Upper and Lower Bounds on the Capacity of Amplitude-Constrained MIMO Channels

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1 Upper and Lower Bounds on the Capacity of Amplitude-Constrained MIMO Channels Alex Dytso, Mario Goldenbaum, Shlomo Shamai Shitz, and H. Vincent Poor Department of Electrical Engineering, Princeton University Department of Electrical Engineering, Technion Israel Institute of Technology Abstract In this work, novel upper and lower bounds for the capacity of channels with arbitrary constraints on the support of the channel input symbols are derived. As an immediate practical application, the case of multiple-input multiple-output channels with amplitude constraints is considered. The bounds are shown to be within a constant gap if the channel matrix is invertible and are tight in the high amplitude regime for arbitrary channel matrices. Moreover, in the high amplitude regime, it is shown that the capacity scales linearly with the minimum between the number of transmit and receive antennas, similarly to the case of average power-constrained inputs. I. INTRODUCTION While the capacity of a multiple-input multiple-output MIMO channel with an average power constraint is well understood [], surprisingly, little is known about the capacity of the more practically relevant case in which the channel inputs are subject to amplitude constraints. The first major contribution to this problem was a seminal work of Smith [] in which it was shown that for the scalar Gaussian noise channel with an amplitude-constrained input the capacity achieving inputs are discrete with finite support. In [3], this result was extended to peak-power-limited quadrature Gaussian channels. Using the approach of [3], in [4] the optimal input distribution was shown to be discrete for MIMO channels with an identity channel matrix and a Euclidian norm constraint on the input vector. Even though the optimal input distribution is known to be discrete, very little is known about the number or the optimal positions of the corresponding constellation points. To the best of our knowledge, the only exception is the work of [5] in which for a scalar Gaussian noise channel it was shown that two point masses are optimal for amplitude values smaller than.67 and three for amplitude values of up to.786. Using a dual capacity expression, in [6] McKellips derived an upper bound on the capacity of a scalar amplitude-constrained channel that is asymptotically tight in the high amplitude regime. By using a clever choice of an auxiliary channel output distribution in the dual capacity expression, the authors of [7] sharpened McKellips bound and extended it to parallel MIMO channels with a Euclidian norm constraint on the input. The scalar version of the upper bound in [7] has been further sharpened in [8] by yet another choice of auxiliary output distribution. In [9], asymptotic lower and upper bounds for a MIMO system were presented and the gap between the bounds was specified. In this work, we make progress on this open problem by deriving several new upper and lower bounds that hold for channels with arbitrary constraints on the support of the input distribution. We then apply them to the special case of MIMO channels with amplitude-constrained inputs. This work was supported in part by the U. S. National Science Foundation under Grants CCF and CNS , by the German Research Foundation under Grant GO 669/-, and by the European Union s Horizon 00 Research And Innovation Programme, grant agreement no

2 A. Contributions and Paper Outline Our contributions and paper outline are as follows. The problem is stated in Section II. In Section III, we derive upper and lower bounds on the capacity of a MIMO channel with an arbitrary constraint on the support of the input. In Section IV, we evaluate the performance of our bounds by studying MIMO channels with invertible channel matrices. In particular, in Theorem 8 it is shown that our upper and lower bounds are within nlogρ bits, where ρ is the packing efficiency and n is the number of antennas. For diagonal channel matrices, it is shown in Theorem 9 that the Cartesian product of pulse-amplitude modulation PAM constellations achieves the capacity to within.64n bits. Section V is devoted to MIMO channels with arbitrary channel matrices. It is shown that in the high amplitude regime, similarly to the average power-constrained channel, the capacity scales linearly with the minimum of the number of transmit and receive antennas. Section VI concludes the paper. B. Notation Vectors are denoted by bold lowercase letters, random vectors by bold uppercase letters, and matrices by bold uppercase sans serif letters e.g., x, X, X. For any deterministic vector x R n, n N, we denote the Euclidian norm of x by x. For some X suppx R n and any p > 0 we define X p p := n E[ X p ], where suppx denotes the support of X. Note that for p, the quantity in defines a norm. The norm of a matrix H R n n is defined as Hx H := sup x:x 0 x. Let S be a subset of R n. Then, VolS := dx denotes its volume. Let R + := {x R : x 0}. We define an n-dimensional ball or radius r R + centered at x R n as the set B x r := {y : x y r}. Recall that for any x R n and r R +, Vol B x r π n = Γ. n +rn For any matrix H R k n and some S R n we define Note that for an invertible H R n n we have HS := {y : y = Hx, x S}. VolHS = deth VolS. We define the maximum and minimum radius of a set S R n that contains the origin as r max S := min{r R + : S B 0 r}, r min S := max{r R + : B 0 r S}. For a given vector a = a,...,a n R n + we define Boxa := {x R n : x i a i,i =,...,n} and the smallest box containing a given set S R n as BoxS := inf{boxa : S Boxa}, respectively. Finally, all logarithms are taken to the base, log + x := max{logx,0}, Qx, x R, denotes the Q-function, and δ x y the Kronecker delta, which is one for x = y and zero otherwise. S

3 3 II. PROBLEM STATEMENT Consider a MIMO system with n t N transmit and n r N receive antennas. The corresponding n r -dimensional channel output for a single channel use is of the form Y = HX+Z, for some fixed channel matrix H R nr nt. Here and hereafter, we assume Z N0,I nr is independent of the channel input X R nt and H is known to both the transmitter and the receiver, where I nr denotes the n r n r identity matrix. Now, in all that follows let X R nt be a convex and compact channel input space that contains the origin i.e., the length-n t zero vector and let F X denote the cumulative distribution function of X. As of the writing of this paper, the capacity CX,H := max IX;HX+Z F X :X X of this channel is unknown and we are interested in finding novel lower and upper bounds. Even though most of the results in this paper hold for arbitrary X, we are mainly interested in the two most important special cases: i per-antenna amplitude constraints; that is, X = Boxa for some given a = A,...,A nt R nt +, ii n t -dimensional amplitude constraint; that is, X = B 0 A for some given A R +. Remark. Note that determining the capacity of a MIMO channel with average per-antenna power constraints is also still an open problem and has been solved for some special cases only [0] []. A. Upper Bounds III. UPPER AND LOWER BOUNDS ON THE CAPACITY To establish our first upper bound on, we need the following result [3, Th. ]: Lemma. Maximum Entropy Under p-th Moment Constraint Let n N and p 0, be arbitrary. Then, for any U R n such that hu < and U p <, we have hu nlog k n,p n p U p, where k n,p := πe p p p Γ n n + n p Γ n + n Theorem. Moment Upper Bound For any channel input space X and any fixed channel matrix H, we have CX,H C M X,H := inf p>0 n rlog where x HX is chosen such that x = r max HX. k nr,p πe Proof: Expressing in terms of differential entropies results in. n p r x+z p, CX,H = max hhx+z hz F X :X X a max n k nr,p rlog n p F X :X X πe r HX+Z p b = n r log k nr,p πe Considering a real-valued channel model is without loss of generality. n p r max F X :X X HX+Z p, 3

4 4 where a follows from Lemma with the fact that hz = nr logπe and b from the monotonicity of the logarithm. Now, notice that HX+Z p is linear in F X and therefore it attains its maximum at an extreme point of the set F X := {F X : X X} i.e., the set of all cumulative distribution functions of X. As a matter of fact [4], the extreme points of F X are given by the set of degenerate distributions on X ; that is, {F X y = δ x y,y X} x X. This allows us to conclude max HX+Z p = max Hx+Z p. F X :X X x X Observe that the Euclidian norm is a convex function, which is therefore maximized at the boundary of the set HX. Combining this with 3 and taking the infimum over p > 0 completes the proof. The following theorem provides two alternative upper bounds that are based on duality arguments. Theorem. Duality Upper Bounds For any channel input space X and any fixed channel matrix H c nr d+ Vol B 0 d where and CX,H C Dual, X,H := log d := r max HX, c nr d := n r CX,H C n r Dual, X,H := where a = A,...,A nr such that Boxa = BoxHX. nr i πe nr Γ nr nr Γ n r d i,, 4 log + A i, 5 πe Proof: Using duality bounds, it has been shown in [7] that for any centered n-dimensional ball of radius r R + c n r+ Vol B 0 r where c n r := n Now, observe that n i max IX;X+Z log F X :X B 0 r Γ n n Γ n ri. πe n CX,H = max F X :X X hhx+z hhx+z HX = max F X :X X IHX;HX+Z, 6 = max I X; X+Z 7 F X: X HX a max F X: X B 0 d,d:=r maxhx b log I X; X+Z c nr d+ Vol B 0 d πe nr Here, a follows from enlarging the optimization domain and b from using the upper bound in 6. This proves 4..

5 5 In order to show the upper bound in 5, we proceed with an alternative upper bound to 7: CX,H = max I X; X+Z F X: X HX a b max I X; X+Z F X: X BoxHX n r max F X: X BoxHX c n r = d I X i ; X i +Z i max I X i ; X i +Z i F Xi : X i A i n r log + A i, πe where the inequalities follow from: a enlarging the optimization domain; b single-letterizing the mutual information; c choosing individual amplitude constraints A,...,A nr =: a R nr + such that Boxa = BoxHX; and d using the upper bound in 6 for n =. This concludes the proof. B. Lower Bounds A classical approach to bound a mutual information from below is to use the entropy power inequality EPI. Theorem 3. EPI Lower Bounds For any fixed channel matrix H and any channel input space X with X absolutely continuous, we have n r CX,H C EPI X,H := max + F X :X X log nr hhx. 8 πe Moreover, if n t = n r = n, H R n n is invertible, and X is uniformly distributed over X, then CX,H C EPI X,H := n log + deth nvolx n. 9 πe Proof: By means of the EPI we conclude nr hhx+z nr hhx + nr hz, max nr CX,H +πe nr hhx F X :X X, 0 which finalizes the proof of the lower bound in 8. To show the lower bound in 9, all we need is to recall that hhx = hx+log deth, which is maximized for X uniformly distributed over X. But if X is uniformly drawn from X, we have n hhx = VolHX n = deth n VolX n, which completes the proof. The results in [] [4] suggest that the channel input distribution that maximizes might be discrete. Therefore, there is a need for lower bounds that, unlike the bounds in Theorem 3, rely on discrete inputs.

6 6 Remark. We note that the problem of finding the optimal input distribution of a general MIMO channel with an amplitude constraint is still open. The technical difficulty relies on the fact that the identity theorem from complex analysis, a key tool in the method developed by Smith [] for the scalar case and later used by [5] for the MIMO channel, does not extend to R n and C n. The interested reader is referred to [6] for a detailed discussion on this issue with examples of why the identity theorem fails in the MIMO setting. Theorem 4. Ozarow-Wyner Type Lower Bound Let X D suppx D R nt be a discrete random vector of finite entropy, g : R nr R nt a measurable function, and p > 0. Furthermore, let K p be a set of continuous random vectors, independent of X D, such that for every U K p we have hu <, U p <, and suppu+x i suppu+x j = for all x i,x j suppx D, i j. Then, where with and k nt,p as defined in Lemma, respectively. CX,H C OW X,H := [HX D gap] +, gap := inf G,p U,X D,g+G,p U U K p g measurable p>0 U+XD gy p G,p U,X D,g := n t log, U p G,p U := n t log k n t,pn p t U p, 3 n t hu Proof: The proof is identical to [3, Th. ]. In order to make the manuscript more self-contained, we repeat it here. Let U and X D be statistically independent. Then, the mutual information IX D ;Y can be lower bounded as IX D ;Y a IX D +U;Y = hx D +U hx D +U Y b = HX D +hu hx D +U Y. 4 Here, a follows from the data processing inequality as X D +U X D Y forms a Markov chain in that order and b from the assumption in. By using Lemma, we have that the last term in 4 can be bounded from above as hx D +U Y n t log k nt,pn pt X D +U gy p. Combining this expression with 4 results in IX D ;Y HX D G,p U,X D,g+G,p U, with G,p and G,p as defined in and 3, respectively. Maximizing the right-hand side over all U K p, measurable functions g :R nr R nt, and p > 0 provides the bound. Interestingly, the bound of Theorem 4 holds for arbitrary channels and the interested reader is referred to [3] for details.

7 7 We conclude the section by providing a lower bound that is based on Jensen s inequality and holds for arbitrary inputs. Theorem 5. Jensen s Inequality Lower Bound For any channel input space X and fixed channel matrix H, we have nr [ ] CX,H C Jensen X,H := max F X :X X log+ E exp HX X, e 4 where X is an independent copy of X. Proof: In order to show the lower bound, we follow an approach of [7]. Note that by Jensen s inequality hy = E[logf Y Y] loge[f Y Y] = log f Y yf Y ydy. 5 R nr Now, evaluating the integral in 5 results in f Y yf Y ydy = R π nr nr a = π nre b = π nre c = [ R nr E [ E nr π nr [ [ R nr e e y HX e HX HX 4 e HX X 4 ] [ E e y HX y HX + y HX dy R nr e y ] ] dy ] HX X dy ], 6 where a follows from the independence of X and X and Tonelli s theorem, b from completing a square, and c from the fact that e y HX X R nr dy = e y dy = π nr R nr. Finally, combining 5 and 6, subtracting hz = nr logπe, and maximizing over F X proves the result. IV. INVERTIBLE CHANNEL MATRICES Consider the case of n t = n r = n antennas with H R n n being invertible. In this section, we evaluate some of the lower and upper bounds given in the previous section for the special case of H being also diagonal and then characterize the gap to the capacity for arbitrary invertible channel matrices. A. Diagonal Channel Matrices Suppose the channel inputs are subject to per-antenna or an n-dimensional amplitude constraint. Then, the duality upper bound C Dual, X,H of Theorem is of the following form. Theorem 6. Upper Bounds Let H = diagh,...,h nn R n n be fixed. If X = Boxa for some a = A,...,A n R n +, then n C Dual, Boxa,H = log + h ii A i. 7 πe Moreover, if X = B 0 A for some A R +, then C Dual, B 0 A,H = n log + h ii A. 8 n πe

8 8 Proof: The bound in 7 immediately follows from Theorem by observing that BoxHBoxa = BoxHa. The bound in 8 follows from Theorem by the fact that Box HB 0 A Box HBox B 0 A = Boxh, where h := A n h,..., h nn. This concludes the proof. For an arbitrary channel input space X, the EPI lower bound of Theorem 3 and Jensen s inequality lower bound of Theorem 5 evaluate to the following. Theorem 7. Lower Bounds Let H = diagh,...,h nn R n n be fixed and X arbitrary. Then, n C Jensen X,H = log +, 9 e ψh,b where with b := B,...,B n and ϕ : R + R +, ψh,b := min b X n ϕ h ii B i ϕx := x e x + πx Q x, 0 and C EPI X,H = n log +VolX n n h ii n. πe Proof: For some given values B i R +, i =,...,n, let the i-th component of X = X,...,X n be independent and uniformly distributed over the interval [ B i,b i ]. Thus, the expected value appearing in the bound of Theorem 5 can be written as [ E e HX X 4 ] [ = E n e h ii X i X i 4 ] [ n = E ] e h ii X i X i 4 = n [ E e h ii X i X i 4 ]. Now, if X is an independent copy of X, it can be shown that the expected value at the right-hand side of is of the explicit form [ ] E e h ii x i x i 4 = ϕ h ii B i with ϕ as defined in 0. Finally, optimizing over all b = B,...,B n X results in the bound 9. The bound in follows by inserting deth = n h ii into 9, which concludes the proof. In Fig., the upper bounds of Theorems and 6 and the lower bounds of Theorem 7 are depicted for a diagonal MIMO channel with per-antenna amplitude constraints. It turns out that the moment upper bound and the EPI lower bound perform well in the small amplitude regime while the duality upper bound and Jensen s inequality lower bound perform well in the high amplitude regime. B. Gap to the Capacity Our first result bounds the gap between the capacity and the lower bound in 9. Theorem 8. Let H R n n be of full rank and Then, ρx,h := Vol B 0 r max HX VolHX CX,H C EPI X,H n log πn n ρx,h n..

9 9 Rate in bits/s/hz Moment upper bound, CM Duality upper bound, CDual, Jensen s lower bound, C Jensen EPI lower bound, C EPI Amplitude Constraint, A, in db Fig.. Comparison of the upper and lower bounds of Theorems, 6, and 7 evaluated for a MIMO system with per-antenna amplitude constraints A = A = A i.e., a = A,A and channel matrix H = as Proof: For notational convenience let the volume of an n-dimensional ball of radius r > 0 be denoted V n r := Vol B 0 r = V n r n = π n r n Γ n +. Now, observe that by choosing p =, the upper bound of Theorem can further be upper bounded as k n, C M X,H nlog n πe x+z a = n log n E[ x+z ] b = n log + n x, πe where a follows since k n, = and b since n E[ Z ] = n. Therefore, the gap between 9 and the moment upper bound of Theorem can be upper bounded as follows: C M X,H C EPI X,H = n log + n x a = n log = n log + VolHX n πe + Vn x n n V n + VolHX n πe + n b n log n πe ρx,hvolhx V n + VolHX n πe ρx,h V n n n

10 0 A A X Fig.. Example of a pulse-amplitude modulation constellation with N = 4 points and amplitude constraint A i.e., PAM4, A, where := A/N denotes half the Euclidean distance between two adjacent constellation points. In case N is odd, 0 is a constellation point. c n log πn n ρx,h n Here, a is due to the fact that x is the radius of an n-dimensional ball, b follows from the inequality +cx c for c and x R n +x +, and c follows from using Stirling s approximation to obtain V n n + eπ n n. The term ρx,h is referred to as the packing efficiency of the set HX. In the following proposition, we present the packing efficiencies for important special cases. Proposition. Packing Efficiencies Let H R n n be of full rank, A R +, and a := A,...,A n R n +. Then, ρ B 0 A,I n =, 3. ρ B 0 A,H = H n deth, 4 ρ π n Boxa,I n = Γ a n n n + A, 5 i ρ Boxa,H Γ n π n H n a n + deth n A i Proof: The packing efficiency 3 follows immediately. Note that r max HB0 A = max Hx = H A. x B 0 A. 6 Thus, as H is assumed to be invertible we have VolHB 0 A = deth VolB 0 A, which results in 4. To show 5, observe that Vol B 0 rmax In Boxa = Vol B 0 a = π n Γ n + a n. The proof of 5 is concluded by observing thatvoli n Boxa = n A i. Finally, observe thatboxa B 0 a implies r max HBoxa r max HB 0 a so that ρ H,Boxa Vol B 0 H a n Vol HBoxa = π Γ H n a n n + deth n A, i which is the bound in 6. We conclude this section by characterizing the gap to the capacity when H is diagonal and the channel input space is the Cartesian product of n PAM constellations. In this context, PAMN,A refers to the set of N N equidistant PAM-constellation points with amplitude constraint A R + see Fig. for an illustration, whereas X PAMN, A means that X is uniformly distributed over PAMN, A [3].

11 Theorem 9. Let H = diagh,...,h nn R n n be fixed and X = X,...,X n. Then, if X i PAMN i,a i, i =,...,n, for some given a = A,...,A n R n +, it holds that where N i := + A i h ii πe and C Dual, Boxa,H C OW Boxa,H c n bits, 7 c := log+ πe + log 6 log + 6 πe.64. Moreover, if X i PAMN i,a, i =,...,n, for some given A R +, it holds that C Dual, B 0 A,H C OW B 0 A,H c n bits, 8 where N i := + A h ii n πe. Proof: Since the channel matrix is diagonal, letting the channel input X be such that its elements X i, i =,...,n, are independent we have that n IX;HX+Z = IX i ;h ii X i +Z i. Let X i PAMN i,a i with N i := + A i h ii πe and observe that half the Euclidean distance between any pair of adjacent points in PAMN i,a i is equal to i := A i /N i see Fig., i =,...,n. In order to lower bound the mutual information IX i ;h ii X i +Z i, we use the bound of Theorem 4 for p = and n t =. Thus, for some continuous random variable U that is uniformly distributed over the interval [ i, i and independent of X i we have that IX i ;h ii X i +Z i HX i log πe 6 [ E U log +Xi gy i ]. 9 E[U ] Now, note that the entropy term in 9 can be lower bounded as HX i = log + A i h ii log + A i h ii +log, 30 πe πe where we have used that x x for every x. On the other hand, the last term in 9 can be upper bounded by upper bounding its argument as follows: E [ U +X i gy i ] a = + 3E[ X i gy i ] E[U ] i b + 3E[Z i ]N i A i h ii = + 3N i A i h ii c 3 Ai h ii πe + A i h ii = + 6 πe. 3 Here, a follows from using that X i and U are independent and E[U ] = i, b from using the estimator 3 gy i = h ii Y i, and c from N i = + A i h ii πe + A i h ii πe. Combining 9, 30, and 3 results in the gap 7. The proof of the gap in 8 follows along similar lines, which concludes the proof.

12 V. ARBITRARY CHANNEL MATRICES For a MIMO channel with an arbitrary channel matrix and an average power constraint, the capacity is achieved by a singular value decomposition SVD of the channel matrix i.e., H = UΛV T and considering the equivalent channel model Ỹ = Λ X+ Z, where Ỹ := UT Y, X := V T X, and Z := U T Z, respectively. To provide lower bounds for channels with amplitude constraints and SVD precoding, we need the following lemma. Lemma. For any given orthogonal matrix V R nt nt and constraint vector a = A,...,A nt R nt + there exists a distribution F X of X such that X = V T X is uniformly distributed over Boxa. Moreover, the components X,..., X nt of X are mutually independent with X i uniformly distributed over [ A i,a i ], i =,...,n t. Proof: Suppose that X is uniformly distributed over Boxa; that is, the density of X is of the form f X x = Vol, x Boxa. Boxa Since V is orthogonal, we have V X = X and by the change of variable theorem for x VBoxa f X x = detv f XV T x = Therefore, such a distribution of X exists. detv Vol Boxa = Vol Boxa Theorem 0. Lower Bounds with SVD Precoding Let H R nr nt be fixed, n min := minn r,n t, and X = Boxa for some a = A,...,A nt R nt +. Furthermore, let σ i, i =,...,n min, be the i-th singular value of H. Then, n min C Jensen Boxa,H = log + 3 e ψh,b and C EPI Boxa,H = n min log + n min A iσ i πe n min, 33 where with b := B,...,B nt and ϕ as defined in 0. n min ψh,b := min ϕσ i B i b Boxa Proof: Performing the SVD, the expected value in Theorem 5 can be written as [ ] [ ] [ ] [ E e HX X 4 = E e UΛVT X X 4 = E e ΛVT X X 4 = E e Λ X X 4 By Lemma there exists a distribution F X such that the components of X are independent and uniformly distributed. Since Λ is a diagonal matrix, we can use Theorem 7 to arrive at 3. ].

13 3 Rate in bits/s/hz 3 Moment upper bound Jensen s lower bound + SVD EPI lower bound + SVD Amplitude Constraint, A, in db Fig. 3. Comparison of the upper bound in Theorem to the lower bounds of Theorem 0 for a 3 MIMO system with amplitude constraints A = A = A 3 = A i.e., a = A,A,A and channel matrix H = ,0.0357, Note that by Lemma there exists a distribution on X such that X is uniform over Boxa R nt and Λ X is uniform over ΛBoxa R n min, respectively. Therefore, by the EPI lower bound given in 8 we obtain + n min hλ X C EPI Boxa,H = n min log πe = n min log + Vol ΛBoxa n min πe = n min log + n min A i n min n min σ i πe which is exactly the expression in 33. This concludes the proof. Remark 3. Notice that choosing the optimal b for the lower bound 3 is an amplitude allocation problem, which is reminiscent of waterfilling in the average power constraint case. It would be interesting to study whether the bound in 3 is connected to what is called mercury waterfilling in [8], [9]. In Fig. 3, the lower bounds of Theorem 0 are compared to the moment upper bound of Theorem for the special case of a 3 MIMO channel. Similarly to the example presented in Fig., the EPI lower bound performs well in the low amplitude regime while Jensen s inequality lower bound performs well in the high amplitude regime. We conclude this section by showing that for an arbitrary channel input space X, in the large amplitude regime the capacity pre-log is given by minn r,n t. Theorem. Let X be arbitrary and H R nr nt fixed. Then, lim r min X CX,H = minn r,n t. log + r minx πe Proof: Notice that there always exists a R nt + and c R + such that Boxa X cboxa. Thus, without loss generality we can consider X = Boxa, a = A,...,A, for sufficiently large A R +. To n min,

14 4 prove the result we therefore start with enlarging the constraint set of the bound in 5: Box HBoxa B 0 rmax HBoxa B 0 rmax HB0 n t A = B 0 rmax UΛV T B 0 n t A = B 0 rmax UΛB0 n t A = B 0 rmax ΛB0 n t A where r := n t A nmin σ i and a := r in 5 it follows that Moreover, n r CBoxa, H lim A B 0 r Boxa, nmin,..., log + A i n min log + πe CBoxa, H n min lim log + A πe A Next, using the EPI lower bound in 33, we have that lim A This concludes the proof. C Boxa,Λ EPI = n min lim log + A πe A r nmin R n min +. Therefore, by using the upper bound nt A nmin σ i. πe nmin log + nmin nta σ i πe nmin = n min. log + A πe log + A n min σ i πe n min = n min. log + A πe VI. CONCLUSION In this work, we have focused on studying the capacity of MIMO systems with bounded channel input spaces. Several new upper and lower bounds have been proposed and it has been shown that the lower and upper bounds are tight in the high amplitude regime. An interesting direction for future work is to determine the exact scaling in the massive MIMO regime i.e., n min. Another interesting future direction is to study generalizations of our techniques to MIMO wireless optical channels [0]. REFERENCES [] I. E. Telatar, Capacity of multi-antenna Gaussian channels, Europ. Trans. Telecommu., vol. 0, no. 6, pp , Nov./Dec [] J. G. Smith, The information capacity of amplitude- and variance-constrained scalar Gaussian channels, Inf. Control, vol. 8, no. 3, pp. 03 9, Apr. 97. [3] S. Shamai Shitz and I. Bar-David, The capacity of average and peak-power-limited quadrature Gaussian channels, IEEE Trans. Inf. Theory, vol. 4, no. 4, pp , Jul [4] B. Rassouli and B. Clerckx, On the capacity of vector Gaussian channels with bounded inputs, IEEE Trans. Inf. Theory, vol. 6, no., pp , Dec. 06. [5] N. Sharma and S. Shamai Shitz, Transition points in the capacity-achieving distribution for the peak-power limited AWGN and free-space optical intensity channels, Probl. Inf. Transm., vol. 46, no. 4, pp , 00. [6] A. L. McKellips, Simple tight bounds on capacity for the peak-limited discrete-time channel, in Proc. IEEE Int. Symp. Inf. Theory ISIT, Chicago, IL, Jul. 004, pp [7] A. Thangaraj, G. Kramer, and G. Böcherer. 05 Capacity bounds for discrete-time, amplitude-constrained, additive white Gaussian noise channels. [Online]. Available:

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