water filling capacity analysis in large mimo systems
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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
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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
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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.
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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
Fig. 4. Convergence of the water-filling capacity with M = 100.
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
Fig. 5. Convergence of the water-filling capacity with M = 1000.
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
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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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