Per-Antenna Constant Envelope Precoding for
Large Multi-User MIMO Systems
Saif Khan Mohammed and Erik G. Larsson
Linköping University Post Print
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Saif Khan Mohammed and Erik G. Larsson, Per-Antenna Constant Envelope Precoding for
Large Multi-User MIMO Systems, 2013, IEEE Transactions on Communications, (61), 3,
1059-1071.
http://dx.doi.org/10.1109/TCOMM.2013.012913.110827
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-93866
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. XX, XX XXXX 1
Per-antenna Constant Envelope Precoding for Large
Multi-User MIMO Systems
Saif Khan Mohammed and Erik G. Larsson
Abstract—We consider the multi-user MIMO broadcast chan-
nel with M single-antenna users and N transmit antennas under
the constraint that each antenna emits signals having constant
envelope (CE). The motivation for this is that CE signals facilitate
the use of power-efficient RF power amplifiers. Analytical and
numerical results show that, under certain mild conditions on
the channel gains, for a fixed M , an array gain is achievable
even under the stringent per-antenna CE constraint. Essentially,
for a fixed M, at sufficiently large N the total transmitted
power can be reduced with increasing N while maintaining a
fixed information rate to each user. Simulations for the i.i.d.
Rayleigh fading channel show that the total transmit power can
be reduced linearly with increasing N (i.e., an O(N ) array gain).
We also propose a precoding scheme which finds near-optimal
CE signals to be transmitted, and has O(MN ) complexity. Also,
in terms of the total transmit power required to achieve a fixed
desired information sum-rate, despite the stringent per-antenna
CE constraint, the proposed CE precoding scheme performs close
to the sum-capacity achieving scheme for an average-only total
transmit power constrained channel.
Index Terms—Multi-user, constant envelope, per-antenna,
large MIMO, GBC.
I. INTRODUCTION
We consider a Gaussian Broadcast Channel (GBC), wherein
a base station (BS) having N antennas communicates with M
single-antenna users in the downlink. Large antenna arrays at
the BS has been of recent interest, due to their remarkable
ability to suppress multi-user interference (MUI) with very
simple precoding techniques [1]. Specifically, under an average
only total transmit power constraint (APC), for a fixed M , a
simple matched-filter precoder has been shown to achieve total
MUI suppression in the limit as N → ∞ [2]. Additionally, due
to the inherent array power gain property
1
, large antenna arrays
are also being considered as an enabler for reducing power
consumption in wireless communications, especially since the
operational power consumption at BS is becoming a matter of
world-wide concern [4], [5].
Despite the benefits of large antenna arrays at the BS,
practically building them would require cheap and power-
efficient RF power amplifiers (PA’s). In conventional BS,
Manuscript received Dec. 5, 2011; revised June 8, 2012 and Sept. 17, 2012;
accepted Oct. 23, 2012. The editor coordinating the review of this paper and
approving it for publication was Ali Ghrayeb.
The authors are with the Communication Systems Division, Dept. of
Electrical Engineering (ISY), Link
¨
oping University, Link
¨
oping, Sweden. This
work was supported by the Swedish Foundation for Strategic Research (SSF),
ELLIIT. The work of Saif Khan Mohammed was partly supported by the
Center for Industrial Information Technology at ISY, Link
¨
oping University
(CENIIT). Parts of the results in this paper were presented at IEEE ICASSP
2012 [15]. Also, the simpler special case of M = 1 (i.e., single-user) has
been studied by us in much greater detail in [16].
1
Under an APC constraint, for a fixed M and a fixed desired information
sum-rate, the required total transmit power decreases with increasing N [3].
power-inefficient PA’s account for about 40-50 percent of
the total operational power consumption [5]. With current
technology, power-efficient RF components are generally non-
linear. The type of transmitted signal that facilitates the use of
most power-efficient/non-linear RF components, is a constant
envelope (CE) signal. In this paper, we therefore consider
a GBC, where the amplitude of the signal transmitted from
each BS antenna is constant and independent of the channel
realization. We only consider the discrete-time complex base-
band equivalent channel model, where we aim to restrict the
discrete-time per-antenna channel input to have no amplitude
variations. Compared to precoding methods which result in
large amplitude-variations in the discrete-time channel input,
the CE precoding method proposed in this paper is expected
to result in continuous-time transmit signals which have a
significantly improved peak-to-average-power-ratio (PAPR).
However, this does not necessarily mean that the proposed
precoding method will result in continuous-time transmit
signals having a perfectly constant envelope. Generation of
perfectly constant envelope continuous-time transmit signals
constitutes future work for us. One possible method to generate
almost constant-envelope continuous-time signals could be to
constrain the phase variation between consecutive constant
amplitude baseband symbols of the discrete-time channel
input.
Since the per-antenna CE constraint is much more restrictive
than APC, in this paper we investigate as to whether MUI
suppression and array power gain can still be achieved under
the stringent per-antenna CE constraint. To the best of our
knowledge, there is no reported work which addresses this
question. Most reported work on per-antenna communication
consider an average-only or a peak-only power constraint (see
[6], [7] and references therein). In this paper, firstly, we derive
expressions for the MUI at each user under the per-antenna CE
constraint, and then propose a low-complexity CE precoding
scheme with the objective of minimizing the MUI energy at
each user. For a given vector of information symbols to be
communicated to the users, the proposed precoding scheme
chooses per-antenna CE transmit signals in such a way that
the MUI energy at each user is small (i.e., of the same order
or less than the variance of the additive white Gaussian noise).
Throughout the paper, we assume that such large antenna
systems will not operate in a regime where the MUI energy is
significantly larger than the AWGN variance, since it is highly
power-inefficient to do so [8].
Secondly, under certain mild channel conditions (including
i.i.d. fading), using a novel probabilistic approach, we ana-
lytically show that, MUI suppression can be achieved even
under the stringent per-antenna CE constraint. Specifically,
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. XX, XX XXXX 2
for a fixed M and fixed user information symbol alphabets,
an arbitrarily low MUI energy can be guaranteed at each user,
by choosing a sufficiently large N. Our analysis further reveals
that, with a fixed M and increasing N , the total transmitted
power can be reduced while maintaining a constant signal-to-
interference-and-noise-ratio (SINR) level at each user.
Thirdly, through simulation, we confirm our analytical
observations for the i.i.d. Rayleigh fading channel. For the
proposed CE precoder, we numerically compute an achievable
ergodic information sum-rate, and observe that, for a fixed
M and a fixed desired ergodic sum-rate, the required total
transmit power reduces linearly with increasing N (i.e., an
O(N) array power gain is achieved under the per-antenna CE
constraint). We also observe that, to achieve a given desired
ergodic information sum-rate, compared to the optimal GBC
sum-capacity achieving scheme under APC, the extra total
transmit power required by the proposed CE precoding scheme
is small (roughly 2.0 dB for sufficiently large N).
Notation: C and R denote the set of complex and real
numbers. |x|, x
∗
and arg(x) denote the absolute value,
complex conjugate and argument of x ∈ C respectively.
khk
2
∆
=
P
i
|h
i
|
2
denotes the squared Euclidean-norm of
h = (h
1
, ··· , h
N
) ∈ C
N
. E[·] denotes the expectation
operator. Abbreviations: r.v. (random variable), bpcu (bits-per-
channel-use), p.d.f. (probability density function).
II. SYSTEM MODEL
Let the complex channel gain between the i-th BS antenna
and the k-th user be denoted by h
k,i
. The vector of channel
gains from the BS antennas to the k-th user is denoted by
h
k
= (h
k,1
, h
k,2
, ··· , h
k,N
)
T
. H ∈ C
M×N
is the channel
gain matrix with h
k,i
as its (k, i)-th entry. Let x
i
denote the
complex symbol transmitted from the i-th BS antenna. Further,
let P
T
denote the average total power transmitted from all
the BS antennas. Under APC, we must have E[
P
N
i=1
|x
i
|
2
] =
P
T
, whereas under the per-antenna CE constraint we have
|x
i
|
2
= P
T
/N , i = 1, 2, ··· , N which is clearly a more
stringent constraint compared to APC. Further, due to the
per-antenna CE constraint, it is clear that x
i
is of the form
x
i
=
p
P
T
/Ne
jθ
i
, where θ
i
is the phase of x
i
. Under CE
transmission, the symbol received by the k-th user is therefore
given by
y
k
=
r
P
T
N
N
X
i=1
h
k,i
e
jθ
i
+ w
k
, k = 1, 2, . . . , M (1)
where w
k
∼ CN(0, σ
2
) is the AWGN noise at the k-th
receiver. For the sake of notation, let Θ = (θ
1
, ··· , θ
N
)
T
denote the vector of transmitted phase angles. Let u =
(
√
E
1
u
1
, ··· ,
√
E
M
u
M
)
T
be the vector of scaled information
symbols, with u
k
∈ U
k
denoting the information symbol
to be communicated to the k-th user. Here U
k
denotes the
unit average energy information alphabet of the k-th user.
E
k
, k = 1, 2, . . . , M denotes the information symbol energy
for each user. Also, let U
∆
=
√
E
1
U
1
×
√
E
2
U
2
×···×
√
E
M
U
M
.
Subsequently, in this paper, we are interested in scenarios
where M is fixed and N is allowed to increase. Also, through-
out this paper, for a fixed M , the alphabets U
1
, ··· , U
M
are
also fixed and do not change with increasing N .
We stress that CE transmission is entirely different from
equal gain transmission (EGT). We explain this difference
for the simple single-user scenario (M = 1). In EGT a unit
average energy complex information symbol u is communi-
cated to the user by transmitting x
i
= w
i
u from the i-th
transmit antenna (with |w
1
| = ··· = |w
N
| =
p
P
T
/N), and
therefore the amplitude of the signal transmitted from each
antenna is not constant but varies with the amplitude of u
(|x
i
| =
p
P
T
/N|u|). In contrast, the CE precoding method
proposed in this paper (Section III-B) transmits a constant
amplitude signal from each antenna (i.e.,
p
P
T
/Ne
jθ
i
from
the i-th antenna), where the transmit phase angles θ
1
, ··· , θ
N
are chosen in such a way that the noise-free received signal is
a known constant times the desired information symbol u.
III. MUI ANALYSIS AND THE PROPOSED CE PRECODER
For any given information symbol vector u to be commu-
nicated, with Θ as the transmitted phase angle vector, using
(1) the received signal at the k-th user can be expressed as
y
k
=
p
P
T
p
E
k
u
k
+
p
P
T
s
k
+ w
k
,
s
k
∆
=
P
N
i=1
h
k,i
e
jθ
i
√
N
−
p
E
k
u
k
(2)
where
√
P
T
s
k
is the MUI term at the k-th user. In this section
we aim to get a better understanding of the MUI energy level
at each user, for any general CE precoding scheme where
the signal transmitted from each BS antenna has constant
envelope. Towards this end, we firstly study the range of values
taken by the noise-free received signal at the users (scaled
down by
√
P
T
). This range of values is given by the set
M(H)
∆
=
n
v = (v
1
, ··· , v
M
) ∈ C
M
v
k
=
P
N
i=1
h
k,i
e
jθ
i
√
N
,
θ
i
∈ [−π, π) , i = 1, . . . , N
o
(3)
For any vector v = (v
1
, v
2
, ··· , v
M
)
T
∈ M(H), from (3) it
follows that there exists a Θ
v
= (θ
v
1
, ··· , θ
v
N
)
T
such that v
k
=
P
N
i=1
h
k,i
e
jθ
v
i
√
N
, k = 1, 2, . . . , M . This sum can be expressed
as a sum of N/M terms (without loss of generality let us
assume that N/M is integral only for the argument presented
here)
v
k
=
N/M
X
q=1
v
q
k
,
v
q
k
∆
=
qM
X
r=(q−1)M +1
h
k,r
e
jθ
v
r
/
√
N , q = 1, . . . ,
N
M
.
(4)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. XX, XX XXXX 3
From (4) it follows that M(H) can be expressed as a direct-
sum of N/M sets, i.e.
M(H) = M
H
(1)
⊕ M
H
(2)
⊕ ··· ⊕ M
H
(N/M)
M
H
(q)
∆
=
n
v = (v
1
, ··· , v
M
) ∈ C
M
v
k
=
P
M
i=1
h
k,(q−1)M +i
e
jθ
i
√
N
, θ
i
∈ [−π, π)
o
, q = 1, . . . , N/M (5)
where H
(q)
is the sub-matrix of H containing only the
columns numbered (q − 1)M + 1, (q − 1)M + 2, ··· , qM.
M
H
(q)
⊂ C
M
is the dynamic range of the received
noise-free signals when only the M BS antennas numbered
(q−1)M+1, (q−1)M+2, ··· , qM are used and the remaining
N − M antennas are inactive. If the statistical distribution of
the channel gain vector from a BS antenna to all the users is
identical for all the BS antennas (as in i.i.d. channels), then,
on an average the sets M
H
(q)
, q = 1, . . . , N/M would all
have similar topological properties. Since, M(H) is a direct-
sum of N/M topologically similar sets, it is expected that for
a fixed M, on an average the region M (H) expands with
increasing N . Specifically, for a fixed M and increasing N,
the maximum Euclidean length of any vector in M(H) grows
as O(
√
N), since M(H) is a direct-sum of O(N ) topolog-
ically similar sets (M(H
(q)
) , q = 1, 2, . . . , N/M) with the
maximum Euclidean length of any vector in M
H
(q)
being
O(1/
√
N) (note that in the definition of M
H
(q)
in (5),
each component of any vector v ∈ M
H
(q)
is scaled down
by
√
N). Also, for a fixed M and increasing N , since M(H)
is a direct-sum of N/M similar sets, it is expected that the
set M(H) becomes increasingly dense (i.e., the number of
elements of M(H) in a fixed volume in C
M
is expected to
increase with increasing N ). The above discussion leads us to
the following results in Section III-A and III-C.
A. Diminishing MUI with increasing N, for fixed M and fixed
E
k
(k = 1, . . . , M)
For a fixed M and fixed E
k
, the information alphabets and
the information symbol energies are fixed. However, since
increasing N (with fixed M ) is expected to enlarge the set
M(H) and make it increasingly denser, it is highly probable
that at sufficiently large N, for any fixed information symbol
vector u = (
√
E
1
u
1
, ··· ,
√
E
M
u
M
)
T
∈ U there exists a
vector v ∈ M(H) such that v is very close to u in terms
of Euclidean distance. This then implies that, with increasing
N and fixed M, for any u ∈ U there exists a transmit phase
angle vector Θ such that the sum of the MUI energy for all
users is small compared to the AWGN variance at the receiver.
Hence, for a fixed M and fixed E
k
, it is expected that the MUI
energy for each user decreases with increasing N.
This is in fact true, as we prove it formally for channels
satisfying the following mild conditions. Specifically for a
fixed M, we consider a sequence of channel gain matrices
{H
N
}
∞
N=M
satisfying
lim
N→∞
|h
(N)
k
H
h
(N)
l
|
N
= 0 , k 6= l (As.1)
lim
N→∞
P
N
i=1
|h
(N)
k
1
,i
||h
(N)
l
1
,i
||h
(N)
k
2
,i
||h
(N)
l
2
,i
|
N
2
= 0 , (As.2)
lim
N→∞
kh
(N)
k
k
2
N
= c
k
, (As.3)
k, l, k
1
, l
1
, k
2
, l
2
∈ (1, 2, . . . , M ) (6)
where c
k
are positive constants, h
(N)
k
denotes the k-th row
of H
N
and h
(N)
k,i
denotes the i-th component of h
(N)
k
. From
the law of large numbers, it follows that i.i.d. channels
satisfy these conditions with probability one [13]. Physical
measurements of the channel characteristics with large antenna
arrays at the BS have revealed closeness to the i.i.d. fading
model, as long as the BS antennas are sufficiently spaced apart
(usually half of the carrier wavelength) [14], [1].
Theorem 1: For a fixed M and increasing N, consider a
sequence of channel gain matrices {H
N
}
∞
N=M
satisfying the
mild conditions in (6). For any given fixed finite alphabet
U (fixed E
k
, k = 1, . . . , M ) and any given ∆ > 0, there
exist a corresponding integer N ({H
N
}, U, ∆) such that with
N ≥ N({H
N
}, U, ∆) and H
N
as the channel gain matrix,
for any u ∈ U to be communicated, there exist a phase
angle vector Θ
u
N
(∆) = (θ
u
1
(∆), ··· , θ
u
N
(∆))
T
which when
transmitted, results in the MUI energy at each user being upper
bounded by 2∆
2
, i.e.
P
N
i=1
h
(N)
k,i
e
jθ
u
i
(∆)
√
N
−
p
E
k
u
k
2
≤ 2∆
2
, k = 1, . . . , M. (7)
Proof – The proof relies on technical results in Theorem 3
(stated and proved in Appendix A) and Theorem 2 (stated and
proved below). All these results assume a fixed M (number
of user terminals) and increasing N (number of BS antennas).
These results are stated for a fixed sequence of channel
matrices {H
N
}
∞
N=M
, fixed information alphabets U
1
, ··· , U
M
and fixed information symbol energy E
1
, ··· , E
M
. Further,
the sequence of channel matrices {H
N
}
∞
N=M
is assumed to
satisfy the conditions in (6) and the information alphabets
are assumed to be finite/discrete. The proofs use a novel
probabilistic approach, treating the transmitted phase angles
as random variables. We now present the proof of Theorem 1.
Let us consider a probability space with the transmitted
phase angles θ
i
, i = 1, 2, . . . , N being i.i.d. r.v’s uniformly
distributed in [−π , π). For a given sequence of channel
matrices {H
N
}, we define a corresponding sequence of r.v’s
{z
N
}, with z
N
∆
= (z
I
(N)
1
, z
Q
(N)
1
, . . . , z
I
(N)
M
, z
Q
(N)
M
) ∈ R
2M
,
where we have
z
I
(N)
k
∆
= Re
P
N
i=1
h
(N)
k,i
e
jθ
i
√
N
, z
Q
(N)
k
∆
= Im
P
N
i=1
h
(N)
k,i
e
jθ
i
√
N
, k = 1, . . . , M. (8)
From Theorem 3 it follows that, for any channel sequence
{H
N
} satisfying the conditions in (6), as N → ∞ (with
fixed M), the corresponding sequence of r.v’s {z
N
} converges
in distribution to a 2M -dimensional real Gaussian random
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. XX, XX XXXX 4
B
∆
(u)
∆
=
(
b = (b
I
1
, b
Q
1
, ··· , b
I
M
, b
Q
M
)
T
∈ R
2M
|b
I
k
−
p
E
k
u
I
k
| ≤ ∆ , |b
Q
k
−
p
E
k
u
Q
k
| ≤ ∆ , k = 1, 2, . . . , M
)
(9)
vector X = (X
I
1
, X
Q
1
, ··· , X
I
M
, X
Q
M
)
T
with independent
zero-mean components and var(X
I
k
) = var(X
Q
k
) = c
k
/2 , k =
1, 2, . . . , M. For a given u = (
√
E
1
u
1
, ··· ,
√
E
M
u
M
)
T
∈ U,
and ∆ > 0, we next consider the box B
∆
(u) defined in (9) (at
the top of this page), where u
I
k
∆
= Re(u
k
) , u
Q
k
∆
= Im(u
k
).
The box B
∆
(u) contains all those vectors in R
2M
whose
component-wise displacement from u is upper bounded by ∆.
Using the fact that z
N
converges in distribution to a Gaussian
r.v. with R
2M
as its range space, in Theorem 2 it is shown
that, for any ∆ > 0, there exist an integer N({H
N
}, U, ∆),
such that for all N ≥ N({H
N
}, U, ∆)
Prob(z
N
∈ B
∆
(u)) > 0 , ∀u ∈ U. (10)
Since the probability that z
N
lies in the box B
∆
(u) is strictly
positive for all u ∈ U, from the definitions of B
∆
(u) in (9)
and z
N
in (8) it follows that, for any u ∈ U there exist a phase
angle vector Θ
u
N
(∆) = (θ
u
1
(∆), ··· , θ
u
N
(∆))
T
such that
Re
P
N
i=1
h
(N)
k,i
e
jθ
u
i
(∆)
√
N
−
p
E
k
u
I
k
≤ ∆ ,
Im
P
N
i=1
h
(N)
k,i
e
jθ
u
i
(∆)
√
N
−
p
E
k
u
Q
k
≤ ∆ (11)
for all k = 1, 2, ··· , M, which then implies (7).
Since Theorem 1 is valid for any ∆ > 0 and (7) holds
for all N ≥ N({H
N
}, U, ∆), we can satisfy (7) for any
arbitrarily small ∆ > 0 by having N ≥ N({H
N
}, U, ∆)
i.e., a sufficiently large N. Hence, the MUI energy at each
user can be guaranteed to be arbitrarily small by having
a sufficiently large N . Theorem 1 therefore motivates us to
propose precoding techniques which can achieve small MUI
energy levels.
An essential part of the proof for Theorem 1 was the
positivity of the box event probability Prob(z
N
∈ B
∆
(u)),
when N is sufficiently large. In the following theorem, we
formally state and prove the positivity of the box event
probability.
Theorem 2: For a given channel sequence {H
N
}
∞
N=M
satisfying (6) and a given fixed finite alphabet set U, for any
∆ > 0, there exist a corresponding integer N ({H
N
}, U, ∆),
such that for all N ≥ N({H
N
}, U, ∆) (with fixed M)
Prob(z
N
∈ B
∆
(u)) > 0 , ∀u ∈ U. (12)
where B
∆
(u) is defined in (9).
Proof – We consider the probability that a n-dimensional
real r.v. X = (X
1
, X
2
, ··· , X
n
) lies in a n-dimensional
box centered at α = (α
1
, . . . , α
n
) ∈ R
n
and denoted by
C(∆, α) =
(x
1
, x
2
, ··· , x
n
) ∈ R
n
|α
k
− ∆ ≤ x
k
≤
α
k
+ ∆ , k = 1, 2, . . . , n
. For notational convenience, we
refer to α
k
+ ∆ and α
k
−∆ as the corresponding “upper” and
“lower” limits for the k-th coordinate. The probability that X
lies in the box C(∆, α) is given by the expansion
Prob(X ∈ C(∆, α)) =
n
X
k=0
(−1)
k
T
k
(∆, α) (13)
where T
k
(∆, α) is the probability that the r.v.
(X
1
, X
2
, ··· , X
n
) belongs to a sub-region of
(x
1
, ··· , x
n
) ∈ R
n
| x
l
≤ α
l
+ ∆ , l = 1, 2, . . . , n
,
where exactly k coordinates are less than their corresponding
“lower” limit and the remaining n − k coordinates are less
than their corresponding “upper” limit. Specifically, T
k
(∆, α)
is given by
2
T
k
(∆, α) =
n
X
i
1
=1
n
X
i
2
=i
1
+1
···
n
X
i
k
=i
k−1
+1
Prob
X
r
≤ α
r
− ∆
∀r ∈ {i
1
, i
2
, ··· , i
k
} ,
X
r
≤ α
r
+ ∆ ∀r /∈ {i
1
, i
2
, ··· , i
k
}
(14)
Using the expansion in (13), the probability of the box event
n
z
N
∈ B
∆
(u)
o
can be expressed as
Prob
z
N
∈ B
∆
(u)
=
Prob
(
√
E
k
u
I
k
− ∆) ≤ z
I
k
(N)
≤ (
√
E
k
u
I
k
+ ∆) ,
(
√
E
k
u
Q
k
− ∆) ≤ z
Q
k
(N)
≤ (
√
E
k
u
Q
k
+ ∆) , k = 1, 2, . . . , M
=
2M
X
k=0
(−1)
k
2M
X
i
1
=1
2M
X
i
2
=i
1
+1
···
2M
X
i
k
=i
k−1
+1
Prob
z
(N)
l
≤
√
E
l
u
l
− ∆
∀l ∈ {i
1
, i
2
, ··· , i
k
} ,
z
(N)
l
≤
√
E
l
u
l
+ ∆ ∀l /∈ {i
1
, i
2
, ··· , i
k
}
(15)
where z
(N)
l
is the l-th component of z
N
(i.e., z
(N)
l
= z
Q
(N)
l/2
for even l, and z
(N)
l
= z
I
(N)
(l+1)/2
for odd l) and u
l
is the l-th
component of the vector (u
I
1
, u
Q
1
, u
I
2
, u
Q
2
, ··· , u
I
M
, u
Q
M
)
T
. For
notational convenience we define
T
(N)
(k, i
1
, i
2
, ··· , i
k
, u, ∆)
∆
=
Prob
z
(N)
l
≤
√
E
l
u
l
− ∆ ∀l ∈ {i
1
, i
2
, ··· , i
k
} ,
z
(N)
l
≤
√
E
l
u
l
+ ∆ ∀l /∈ {i
1
, i
2
, ··· , i
k
}
1 ≤ i
1
< i
2
< ··· < i
k
≤ 2M , 0 ≤ k ≤ 2M. (16)
Let Y = (Y
1
, Y
2
, ··· , Y
2M
) denote a multivariate 2M-
dimensional real Gaussian r.v. with independent zero mean
components and var(Y
2k−1
) = var(Y
2k
) = c
k
/2 , k =
1, 2, . . . , M. From Theorem 3 (Appendix A) it follows
that the c.d.f. of z
N
converges to the c.d.f. of Y
as N → ∞. This convergence in distribution implies
that, for any given arbitrary δ > 0, for each term
2
As an example, for n = 2, we have Prob
α
1
− ∆ ≤ X
1
≤ α
1
+
∆ , α
2
− ∆ ≤ X
2
≤ α
2
+ ∆
= T
0
(∆, α) − T
1
(∆, α) + T
2
(∆, α),
where T
0
(∆, α)
∆
= Prob(X
1
≤ α
1
+ ∆ , X
2
≤ α
2
+ ∆ ), T
2
(∆, α)
∆
=
Prob(X
1
≤ α
1
− ∆ , X
2
≤ α
2
− ∆), and T
1
(∆, α)
∆
= Prob(X
1
≤
α
1
+ ∆ , X
2
≤ α
2
− ∆) + Prob(X
1
≤ α
1
− ∆ , X
2
≤ α
2
+ ∆).