second-order optimal signaling, we canuse a water-filling strategy with water level µ and base level ω ν + i b i − e i and assuming b j − ν = 1 and e j − ν = 0 for j = k − r + 1 , . . . , k . The power level is still function of µ − ω ν + i b i − e i , which is the difference between water level and base level. The functions are now slightlydifferent in (Eq. (28) and (29)). C. High SNR Approximation1) rank ( H e ) = m a : In this case dim ( S b ) = r = 0 . For the power constraint in Eq. (12),we get the following complementary slackness condition µ (cid:32) (cid:88) i : b i >e i ω ν + i p ν + i − P T (cid:33) = 0 . (31)Based on proposition 2, the optimal power p ∗ for maximum secrecy satisfies p ∗ ≤ p (cid:48) . Therefore,at very high SNR when P T → ∞ , secrecy rate for finite-alphabet input is maximized by using afraction of the total available power. The power constraint inequality becomes a strict inequality.In the above complementary slackness condition, we attain µ = 0 . Denoting high SNR powerallocation as p high i , we re-write Eq. (20) as b i mmse (cid:16) b i p high ν + i (cid:17) = e i mmse (cid:16) e i p high ν + i (cid:17) . (32) September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 15
In [ ? ], based on the sub-optimum estimator ˆ s ( y, ρ ) = arg min s k (cid:12)(cid:12) y − √ ρ s k (cid:12)(cid:12) the followingMMSE approximation at high SNR is found:mmse ( ρ ) ≈ K exp (cid:26) − d ρ (cid:27) , (33)in which K is a constant and d is the minimum distance between two signaling points in thediscrete unit variance input constellation.Reference [ ? ] also provided a table containing formula for calculating d of different finite-alphabet constellations. Using Eq. (33), we obtain the following high SNR approximation ofpower allocation p high ν + i = log σ i d b i − e i ) ω ν + i , for i such that b i > e i . (34)In [ ? ], it was observed that, at high SNR regime, the power allocation for parallel Gaussianchannel with finite-alphabet demonstrates a channel inversion characteristic. In other words,stronger channels receive less power allocation. This is because the mutual information of a M -ary constellation cannot exceed log M bits/s/Hz, Thus, there is little incentive to allocate morepower to a channel once the mutual information is near saturation. Instead, additional power isbetter allocated to weaker channels for higher rate. In Eq. (34), we observe a similar channelinversion phenomenon, although in this case, the effective channel ( b i − e i ) ω ν + i . This is in sharpcontrast to both the water-filling power allocation at low SNR regime and the power allocationfor Gaussian input (see Eq. (28)), where the power allocation was proportional to the effectivechannel.
2) rank ( H e ) < m a : Based on Eq. (33), mmse( ρ ) decays exponentially to zero as ρ → ∞ .From Eq. (18), therefore, we find that µ → as P T → ∞ . Hence, for the subset of parallelchannels, i : b i > e i , Eq. (34) will still provide a high SNR approximation of power allocation.The subset of parallel channels j = k − r + 1 , . . . , k are in the subspace S b . For these channels,there are no components in Eve’s subspace and a similar channel inversion style power allocationcan be achieved as presented in [ ? ] with an effective channel ω j . For an M -ary constellation,mutual information for these channels will become close to log M at high SNR.In summary, for the subset of parallel channels i : b i > e i , a channel inversion based powerallocation based on the effective channel b i − e i ω ν + i will be performed, whereas for the set ofparallel channels j = k − r + 1 , . . . , k , a channel inversion type power allocation based on September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 16 effective channel gains /ω j will be performed. A high SNR approximation of the achievablesecrecy rate for the case rank ( H e ) < m a is given by R high s ≈ r log M + (cid:88) i : b i >e i (cid:20) I (cid:18) b i ω ν + i p high ν + i (cid:19) − I (cid:18) e i ω ν + i p high ν + i (cid:19)(cid:21) . (35)VI. N UMERICAL R ESULTS
A. Transmitter with Accurate Eavesdropper CSI
Without any loss of generality, we assume equal noise power level at receivers of both Boband Eve. We also assume that Alice has full CSI of both Bob and Eve. Our numerical resultsaverage over channel realization, where each entry of both Bob’s and Eve’s channel matricesis i.i.d. complex random Gaussian variable with zero mean and unit variance.In Fig. 2, we present numerical test results for a × × MIMOME system. In this test, weensure that each realization of H e is non-singular, i.e., rank ( H e ) = 5 = m a . Since, rank ( H e ) = m a , there is no parallel channel only directed at Bob. In addition to result for the proposed powerallocation (PA) algorithm, we also present results obtained using the water-filling PA of SectionV-A and results from equal power over all channels (uniform PA) [ ? ]. We also present high SNRapproximation results for every tested constellation as well as the low SNR approximation resultfor BPSK. For other constellations (QPSK, 16-QAM and 64-QAM), low SNR approximationgives the same result as the water-filling PA.As seen in Fig. 2, at high SNR, power allocations according to water-filling and uniformstrategies would drop the secrecy rate to almost zero. This result is intuitive. For finite-alphabet,the achievable mutual information at high power approaches the saturation value of log M .For Gaussian input, however, the mutual information or the capacity increases monotonicallywith increasing power. Since both water-filling and uniform strategies assume a Gaussian inputdistribution, at high SNR both schemes would use more power to transmit signals. This strategydrops the secrecy rate asymptotically to zero as the difference in mutual information betweenAlice-to-Bob and Alice-to-Eve narrows with at very high SNR. Fig. 2 also indicates that the highSNR approximation analysis closely matches the secrecy rate at high SNR regime. Similarly, thesecrecy rate at low SNR is also closely approximated by the low-SNR approximation analyticalresult.Fig. 3 presents test results for a × × MIMOME system. Because in this case rank ( H e ) Thus far, our analysis assumes that Alice possesses full channel information of Eve. In practice,however, Eve’s CSI or even the presence of a passive Eve is difficult to determine. Therefore,in this section, we will numerically evaluate the scenario when Alice only has access to partial(statistical) information regarding Eve’s channel state. In particular, let Eve’s channel consistsof H e = ˆ H e + E e . Here, ˆ H e is Eve’s mean CSI known to Alice whereas E e is the CSI uncertainty which is modeledas zero mean white Gaussian noise with variance σ e , i.e., E e ∼ CN ( , σ e I ) .In this case, if Alice performs power allocation based on the known observation ˆ H e bydisregarding the uncertainty, the power allocation may not be optimal. In fact, Alice may even September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 18 lose secrecy. In the following proposition, we present the achievable ergodic secrecy rate for agiven power allocation p . Proposition 4: For a given power allocation p based on the known observation ˆ H e withoutconsidering the uncertainty, the achievable ergodic secrecy rate R sec ( p ) can be given as follows R sec ( p ) = (cid:88) i : b i >e i (cid:34) I (cid:18) b i ω ν + i p ν + i (cid:19) − E (cid:40) I (cid:32) ( e i + ˜ e ν + i ) σ ν + i ω ν + i p ν + i (cid:33)(cid:41)(cid:35) + k (cid:88) j = k − r +1 (cid:20) I (cid:18) ω j p j (cid:19) − E (cid:26) I (cid:18) ˜ e j σ j ω j p j (cid:19)(cid:27)(cid:21) (36)Here, ˜ e i ∼ CN (0 , σ e ω i ) , i = 1 , . . . , k are i.i.d. random. Also, σ i = 1 + σ e (cid:32) k (cid:88) (cid:96) =1 p (cid:96) − p i (cid:33) . Proof: See Appendix D.From Eq. (36) we notice that, data symbols s k − r +1 , . . . , s k - previously only observed by Bob,are also seen by Eve now. In addition, for the symbols s ν + i , . . . , s ν + s - observed by both Boband Eve, an additional uncertainty term ˜ e ν + i is added to the mutual information of Eve.In Fig. 4 and 5, we present test results of achievable ergodic secrecy rate for different valuesof variance of channel uncertainty, for × × and × × MISOME systems, respectively.Here, Alice is only aware of the mean CSI ˆ H e and use this CSI for power control withoutconsidering the uncertainty component E e . We present test results for different values of availabletransmission power. Both Fig. 4 and 5 indicate that the achievable ergodic secrecy rate decreaseswith larger channel uncertainty. In addition, we observe from Fig. 4 that the achievable ergodicsecrecy rate plots for transmission power P T ≥ stays the same. This is because Alice does notrequire all available power to transmit at higher SNR. Hence, the power allocation stays the sameat higher SNR. However, in Fig. 5, we observe that the achievable ergodic secrecy rate in factdecreases with larger transmission power for P T > dB. In this case, rank (cid:16) ˆ H e (cid:17) < m a . WhenAlice only uses this information for power control without considering the uncertainty, Alice willlikely allocate more power to the bank of parallel channels which only has components towardBob. At high SNR, Alice will allocate more powers to these channels. However, as shown inEq. (36), because of channel uncertainty E e , Eve now also possesses components along thesechannels. In other words, Eve can also receive signals from these channels. As SNR grows large,the mutual information for finite alphabet saturates to log M . As a result, Eve can receive nearlyfull data information in these channel and consequently, the achievable ergodic secrecy rate will September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 19 decrease at high SNR.We note that, in order to maximize the achievable ergodic secrecy rate under channel uncer-tainty, one needs to solve the following optimization problemmaximize p s (cid:88) i =1 (cid:34) I (cid:18) b i ω ν + i p ν + i (cid:19) − E (cid:40) I (cid:32) ( e i + ˜ e ν + i ) σ ν + i ω ν + i p ν + i (cid:33)(cid:41)(cid:35) + k (cid:88) j = k − r +1 (cid:20) I (cid:18) ω j p j (cid:19) − E (cid:26) I (cid:18) ˜ e j σ j ω j p j (cid:19)(cid:27)(cid:21) subject to : s (cid:88) i =1 p ν + i + k (cid:88) j = k − r +1 p j ≤ P T . Investigation of this optimization problem is beyond the scope of this work. However, in [ ? ], apower allocation algorithm to maximize the achievable ergodic secrecy rate under arbitrary inputdistribution is provided for a multiple-input single-output and single-eavesdropper scenario. Anextension of the power allocation algorithm provided in [ ? ] can also be used to solve the aboveoptimization problem. VII. C ONCLUSION This work considers the effect of practical finite-alphabet inputs on the secrecy performance ofan MIMOME system. Our investigation led to the application of a precoding matrix to convertthe MIMOME system into a bank of parallel channels so as to reformulate the achievablesecrecy rate problem. We proposed a decentralized dual decomposition and a correspondingpower allocation algorithm to maximize the achievable secrecy rate based on the proposedprecoding for channel transformation. We analyzed the Gaussian input as a special case andprovided a water-filling inspired power allocation strategy. Furthermore, we derived analyticalresults of achievable secrecy rate based on approximations at low and high SNR scenarios. Ourresults show that power allocation strategy based on Gaussian input is far from optimal whenapplied blindly in finite-alphabet input situation and may in fact be very risky by driving thesecrecy rate to zero at higher SNR. A PPENDIX AP ROOF OF P ROPOSITION x can be obtained by solvingthe following optimization problem September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 20 maximize K x I ( x ; y b ) − I ( x ; y e ) subject to : K x (cid:23) , K x = K Hx tr ( K x ) ≤ P T (37)When we apply the precoding matrix W from Eq. (4), the input of the system becomes x = Ws .The objective function of the above optimization problem can be written as follows I ( x ; y b ) − I ( x ; y e ) (38) = I ( s ; H b Ws + n b ) − I ( s ; H e Ws + n e ) (39) = I (cid:16) s ; ˜ H b s + ˜ n b (cid:17) − I (cid:16) s ; ˜ H e s + ˜ n e (cid:17) (40) = (cid:34) s (cid:88) i =1 I (cid:0) b i p k − r − s + i (cid:1) + k (cid:88) j = k − r +1 I ( p j ) (cid:35) − (cid:34) k − r − s (cid:88) (cid:96) =1 I ( p (cid:96) ) + s (cid:88) i =1 I (cid:0) e i p k − r − s + i (cid:1)(cid:35) (41) = k (cid:88) j = k − r +1 I ( p j ) + s (cid:88) i =1 (cid:2) I (cid:0) b i p k − r − s + i (cid:1) − I (cid:0) e i p k − r − s + i (cid:1)(cid:3) − k − r − s (cid:88) (cid:96) =1 I ( p (cid:96) ) . (42)Note that (40) follows from (39) since linear unitary transformation of channel outputs preservesmutual information. Because (40) represents the difference in mutual information between asubset of parallel channels, we can rewrite (40) into the summation form of (41).Since s is a random vector with identity correlation matrix, K x = E (cid:2) xx H (cid:3) = WW H , wehave tr ( K x ) = (cid:80) ki =1 ω i p i . Therefore, we can reformulate the optimization problem in (37) asfollowsmaximize { p i } s (cid:88) i =1 (cid:2) I (cid:0) b i p k − r − s + i (cid:1) − I (cid:0) e i p k − r − s + i (cid:1)(cid:3) − k − r − s (cid:88) i =1 I ( p i ) + k (cid:88) i = k − r +1 I ( p i ) subject to : k (cid:88) i =1 ω i p i ≤ P T (43)Note that the above optimization problem is a distributed power allocation problem for thebank of parallel channels shown in Fig. 1, where the variables { p i } ’s are coupled throughthe total power constraint. Because I ( . ) ≥ , we have optimum power allocation p ∗ i = 0 for i = 1 , . . . , k − r − s . In addition, since I ( γ ) are monotonically increasing in γ , I ( b i p k − r − s + i ) − September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 21 I ( e i p k − r − s + i ) ≤ whenever b i ≤ e i , hence p ∗ i = 0 , for all i with b i ≤ e i . Furthermore, fromFig. 1, we observe that symbols s k +1 , . . . , s m a are transmitted towards the direction of S n and isnot part of the optimization problem (43). Hence, it will be wasteful to expend any transmissionpower to transmit these symbols. Therefore, p ∗ i = 0 for i = k +1 , . . . , m a . By change of variables, p (cid:48) = ω i p i , and then replacing p (cid:48) with p , we obtain the optimization problem in Eq. (12).A PPENDIX BP ROOF OF P ROPOSITION b i ≤ e i , the difference in mutual information between Alice-to-Bob and Alice-to-Evechannel will be negative. Hence, no secrecy is possible. As a result, the optimal power allocationshould be zero.When b i > e i , it is possible to achieve a positive secrecy rate. If, in the solution of theoriginal problem (12), the sum power constraint inequality becomes a strict inequality, then µ = 0 according to the KKT condition [ ? ]. Such condition can occur at high SNR regime.Therefore, the solution of problem (20) will be achieved by the power allocation p (cid:48) that rendersthe MMSE difference function in Eq. (21) zero. If the MMSE difference function admits aunique zero solution, then the solution of Eq. (20) will be optimal, i.e., p ∗ ν + i = p (cid:48) .If on the other hand, the sum power constraint in the original problem admits an equality,then µ > . If the MMSE difference function is monotonically decreasing for ≤ p ≤ p (cid:48) , thenthe function f mmseD ( p, b i , e i , ω ν + i ) − µ will be zero for a power allocation value p ∗ that is lessthan p (cid:48) . Now, if MMSE difference function is strictly monotonically decreasing, then the powerallocation solution will be unique. Therefore, the solution of Eq. (20) will be optimal.A PPENDIX CP ROOF OF P ROPOSITION ρ = b i p and ρ = e i p . First, we will prove that the difference function, g D ( ρ , ρ ) = g ( ρ ) − g ( ρ ) admits a unique zero for p > .Assume that the strictly unimodal function g ( ρ ) is strictly monotonically increasing for ρ ≤ m and strictly monotonically decreasing for ρ > m . When, ρ < ρ ≤ m , the difference function g D ( ρ , ρ ) cannot be zero due to the strictly monotonically increasing nature of the function g ( ρ ) . Similarly, when ρ > ρ > m , the difference function g D ( ρ , ρ ) cannot be zero due to the September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 22 strictly monotonically decreasing nature of the function g ( ρ ) . Therefore, the difference function g D ( ρ , ρ ) can be zero only when ρ < m < ρ .Suppose, for p = p (cid:48) , the difference function g D ( b i p, e i p (cid:48) ) equals zero. For p > p (cid:48) , g ( b i p ) >g ( b i p (cid:48) ) and g ( e i p ) < g ( e i p (cid:48) ) due to the strictly unimodal properties of g ( ρ ) . Hence, when ρ 00 ˜ e . . . . . . ... ... . . . ... ... . . . ... . . . ˜ e k . . . 00 0 . . . . . . ... ... . . . ... ... . . . ... . . . . . . + e . . . ˜ e k . . . e . . . ˜ e k . . . ... ... ... . . . ... ˜ e k ˜ e k . . . . . . e ( k +1)1 ˜ e ( k +1)2 . . . ˜ e ( k +1) k . . . ... ... . . . ... ... . . . ... ˜ e m e ˜ e m e . . . ˜ e m e k . . . (45)Here, ˜ e ij ∼ CN (0 , σ e ω j ) . Based on the above decomposition and replacing p i with p i ω i , the i -th entry of the vector ˜ y e can be expressed as follows ˜ y e i = ( e (cid:48) i + ˜ e i ) 1 √ ω i √ p i s i + g i + n e i (46) September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 23 Here, g i = k (cid:88) (cid:96) =1 ,(cid:96) (cid:54) = i ˜ e i(cid:96) √ ω (cid:96) √ p (cid:96) s (cid:96) and e (cid:48) i can be expressed as e (cid:48) i = , i = 1 , . . . , k − r − se i − ν , i = ν + 1 , . . . , ν + s and ν = k − r − s , i = k − r + 1 , . . . , m e (47)Assume that the eavesdropper is performing conventional decoding on each of these parallelbranch by considering the cross terms as a part of noise. Then the achievable ergodic secrecy ratefor a given power allocation p can be written as in Eq. (36). If in case, Eve employs advanceddecoding scheme, e.g. successive interference cancellation etc. then Eq. (36) will serve as aupper bound on the achievable ergodic secrecy rate. September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 24 Bob +− rk p +− rk s p s srk p −− srk s −− +−− srk p +−− srk s rk p − rk s − k p k s + k p + k s a m p a m s Alice e s e b s b Eve Fig. 1. Precoding matrix W = Ψ a BP / converts MIMOME channel to a bank of parallel channels September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 25 −10 −5 0 5 10 15 20 25 30 35 40012345678 SNR (dB) S e c r e cy R a t e Optimum PAWater−filling PAHigh SNR approx.Low SNR approx.Uniform PA BPSK QPSK 16 QAM64 QAM Gaussian Fig. 2. × × MIMOME system −10 −5 0 5 10 15 20 25 30 35 400246810121416 SNR (dB) S e c r e cy R a t e Uniform PALow SNR approx.High SNR approx.Water−filling PAOptimum PA Gaussian64 QAM16 QAMQPSKBPSK Fig. 3. × × MIMOME system September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 26 −4 −3 −2 −1 E r god i c S e c r e cy R a t e σ e2 P T = −10 dBP T = 0 dBP T = 5 dBP T = 10 dBP T = 20 dBP T = 30 dB Fig. 4. Secrecy Rate of a × × MIMOME system with partial Eve’s CSI −4 −3 −2 −1 σ e2 E r god i c S e c r e cy R a t e P T = −10 dBP T = 0 dBP T = 5 dBP T = 10 dBP T = 20 dBP T = 30 dB Fig. 5. Secrecy Rate of a × × MIMOME system with partial Eve’s CSI September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 27 ρ ρ mm s e ( ρ ) BPSKQPSK16 QAM64 QAM Fig. 6. ρ vs, ρ mmse ( ρ ) plot for BPSK, QPSK, -QAM and -QAM constellations. September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 28 p f D ( p , a ) a = 0.9a = 0.8a = 0.7a = 0.6a = 0.5a = 0.4a = 0.3a = 0.2a = 0.1 (a) p f D ( p , a ) a = 0.99a = 0.98a = 0.97a = 0.96a = 0.95a = 0.94a = 0.93a = 0.92a = 0.91 (b) p f D ( p , a ) a = 0.09a = 0.08a = 0.07a = 0.06a = 0.05a = 0.04a = 0.03a = 0.02a = 0.01 (c)Fig. 7. MMSE difference function f D ( p, a ) for different values of a = e i (cid:30) b i for BPSK September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 29 p f D ( p , a ) a = 0.9a = 0.8a = 0.7a = 0.6a = 0.5a = 0.4a = 0.3a = 0.2a = 0.1 (a) p f D ( p , a ) a = 0.99a = 0.98a = 0.97a = 0.96a = 0.95a = 0.94a = 0.93a = 0.92a = 0.91 (b) p f D ( p , a ) a = 0.09a = 0.08a = 0.07a = 0.06a = 0.05a = 0.04a = 0.03a = 0.02a = 0.01 (c)Fig. 8. MMSE difference function f D ( p, a ) for different values of a = e i (cid:30) b i for QPSK September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 30 p f D ( p , a ) a = 0.9a = 0.8a = 0.7a = 0.6a = 0.5a = 0.4a = 0.3a = 0.2a = 0.1 (a) p f D ( p , a ) a = 0.99a = 0.98a = 0.97a = 0.96a = 0.95a = 0.94a = 0.93a = 0.92a = 0.91 (b) p f D ( p , a ) a = 0.09a = 0.08a = 0.07a = 0.06a = 0.05a = 0.04a = 0.03a = 0.02a = 0.01 (c)Fig. 9. MMSE difference function f D ( p, a ) for different values of a = e i (cid:30) b i for -QAM September 17, 2018 DRAFTUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 4, 2011 31 p f D ( p , a ) a = 0.9a = 0.8a = 0.7a = 0.6a = 0.5a = 0.4a = 0.3a = 0.2a = 0.1 (a) p f D ( p , a ) a = 0.99a = 0.98a = 0.97a = 0.96a = 0.95a = 0.94a = 0.93a = 0.92a = 0.91 (b) p f D ( p , a ) a = 0.09a = 0.08a = 0.07a = 0.06a = 0.05a = 0.04a = 0.03a = 0.02a = 0.01 (c)Fig. 10. MMSE difference function f D ( p, a ) for different values of a = e i (cid:30) b i for -QAM-QAM