Water Filling Capacity Analysis in Large MIMO Systems

8/10/2019 Water Filling Capacity Analysis in Large MIMO Systems

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Water-filling Capacity Analysis in Large MIMO

Systems

Yi Lu, Wei Zhang

School of Electrical Engineering & Telecommunications

The University of New South WalesSydney, NSW 2052, Australia

Email: [email protected]; [email protected]

Abstract—In this paper, we study a water-filling power al-location scheme in large M × N MIMO systems over flatRayleigh fading channels. It is shown that when M = N withsufficiently large M , the channel capacity of the water-fillingscheme almost converges to a constant regardless of channelrandomness. Moreover, it is proved that for the water-fillingscheme, the required channel information at the transmitter inlarge MIMO systems can be greatly reduced without capacityloss. When M ≫ N or M ≪ N , it is shown that allocatingequal power on each eigenchannel is almost as optimal as water-

filling power allocation scheme in channel capacity. Index Terms—Massive MIMO, capacity analysis, water-filling

I. INTRODUCTION

Multiple-input multiple-output (MIMO) technology is an

important advance in wireless communications since it offers

significant increase in channel capacity and communication

reliability without requiring additional bandwidth or transmit

power. Specifically, when the perfect channel state information

(CSI) can be obtained by the receiver (so-called coherent

setting), the channel capacity grows linearly with the minimum

number of transmit and receive antennas [1]. Furthermore, if

the perfect CSI is also known at the transmitter, a namelywater-filling power allocation scheme can be used to maximize

the channel capacity especially when the total transmitter

power (or signal-to-noise ratio) is limited [2]. Unlike equal-

power allocation schemes, the water-filling scheme allocates

different powers to different eigenchannels and better non-

zero eigenchannels will support larger fraction of the entire

data rate [2].

Recently, massive MIMO wireless communication has at-

tracted increasing interests since the growing demands for

higher throughput and reliability can be satisfied [3], [4]. As

the MIMO array becomes larger, the per-antenna transmit

power greatly reduces. Moreover, the random matrix theory

reveals that things which are random before become deter-ministic in large systems, e.g., the singular values of the

large MIMO channel matrix [3], [6]. Meanwhile, large MIMO

channel matrices will also lead to considerable consumption

and inaccuracy on the channel estimation and the channel

feedback [3], [5].

This research was supported under Australian Research Council’s Discovery

Projects funding scheme (project number DP1094194).

In this paper, we consider the water-filling power allocation

scheme for large M × N MIMO systems. Interestingly, when

M = N , there are three major findings based on the capacity

analysis with the water-filling scheme: 1) We prove that

as M → ∞, the channel capacity with the water-filling

scheme converges to a constant related to M and the total

transmit power only, regardless of the channel randomness; 2)

Unlike the conventional water-filling scheme, we show that the

channel information fed back to the transmitter which is usedfor precoder design can be reduced greatly as M is sufficiently

large; 3) We show that by using the limited feedback of

channel information, the maximum channel capacity coincides

with the water-filling capacity and the power allocation is

also as optimal as the water-filling solution. When M ≫ N or M ≪ N , we show that allocating equal power on each

eigenchannel is as optimal as the water-filling scheme in

channel capacity.

The paper is organized as follows. In Section II, system

model is introduced. In Section III, we derive the channel

capacity with the water-filling scheme in large M ×M systems

and show that with limited feedback of channel information

at the transmitter, the capacity converges to the optimal value(i.e., with full channel knowledge feedback). In Section IV, we

analyze the water-filling scheme in large M ×N systems with

M ≫ N or M ≪ N . In Section V, some simulation results

are given to validate our main results. Finally, the paper is

concluded in Section VI.

I I . SYSTEM M ODEL

In this paper, we consider a MIMO system with M an-

tennas at transmitter and N antennas at receiver. The chan-

nel is assumed to be flat Rayleigh fading. Denote s =[s1 s2

· · · sM ]

T as the information symbols where the su-

perscript T represents the transpose of a matrix or a vector.

Before each transmission, a total power of P is allocat-

ed to each information symbol si (i = 1, 2, · · · , M ) as√ P 1s1

√ P 2s2 · · ·

√ P M sM

based on some power alloca-

tion scheme with P =

i P i. Afterwards, the transmitted

symbol vector is precoded by an M × M matrix V̂ as x =V̂Qs, where Q ∈ RM ×M = diag

√ P 1,

√ P 2, · · · , √

P M

is

the power allocation matrix. Thus, the received symbol vector

978-1-4673-6044-9/13/$31.00 ©2013 IEEE 186



Fig. 1. Illustration of generating samples from the cdf.

is given as

y = Hx+w (1)

= H(V̂Qs) + w, (2)

where H ∈ CN ×M is the Rayleigh fading channel matrix with

i.i.d. entries following distribution CN (0, 1) and w ∈ CN is

the white complex Gaussian noise vector with the distribution

CN (0, σ2I).

III. WATER-FILLING C APACITY A NALYSIS IN LARGE

M × M SYSTEMS

With the perfect CSI at the transmitter, the channel capacity

can be maximized by the water-filling scheme which is an

adaptive power allocation depending on the instantaneouschannel information. However, feeding back the instantaneous

H is always inefficient, especially in large MIMO systems. In

this section, we prove that the full channel knowledge is not

necessary at the transmitter in large MIMO systems to obtain

the maximum channel capacity. Moreover, the feedback load

can be pre-determined with large value of M . Lemma 1: Consider an M ×M matrix H with i.i.d. entries

following distribution CN (0, 1). As M → ∞, the empirical

distribution function of the singular values λi’s of Wishart

matrix HHH almost converges to

F (x) =

√ 4M x − x2 − 4M arctan

4M −xx

2πM + 1, (3)

whose density function is given by

f (x) = 1

2πM

4M − x

x , (4)

with x ∈ [0, 4M ].The proof of Lemma 1 is straightforward by following the

general Marchenco-Pastur law in [6].

The following theorem captures the maximum channel

capacity in large M × M systems with reduced feedback of

channel information.

Theorem 1: Consider a large MIMO system with Gaussian

distributed information symbols. Denote H = UΛVH as the

singular value decomposition (SVD) of the channel matrix H

and vm the m-th column of V. As M → ∞, with the feedback

of [v1 v2

· · · vK̄ ] at the transmitter, the channel capacity C

is approximately equal to the water-filling capacity C wf , thatis

C ≈ C wf ≈ K̄ ln( ν̄

σ2) +

K̄ i=1

ln(λ̄i) (nats/s/Hz), (5)

where K̄ denotes the number of water-filled (non-empty) sub-

streams, K̄ and ν̄ are constants satisfying the total power

constraintK̄

i=1

ν̄ − σ2

λ̄i

= P , λ̄i’s are given by

λ̄i = F −1

M − i + 1

M

, i = 1, 2, · · · , M (6)

and F −1 denotes the inverse function of F (x) in (3).

Fig. 1 shows a geometric interpretation of λ̄i in (6). Notethat since ν̄ , K̄ and λ̄i’s are all constants which are dependent

on only the number of antennas M and the transmit power

P in large M × M systems, the channel capacity (i.e., the

transmission rate) is always a constant for given P and M . Remark 1: Theorem 1 shows that in large M ×M systems,

the full CSI is no longer necessary at the transmitter to achieve

the maximum channel capacity, which can greatly reduce the

feedback load. More specifically, if the M × M matrix H or

V is fed back to the transmitter by columns, the feedback load

in large systems can be reduced to only K̄ M

of that by using

the conventional water-filling scheme.

Before we prove Theorem 1, the following proposition is

needed.Proposition 1: Consider an M × M matrix H with i.i.d.

entries following distribution CN (0, 1). As M → ∞, the

singular values λi’s of the Wishart matrix HHH in descending

order almost match the constants in (6), i.e.,

λi ≈ λ̄i (7)

for i = 1, 2, · · · , M .Since all the λi’s follow the pdf and the cdf in (4) and

(3), respectively, it is obvious that λi ≈ λ̄i as M → ∞. Fig.

2 shows the comparison between the constants λ̄i’s and the

singular values λi’s of a random Wishart matrix HHH . We

can see that when M is large, the instantaneous singular values

almost coincide with ¯λi’s, which validates the approximationin Proposition 1.

The proof of Theorem 1 is given with two steps. Firstly, the

channel capacity C is derived based on the limited feedback

channel knowledge. Secondly, we prove that the water-filling

capacity is approximately equal to C as M → ∞.

Proof: Define H = UΛVH as the SVD of the channel

matrix H. We firstly assume that the full matrix V is known

at the transmitter and used as the precoder, i.e., V̂ = V. Let

187



0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Index of the singular values

V a

l u e

Approximate singular values

Instantaneous singular values

3 30 3 32 3 34 3 36 3 38

1200

1210

1220

1230

1240

Fig. 2. Comparison of the approximate singular values λ̄i’s vs. instantaneoussingular values λi’s with M = 1000.

Q = diag√ P 1, √ P 2, · · · , √ P M

be the power allocation

matrix. With M antennas at both the transmitter and receiver,

the channel capacity is given by

C = maxP i:

i P i≤P

lndetI+ HV̂Q2 V̂H HH

(8)

= maxP i:

i P i≤P

M i=1

ln(1 + P iλi

σ2 ) (9)

≈ maxP i:

i P i≤P

M i=1

ln(1 + P iλ̄i

σ2 ). (10)

The approximation is resulted from Proposition 1. By solving

the optimization in (10), we obtain

P̄ i = (ν̄ − σ2

λ̄i)+ (11)

for i = 1, 2, · · · , M , where (x)+ = max(0, x), ν̄ satisfies

M i=1

P̄ i =K̄ i=1

(ν̄ − σ2

λ̄i) = P (12)

and K̄ represents the number of non-zero P̄ i’s (i.e., water-

filled sub-streams). Clearly, P̄ i ≥ P̄ i+1 as λ̄i ≥ λ̄i+1. Hence,

C ≈M i=1

ln(1 +P̄ iλ̄i

σ2 ) (13)

=K̄ i=1

ln(1 +P̄ iλ̄i

σ2 ) (14)

=K̄ i=1

ln(ν̄ ̄λiσ2

) (15)

= K̄ ln( ν̄

σ2) +

K̄ i=1

ln(λ̄i). (16)

Since P j = 0 for j > K̄ , the knowledge of vj’s are not

necessary at the transmitter. Hence, the necessary feedback

channel knowledge for precoder design is [v1 v2 · · · vK̄ ],

rather than the full matrix V assumed at the beginning of this

proof.

On the other hand, with the perfect CSIT, the instanta-

neous channel capacity can be maximized by the water-filling

scheme. With M antennas at both the transmitter and the

receiver, the water-filling capacity is formulated as

C wf = maxP i:

i P i≤P

M i=1

ln(1 + P iλi

σ2 ) (17)

=M i=1

ln(1 + P ∗i λi

σ2 ), (18)

where λi denotes the i-th largest singular value of HHH and

the water-filling solution P ∗i is given as

P ∗i = (ν − σ2

λi)+ (19)

with ν satisfyingi

P ∗i =i

(ν − σ2

λi)+ = P. (20)

According to Proposition 1, it has ν ≈ ν̄ and P ∗i ≈ P̄ i for

i = 1, 2, · · · , M as M → ∞. Thus,

limM →∞

C wf ≈K̄ i=1

ln(1 +P̄ iλ̄i

σ2 ) = C. (21)

IV. WATER-FILLING S CHEME IN L ARGE MIMO SYSTEMS

(M

≫N OR M

≪N )

If H is of the size N × M , as long as M → ∞, N → ∞and M

N = α ≥ 1 (α is a constant), the density function of the

N singular values of HHH is given as

f α(x) =

(x − N a)(N b − x)

2πN x (22)

for x ∈ [N a,N b], where a = (1 − √ α)2 and b = (1 +

√ α)2.

This distribution is also derived from the general Marchenko-

Pastur Law [6]. More specifically, when M N

= α ≫ 1, it hasab → 1, which indicates that all the ordered singular values of

HHH converge to a constant as α → ∞, i.e.,

λi →

N α = M, for i = 1, 2,

· · · , N (23)

and

λN λ1

= N a

N b → 1. (24)

Remark 2: Consider the system model in (1) with H ∈CN ×M , M → ∞, N → ∞ and M ≫ N . Then, allocating

equal power to each eigenchannel is almost as optimal as the

water-filling power allocation in channel capacity.

188



1 2 3 4 54

6

8

10

12

14

16

18

Total power P

C a

p a c i t y

Instantaneous

Approximation

Fig. 3. Convergence of the water-filling capacity with M = 10.

1 2 3 4 560

80

100

120

140

160

180

Total power P

C a p a c i t y

Instantaneous

Approximation


The claim in Remark 2 is obvious by considering the similar

proof of Theorem 1. Since all the λi’s vary little as M ≫ N ,the power allocation almost satisfies

P i ≈ P j , for i = j. (25)

Remark 3 ( [7],Section 6.4.2): Consider the system model

in (1) with H ∈ CN ×M . With large M and N and the perfect

CSI at the transmitter, the channel capacity when M = αN equals to that when M = 1

αN with constant α ≥ 1.

Remark 3 shows that with the perfect CSIT and the water-

filling power allocation, the channel capacity when M ≫ N is equal to that when M ≪ N . Thus, Remark 2 is also valid

for M ≪ N .

1 2 3 4 5700

800

900

1000

1100

1200

1300

1400

1500

1600

Total power P

C a

p a c i t y

Instantaneous

Approximation


2 4 6 8 10 12120

140

160

180

200

220

240

260

Ratio of M/N

C a p a c i t y

Equal•power allocation

Water•filling power allocation

Fig. 6. Capacity comparison between water-filling power allocation andequal-power allocation on each eigenchannel with M = 1200.

V. SIMULATION R ESULTS

In this section, we show some simulation results to validate

Theorem 1 and Remark 2. The noise power σ2 is assumed to

be 1.

Figures 3-5 show the convergence of the water-filling ca-

pacity as M → ∞. The circle marked curves representthe instantaneous water-filling capacity in (18) and the thick

curves represent the convergent capacity in (5). It can be

shown that the instantaneous water-filling capacity in (18)

almost converges to the capacity in (5) as M becomes large,

which validates the convergence and the accuracy in Theorem

1.

Fig. 6 shows the instantaneous channel capacity with

equal-power allocation on each eigenchannel and water-filling

189



scheme given limited total power P = 10−2 and M = 1200. It

can be seen that as M N

becomes large, the capacity with equal-

power allocation on each eigenchannel is almost equal to that

with the water-filling scheme, which validates the conclusion

in Remark 2.

VI. CONCLUSION

In this paper we have derived the channel capacity of

the water-filling power allocation scheme in large M × N systems. When M = N , it has been proved that the channel

capacity with the water-filling scheme converges to a constant

regardless of randomness of MIMO channels for sufficiently

large M . Moreover, we have shown that without capacity

loss, the required feedback of channel information is greatly

reduced compared with the full CSI needed in the conventional

water-filling scheme. When M ≫ N or M ≪ N , we have

shown that the equal-power allocation on each eigenchannel

is almost as optimal as the water-filling scheme in channel

capacity.

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Water Filling Capacity Analysis in Large MIMO Systems