Data-Aided Channel Estimation in Large Antenna
Systems
Junjie Ma, and Li Ping
Department of Electronic Engineering, City University of Hong Kong, Hong Kong
Emails: [email protected], [email protected]
Abstract—This paper is concerned with the uplink in a multicell large antenna system. We study a channel estimation scheme
where partially decoded data is used to estimate the channel. We
show that there are two types of interference components in this
scheme that do not vanish even when the number of antennas
grows to infinity: cross-contamination and self-contamination.
Cross contamination is in principle similar to pilot contamination
in a conventional pilot-based channel estimation scheme, while
self-contamination is unique for the data-aided scheme. The dataaided scheme can effectively suppress the contamination effect
by increasing the data frame length without causing rate loss.
This is confirmed by both analysis and simulation results.
I. I NTRODUCTION
Assume that all other system parameters, such as the
number of users, the average transmission power per user and
rate per user, are fixed in a cellular system. Then, provided
that perfect channel state information (CSI) at the base station
(BS) is available, cross user interference can be completely
suppressed when M → ∞, where M is the number of
antennas at the BS. This fact motivates the research activities
on large (or massive) antenna systems [1]–[9].
Channel estimation is a challenging problem for a large
antenna system [1]. It is shown in [1] that, if CSIR is to
be estimated using non-orthogonal pilot signals, the effect
of interference does not vanish even when M → ∞. This
phenomenon is referred to as “pilot contamination” [1]–[9].
The treatments for pilot contamination have been extensively studied [2]–[6]. However, the asynchronous transmission scheme in [2] may result in strong interference when
two closely located mobile terminals in neighboring cells
are transmitting and receiving at the same time [10]. The
coordinated estimation technique of [3] relies on specific
conditions on channel covariance matrix and the claim of
[4] is only valid for the asymptotic case of infinite antennas.
The singular value decomposition (SVD)-based blind channel
estimation scheme proposed in [5], [6] has high computational
complexity for a large antenna system.
In this paper, we study a scheme in which partially decoded
data are used to aid channel estimation [11], [12]. We analyze
the distortion at the receiver output in this scheme. We
show that there are residual interference terms that remain
bounded away from zero when M → ∞. These residual
interference terms, broadly referred to as “contamination”,
can be divided into two types. The first type, referred to as
cross-contamination, is due to the correlation among the data
signals from different users. This is in principle similar to the
correlation among the pilots in a conventional scheme [1]. The
second type, referred to as self-contamination, results from the
dependency between channel estimate and estimation error.
(Recall that these two are independent in a classical minimum
mean-square-error (MMSE) estimation for a linear Gaussian
model.)
With analysis and simulation, we show that the two types
of contamination may cause considerable error if not treated
properly. We also show that cross-contamination can be reduced by using a long data frame (as long data sequences naturally have low cross correlation), and that self-contamination
can be alleviated by iterative processing. In principle, pilot
contamination can also be mitigated by increasing the pilot
sequence length for the conventional pilot-based scheme.
However, this approach will reduce the effective data rate and
thus not desirable in practice.
II. P RELIMINARIES
A. Linear Model and LMMSE Estimation
Let b (M ×1) and w (M ×1) be two vectors of independent
identically distributed (i.i.d.) random variables with mean zero.
Assume that E[bwH ] = 0, i.e., b and w are uncorrelated.
Consider the following linear model:
a=c·b+w
(1)
where c is a fixed constant scalar. The linear minimum meansquare-error (LMMSE) estimator [13] of b upon observing a
is given by
b̂ = ϕ · a
(2)
where ϕ is a linear combining coefficient. Using (2), we can
write
b=ϕ·a+ξ
(3)
where ξ ≡ b − b̂ is referred to as the estimation error for b̂.
In (2), ϕ can be computed as [13]
ϕ=
c∗ · vb
c∗ · vb
= 2
va
|c| · vb + vw
(4)
where va , vb and vw are, respectively, the variances of the
entries of a, b and w. Also, ξ is an i.i.d zero-mean vector
uncorrelated with b̂. The variance of an entry of ξ is given by
vξ = vb −
|c|2 · vb2
.
|c|2 · vb + vw
Here vξ is also the mean square error (MSE) for b̂.
(5)
B. Gaussian Assumption and Related Properties
We now further assume that both b and w are Gaussian in
(1). The LMMSE estimator in (2) now becomes an MMSE
one [13]. The estimate b̂ and the estimation error ξ are jointly
Gaussian and mutually independent [13]. We can compute the
following quantities involving a and b,
E |aH b|2 = E |aH (ϕ · a + ξ)|2
(6)
= |ϕ|2 · E kak4 + vξ · E kak2 .
The second equality in (6) holds because a and ξ are mutually
independent. (As b̂ = ϕ · a and ξ are mutually independent.)
Applying a similar procedure to E |aH (b − a)|2 leads to:
E |aH (b − a)|2 = E |aH ((ϕ − 1) · a + ξ)|2
= |ϕ − 1|2 · E kak4 + vξ · E kak2 .
(7)
Eqns. (6) and (7) will be useful for discussions on interference
power in Section IV. The expectations in (6) and (7) are
computed as follows. Since a ∼ CN (0, va · I), we have
kak2 ∼ va /2 · χ2 (2M ) where χ2 (2M ) denotes the chi-square
distribution with 2M degrees of freedom. Thus,
E kak2 = va · M ;
(8a)
E kak4 = va2 · (M 2 + M ).
(8b)
III. DATA - AIDED C HANNEL E STIMATION
In this section, we study a data-aided channel estimation
scheme based on certain available a priori data information.
This information could be obtained from the output of a softinput soft-output channel decoder. For simplicity, we assume
that only partially decoded data are used in channel estimation
in this section.
A. System Model
Consider an L-cell system with the following assumptions.
Each cell contains only one user. Each base station (BS)
is equipped with M antennas while each mobile unit is
equipped with only one antenna. We will focus on the uplink
transmission of cell 1. Our task is to estimate the channel
during a frame of J consecutive symbols in time. A frame of
received signals over different antennas at the BS of cell 1 are
denoted as
X
y1m (j) = h1m x1 (j) +
him xi (j) + n1m (j),
(9)
i6=1
m = 1, . . . , M, j = 1, . . . , J
where h1m is the equivalent channel gain from the user in cell
i to the mth antenna of BS 1, xi (j) the jth symbol transmitted
by user i with unit power, n1m (j) the additive white Gaussian
noise sample with mean zero and variance N0 . The equivalent
1/2
channel him is represented as him = βi · gim , where βi includes the large scale fading factor and the transmit power, and
gim ∼ CN (0, 1) is a Rayleigh fading factor. We assume that
{βi } are known at the receiver. We will focus on a quasi-static
channel, in which the channel remains constant in a frame and
change independently from frame to frame. Denote y1 (j) ≡
[y11 (j), y12 (j), . . . , y1M (j)]T , hi ≡ [hi1 , hi2 , . . . , hiM ]T and
n1 (j) ≡ [n11 (j), n12 (j), . . . , n1M (j)]T , we can now rewrite
(9) as
X
y1 (j) = h1 x1 (j) +
hi xi (j) + n1 (j), j = 1, . . . , J. (10)
i6=1
Note that some of the above assumptions are for notational
simplicity. For example, the discussions below can also be
extended to a system with multiple users in a cell.
B. A Priori Data Information
The data symbol x1 (j) is modeled as follows:
x1 (j) = x̄1 (j) + ∆x1 (j)
(11)
where
1) {x̄1 (j)} and {∆x1 (j)} are the known and unknown
parts at the receiver side, respectively;
2) {x̄1 (j)} and {∆x1 (j)} are mutually independent; and
3) both {x̄1 (j)} and {∆x1 (j)} have i.i.d. zero mean random variables with variances given by 1 − vx and vx
respectively.
We will treat {x̄1 (j)} as the samples of a random variable.
In practice, a soft-input soft-output decoder will generate the
log-likelihood ratios (LLRs) related to x1 (j) [14]. Using the
LLRs, x̄1 (j) is computed as the mean of x1 (j). By assumption
3, the average power of x1 (j) is normalized, i.e.,
E |x1 (j)|2 = E |x̄1 (j)|2 + E |∆x1 (j)|2 = 1.
(12)
In (12), the average over the distribution of x̄1 (j) has also
been taken.
Similar modeling of a priori information has been widely
used in turbo-type iterative signal processing, e.g., [11], [15].
C. Channel Estimation
We first combine the J time samples {y1 (j), j =
1, 2, . . . , J} as follows,
z=
J
X
x̄∗1 (j)y1 (j).
(13)
j=1
For notational brevity, denote x̄1 ≡[x̄1 (1), . . . , x̄1 (J)]T ,
∆x1 ≡[∆x1 (1), . . . , ∆x1 (J)]T and xi ≡[xi (1) . . . , xi (J)]T .
Substituting (9) into (13), we have
z = (x̄H
1 x1 ) · h1 +
J
X
X
(x̄H
x
)
·
h
+
x̄∗1 (j)n1 (j)
i
1 i
j=1
i6=1
2
= kx̄1 k · h1 + ñ,
ñ ≡ (x̄H
1 ∆x1 )h1 +
(14)
X
(x̄H
1 xi )hi +
i6=1
J
X
x̄∗1 (j)n1 (j).
j=1
The LMMSE estimator of h1 is given by
ĥ1 = θ · z
(15a)
where ϕi · ĥ1 and ξi are independent. Substituting (19) into
(17), we have
where, based on (2) and (4), we have
θ=
kx̄1
k2
β1
P
.
· β1 + vx · β1 + i6=1 βi + N0
(15b)
Note that ñ and h1 in (14) are not mutually independent,
H
and ñ is not Gaussian (as (x̄H
1 ∆x1 )h1 and (x̄1 xi )hi are
not Gaussian). Therefore the standard linear Gaussian model
discussed in Section II-B does not apply to (14). The channel estimate ĥ1 in (15a) and the estimation error h1 − ĥ1
are not independent. This fact results in the so-called selfcontamination, as will be discussed in Section IV-C.
ξi · xi (j),
Ii = ϕi · kĥ1 k2 · xi (j) + ĥH
{z
} | 1 {z
}
|
Ii0
2
E
|Ii00 |2
To maintain low receiver complexity, a simple matched filter
(MF) detector is employed:
+
i6=1
self-interference I1
ĥH
1 hi
|
· xi (j) + ĥH
n1 (j) .
{z
} | 1 {z }
cross-interference Ii
(16)
noise N
B. Cross-interference Power
Let us first discuss the following interference term inside
the summation in (16)
i 6= 1.
(17)
We now compute the average power of Ii . Combining (14)
and (15a), we have
ĥ1 = θ · (x̄H
1 xi ) · hi + ηi
(18a)
where
ηi ≡
X
k6=i
θ · (x̄H
1 xk )hk +
J
X
θ · x̄∗1 (j)n1 (j).
= |ϕi | ·
· (M + M ),
= ei · E kĥ1 k2
i 6= 1;
(21b)
(21c)
i 6= 1
(21d)
ϕi =
∗
θ · (x̄H
1 xi ) · βi
,
vĥ1
(22a)
2
2
θ2 · |x̄H
1 xi | · βi
,
vĥ1
L
X
2
2
|x̄H
x
|
·
β
+
k
x̄
k
·
N
= θ2 ·
.
k
k
1
0
1
ei = β i −
(22b)
(22c)
k=1
In the above, kĥ1 k2 x1 (j) is the desired signal,
P
H
i6=1 ĥ1 hi · xi (j) represent the cross-interference from
other cells. Since the detection is based on ĥ1 , we treat
ĥH
1 (h1 − ĥ1 ) · x1 (j) as interference although it contains a part
of the desired signal x1 (j). We will refer to ĥH
1 (h1 −ĥ1 )·x1 (j)
as self-interference.
Ii ≡ ĥH
1 hi · xi (j),
(21a)
2
where ϕi , ei (the variance of ξi ) and vĥ1 (the variance of ĥ1 )
are respectively given by,
vĥ1
H
2
ĥH
1 y1 (j) = kĥ1 k · x1 (j) + ĥ1 (h1 − ĥ1 ) · x1 (j)
{z
} |
{z
}
|
signal S
vĥ2
1
= ei · vĥ1 · M,
A. Data Detection
X
(20)
Ii00
Using (8), we have
E |Ii0 |2 = |ϕi |2 · E kĥ1 k4
IV. DATA D ETECTION
In this section, we study the impact of channel estimation
error on the performance of data detection. For ease of
analysis, we assume that the receiver is based on a simple
matched filter (MF) detector.
i 6= 1.
(18b)
j=1
By assuming that {xi , ∀i} and x̄1 are given and fixed, (18)
becomes a standard linear Gaussian model. Based on the results in Section II-B, hi can be decomposed into the following
two parts:
hi = ϕi · ĥ1 + ξi
(19)
(The notation “E” in (21) represents expectation conditioned
on {xi , ∀i} and x̄1 . In the following, “E” could denote both
conditional and unconditional expectation, the meaning will
be clear from the context). We can also see from (16) that the
average signal power is given by
h
i
E |S|2 ≡ E kĥ1 k4 = vĥ2 · (M 2 + M ).
(23)
1
From (21a) and (23), both the power of Ii0 and S grow
with M in the order of O(M 2 ). Thus, Ii0 becomes a limiting
interference component when M → ∞. This is similar to the
pilot-contamination effect in a pilot based channel estimation
scheme [1]. In the following discussions, we will refer to Ii0
as “cross-contamination”.
In the special case when M and J
are large, we have the
following approximation for E |Ii0 |2
E |Ii0 |2 ≈
βi2
M2
·
,
1 − vx J
i 6= 1.
(24)
Due to space limitation, we put the detailed
of (24)
derivations
in [16]. We can also see from (24) that E |Ii0 |2 reduces as vx
decreases,
as can be expected. Furthermore, in [16] we show
that E |Ii00 |2 can be approximated as
E |Ii00 |2 ≈ βi · β1 · M, i 6= 1.
(25)
C. Self-interference Power
Applying a similar procedure as (19), we can decompose
h1 into two mutually independent parts as
h1 = ϕ1 · ĥ1 + ξ1 .
(26)
Then I1 can be rewritten as
I1 ≡ ĥH
h
−
ĥ
· x1 (j)
1
1
1
40
30
(27)
I100
The average power of I10 and I100 are given respectively as
E |I10 |2 = |ϕ1 − 1|2 · vĥ2 · (M 2 + M ),
(28a)
1
00 2 (28b)
E |I1 | = e1 · vĥ1 · M,
Average power (dB)
= (ϕ1 − 1) · kĥ1 k2 · x1 (j) + ĥH
ξ1 · x1 (j) .
|
{z
} | 1 {z
}
I10
cross-contamination, accurate
cross-contamination, approximate
self-contamination, accurate
self-contamination, approximate
(29)
J=1
10
J = 32
J = 64
J = 128
024
J = 1024
0
where ϕ1 and e1 can be obtained from (22a) and (22b) by
setting i = 1. For a large J and
M , we derive
in [16] the
following approximations for E |I10 |2 and E |I100 |2 ,
vx · β12 M 2
·
,
E |I10 |2 ≈
1 − vx J
00 2 E |I1 | ≈ 0.
J=3
2
J=6
J=1 4
28
20
-10 0
10
-1
-2
10
10
vx
Fig. 1. Average power of cross-contamination and self-contamination for
M = 128 and different J, SNRtarget = Ptarget /N0 = 0 dB.
(30)
Similar to (21), the power of I10 is also proportional to M 2 .
We refer to this effect as “self-contamination”. As discussed
in Section III-C, ĥ1 and h1 − ĥ1 are not independent. This
dependency essentially causes the self-contamination effect.
This effect does not exist in a conventional pilot-based scheme
[7], [8] where pilots are assumed known at the receiver.
D. Numerical Results
Consider a 7-cell cellular system with normalized cell
radius. The users are assumed to be uniformly randomly
located. Assume a fourth-power path-loss attenuation law. Let
us focus on a link from a user in cell i to the BS in cell j.
The path-loss of this link is given by γ · d−4
i→j , where γ is
a constant and di→j the distance of this link. For simplicity,
log-normal shadowing is not considered.
We adopt a power control policy [17] such that the receive
power of a user to its own BS is Ptarget . The transmit power
of this user is then Ptarget /(γ · d−4
i→i ). This implies that βi is
given by
Ptarget
−4
βi = di→1 ·
.
d−4
i→i
The average power of cross-contamination and selfcontamination against vx are plotted in Fig. 1 for different
J values. The number of antennas is fixed to be M = 128.
The solid lines are obtained by averaging (21a) and (28a)
over all possible {xi }, x̄1 and the distributions of {βi }, using
Monte Carlo simulations. For reference, the approximations
using (24) and (29) are also included.
From Fig. 1, we can see that the approximations are
reasonably accurate except when vx ≈ 1 and J is small.
The inaccuracy
is due to the assumption β1 · J · (1 − vx ) P
vx ·β1 + k6=1 βk +N0 in deriving (24) and (29). (See (38), (51)
in [16]). This approximation is loose when vx ≈ 1 and J is
relatively small. Moreover, we have the following observations
from Fig. 1:
• When vx is relatively large, both cross-contamination and
self-contamination are serious.
•
•
For a fixed J, both cross-contamination and selfcontamination decrease as vx becomes smaller. When
vx → 0, cross-contamination converges to a constant
while self-contamination vanishes completely (see (24),
(29)). In other words, when the a priori data information
is accurate, self-contamination is negligible.
Both cross-contamination and self-contamination reduce
as J becomes larger.
1
accurate
approximate
0.8
0.6
0.4
J=3
2
J=6
4
0.2
0 0
10
J=1
28
J = 1024
-1
10
-2
10
vx
Fig. 2. The power ratio of contamination for M = 128 and different J.
SNRtarget = 0 dB. {βi } are generated in the same way as in Fig. 1.
To quantify the contamination effect in a data-aided scheme,
we define the following ratio:
power (self-contamination + cross-contamination)
.
power (self-interference + cross-interference + noise)
(31)
The plots of δ against vx are given in Fig. 2 for different J
values. The approximate results based on (24)-(25), (29)-(30),
and (56) in [16] are also given for comparison. Again, we can
see that the contamination effect becomes marginal when J is
sufficiently large.
δ=
h
i
E kĥ1 k4
i
i
h
i P
h
SIN Rx = h
H
2 + E kĥ k2 · N
2 +
E |ĥH
0
1
1 (h1 − ĥ1 )|
i6=1 E |ĥ1 hi |
SIN Rxapp =
(32)
β12 · M 2
vx ·β12
1−vx
·
M2
J
+
P
βi2
i6=1
1−vx
E. SINR Performance
From (16), the SINR contained in the output of the MF
detector, denoted as SIN Rx , is defined in (32), as shown at
the top of the next page. Furthermore, using (24)-(25), (29)(30), and also (55)-(56) in [16], we have the approximate SINR
expression SIN Rxapp for a large M and J in (33).
In Fig. 3, SIN Rx and SIN Rxapp are plotted for various M
and J. We can see that the SINR grows as J and M increase.
Moreover, SIN Rxapp is an accurate approximation of SIN Rx ,
except when vx ≈ 1.
22
·
M2
J
(33)
+ β i · β 1 · M + β 1 · N0 · M
V. I TERATIVE C HANNEL E STIMATION AND S IGNAL
D ETECTION
In this section, we discuss an iterative joint channel estimation and data detection process [11], [12] which gradually
improves the system performance.
A. Transmitter Structure
The transmitter structure for the user in cell 1 is illustrated
in Fig. 4(a). We assume that one transmitted codeword spans
several coherence blocks. These coherence blocks may be
transmitted consecutively in time, or concurrently over different OFDM sub-carriers. In each coherence block, the first J
20
M = J = 128
18
b1
SINR (dB)
16
M = J = 64
14
(a) transmitter
M = J = 32
12
x̂1
10
Data
detector
Decoder
8
x1 , v x
accurate
approximate
6
4
2 0
10
 y  j 
p
1
Fig. 4. Transceiver structure for user 1.
F. Discussions
From Eqn. (33), we can see that:
app
• When J is fixed and M increases, SIN Rx
is bounded.
When M → ∞, the limiting value of SIN Rxapp is
vx
1−vx
Channel
Eestimator
1
10
Fig. 3. Average SINR of the data aided channel estimation scheme for
different M and J. SNRtarget = 0 dB. {βi } are generated in the same
way as in Fig. 1.
J · β12
P
· β12 + i6=1
ĥ1
 y  j 
(b) receiver
-2
-1
10
vx
SIN Rxapp ≈
(x1, p1)
Pilot
insertion
x1
Encoder
1
1−vx
· βi2
.
(34)
In this case, contamination is dominant and other interferences are negligible.
app
• When both J and M increase, SIN Rx
can be unbounded. In this case, the power of contamination and
other interferences is of the same order.
Also, note from (33) that SIN Rx is a function of vx . In the
next section, we will introduce a practical iterative processing
scheme where vx can be gradually reduced.
symbols are data symbols and the other Jp symbols are pilots.
At the transmitter side, the input binary information sequence
b1 is first processed by the encoder (which includes forward
error coding, random interleaving and signal mapping) to get
the data symbol x1 (j). The data symbols x1 is multiplexed
with the random pilot symbols p1 and transmitted through
the antennas. The received signals corresponding to data and
pilot transmissions are represented by y1 (j), j = 1, . . . , J and
y1p (j), j = J + 1, . . . , J + Jp respectively. We set Jp = 1 for
our simulations.
The receiver structure is shown in Fig. 4(b), where iterative channel estimation and signal detection is adopted. The
channel estimator and the signal detector have been discussed
in Section III and Section IV respectively. Conventioal pilot
based channel estimation is performed in the first iteration.
In subsequent iterations, data-aided channel estimation will be
employed to refine the channel estimation. For illustration purpose, we assumed that only a priori data information is used
to estimate the channel. Here, we adopt a general LMMSE
estimator which combines the information from pilot and data
[15]. The processings of the channel estimator, data detector
and decoder are executed iteratively until convergence.
B. Simulation Results
0
ACKNOWLEDGMENT
10
=1
= 100
This work was supported by a grant from the University
Grants Committee (Project No. AoE/E-02/08) of the Hong
Kong Special Administrative Region, China. The authors
would also like to thank Alcatel-Lucent Shanghai Bell Company Ltd for supporting the research work leading to this
paper.
-1
10
conventional pilot-based
SVD blind estimation
=1
= 100
-2
BER
in this scheme: cross-contamination and self-contamination.
Both analysis and simulation demonstrates that the data-aided
scheme can effectively suppress the contamination effect and
achieve improved performance in large antenna systems.
10
-3
=
10
1
=1
00
perfect CSI
data-aided: 4th iteration
R EFERENCES
-4
10
-4
-2
0
2
4
6
SNR (dB)
Fig. 5. BER performances of the pilot based channel estimation scheme, the
data-aided scheme and the SVD blind scheme. L = 7 and M = 128. β1 = 1,
βi 6= 1 for i 6= 1. J = 63, Jp = 1. A codeword spans 64 coherence blocks.
The rate-1/2 (23, 35)8 convolutional code is employed with Gray-mapped
256-QAM modulation.
The BER performances of the above iterative scheme, the
conventional pilot based one and the SVD blind estimation
scheme [6] are demonstrated in Fig. 5. In the simulations, for
simplicity, we set β1 = 1 and {βi = 0.1, i 6= 1}. This model
has been previously used in [7], [8]. Define
power of pilot symbol
ρ=
.
power of data symbol
We consider two different values for ρ: ρ = 1 and ρ = 100.
From Fig. 5, we can see that the BER performances for both
ρ values are very poor for the conventional pilot-only scheme.
As the problem is caused by the correlation among pilots,
increasing pilot power alone (even to an extremely large value
of ρ = 100) cannot solve the problem.
On the other hand, the data-aided channel estimation technique can improve the BER performance drastically. After
only 4 iterations, the performance is reasonably close to the
benchmark scheme with perfect CSI. It is also interesting
to see that the performance difference between ρ = 1 and
ρ = 100 is marginal in the region of BER< 10−4 , indicating
that the pilot power does not need to be too large in practice.
Clearly, the iterative scheme is very effective in treating
contamination.
From the figure, we can also see that the performance of the
blind SVD scheme is much better than that of the conventional
pilot only scheme, but still worse that data-aided one in the
high SNR region. Moreover, the complexity of the SVD blind
scheme is cubic of M , while that of the data-aided scheme is
linear.
VI. C ONCLUSION
We analyzed the performance of the data-aided channel estimation scheme in a multi-cell large antenna system. We showed that there are two types of contamination
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A PPENDIX A
A PPROXIMATIONS OF THE I NTERFERENCE , S IGNAL AND N OISE P OWER
The power of cross and self interferences for fixed {xi } and x̄1 are given in (21) and (28) respectively. We need to further
average (21) and (28) over the distribution of {xi } and x̄1 . In what follows, we will derive simple approximate expressions
for the average interference, signal and noise power when both J and M are large.
A. Cross-interference Power
Substituting (22c) into (22a) and (22b), we have
2 2
· βi ,
|ϕi |2 · vĥ2 = θ2 · x̄H
1 xi
1
X
2
x̄H
· βk + kx̄1 k2 · N0 · βi .
ei · vĥ1 = θ2 ·
1 xk
(35a)
(35b)
k6=i
Averaging (35) over {xi , ∀i} (conditioned on x̄1 ) and substituting in the definition of θ in (15b), we have
h
i
kx̄1 k2 · βi2 · β12
E{xi } |ϕi |2 · vĥ2 = 2 ,
P
1
kx̄1 k2 · β1 + vx · β1 + k6=1 βk + N0
P
h
i
kx̄1 k2 · β1 + vx · β1 + k6=1,k6=i βk + N0 · kx̄1 k2 · βi · β12
E{xi } ei · vĥ1 =
.
2
P
kx̄1 k2 · β1 + vx · β1 + k6=1 βk + N0
(36a)
(36b)
From assumption 3 in Section III-B, x̄1 is also random and the entries are i.i.d. with mean zero and variance 1 − vx . By the
law of large numbers,
1
a.s.
· kx̄1 k2 −−−−→ 1 − vx .
(37)
J→∞
J
a.s.
where −−−−→ denotes almost sure convergence as J tends to infinity. Approximating kx̄1 k2 by J · (1 − vx ) and noting that
J→∞
P
β1 · J · (1 − vx ) vx · β1 + k6=1 βk + N0 for a large J, we have
i
h
βi2
E{xi },x̄1 |ϕi |2 · vĥ2 ≈
, i 6= 1,
1
J · (1 − vx )
h
i
E{xi },x̄1 ei · vĥ1 ≈ βi · β1 , i 6= 1.
(38a)
(38b)
Combining (38) and (21), we finally arrive at the following approximate expressions for E[|Ii0 |2 ] and E[|Ii00 |2 ] when both M
and J are large.
βi2
M2
E |Ii0 |2 ≈
·
, i 6= 1,
1 − vx J
00 2 E |Ii | ≈ βi · β1 · M, i 6= 1.
(39)
B. Self-interference Power
We first consider I100 given in (28b). The expression for e1 · vĥ1 is given in (35b) by setting i = 1. Similar to the treatment
of Ii00 , i 6= 1, taking average over {xi , ∀i} (conditioned on x̄1 ), we have
P
2
3
k6=1 βk + N0 · kx̄1 k · β1
E{xi } e1 · vĥ1 =
(40)
2 .
P
kx̄1 k2 · β1 + vx · β1 + k6=1 βk + N0
Again, approximating kx̄1 k2 by J · (1 − vx ), we have
h
i
1
.
=O
J
M
J
E{xi },x̄1 e1 · vĥ1
(41)
Combining (28b) and (41), we have
E |I100 |2 = O
≈ 0.
(42)
In the above, E |I100 |2 is approximated as zero since it is of a smaller order compared with E |I10 |2 , as will be seen from
the following discussions.
We now consider the average power of I10 in (28a), given by
E |I10 |2 = |ϕ1 − 1|2 · vĥ2 · (M 2 + M ).
(43)
1
I10
Ii0 ,
Ii00
I10 .
The average of
over {xi , ∀i} is not so easy, as compared with that of
and
We next first approximate
simpler form by using the law of large numbers. To this end, combining (22a), (22c) and (15b), we have
P
1
2
·
k
x̄
k
·
β
+
v
·
β
+
β
+
N
1
1
x
1
0
k6=1 k
J
x̄H
1 x1
·
.
ϕ1 =
P
2
2
1
1
1
H
H
2
J }
· β1 + k6=1 J 2 · x̄1 xk · βk + J · kx̄1 k · N0 | {z
J 2 · x̄1 x1
|
{z
}
B
I10
into a
(44)
A
In the following, we will derive the asymptotic value for the term A as J → ∞. The derivations are based on the following
results:
1
a.s.
· kx̄1 k2 −−−−→ 1 − vx ,
(45a)
J→∞
J
a.s.
1
· x̄H
−−−→ 0,
(45b)
1 ∆x1 −
J→∞
J
a.s.
1
· x̄H
−−−→ 0,
(45c)
1 xi −
J→∞
J
1 H 2 a.s.
· x̄1 x1 −−−−→ (1 − vx )2 .
(45d)
J→∞
J2
2 as
Eqns. (45a)-(45c) are direct applications of the law of large numbers. To prove (45d), we first rewrite J −2 · x̄H
1 x1
1 H 2
1
· x̄1 x1 = 2 x̄H
(x̄1 + ∆x1 )(x̄1 + ∆x1 )H x̄1
J2
J 1
1 = 2 kx̄1 k4 + kx̄1 k2 · x̄H
1 ∆x1
J
2 H
.
+ kx̄1 k2 · ∆xH
1 x̄1 + x̄1 ∆x1
The result in (45d) can then be verified by applying (45a)-(45c) to (46).
Combining (45) and (44), we have
1
a.s.
.
A −−−−→
J→∞ 1 − vx
Substituting (15b) into (22c), vĥ1 is rewritten as
P
L
1
H
2
2
·
|
x̄
x
|
·
β
+
k
x̄
k
·
N
· β12
2
k
k
1
0
1
k=1
J
vĥ1 =
2 .
P
1
2·β +v ·β +
·
k
x̄
k
β
+
N
1
1
x
1
0
k6=1 k
J2
Using (45), it can be shown that
a.s.
vĥ1 −−−−→ β1
J→∞
Now keeping the term B and substituting in the asymptotic values of A and vĥ1 in (47) and (49), we have
2
1
x̄H x1
· 1
− 1 · β12
|ϕ1 − 1|2 · vĥ2 ≈ 1
1 − vx
J
2
1
kx̄1 k2 + x̄H
1 ∆x1
≈
·
− 1 · β12
1 − vx
J
2
1
x̄H
1 ∆x1 2
≈
·
· β1
1 − vx
J
where we have used (45a) in the last step. By taking expectation of the above expression, we finally have
i
h
vx · β12
E{x1 },x̄1 |ϕ1 − 1|2 · vĥ2 ≈
.
1
J · (1 − vx )
Combining (51) and (28a), we have
vx · β12 M 2
E |I10 |2 ≈
·
.
1 − vx J
(46)
(47)
(48)
(49)
(50a)
(50b)
(50c)
(51)
(52)
C. Signal and Noise Power
Conditioned on {x1 } and x̄1 , the signal and noise power are respectively given by (see (16) and (8))
i
h
E[|S|2 ] ≡ E kĥ1 k4 = vĥ2 · (M 2 + M ),
i1
h
2
2
E[|N | ] ≡ N0 · E kĥ1 k = vĥ1 · N0 · M.
(53)
(54)
a.s.
As shown in (49), vĥ1 −−−−→ β1 . We then approximate the signal and noise power as follows for a large M and J
J→∞
E[|S|2 ] ≈ β12 · M 2 ,
(55)
E[|N |2 ] ≈ β1 · N0 · M.
(56)