An Outage Probability Comparison
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Receive Antenna Array Strategies in Fading and Interference: An Outage Probability Comparison Juan M. Romero-Jerez, Member, IEEE, and Andrea J. Goldsmith, Fellow, IEEE
Abstract— We explore tradeoffs between different reception strategies of multiple receive antennas in fading channels with co-channel interference (CCI). Our tradeoff analysis is based on outage probability. We assume the signal from the desired user at the receive antenna array to be affected by Rice, Nakagami or Rayleigh fading, while CCI signals are assumed to experience Rayleigh fading. We provide closed-form analytical expressions for the outage probability of different diversity schemes, such as maximal ratio combining (MRC) and optimum combining (OC), and also for interference cancellation (IC) based on antenna beamsteering, which steers nulls in the array radiation pattern in the direction of the strongest interferers. Our analysis provides a unified framework for studying outage probability of different multiple-antenna reception strategies, and our closedform expressions are easily computed, facilitating a performance comparison under a range of operating conditions. Our numerical results show that IC yields significantly better performance than MRC if the system is interference-limited and the number of dominant interferers is lower than the number of receive antennas, or when the output SINR is low. In addition, when the background noise can be neglected, we provide numerical results for MRC and OC when the signals from the desired user at the different receive antennas are arbitrarily distributed. Index Terms— Antenna Array, Diversity, Interference Cancellation, Beamsteering, Outage Probability
I. I NTRODUCTION One of the key techniques to combat the impairments of the wireless communication channel is the use of adaptive antenna arrays. In particular, the detrimental effect of multipath fading can be reduced by using diversity combining techniques whereby the received signals at every antenna are properly weighted and added to improve the output signalto-noise ratio (SNR) [1], [2]. In order for linear combining techniques to perform optimally, the antenna elements must be uncorrelated. In flat-fading channels, maximal ratio combining (MRC) diversity is well known to be optimum in the sense of maximizing the output SNR. If the desired signal is affected by both co-channel interference (CCI) and flat-fading, the diversity combining technique that maximizes the output signal-tointerference-plus-noise ratio (SINR) is the so-called optimum combining (OC) [2], [3], therefore, OC aways has the same or better SINR as MRC in channels with fading and/or CCI. The drawback of OC is a higher implementation complexity because the channel state of the desired and all CCI signals must be known at every receive antenna. Therefore, even under the presence of interference, MRC may be preferable in certain scenarios due to its reduced complexity, as only the channel state of the desired user must be known at the receiver. Juan M. Romero Jerez is with the University of M´alaga, Spain. Andrea J. Goldsmith is with Stanford University, CA.
Another way of using multiple antennas is to adaptively modify the radiation pattern of the array to either enhance the received power of the desired signal or reduce the interference by placing nulls in the direction of the dominant interferers [4]. Contrary to the case of diversity combining, in this case, the elements of the antenna array must be close to each other so that the antennas can be highly correlated, thereby reducing the angular spread of the received signals and allowing for a better estimation of the angle of arrival (AoA) of the signal sources [5]. This technique, known as beamforming, does not provide diversity gain, but it may be preferable in the low SINR regime when there are a few strong dominant interferers. In addition, it is simpler to implement than OC because only one parameter, the AoA, is required for each source. Though OC is the optimum technique in the presence of interference and simultaneously provides diversity and interference cancellation, its implementation complexity may preclude its use in real systems, especially when the number of antennas and interferers is high. Therefore, a performance comparison of the commonly used techniques: MRC (to provide only diversity) and IC (to achieve the cancellation of the strongest interferers), along with the best performance achieved by OC in channels with fading and interference, is important for wireless system design, and is the main focus of this work. Outage probability is a common metric to measure the performance of wireless communication systems. We define an outage as the event that the SINR falls below a predefined threshold. In this work we present new closed-form expressions generalizing previous results on outage probability of wireless systems in interference environments using MRC, OC and an idealized beamforming algorithm able to cancel the strongest CCI signals (null-steering). In addition, we provide a comparison among these different techniques. We assume that all CCI signals undergo flat Rayleigh fading, while the signal of interest (SOI) can experience Rice, Nakagami or Rayleigh fading. All our closed-form outage probability expressions are easily computable, as they are expressed using wellknown tabulated functions typically available in most software computation packages such as Matlab. Several recent works have investigated the outage probability of the antenna processing schemes studied in this paper. Specifically, in [6], [7], [8] results are presented for the outage probability of MRC under Rayleigh interference considering that the average powers associated with CCI signals are the same or are completely different. The case of arbitrary interference power is studied in [9] for a SOI under Nakagami, Rice or Rayleigh fading, but the background noise is neglected. Results for arbitrary power interference and noise
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are presented in [10], though only for a SOI under Rayleigh fading. In this paper, we extend previous results of MRC under CCI by allowing the SOI to be arbitrarily distributed and the CCI signals to have arbitrary average powers, while simultaneously considering background noise. For interference limited systems (negligible background noise) we derive novel expressions allowing the SOI to be arbitrarily distributed at every receive antenna. This is typical of propagation environments where the desired user does not experience the same propagation conditions at all the receive antennas, e.g., there could be a line of sight (LOS) signal component for some antennas and not for others. This might occur in practice in a cellular system with macrodiversity combining across multiple base stations. As in the case of MRC, and in order to provide a fair comparison, we also generalize previous results of OC under Rayleigh-distributed CCI signals by allowing the signal from the desired user at every antenna to be arbitrarily distributed and to have arbitrary average powers. Previous analytical results for outage probability of OC diversity include [11] for a Rayleigh fading SOI and [6], [7], [12] for a Ricean fading SOI, and the work of Zhang and Cui [13], where a very simple and general result for the outage probability is presented considering different fading scenarios for the SOI with arbitrary correlation. However, none of these works consider the more general case of an arbitrary fading distribution on the desired users’ signals at every receive antenna, as we consider here. For beamforming antenna array systems, most analytical results on outage probability consider all CCI signals to be identically distributed [14], [15]. In [16] the outage probability is computed for arbitrary fading distributions of interferers, but the expressions must be computed by performing numerical integration. An interesting framework for Rayleigh-distributed CCI signals is presented in [17], where a closed-form expression of the outage probability is derived for, in principle, arbitrary average CCI signal powers. However, the results are restricted to the case where the terms of the partial fraction expansion of the moment generating function (MGF) of the interference have a multiplicity order equal to one. In our analysis we remove that restriction and present results for the case of completely arbitrary Rayleigh-distributed CCI signals with arbitrary powers and an arbitrarily distributed SOI. The remainder of this paper is organized as follows. In Sections II, III, and IV we present, respectively, the system model and the outage probability expressions for, respectively, MRC, OC and ideal null-steering. Numerical results and a performance comparison among the different techniques are presented in Section V. Finally, discussion and concluding remarks are given in Section VI. II. O UTAGE P ROBABILITY OF MRC S YSTEMS WITH I NTERFERENCE A. System Model and Interference Statistics We consider a wireless communication system with N uncorrelated receive antennas where the received signal from the desired user at every antenna is assumed to be corrupted by L interference signals. In addition, and unless stated otherwise,
the desired signal at each antenna is corrupted by additive white Gaussian noise (AWGN). The received noise vector is assumed to be zero mean with covariance matrix σ 2 IN ×N , where IN ×N denotes the N × N identity matrix. The desired and interference signals undergo slow flat fading and a coherent receiver is employed which is assumed to have perfect knowledge of the instantaneous channel state. The signals at every receive antenna from the desired user are assumed to experience Nakagami, Rice or Rayleigh fading. On the other hand, a given interfering user experiences independent identically-distributed (i.i.d.) Rayleigh fading at the N-element array, and the power of every interfering user is assumed to be arbitrary. Let c s = [cs1 . . . csN ]T and ci = [ci1 . . . ciN ]T denote, respectively, the channel gain vectors of the desired and the i-th interfering user at the antenna array, where the superscript T refers to vector transposition. Then, the received baseband N-element signal vector can be written as: r = cs bs +
L Pi ci bi + n,
(1)
i=1
where Pi is the mean power of the i-th interferer at each antenna, n is the N-dimensional received noise vector, and bs and bi are, respectively the transmitted symbols from the desired and i-th interfering user. For simplicity, it is assumed that bs = 1 and bs = 1 ∀i, 1 ≤ i ≤ L. The elements of ci are assumed to be complex Gaussian random variables with zero mean and unit variance, whereas for the received signal from the desired user at antenna n it is assumed that |csn | follows either a Nakagami, Rice or Rayleigh distribution 2 satisfying E[|csn | ] = Psn . That is, for the general case, and unless stated otherwise, we allow the elements of c s to be independent non-identically-distributed (i.n.d.) random variables with arbitrary average squared mean. In an MRC receiver, the antenna array elements are weighted by the channel gains associated with the desired user, yielding the following output signal: 2
yR = cH s r = |cs | bs +
L H Pi cH s ci bi + cs n,
(2)
i=1
where the superscript H denotes Hermitian transposition. It follows that the received SINR after combining will be: γ = L i=1
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|cs |
2
2
2 Pi |cH s ci | + |cs | σ
= L i=1
|cs |
2 2
Pi |vi | + σ 2
, (3)
where vi = cH s ci / |cs |. It is demonstrated in [6] that if the elements of ci are i.i.d. zero mean unit variance complex Gaussian random variables, then v i and cs are mutually independent and v i is a zero mean unit variance complex 2 Gaussian random variable. It follows that |v i | is exponentially distributed with unit mean. We can write the received SINR as follows: X γ= , (4) Y + σ2 N L √ 2 2 2 where X = |cs | = n=1 |csn | and Y = i=1 Pi |vi | , being X and Y independent. Note that Y is the sum of L independent exponential random variables with arbitrary mean
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powers. Let us divide the total number of interferes L into J groups, where every interferer in a group has the same mean power Pi . Consider that there are n i interferers in a given group with mean power P i . Then, the pdf of Y can be shown to be given by [9], [18]: fY (y) =
ni J
Aij
i=1 j=1
y ni −j (ni − j)!Pini −j+1
where Aij are coefficients given by: Aij
=
(−1)
j−1
J
ΩA k=1,k=i
e−y/Pi ,
nk + qk − 1 nk − 1
Pkqk Pink × , (Pi − Pk )nk +qk
where Ψn (s) is the MGF of the received signal power at antenna n. To calculate the k-th order derivative of Ψ n (s) we apply the Leibniz derivative rule, yielding N I dk ΨX (s) dk = k Ψn (s) dsk ds i=1 k! (q ) (q ) (q ) = Ψ 1 (s)Ψ2 2 (s) · · · ΨN N (s) q1 !q2 ! . . . qN ! 1
(5)
ΩX
= k!
N (q ) Ψn n (s) , qn ! n=1
(13)
ΩX
(6)
where ΩA is the set of (J − 1)-tuples such that Ω A = J (q1 , . . . qJ ) : qk ∈ N, qi = 0, k=1 qk = j − 1 . The cdf of Y will be: k n J ni i −j y 1 Aij e−y/Pi . (7) FY (y) = 1 − k! P i i=1 j=1
where ΩX is the set of N-tuples such that Ω X = N (q ) (q1 , q2 . . . qN ) : qn ∈ N, n=1 qn = k , and Ψn n (s) is the qn -th order derivative of Ψ n (s). Substituting (13) into (11) we obtain the general expression of P out for arbitrary average power CCI signals:
k=0
=
Pout
For the case in which all CCI signals have the same average power PI , (7) simplifies to: L−1 1 y k −y/PI FY (y) = 1 − e . (8) k! PI
ni J
Aij
i=1 j=1
×
n i −j
1
k=0
(βPi )k
N
1
n) Ψ(q (s) .
n q ! s=−1/βPi n n=1
(14)
ΩX
k=0
B. Outage Probability Calculation of Interference-Limited Systems In an interference-limited system, the background noise is assumed to be negligible and it is shown in Appendix I that the probability of outage, defined as the probability that the output SIR falls below a predefined threshold β, can be calculated as follows: ∞ Pout = 1 − FY (x/β)fX (x)dx, (9) 0
where FY (·) is the cdf of the output interference and f X (·) is the pdf of the output signal from the desired user, with X and Y defined in (4). Introducing (7) into (9), we can write: n ni J i −j 1 1 ∞ k −x/βPi Pout = Aij x e fX (x)dx. k k! 0 i=1 j=1 k=0 (βPi ) (10) Note that the integral in (10) is in fact the k-th order derivative of the moment generating function (MGF) of X, Ψ X (s), with s = −1/βPi , and therefore we have:
n ni J i −j 1 1 dk ΨX (s)
Pout = Aij . k k! dsk s=−1/βPi i=1 j=1 k=0 (βPi ) (11) As X is a sum of independent random variables, its MGF will be given by the product of the MGF of the individual terms in the sum, that is, ΨX (s) =
N n=1
Ψn (s),
(12)
If all CCI signals have the same average power P I , (14) simplifies to
Pout =
L−1 k=0
N
1 1
(qn ) Ψ (s) . (15)
n (βPI )k q ! s=−1/βPI n n=1 ΩX
Note that our results include the case where the signal power of the desired user at every receive antenna may have a different distribution. Thus (14) and (15) can be evaluated for any distribution for which the n-th order derivative of the MGF of the fading power distribution is known. In Table I the n-th order derivative for Nakagami, Rice and Rayleigh fading are shown. In particular, when the distribution of the received power at every antenna is i.n.d. Nakagami and the CCI signals have arbitrary average powers, using the Nakagami entry in Table I, (14) becomes:
Pout
=
ni J
Aij
i=1 j=1
×
n i −j
1
k=0
(βPi )k
N ΩX
n Psqnn mm n Γ (mn + qn ) . (mn + Psn /(βPi ))mn +qn Γ (mn ) qn ! n=1
(16) For the case of i.n.d. Ricean fading for the received signals of the desired user at every antenna and CCI signals with
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TABLE I S TATISTICS OF S IGNAL P OWER FOR C OMMON FADING C HANNELS Type of Fading
PDF, CDF, MGF and n-th order derivative of the MGF of the signal power
Rayleigh
f (x) =
1 p
exp − x , where p = E[x] p
F (x) = 1 − ex/p Ψ(s) =
1 1−sp
Ψ(n) (s) = f (x) =
Nakagami
m p
FX (x) =
n!pn (1−sp)n+1
m
1 xm−1 Γ(m)
Ψ(n) (s) = f (x) =
1+K p
1+K x p
exp −K − √
1+K 1+K−sp
Ψ(n) (s) =
−m
pn mm Γ(m+n) (m−sp)n+m Γ(m)
F (x) = 1 − Q Ψ(s) =
2K,
exp
Pout =
i=1 j=1
Aij
n i −j
Kps 1+K−sp
pn (1+K)(n!)2 (1+K−sp)n+1
exp
1 k
k=0
(βPi )
N
Psqnn (1 + Kn )qn ! (1 + Kn + Psn /(βPi ))qn +1 ΩX i=1 (17) −Kn Psn /(βPi ) × exp (1 + Kn + Psn /(βPi )) l qn (1 + Kn )Kn 1 . × (l!)2 (qn − l)! 1 + Kn + Psn /(βPi ) ×
I0
2
(1+K)K x p
2 1+K x , where K ≥ 0 p
arbitrary powers, using (14) we have ni J
1 2
γ(m,xm/p) Γ(m)
p Ψ(s) = 1 − s m
Rice
exp − xm , where m ≥ p
l=0
It can be shown that when the elements of X are i.i.d, (16) and (17) collapse, respectively, to [9, eq. (21)] and [9, eq. (18)].
Kps 1+K−sp
n 1 l=0 (l!)2 (n−l)!
(1+K)K 1+K+−sp
l
leading to close form expressions of the outage probability, as we show below. From (7) the following equality holds: ni J 2 x 2 −σ Aij e−(x/β−σ )/Pi = 1 − FY β i=1 j=1 k n i −j x 1 1 2 −σ × .(20) k! Pik β k=0
Substituting (20) into (19) and using straightforward manipulation, we obtain: Pout
= FX (βσ 2 ) +
ni n J i −j k i=1 j=1 k=0 l=0
C. Outage Probability Calculation of Systems with NonNegligible Background Noise When the noise power at the receiver is comparable to the interferer powers its effect cannot be neglected, and an outage will be considered to occur when the received SINR is below a predefined threshold β, that is: X Pout = P < β . (18) Y + σ2 It is shown in Appendix I that the outage probability can be calculated as follows: ∞ x − σ 2 fX (x)dx. Pout = FX (βσ 2 ) + 1 − FY β βσ2 (19) This particular expression is suitable for our purpose because it is given as a function of an integral of the cdf of Y, thus
2
Aij
eσ /Pi (−σ 2 )k−l l!(k − l)!Pik β l
× GX (l, −1/(βPi ), βσ 2 ), where we define GX (n, s, Λ) =
∞
Λ
xn esx fX (x)dx.
(21)
(22)
Note that (22) can be regarded as the incomplete n-th order derivative of the MGF of X. Therefore its convergence is guaranteed as long as this derivative exists, which is the case in Nakagami, Rice and Rayleigh fading. 1) Nakagami-Distributed Desired User: When the received signal from the desired user at every antenna element undergoes i.i.d. Nakagami fading, the power X of the desired signal after combining will be given by a Gamma distribution, with pdf mN m 1 fX (x) = xmN −1 e−xm/Ps . (23) Ps Γ(m)
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The corresponding cdf will be FX (x) =
γ(mN, xm/Ps ) , Γ(mN )
the case in (28) which, therefore, can be expressed in closedform. Specifically, following [20], we can write (24) QN +2n,N −1 (a, b) =
x
where γ(α, x) = 0 z α−1 e−z dz denotes the incomplete Gamma function. Introducing (23) into (22), the evaluation of the integral yields: GX (n, s, Λ) =
(m/Ps )mN (m/Ps − s)mN +n Γ (mN + n, (m/Ps − s) Λ) , (25) × Γ(mN )
∞ where Γ(α, x) = x z α−1 e−z dz denotes the complementary incomplete Gamma function. Then, by substituting (24) and (25) into (21), the outage probability for a desired user with i.i.d. Nakagami fading can be calculated. 2) Ricean-Distributed Desired User: When the received signal at every antenna experiences i.i.d. Ricean fading, the received signal power X after combination follows a noncentral chi-squared distribution with pdf fX (x)
=
N2+1 x N2−1 1+K 1+K e−(KN + Ps x) Ps KN ⎛ ⎞ (1 + K)KN ⎠ × IN −1 ⎝2 x , (26) Ps
where In (x) is the modified Bessel function of the first kind and order n [19]. The corresponding cdf of X can be expressed as √ 1+K 2KN, 2 x , (27) FX (x) = 1 − QN Ps ∞ 2 2 where Qn (a, b) = b a1−n xn e−(x +a )/2 In−1 (ax)dx denotes the n-order Marcum-Q function. Introducing (26) into (22), and solving the integral we obtain
N2+1 n 1+K Ps 1 + K − sPs 2(1 + K − sPs ) 1−N sKN Ps × (2KN ) 2 exp 1 + K − sPs ⎛ ⎞ (1 + K)KN 1+K × QN +2n,N −1 ⎝ 2 , 2 − s Λ⎠ , 1 + K − sPs Ps
GX (n, s, Λ) =
(28) ∞
2
2
where Qp,q (a, b) = b xp e−(x +a )/2 Iq (ax)dx denotes the Nuttal-Q function. This function is not considered a tabulated function1, however, the Nuttal-Q function can be expressed in terms of the more common Marcum-Q function and modified Bessel function of the first kind (available in most software computation packages) when p + q is odd [20]. This is exactly 1 That is, it is not typically included in software computation packages, such as Matlab.
n+1
cl (n)aN −1+2(l−1) QN −1+l (a, b)
l=1
2 n a + b2 Pn, l (b2 )al−1 bN +l IN +l−2 (ab) + exp − 2 l=1 (29) with
cl (n) = 2
n−l+1
and 2
Pn, l (b ) =
n−l
2
n−l−j
j=0
n n−l+1 n−1−j n−l−j
(N − 1 + n)! (N + l − 2)! (N − 1 + n)! 2j b . (N − 1 + l + j)!
Substituting now (27) and (28) into (21), the outage probability for an i.i.d. Rice desired user can be calculated. 3) Rayleigh-Distributed Desired User: When the desired signal at every receive antenna experiences Rayleigh fading, it is possible to find a closed-form expression for the outage probability for arbitrary powers of the desired signal at every antenna (i.e, the desired user experiences i.n.d. Rayleigh fading). Let us divide the number of antennas N into J s groups, where for a given group the average received power from the desired user is the same. Consider that there are n sh antennas in a given group receiving an average power from the desired user Psh . Then, the pdf of X will be given by: fX (x) =
nsh Js h=1 t=1
Bht
xnsh −t −x/Psh , nsh −t+1 e (nsh − t)!Psh
(30)
where Bht are coefficients that can be calculated as Js nsk + qk − 1 t−1 Bht = (−1) nsk − 1 ΩB k=1,k=h
qk nsk Psk Psh × , (Psh − Psk )nsk +qk
(31)
where ΩB is the set of (Js − 1)-tuples such that ΩB = s (q1 , . . . qJs ) : qk ∈ N, qh = 0, Jk=1 qk = t − 1 . The corresponding cdf for the random variable X is given by k nsh n Js sh −t x 1 Bht e−x/Psh . (32) FX (x) = 1 − k! Psh t=1 h=1
k=0
Now, from (30) and (22), solving the integral we obtain: nsh Js
(nsh − t + n)! (nsh − t)! h=1 t=1
l 1 Psh × exp −Λ − s × (1 − sPsh )nsh −t+n+1 Psh
nsh −t+l u 1 1 × −s . Λ u! Psh u=0
GX (n, s, Λ) =
Bht
(33)
Now, substituting (32) and (33) into (21), the outage probability for i.n.d. Rayleigh fading of the desired user can be
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calculated. However, in this case, a more compact and simpler solution for the outage probability (with less nested equations in the final expression) can be obtained using an alternative approach. Let us rewrite the outage probability definition as follows: X < β = P (X < β(Y + σ 2 )). (34) Pout = P Y + σ2 From (34), conditioning on the random variable Y, with the help of (32) and after some manipulation, we can write the outage probability to be Pout |Y =y = 1 − ×
nsh Js
h=1 t=1 n sh −t k=0
β Psh
Bht e−βσ
2
e
1 y n (σ 2 )k−n . n!(k − n)! n=0 (35)
To find an expression for the outage probability, we can introduce the pdf of Y, given in (5), into (36) and solve the integral. However, a simpler expression is obtained by noting that the integral in (36) is in fact the n-th order derivative of the MGF of the random variable Y, Ψ Y (s), with s = −β/Psh . Thus, we can write k nsh n Js sh −t 2 β Bht e−βσ /Psh Pout = 1 − Psh h=1 t=1 k=0 (37)
k n
1 2 k−n d ΨY (s)
(σ ) × . n!(k − n)! dsn s=−β/Psh n=0 Since Y is a sum of exponential random variables, it follows that its MGF is given by 1 , (1 − sPi )ni i=1
=
1−
nsh Js
Bht e−βσ
2
/Psh
h=1 t=1 n sh −t
k k β 1 (σ 2 )k−n × Psh (k − n)! n=0 k=0 J Piqi ni + qi − 1 × . qi (1 + βPi /Psh )ni +qi (40)
k k
J
Pout
ΩY i=1
/Psh −βy/Psh
The unconditioned outage probability can be obtained by integrating (35) over all possible values of Y, that is: k nsh n Js sh −t 2 β Pout = 1 − Bht e−βσ /Psh Psh h=1 t=1 k=0 k ∞ 1 (σ 2 )k−n × y n e−βy/Psh fY (y)dy. n!(k − n)! 0 n=0 (36)
ΨY (s) =
such that Ω Y = where ΩY is the set of J-tuples J (q1 , . . . qJ ) : qi ∈ N, i=1 qi = n . Finally, introducing (39) into (37), the outage probability will be:
(38)
where J and n i were defined in Section II-A. By the application of the Leibniz derivative rule, we can write: J dn ΨY (s) 1 dn = dsn dsn i=1 (1 − sPi )ni J 1 1 dqi = n! qi ! dsqi (1 − sPi )ni ΩY i=1 J Piqi ni + qi − 1 , = n! qi (1 − Pi s)ni +qi i=1 ΩY
(39)
III. O UTAGE PROBABILITY OF O PTIMUM C OMBINING S YSTEMS We consider a system with N uncorrelated receive antennas where the received signal at every antenna from the desired user is assumed to be corrupted by L interference signals, satisfying L ≥ N (overloaded system). We follow in this section an approach similar to [13] and assume that all interferers experience i.i.d. Rayleigh fading with the same average power P I and the system is assumed to be interference limited (negligible background noise). In an OC receiver, the received signal vector is weighted to maximize the resulting SIR. The output signal after weighting will be L wH ci bi , yR = w cs bs + PI H
(41)
i=1
where cs , ci , bs and bi are defined as in Section II-A. It is well known that the weight vector that maximizes the SIR is given by [21]: w = gR−1 cs , (42) where g is an arbitrary constant and R is a Wishart distributed matrix given by L R= ci cH (43) i , i=1
which results in the following conditional output SIR: γ=
1 H −1 cs R cs . PI
(44) 2
Let us define the random variables X = |c s | = N −1 2 H −1 cs . n=1 |csn | , as in Section II-A, and ξ = cs R The outage probability, defined as the probability that the received SIR falls below a predefined threshold β, can be written as: Pout = P (γ < β) = P (ξ > 1/βPI ) = 1 − Fξ (1/βPI ). (45) It is shown in [13] that the random variable ξ, conditioned on X, is Gamma distributed with pdf f ξ|X=x (z) =
z L−N xL−N +1 e−xz , (L − N )!
(46)
7
and it is straightforward that the conditional cdf of ξ will be F ξ|X=x (z) = 1 − e−xz
L−N k=0
1 (xz)k . k!
(47)
The unconditional cdf of ξ is obtained by simple averaging (47) over all possible values of X, yielding: L−N 1 ∞ zk Fξ (z) = 1 − xk e−xz fX (x)dx. (48) k! 0 k=0
And noting that the integral in (48) is in fact the k-th order derivative of the MGF of X parametrized by s = −z, we can write
L−N 1 dk ΨX (s)
zk . (49) Fξ (z) = 1 − k! dsk s=−z k=0
We now introduce in (49) the kth-order derivative of the MGF of X, which is given in (13), we obtain L−N N
1
n) Ψ(q Fξ (z) = 1 − zk . (50) n (s)
q ! s=−z n=1 n k=0
ΩX
From (45) and (50) it is clear that the final expression for the outage probability when the received desired signal at every antenna is arbitrarily distributed will be L−N N
1 1
(qn ) Pout = Ψ (s) . (51)
n (βPI )k q ! s=−1/βPI n=1 n ΩX
k=0
Note that (51) is in fact the same result obtained in (15) for interference-limited MRC systems for equal average power of the CCI signals by just changing in (15) the number of interferers L to L − N + 1. That is, OC diversity with equal average CCI signal powers for overloaded systems achieves the cancellation of N − 1 interferers, while still retaining the diversity improvement of MRC under interference. This fact was previously known to hold when the signals at every antenna from the desired user are i.i.d. [2], [7], but it is now proved to hold even for arbitrary distribution of the signals from the desired user at every receive antenna. In particular, when the distribution of the received power at every antenna from the desired user is i.n.d. Nakagami, from (51) we can write the outage probability as Pout =
L−N k=0
×
1 (βPI )k
N ΩX
n Psqnn mm n Γ (mn + qn ) , (mn + Psn /(βPi ))mn +qn Γ (mn ) qn ! n=1
(52)
while for the case of i.n.d. Ricean fading for the received signals from the desired user at every antenna we have Pout
=
L−N
Psqnn (1 + Kn )qn ! 1 (βPI )k (1 + Kn + Psn /(βPi ))qn +1 k=0 ΩX i=1 −Kn Psn /(βPi ) × exp (1 + Kn + Psn /(βPi )) l q n (1 + Kn )Kn 1 × . (l!)2 (qn − l)! 1 + Kn + Psn /(βPi ) N
l=0
(53)
It can be shown that if the received signals at every antenna from the desired user have the same average powers and parameter K (i.e., i.i.d. Ricean fading), (53) collapses to [7, eq. (8)]. Note that our results can be applied to macrodiversity combining systems, and are given in a complete analytical form and no simulations are needed. In particular, our results are exact, and don’t require numerical integration or hypergeometric functions (which are in fact infinite series that are typically truncated in calculations). IV. O UTAGE P ROBABILITY OF A N ULL -S TEERING B EAMFORMER A. System Model and Interference Statistics In this section we consider that the N antennas at the receive end are employed for interference cancellation. The optimum weights for a linear beamformer are also given by (42)-(43). However, there is a fundamental difference with respect to OC, as in this case, cs and ci are the array response vectors for the desired an the i-th interfering signals, respectively. Note that only the angle of arrival (AoA) of a signal is needed to calculate the array response [5], that is, only one parameter per interferer must be known at the receiver in this case. In contrast, for OC, N parameters per interferer (the channel gain at every antenna) must be known at the receiver to calculate the antenna weights. Instead of calculating the outage probability starting from the optimum antenna weights, which will be a function of the particular antenna array layout and the AoA of the different receive signals, we will use an alternative approach that will allow us to obtain closed form expressions for the outage probability. An ideal null-steering algorithm is assumed which is able to perfectly cancel the N −1 interferers with the highest powers while not having a detrimental effect on the desired signal. However, we consider that not all interferers can be canceled, that is, for L total interferers, if L > N − 1, then L − N + 1 interference signals remain after cancellation. In addition to interference, the desired signal at each antenna is corrupted by additive white Gaussian noise (AWGN). The signals at the receive end from the desired user is assumed to experience Nakagami, Rice or Rayleigh fading. On the other hand, we assume that there are L interfering users experiencing Rayleigh fading, and the average power P i for i = 1 . . . L of every interfering user at the N-element array is assumed to be arbitrary. Let the set {yk }L k=1 denote the instantaneous interference signal powers. The order statistics y k;L are obtained by arranging the values of the set in increasing order of magnitude y1:L ≤ y2:L ≤ . . . yL:L . The pdf of the order statistics is given by [22] [17]: fy1:L ≤y2:L ≤...yL:L (x1 , x2 . . . xL ) =
L
fσ(i) (xi ), (54)
σ∈SL i=1
where σ = {σ(1)σ(2) . . . σ(L)}, and S L denotes the set of the L! permutations of the integers {1, 2, . . . , L}. As CCI signals undergo Rayleigh fading, we have that f σ(i) (xi ) = M 1 −xi /Pσ(i) . Let Y = i=1 yi:L denote the remaining Pσ(i) e
8
interference after interference cancellation corresponding to the sum of the M = L − N + 1 weakest signal powers. The MGF of Y can be calculated as follows: M ∞ ΨY (s) = E exp s yi:L = fσ(L) (xL )dxL
i=1
∞
xL−1
σ∈SL
··· × fσ(M+1) (xM+1 )dxM+1 ∞ xM × esxM fσ(M) (xM )dxM xM −1 ∞ ∞ sx2 ··· × e fσ(2) (x2 )dx2 esx1 fσ(1) (x1 )dx1 . (55)
It is shown in [17] that the solution of (55) can be written as
θ 0 L−M
σ∈SL
l=1
M
θl
k=1
1 , θL−M+k − ks
(56)
L k−1 where θ 0 = i=1 1/Pi and θk = i=0 1/Pσ(L−i) . We can rewrite (56) in a more convenient form as ΨY (s) =
θ 0 L
M
l=1 θl k=1
σ∈SL
1 , 1 − αk s
(57)
θ 0
L
σ∈SL
Jα
l=1 θl i=1
ΨY (s) =
θ 0
L
σ∈SL
1 . (1 − αi s)nαi
(58)
nαi Jα
l=1 θl i=1 j=1
δij
1 , (59) (1 − αi s)nαi −j+1
where the δij are coefficients given by δij
= (−1)j−1 ×
L
σ∈SL
l=1 θl i=1 j=1
Jα
Ωα k=1,k=i qk nk αk αi , (αi − αk )nk +qk
nk + qk − 1 nk − 1
δij
y nαi −j (ni −
nα −j+1 e j)!αi i
−y/αi
.
(61)
The cdf of Y will be θ 0 FY (y) = 1 − L σ∈SL
×
nαi Jα
l=1 θl
δij e
−y/αi
i=1 j=1
1 y k . (62) k! αi
nαi −j k=0
B. Outage Probability Calculation The outage probability can be calculated using the same framework as that presented in Section II-C. In particular, we have to solve (19), where in this case X is the power of the desired signal undergoing Rice, Rayleigh or Nakagami fading, and Y is the remaining interference power after cancellation, which is distributed according to (61). The general expression obtained after solving yields Pout = FX (βσ ) +
σ∈SL
θ 0
L
nαi nαi −j k Jα
l=1 θl i=1 j=1 k=0 l=0
δij
2
The partial fraction decomposition in (58) yields
nαi Jα
θ 0
2
where αk = k/θL−M+k , for k = 1 . . . M . To calculate the pdf of Y we first achieve a partial fraction decomposition of (57). To do so, and in order that our results can be completely general, we first must assume that the coefficients α k may have an order of multiplicity higher than one 2 . Therefore, we divide the M coefficients α k into Jα groups with n αi elements each, where all the coefficients in a given group take the same Jα nαi = M , and (57) can be rewritten value αi . That is: i=1 as: ΨY (s) =
fY (y) =
0
x1
ΨY (s) =
shown that, by inverse Laplace transformation of (59), the pdf of Y can be written as
×
eσ /αi (−σ 2 )k−l GX (l, −1/(βαi ), βσ 2 ), l!(k − l)!αki β l
(63)
where the distribution function of X, F X (·), is given in Table I for the different types of fading of the desired signal considered here, and G X (n, s, Λ) is defined as in (22). Introducing in (22) the density functions f X (·) given in Table I leads to the following results when the desired signal experiences Nakagami fading: (m/Ps )m Γ (m + n, (m/Ps − s) Λ) , m+n (m/Ps − s) Γ(m) (64) and the following result under Ricean fading: GX (n, s, Λ) =
1+K GX (n, s, Λ) = 1 + K − sPs n Ps sKPs × exp 2(1 + K − sPs ) 1 + K − sPs ⎛ ⎞ 1 + K (1 + K)K × Q2n+1, 0 ⎝ 2 , 2 − s Λ⎠ , 1 + K − sPs Ps (65)
(60)
where Ωα is the set of (Jα − 1)-tuples such that Ωα = Jα (q1 , . . . qJα ) : qk ∈ N, qi = 0, k=1 qk = j − 1 . It can be 2 As a simple example, consider that we have a systems with L=4, M=3, and the powers are given by (P1 , P2 , P3 , P4 ) = (1/3, 1, 1, 1). It is easy to show that the coefficient will be (α1 , α2 , α3 ) = (1/2, 2/3, 1/2), i.e., we obtain a coefficient with value 1/2 and multiplicity 2.
where Q2n+1, 0 (a, b) is calculating using (29) with N = 1. When the desired user experiences Rayleigh fading we have
1 n!Psn exp −Λ − s GX (n, s, Λ) = (1 − sPs )n+1 Ps u
(66) n 1 1 × −s . −Λ u! Ps u=0 However, for the Rayleigh case, the use of the alternative approach for the outage probability calculation presented in
9
0
0
10
10
Ld=4 Ld=5 −1
−1
10
10
−2
−2
10
10
−3
Outage probability
Outage probability
−3
10
−4
10
−5
10
−6
10
IC
−4
10
MRC −5
10
−6
10
10 N=2,3,4,5
−7
−7
10
−8
10 −20
10 MRC IC −15
−8
−10
−5
0
5
10
15
20
Normalized Average SINR (dB)
Section II-C.3 will result in a much simpler expression for Pout . From (34) and conditioning on the remaining interference after cancellation Y, it is straightforward to show that Pout |Y =y = P (X < β(y + σ 2 )) = 1 − e−β(y+σ
2
)/Ps
. (67)
The unconditioned outage probability can be calculated by simply integrating (67) over all possible values of the random variable Y, that is: ∞ 2 Pout = 1 − e−βσ /Ps e−βy/Ps fY (y)dy. (68) 0
Note that the integral in (68) is in fact the MGF of Y with s = −β/Ps . Therefore, with the help of (57) the resulting outage probability when the desired signal experiences Rayleigh fading will be: 2
/Ps
σ∈SL
θ 0
M
l=1 θl
k=1
L
−15
−10
−5
0
5
10
15
20
Normalized Average SINR (dB)
Fig. 1. Outage probability comparison between MRC and IC for different number of receive antennas. SOI is Rayleigh distributed. L=6. Ld =3. INR=90 dB. R=60 dB.
Pout = 1 − e−βσ
10 −20
1 . (69) 1 + αk β/Ps
V. N UMERICAL R ESULTS In this section numerical results are presented for the outage probability of a multiuser wireless system employing N antennas at the receive end. The N antennas at reception are employed to provide interference cancellation (IC), Maximal Ratio Combining (MRC) or Optimum Combining (OC). All the results in the figures below have been obtained analytically by the direct computation of expressions (16), (17), (21)-(29), (52), (53) and (63)-(66). For MRC and OC, the computational complexity required to calculate our expressions of P out will be primarily determined by the complexity of the required summations over integer partitions. We demonstrate in Appendix II that this complexity is in general exponential. In particular, the complexity to calculate the coefficients A ij given in (6) grows exponentially with J, the number of different interfering powers. However, for the case when every interferer has a different average power (i.e. n i = 1 for all i) the
Fig. 2. Effect of number of dominant interferers on outage probability. SOI is Rayleigh distributed. L=6. N=5. R=60 dB. INR=90 dB.
complexity of these coefficients is linear. Also, the complexity of the MGF derivative in (13) and the coefficients B hl in (31) grow exponentially with the number of antennas. For IC the complexity of our expressions is primarily determined by a summation over a set of permutations, and so the expressions show in this case a factorial complexity with the number of interfering users. Note however that a practical system will generally have a small number of antennas and/or dominant interferers, for which this complexity is reasonable. In the results we show, we assume that the system is affected by L interferers undergoing Rayleigh fading. The signal of interest (SOI) is affected by Rice, Nakagami or Rayleigh fading. In some of the figures we show, we consider that there are Ld dominant interferers out of the L total interferers. Every dominant interferer is assumed to be received with an average power R dB higher than a non-dominant interferer. The figures shown in this section present results for the outage probability as a function of the average SINR per antenna normalized by β, which is the threshold SINR. The average SINR per antenna is defined as: N Ps 1/N SIN R = L n=1 n , 2 i=1 Pi + σ where Psn is the mean received power from the SOI at antenna n and Pi is the mean received power from the i-th interferer. In Fig. 1 we provide a comparison between MRC and IC when the number of antennas increases from 2 to 5. We consider L = 6 interferers with L d = 3 dominant interferes which have a power 60 dB higher then the non-dominant interferers. In addition the total interference to noise ratio (INR) is assumed to be 90 dB. For MRC the results show the typical performance improvement with the number of antennas, with an increase of the slope of the curves and diminishing returns as the antenna number increases. However, for IC the situation is very different: N antennas will cancel N − 1 interferers, so when N increases from 2 to 3 there is a performance improvement as an additional interferer is
10
0
0
10
10 INR=90dB INR=45dB
−1
10
10
−2
−2
10
10
MRC
−3
−4
Outage probability
Outage probability
−3
10
10
IC
−5
10
−6
10
−4
10
−5
10
N=2,3,4
−6
10
10
−7
−7
10
10
−8
10 −20
OC MRC IC
−1
−8
−15
−10
−5
0
5
10
15
10 −20
20
−10
0
Normalized Average SINR (dB)
10
20
30
40
Normalized Average SIR (dB)
Fig. 3. Effect of INR on outage probability. SOI is Rayleigh distributed.L=6. Ld =4. N=5. R=60 dB.
Fig. 5. Outage probability comparison for OC, MRC and IC for different number of receive antennas. SOI is Rayleigh distributed. L=4. No noise. Equal power interferers.
0
10
R=60dB R=35dB R=10dB
−1
0
10
OC MRC IC
10
−1
10 −2
10
−2
10 MRC
10
−3
IC
Outage probability
Outage probability
−3
−4
10
−5
10
10
−4
10
−5
10
−6
10
−6
10
N=2,3,4
−7
10
−7
10 −8
10 −20
−15
−10
−5
0
5
10
15
20 −8
Normalized Average SINR (dB)
10 −20
−10
0
10
20
30
40
Normalized Average SIR (dB)
Fig. 4. Effect of R on outage probability. SOI is Rayleigh distributed. L=6. Ld =4. N=5. INR=90 dB
canceled. However, there is a dramatic improvement when the number of antennas increases from 3 to 4. This is due to the fact that when N = 4 all 3 dominant interferers are canceled, and therefore the interference is only due to low power signals, resulting in an improvement of the outage probability of more than 4 orders of magnitude. In Fig. 2 we can see that when N is fixed to 5 antenna elements, the performance of IC improves dramatically when the number of dominant interferers decreases from 5 to 4, because in the latter case all these dominant interferers are canceled. MRC will be better than IC only if the average received SINR is very high, but that is not an effective design criterion from a system perspective. Also note that the impact of the number of dominant interferers on MRC is almost negligible as the total average interference remains constant for a given average SINR and SOI power, and MRC only improves the performance by improving the output SOI. In
Fig. 6. Outage probability comparison for OC, MRC and IC for different number of receive antennas. SOI is Rice distributed with K=7 dB. L=4. No noise. Equal power interferers.
addition, both Fig 1. and Fig. 2 show that even when the dominant interferers are not canceled, IC is better than MRC in the low SINR regime. Fig. 3 shows the effect of increasing the noise in relation to the total interference (i.e. decreasing the INR). The effect is, as expected, a much higher outage probability for IC, as the noise cannot be canceled, while the impact on MRC is nearly negligible due to the reason mentioned above. Fig. 4 shows the effect of decreasing the power of the dominant interferers with respect to the power of the nondominant interferers (i.e., decreasing R). It can be observed that the higher the relative power of dominant interferers, the better the performance of IC. Again, the effect on MRC is very small. Figs 5, 6 and 7 show a performance comparison of MRC, IC
11
0
0
10
10 OC MRC IC
−1
OC MRC −1
10
10
−2
−2
10
10
K=0 (Rayleigh)
−3
Outage probability
Outage probability
−3
10
−4
10
−5
10
10
−4
10
−5
10
N=2,3,4
−6
10
K=10dB
−6
10
K=15dB −7
10
−7
10
−8
10 −20
Constant SOI
−8
−15
−10
−5
0
5
10
15
20
Normalized Average SIR (dB)
10 −10
−8
−6
−4
−2
0
2
4
6
8
10
Normalized Average SIR (dB)
Fig. 7. Outage probability comparison for OC, MRC and IC for different number of receive antennas. SOI is Nakagami distributed with m=10. L=4. No noise. Equal power interferers.
Fig. 9. Effect of parameter K on outage probability. SOI is Rayleigh distributed in all antennas except one, which receives a RIcean distributed SOI. L=10. N=4. No noise. Equal power interferers.
0
10
OC MRC −1
10
−2
10
Outage probability
−3
10
−4
10
MRC, N=1 −5
10
−6
10
−7
10
D=40, 20, 0 dB −8
10 −20
−10
0
10
20
30
40
Normalized Average SIR (dB)
Fig. 8. Effect of D on outage probability. SOI is Nakagami distributed with m=1.5. L=10. N=4. No noise. Equal power interferers.
and OC for a SOI undergoing Rayleigh, Rice and Nakagami fading respectively. All interferers are assumed to be received with the same average power and the noise is neglected. The curves show that IC is only better for low average SIR, and the benefit of IC with respect to MRC for low average SIR increases as the channel of the SOI improves. This figures also show that OC is the best technique, because it achieves the same cancellation as IC while at the same time achieves the same diversity as MRC (note that the OC and MRC curves are parallel). Fig. 8 shows the effect of unbalanced average power from the desired user at the different antenna elements on MRC and OC. It is assumed that all antenna elements receive signals with Nakagami fading from the desired user, but one of the antennas receive an average power D dB higher than
the average power received at any other antenna. It can be seen that an unbalanced received power results in a worse performance for a given average SIR. Specifically, a higher average SIR is required to achieve the diversity gain as the power in the selected antenna increases with respect to the rest. This can be seen from the fact that a higher SIR is required to achieve a high slope in the curve. In the case of asymptotically high factor D, all the power is received in one antenna, and no diversity gain is achieved. Note also that OC has the same slope at every point as MRC, but a lower outage probability, due to interference cancellation. Fig. 9 also shows the effect of power imbalance from the desired user but from a different perspective. In this case all the receive antennas have the same power from scattered signals, that is, from NLOS components of the transmitted signal. In addition, one of the antennas has a LOS component which is received with a power K times higher than the NLOS component of that antenna and the rest of antennas. In other words, all antennas except one receive a Rayleigh-distributed signal, while one antenna receives a Ricean-distributed signal with parameter K and a total average power (1 + K) times higher than the power received in the rest of antennas. Our results show a performance improvement as the LOS component increases. In the limiting case (K = ∞) the performance is equivalent to a single receive antenna with a constant received signal from the desired user. VI. C ONCLUSION We provide general and easily computable closed-form expressions for the outage probability of wireless systems employing different array processing techniques. In particular, we present new expressions for the outage probability of MRC, OC and ideal null-steering beamforming (which provides interference cancellation, IC) in the presence of CCI and general signal fading. Our analysis of MRC allows an arbitrary number of antennas and interfering users. The
12
developed expressions for IC require the number of antennas to be equal or lower than the number of interferers. However, since we may consider interferers with arbitrary powers, we can assign a negligible power to some of the interferers, and therefore the comparative analysis presented here is suitable for the study of underloaded systems, i.e., systems with more antennas than dominant interferers, as well as for overloaded systems, when the number of dominant interferers is equal or higher than the number of antennas. Our results show that, for underloaded systems, the null steering beamformer is able to cancel all dominant interferers and its performance is better than MRC except for a very high average SINR. In contrast, for overloaded systems, or when the background noise is high with respect to the interference, a higher benefit is obtained if MRC is used at reception, except for the case in which the received average SINR is low. The SINR threshold below which IC performs better than MRC is a function of the channel of the desired user and the background noise. When the channel of the desired user improves, a higher benefit is obtained for low average SINR by using a beamformer. On the other hand, the background noise has a higher detrimental effect on the beamformer, as the noise cannot be canceled. For interference-limited system (negligible background noise), expressions have also been derived for the performance analysis of OC, and comparative results have been presented between this technique, MRC, and IC. As expected, OC achieves the lowest outage probability due to its ability to offer both interference cancellation and diversity gain simultaneously, but at a significantly higher implementation complexity. A comparative analysis has also been presented between OC and MRC when the desired user signals at every receive antenna are arbitrarily distributed for interference-limited overloaded systems with equal power interferers. Our results show that OC performs the same as MRC with N-1 interferers canceled. This fact was known to hold for i.i.d. signals from the desired user at every antenna, and has now been proved for the i.n.d. case. A PPENDIX I A N E XPRESSION FOR C ALCULATING THE O UTAGE P ROBABILITY The outage probability is defined as the event that the SINR falls below a predefined threshold β, that is X < β , (70) Pout = P Y + σ2
only take positive values, it is obvious that we will have: Pout |X=x 1, M > 1. (79)
S(1, M ) = M ∀ M > 0 ;
S(K, 1) = 1 ∀ K > 0
13
By computing (79) iteratively we can write: S(K, M ) = M +
M−1
(M − m) · Pm (K)
(80)
m=1
where Pm (K) =
i1 K−1
im−1
···
i1 =1 i2 =1
1.
(81)
im =1
From (79) it is clear that the complexity of S(K, M ) is linear when K = 1, and thus we canwrite S(1, M ) ∈ O(M ). For K > 1, taking into account that ni=1 ai ∈ O(nk+1 ) given that ai ∈ O(nk ), it is clear from (81) that P m (K) ∈ O(K m ). As a simple check of this result, the exact expressions of Pm (K) for m=1, 2 and 3 are: P 1 (K) = K − 1, P2 (K) = K(K − 1)/2 and P3 (K) = (K(K − 1)(2K − 1) + 3K(K − 1))/12, and these expressions show the derived complexity. Noticing that (80) has a summation of terms of exponential complexity with increasing exponent, we can finally write S(K, M ) ∈ O(K M−1 ). R EFERENCES [1] A.J. Goldsmith, Wireless Communications, Cambridge University Press, 2005. [2] M.K Simon, M.-S Alouini, Digital Communications over Fading Channels, Hoboken, N.J, Wiley 2005. [3] J.H. Winters, ”Optimum Combining in Digital Mobile Radio with Cochannel Interference”, IEEE Journal on Selected Areas in Comm, vol. 2, no. 4, pp. 528 - 539, July 1984 [4] L.C. Godara, ”Applications of Antenna Arrays to Mobile Communications, Part I: Performance Improvement, Feasibility, and System Considerations”, Proc. IEEE, vol. 85, no. 7, pp. 1031 -1060, July 1997 [5] R. Vaughan J.B. Andersen, Channels, Propagations and Antennas for Mobile Communications, The Institution of Electrical Engineers, London, U.K., 2003. [6] A. Shah and A.M. Haimovich, ”Performance Analysis of Maximal ratio Combining and Comparison with Optimum combining for Mobile Radio Communications with Co-channel Interference”, IEEE Trans. Veh. Technol., vol. 49, pp. 1454-1463, July 2000. [7] C. Chayawan and V.A Aalo, ”On the Outage Probability of Optimum Combining and Maximal Ratio Combining Schemes in an InterferenceLimited Rice Fading Channel,” IEEE Trans. Comm., vol. 50, no 4, pp. 532-535, April 2002. [8] V.A. Aalo and C. Chayawan, ”Outage probability of cellular radio systems using maximal ratio combining in Rayleigh fading channel with multiple interferers”, Electronics Letters, vol. 36, no. 15, July 2000. [9] X.W. Cui, Q.T. Zhang, Z.M. Feng; ”Outage performance for Maximal Ratio Combiner in the Presence of Unequal-Power Co-channel Interferers,” IEEE Communications Letters, vol. 8, no. 5, May 2004. [10] J.P. Pe˜na-Martin and J. M. Romero-Jerez, ”Outage Probability with MRC in presence of multiple interferers under Rayleigh fading channels”, Electronics Letters, vol. 40, no. 14, July 2004. [11] A. Shah and A.M. Haimovich, ”‘Performance Analysis of Optimum Combining in Wireless Communications with Rayleigh Fading and Cochannel Interference,”’ IEEE Trans. Commun., vol. 46, pp. 473479, Apr. 1998. [12] M. Kang, M.-S. Alouini and L.Yang, ”Outage Probability and Spectrum Efficiency of Cellular Mobile Radio Systems with Smart Antennas,” IEEE Trans. Commun., vol. 50, pp. 18711877, Dec. 2002. [13] Q.T Zhang and X.W. Cui, ”Outage Probability for Optimum Combining of Arbitrarily Faded Signals in the Presence of Correlated Rayleigh Interferers,” IEEE Trans. Veh. Technol., vol. 53, no. 4, pp. 1043-1051, July 2004. [14] M.O. Hasma, M.-S. Alouini A. Bastami and E.S Ebbini, ”Performance Analysis of Cellular Mobile Radio Systems with Succesive Co-channel Interference Cancellation,” IEEE Trans. Wireless Comm., vol. 2, no. 1, pp. 29-49, January 2003. [15] R. Mostafa, A. Annamalai and J.H. Reed, ”Performance Evaluation of Cellular Mobile Radio Systems with Interference Nulling of Dominant Interferers,” IEEE Trans. Comm., vol. 52, no. 2, pp. 326-335, February 2004.
[16] K.K. Bae, A. Annamalai and W.H. Tranter, ”Outage Probability of Cellular Radio Networks with Partial Cancellation of Independent but Non-Identically Distributed Cochannel Interferers,” Proc. IEEE Vehicular Technology Conference, Los Angeles, CA, September 2004. [17] G. Sarfraz and A. Annamalai, ”Performance Evaluation of Cellular Mobile Radio Systems with Successive Cancellation of Non-identically Distributed Co-channel Interferers in a Rayleigh Fading Environment,” Proc. IEEE Wireless Communications and Networking Conference, pp. 579-584, New Orleans, LA, March 2003 [18] J.M. Romero-Jerez J.P. Pena-Martin, G. Aguilera and A. Goldsmith, ”Performance of MIMO MRC Systems with Co-Channel Interference,” Proc. IEEE International Conference on Communications, Istambul, Turkey, June 2006. [19] I. Gradshteyn and I. Ryzhik, Tables of integrals, series and products; San Diego CA: Academic, 1994. [20] M.K. Simon ”The Nuttall Q Function-Its Relation to the Marcum Q Function and Its Application in Digital Communication Performance Evaluation,” IEEE Trans. Comm., vol. 50, no11, pp. 1712-1715, November 2002. [21] R.A. Monzingo and T.A. Miller, Introduction to Adaptive Arrays, New York, Wiley, 1980. [22] A. Annamalai, G.K. Deora, C. Tellambura, ”Theoretical Diversity Improvement in GSC(N,L) Receiver with Nonidentical Fading Statistical,” IEEE Trans. Comm., vol. 53, no. 6, pp. 1027-1035, June 2005.
Receive Antenna Array Strategies in Fading and Interference: An Outage Probability Comparison Juan M. Romero-Jerez, Member, IEEE, and Andrea J. Goldsmith, Fellow, IEEE
Abstract— We explore tradeoffs between different reception strategies of multiple receive antennas in fading channels with co-channel interference (CCI). Our tradeoff analysis is based on outage probability. We assume the signal from the desired user at the receive antenna array to be affected by Rice, Nakagami or Rayleigh fading, while CCI signals are assumed to experience Rayleigh fading. We provide closed-form analytical expressions for the outage probability of different diversity schemes, such as maximal ratio combining (MRC) and optimum combining (OC), and also for interference cancellation (IC) based on antenna beamsteering, which steers nulls in the array radiation pattern in the direction of the strongest interferers. Our analysis provides a unified framework for studying outage probability of different multiple-antenna reception strategies, and our closedform expressions are easily computed, facilitating a performance comparison under a range of operating conditions. Our numerical results show that IC yields significantly better performance than MRC if the system is interference-limited and the number of dominant interferers is lower than the number of receive antennas, or when the output SINR is low. In addition, when the background noise can be neglected, we provide numerical results for MRC and OC when the signals from the desired user at the different receive antennas are arbitrarily distributed. Index Terms— Antenna Array, Diversity, Interference Cancellation, Beamsteering, Outage Probability
I. I NTRODUCTION One of the key techniques to combat the impairments of the wireless communication channel is the use of adaptive antenna arrays. In particular, the detrimental effect of multipath fading can be reduced by using diversity combining techniques whereby the received signals at every antenna are properly weighted and added to improve the output signalto-noise ratio (SNR) [1], [2]. In order for linear combining techniques to perform optimally, the antenna elements must be uncorrelated. In flat-fading channels, maximal ratio combining (MRC) diversity is well known to be optimum in the sense of maximizing the output SNR. If the desired signal is affected by both co-channel interference (CCI) and flat-fading, the diversity combining technique that maximizes the output signal-tointerference-plus-noise ratio (SINR) is the so-called optimum combining (OC) [2], [3], therefore, OC aways has the same or better SINR as MRC in channels with fading and/or CCI. The drawback of OC is a higher implementation complexity because the channel state of the desired and all CCI signals must be known at every receive antenna. Therefore, even under the presence of interference, MRC may be preferable in certain scenarios due to its reduced complexity, as only the channel state of the desired user must be known at the receiver. Juan M. Romero Jerez is with the University of M´alaga, Spain. Andrea J. Goldsmith is with Stanford University, CA.
Another way of using multiple antennas is to adaptively modify the radiation pattern of the array to either enhance the received power of the desired signal or reduce the interference by placing nulls in the direction of the dominant interferers [4]. Contrary to the case of diversity combining, in this case, the elements of the antenna array must be close to each other so that the antennas can be highly correlated, thereby reducing the angular spread of the received signals and allowing for a better estimation of the angle of arrival (AoA) of the signal sources [5]. This technique, known as beamforming, does not provide diversity gain, but it may be preferable in the low SINR regime when there are a few strong dominant interferers. In addition, it is simpler to implement than OC because only one parameter, the AoA, is required for each source. Though OC is the optimum technique in the presence of interference and simultaneously provides diversity and interference cancellation, its implementation complexity may preclude its use in real systems, especially when the number of antennas and interferers is high. Therefore, a performance comparison of the commonly used techniques: MRC (to provide only diversity) and IC (to achieve the cancellation of the strongest interferers), along with the best performance achieved by OC in channels with fading and interference, is important for wireless system design, and is the main focus of this work. Outage probability is a common metric to measure the performance of wireless communication systems. We define an outage as the event that the SINR falls below a predefined threshold. In this work we present new closed-form expressions generalizing previous results on outage probability of wireless systems in interference environments using MRC, OC and an idealized beamforming algorithm able to cancel the strongest CCI signals (null-steering). In addition, we provide a comparison among these different techniques. We assume that all CCI signals undergo flat Rayleigh fading, while the signal of interest (SOI) can experience Rice, Nakagami or Rayleigh fading. All our closed-form outage probability expressions are easily computable, as they are expressed using wellknown tabulated functions typically available in most software computation packages such as Matlab. Several recent works have investigated the outage probability of the antenna processing schemes studied in this paper. Specifically, in [6], [7], [8] results are presented for the outage probability of MRC under Rayleigh interference considering that the average powers associated with CCI signals are the same or are completely different. The case of arbitrary interference power is studied in [9] for a SOI under Nakagami, Rice or Rayleigh fading, but the background noise is neglected. Results for arbitrary power interference and noise
2
are presented in [10], though only for a SOI under Rayleigh fading. In this paper, we extend previous results of MRC under CCI by allowing the SOI to be arbitrarily distributed and the CCI signals to have arbitrary average powers, while simultaneously considering background noise. For interference limited systems (negligible background noise) we derive novel expressions allowing the SOI to be arbitrarily distributed at every receive antenna. This is typical of propagation environments where the desired user does not experience the same propagation conditions at all the receive antennas, e.g., there could be a line of sight (LOS) signal component for some antennas and not for others. This might occur in practice in a cellular system with macrodiversity combining across multiple base stations. As in the case of MRC, and in order to provide a fair comparison, we also generalize previous results of OC under Rayleigh-distributed CCI signals by allowing the signal from the desired user at every antenna to be arbitrarily distributed and to have arbitrary average powers. Previous analytical results for outage probability of OC diversity include [11] for a Rayleigh fading SOI and [6], [7], [12] for a Ricean fading SOI, and the work of Zhang and Cui [13], where a very simple and general result for the outage probability is presented considering different fading scenarios for the SOI with arbitrary correlation. However, none of these works consider the more general case of an arbitrary fading distribution on the desired users’ signals at every receive antenna, as we consider here. For beamforming antenna array systems, most analytical results on outage probability consider all CCI signals to be identically distributed [14], [15]. In [16] the outage probability is computed for arbitrary fading distributions of interferers, but the expressions must be computed by performing numerical integration. An interesting framework for Rayleigh-distributed CCI signals is presented in [17], where a closed-form expression of the outage probability is derived for, in principle, arbitrary average CCI signal powers. However, the results are restricted to the case where the terms of the partial fraction expansion of the moment generating function (MGF) of the interference have a multiplicity order equal to one. In our analysis we remove that restriction and present results for the case of completely arbitrary Rayleigh-distributed CCI signals with arbitrary powers and an arbitrarily distributed SOI. The remainder of this paper is organized as follows. In Sections II, III, and IV we present, respectively, the system model and the outage probability expressions for, respectively, MRC, OC and ideal null-steering. Numerical results and a performance comparison among the different techniques are presented in Section V. Finally, discussion and concluding remarks are given in Section VI. II. O UTAGE P ROBABILITY OF MRC S YSTEMS WITH I NTERFERENCE A. System Model and Interference Statistics We consider a wireless communication system with N uncorrelated receive antennas where the received signal from the desired user at every antenna is assumed to be corrupted by L interference signals. In addition, and unless stated otherwise,
the desired signal at each antenna is corrupted by additive white Gaussian noise (AWGN). The received noise vector is assumed to be zero mean with covariance matrix σ 2 IN ×N , where IN ×N denotes the N × N identity matrix. The desired and interference signals undergo slow flat fading and a coherent receiver is employed which is assumed to have perfect knowledge of the instantaneous channel state. The signals at every receive antenna from the desired user are assumed to experience Nakagami, Rice or Rayleigh fading. On the other hand, a given interfering user experiences independent identically-distributed (i.i.d.) Rayleigh fading at the N-element array, and the power of every interfering user is assumed to be arbitrary. Let c s = [cs1 . . . csN ]T and ci = [ci1 . . . ciN ]T denote, respectively, the channel gain vectors of the desired and the i-th interfering user at the antenna array, where the superscript T refers to vector transposition. Then, the received baseband N-element signal vector can be written as: r = cs bs +
L Pi ci bi + n,
(1)
i=1
where Pi is the mean power of the i-th interferer at each antenna, n is the N-dimensional received noise vector, and bs and bi are, respectively the transmitted symbols from the desired and i-th interfering user. For simplicity, it is assumed that bs = 1 and bs = 1 ∀i, 1 ≤ i ≤ L. The elements of ci are assumed to be complex Gaussian random variables with zero mean and unit variance, whereas for the received signal from the desired user at antenna n it is assumed that |csn | follows either a Nakagami, Rice or Rayleigh distribution 2 satisfying E[|csn | ] = Psn . That is, for the general case, and unless stated otherwise, we allow the elements of c s to be independent non-identically-distributed (i.n.d.) random variables with arbitrary average squared mean. In an MRC receiver, the antenna array elements are weighted by the channel gains associated with the desired user, yielding the following output signal: 2
yR = cH s r = |cs | bs +
L H Pi cH s ci bi + cs n,
(2)
i=1
where the superscript H denotes Hermitian transposition. It follows that the received SINR after combining will be: γ = L i=1
4
|cs |
2
2
2 Pi |cH s ci | + |cs | σ
= L i=1
|cs |
2 2
Pi |vi | + σ 2
, (3)
where vi = cH s ci / |cs |. It is demonstrated in [6] that if the elements of ci are i.i.d. zero mean unit variance complex Gaussian random variables, then v i and cs are mutually independent and v i is a zero mean unit variance complex 2 Gaussian random variable. It follows that |v i | is exponentially distributed with unit mean. We can write the received SINR as follows: X γ= , (4) Y + σ2 N L √ 2 2 2 where X = |cs | = n=1 |csn | and Y = i=1 Pi |vi | , being X and Y independent. Note that Y is the sum of L independent exponential random variables with arbitrary mean
3
powers. Let us divide the total number of interferes L into J groups, where every interferer in a group has the same mean power Pi . Consider that there are n i interferers in a given group with mean power P i . Then, the pdf of Y can be shown to be given by [9], [18]: fY (y) =
ni J
Aij
i=1 j=1
y ni −j (ni − j)!Pini −j+1
where Aij are coefficients given by: Aij
=
(−1)
j−1
J
ΩA k=1,k=i
e−y/Pi ,
nk + qk − 1 nk − 1
Pkqk Pink × , (Pi − Pk )nk +qk
where Ψn (s) is the MGF of the received signal power at antenna n. To calculate the k-th order derivative of Ψ n (s) we apply the Leibniz derivative rule, yielding N I dk ΨX (s) dk = k Ψn (s) dsk ds i=1 k! (q ) (q ) (q ) = Ψ 1 (s)Ψ2 2 (s) · · · ΨN N (s) q1 !q2 ! . . . qN ! 1
(5)
ΩX
= k!
N (q ) Ψn n (s) , qn ! n=1
(13)
ΩX
(6)
where ΩA is the set of (J − 1)-tuples such that Ω A = J (q1 , . . . qJ ) : qk ∈ N, qi = 0, k=1 qk = j − 1 . The cdf of Y will be: k n J ni i −j y 1 Aij e−y/Pi . (7) FY (y) = 1 − k! P i i=1 j=1
where ΩX is the set of N-tuples such that Ω X = N (q ) (q1 , q2 . . . qN ) : qn ∈ N, n=1 qn = k , and Ψn n (s) is the qn -th order derivative of Ψ n (s). Substituting (13) into (11) we obtain the general expression of P out for arbitrary average power CCI signals:
k=0
=
Pout
For the case in which all CCI signals have the same average power PI , (7) simplifies to: L−1 1 y k −y/PI FY (y) = 1 − e . (8) k! PI
ni J
Aij
i=1 j=1
×
n i −j
1
k=0
(βPi )k
N
1
n) Ψ(q (s) .
n q ! s=−1/βPi n n=1
(14)
ΩX
k=0
B. Outage Probability Calculation of Interference-Limited Systems In an interference-limited system, the background noise is assumed to be negligible and it is shown in Appendix I that the probability of outage, defined as the probability that the output SIR falls below a predefined threshold β, can be calculated as follows: ∞ Pout = 1 − FY (x/β)fX (x)dx, (9) 0
where FY (·) is the cdf of the output interference and f X (·) is the pdf of the output signal from the desired user, with X and Y defined in (4). Introducing (7) into (9), we can write: n ni J i −j 1 1 ∞ k −x/βPi Pout = Aij x e fX (x)dx. k k! 0 i=1 j=1 k=0 (βPi ) (10) Note that the integral in (10) is in fact the k-th order derivative of the moment generating function (MGF) of X, Ψ X (s), with s = −1/βPi , and therefore we have:
n ni J i −j 1 1 dk ΨX (s)
Pout = Aij . k k! dsk s=−1/βPi i=1 j=1 k=0 (βPi ) (11) As X is a sum of independent random variables, its MGF will be given by the product of the MGF of the individual terms in the sum, that is, ΨX (s) =
N n=1
Ψn (s),
(12)
If all CCI signals have the same average power P I , (14) simplifies to
Pout =
L−1 k=0
N
1 1
(qn ) Ψ (s) . (15)
n (βPI )k q ! s=−1/βPI n n=1 ΩX
Note that our results include the case where the signal power of the desired user at every receive antenna may have a different distribution. Thus (14) and (15) can be evaluated for any distribution for which the n-th order derivative of the MGF of the fading power distribution is known. In Table I the n-th order derivative for Nakagami, Rice and Rayleigh fading are shown. In particular, when the distribution of the received power at every antenna is i.n.d. Nakagami and the CCI signals have arbitrary average powers, using the Nakagami entry in Table I, (14) becomes:
Pout
=
ni J
Aij
i=1 j=1
×
n i −j
1
k=0
(βPi )k
N ΩX
n Psqnn mm n Γ (mn + qn ) . (mn + Psn /(βPi ))mn +qn Γ (mn ) qn ! n=1
(16) For the case of i.n.d. Ricean fading for the received signals of the desired user at every antenna and CCI signals with
4
TABLE I S TATISTICS OF S IGNAL P OWER FOR C OMMON FADING C HANNELS Type of Fading
PDF, CDF, MGF and n-th order derivative of the MGF of the signal power
Rayleigh
f (x) =
1 p
exp − x , where p = E[x] p
F (x) = 1 − ex/p Ψ(s) =
1 1−sp
Ψ(n) (s) = f (x) =
Nakagami
m p
FX (x) =
n!pn (1−sp)n+1
m
1 xm−1 Γ(m)
Ψ(n) (s) = f (x) =
1+K p
1+K x p
exp −K − √
1+K 1+K−sp
Ψ(n) (s) =
−m
pn mm Γ(m+n) (m−sp)n+m Γ(m)
F (x) = 1 − Q Ψ(s) =
2K,
exp
Pout =
i=1 j=1
Aij
n i −j
Kps 1+K−sp
pn (1+K)(n!)2 (1+K−sp)n+1
exp
1 k
k=0
(βPi )
N
Psqnn (1 + Kn )qn ! (1 + Kn + Psn /(βPi ))qn +1 ΩX i=1 (17) −Kn Psn /(βPi ) × exp (1 + Kn + Psn /(βPi )) l qn (1 + Kn )Kn 1 . × (l!)2 (qn − l)! 1 + Kn + Psn /(βPi ) ×
I0
2
(1+K)K x p
2 1+K x , where K ≥ 0 p
arbitrary powers, using (14) we have ni J
1 2
γ(m,xm/p) Γ(m)
p Ψ(s) = 1 − s m
Rice
exp − xm , where m ≥ p
l=0
It can be shown that when the elements of X are i.i.d, (16) and (17) collapse, respectively, to [9, eq. (21)] and [9, eq. (18)].
Kps 1+K−sp
n 1 l=0 (l!)2 (n−l)!
(1+K)K 1+K+−sp
l
leading to close form expressions of the outage probability, as we show below. From (7) the following equality holds: ni J 2 x 2 −σ Aij e−(x/β−σ )/Pi = 1 − FY β i=1 j=1 k n i −j x 1 1 2 −σ × .(20) k! Pik β k=0
Substituting (20) into (19) and using straightforward manipulation, we obtain: Pout
= FX (βσ 2 ) +
ni n J i −j k i=1 j=1 k=0 l=0
C. Outage Probability Calculation of Systems with NonNegligible Background Noise When the noise power at the receiver is comparable to the interferer powers its effect cannot be neglected, and an outage will be considered to occur when the received SINR is below a predefined threshold β, that is: X Pout = P < β . (18) Y + σ2 It is shown in Appendix I that the outage probability can be calculated as follows: ∞ x − σ 2 fX (x)dx. Pout = FX (βσ 2 ) + 1 − FY β βσ2 (19) This particular expression is suitable for our purpose because it is given as a function of an integral of the cdf of Y, thus
2
Aij
eσ /Pi (−σ 2 )k−l l!(k − l)!Pik β l
× GX (l, −1/(βPi ), βσ 2 ), where we define GX (n, s, Λ) =
∞
Λ
xn esx fX (x)dx.
(21)
(22)
Note that (22) can be regarded as the incomplete n-th order derivative of the MGF of X. Therefore its convergence is guaranteed as long as this derivative exists, which is the case in Nakagami, Rice and Rayleigh fading. 1) Nakagami-Distributed Desired User: When the received signal from the desired user at every antenna element undergoes i.i.d. Nakagami fading, the power X of the desired signal after combining will be given by a Gamma distribution, with pdf mN m 1 fX (x) = xmN −1 e−xm/Ps . (23) Ps Γ(m)
5
The corresponding cdf will be FX (x) =
γ(mN, xm/Ps ) , Γ(mN )
the case in (28) which, therefore, can be expressed in closedform. Specifically, following [20], we can write (24) QN +2n,N −1 (a, b) =
x
where γ(α, x) = 0 z α−1 e−z dz denotes the incomplete Gamma function. Introducing (23) into (22), the evaluation of the integral yields: GX (n, s, Λ) =
(m/Ps )mN (m/Ps − s)mN +n Γ (mN + n, (m/Ps − s) Λ) , (25) × Γ(mN )
∞ where Γ(α, x) = x z α−1 e−z dz denotes the complementary incomplete Gamma function. Then, by substituting (24) and (25) into (21), the outage probability for a desired user with i.i.d. Nakagami fading can be calculated. 2) Ricean-Distributed Desired User: When the received signal at every antenna experiences i.i.d. Ricean fading, the received signal power X after combination follows a noncentral chi-squared distribution with pdf fX (x)
=
N2+1 x N2−1 1+K 1+K e−(KN + Ps x) Ps KN ⎛ ⎞ (1 + K)KN ⎠ × IN −1 ⎝2 x , (26) Ps
where In (x) is the modified Bessel function of the first kind and order n [19]. The corresponding cdf of X can be expressed as √ 1+K 2KN, 2 x , (27) FX (x) = 1 − QN Ps ∞ 2 2 where Qn (a, b) = b a1−n xn e−(x +a )/2 In−1 (ax)dx denotes the n-order Marcum-Q function. Introducing (26) into (22), and solving the integral we obtain
N2+1 n 1+K Ps 1 + K − sPs 2(1 + K − sPs ) 1−N sKN Ps × (2KN ) 2 exp 1 + K − sPs ⎛ ⎞ (1 + K)KN 1+K × QN +2n,N −1 ⎝ 2 , 2 − s Λ⎠ , 1 + K − sPs Ps
GX (n, s, Λ) =
(28) ∞
2
2
where Qp,q (a, b) = b xp e−(x +a )/2 Iq (ax)dx denotes the Nuttal-Q function. This function is not considered a tabulated function1, however, the Nuttal-Q function can be expressed in terms of the more common Marcum-Q function and modified Bessel function of the first kind (available in most software computation packages) when p + q is odd [20]. This is exactly 1 That is, it is not typically included in software computation packages, such as Matlab.
n+1
cl (n)aN −1+2(l−1) QN −1+l (a, b)
l=1
2 n a + b2 Pn, l (b2 )al−1 bN +l IN +l−2 (ab) + exp − 2 l=1 (29) with
cl (n) = 2
n−l+1
and 2
Pn, l (b ) =
n−l
2
n−l−j
j=0
n n−l+1 n−1−j n−l−j
(N − 1 + n)! (N + l − 2)! (N − 1 + n)! 2j b . (N − 1 + l + j)!
Substituting now (27) and (28) into (21), the outage probability for an i.i.d. Rice desired user can be calculated. 3) Rayleigh-Distributed Desired User: When the desired signal at every receive antenna experiences Rayleigh fading, it is possible to find a closed-form expression for the outage probability for arbitrary powers of the desired signal at every antenna (i.e, the desired user experiences i.n.d. Rayleigh fading). Let us divide the number of antennas N into J s groups, where for a given group the average received power from the desired user is the same. Consider that there are n sh antennas in a given group receiving an average power from the desired user Psh . Then, the pdf of X will be given by: fX (x) =
nsh Js h=1 t=1
Bht
xnsh −t −x/Psh , nsh −t+1 e (nsh − t)!Psh
(30)
where Bht are coefficients that can be calculated as Js nsk + qk − 1 t−1 Bht = (−1) nsk − 1 ΩB k=1,k=h
qk nsk Psk Psh × , (Psh − Psk )nsk +qk
(31)
where ΩB is the set of (Js − 1)-tuples such that ΩB = s (q1 , . . . qJs ) : qk ∈ N, qh = 0, Jk=1 qk = t − 1 . The corresponding cdf for the random variable X is given by k nsh n Js sh −t x 1 Bht e−x/Psh . (32) FX (x) = 1 − k! Psh t=1 h=1
k=0
Now, from (30) and (22), solving the integral we obtain: nsh Js
(nsh − t + n)! (nsh − t)! h=1 t=1
l 1 Psh × exp −Λ − s × (1 − sPsh )nsh −t+n+1 Psh
nsh −t+l u 1 1 × −s . Λ u! Psh u=0
GX (n, s, Λ) =
Bht
(33)
Now, substituting (32) and (33) into (21), the outage probability for i.n.d. Rayleigh fading of the desired user can be
6
calculated. However, in this case, a more compact and simpler solution for the outage probability (with less nested equations in the final expression) can be obtained using an alternative approach. Let us rewrite the outage probability definition as follows: X < β = P (X < β(Y + σ 2 )). (34) Pout = P Y + σ2 From (34), conditioning on the random variable Y, with the help of (32) and after some manipulation, we can write the outage probability to be Pout |Y =y = 1 − ×
nsh Js
h=1 t=1 n sh −t k=0
β Psh
Bht e−βσ
2
e
1 y n (σ 2 )k−n . n!(k − n)! n=0 (35)
To find an expression for the outage probability, we can introduce the pdf of Y, given in (5), into (36) and solve the integral. However, a simpler expression is obtained by noting that the integral in (36) is in fact the n-th order derivative of the MGF of the random variable Y, Ψ Y (s), with s = −β/Psh . Thus, we can write k nsh n Js sh −t 2 β Bht e−βσ /Psh Pout = 1 − Psh h=1 t=1 k=0 (37)
k n
1 2 k−n d ΨY (s)
(σ ) × . n!(k − n)! dsn s=−β/Psh n=0 Since Y is a sum of exponential random variables, it follows that its MGF is given by 1 , (1 − sPi )ni i=1
=
1−
nsh Js
Bht e−βσ
2
/Psh
h=1 t=1 n sh −t
k k β 1 (σ 2 )k−n × Psh (k − n)! n=0 k=0 J Piqi ni + qi − 1 × . qi (1 + βPi /Psh )ni +qi (40)
k k
J
Pout
ΩY i=1
/Psh −βy/Psh
The unconditioned outage probability can be obtained by integrating (35) over all possible values of Y, that is: k nsh n Js sh −t 2 β Pout = 1 − Bht e−βσ /Psh Psh h=1 t=1 k=0 k ∞ 1 (σ 2 )k−n × y n e−βy/Psh fY (y)dy. n!(k − n)! 0 n=0 (36)
ΨY (s) =
such that Ω Y = where ΩY is the set of J-tuples J (q1 , . . . qJ ) : qi ∈ N, i=1 qi = n . Finally, introducing (39) into (37), the outage probability will be:
(38)
where J and n i were defined in Section II-A. By the application of the Leibniz derivative rule, we can write: J dn ΨY (s) 1 dn = dsn dsn i=1 (1 − sPi )ni J 1 1 dqi = n! qi ! dsqi (1 − sPi )ni ΩY i=1 J Piqi ni + qi − 1 , = n! qi (1 − Pi s)ni +qi i=1 ΩY
(39)
III. O UTAGE PROBABILITY OF O PTIMUM C OMBINING S YSTEMS We consider a system with N uncorrelated receive antennas where the received signal at every antenna from the desired user is assumed to be corrupted by L interference signals, satisfying L ≥ N (overloaded system). We follow in this section an approach similar to [13] and assume that all interferers experience i.i.d. Rayleigh fading with the same average power P I and the system is assumed to be interference limited (negligible background noise). In an OC receiver, the received signal vector is weighted to maximize the resulting SIR. The output signal after weighting will be L wH ci bi , yR = w cs bs + PI H
(41)
i=1
where cs , ci , bs and bi are defined as in Section II-A. It is well known that the weight vector that maximizes the SIR is given by [21]: w = gR−1 cs , (42) where g is an arbitrary constant and R is a Wishart distributed matrix given by L R= ci cH (43) i , i=1
which results in the following conditional output SIR: γ=
1 H −1 cs R cs . PI
(44) 2
Let us define the random variables X = |c s | = N −1 2 H −1 cs . n=1 |csn | , as in Section II-A, and ξ = cs R The outage probability, defined as the probability that the received SIR falls below a predefined threshold β, can be written as: Pout = P (γ < β) = P (ξ > 1/βPI ) = 1 − Fξ (1/βPI ). (45) It is shown in [13] that the random variable ξ, conditioned on X, is Gamma distributed with pdf f ξ|X=x (z) =
z L−N xL−N +1 e−xz , (L − N )!
(46)
7
and it is straightforward that the conditional cdf of ξ will be F ξ|X=x (z) = 1 − e−xz
L−N k=0
1 (xz)k . k!
(47)
The unconditional cdf of ξ is obtained by simple averaging (47) over all possible values of X, yielding: L−N 1 ∞ zk Fξ (z) = 1 − xk e−xz fX (x)dx. (48) k! 0 k=0
And noting that the integral in (48) is in fact the k-th order derivative of the MGF of X parametrized by s = −z, we can write
L−N 1 dk ΨX (s)
zk . (49) Fξ (z) = 1 − k! dsk s=−z k=0
We now introduce in (49) the kth-order derivative of the MGF of X, which is given in (13), we obtain L−N N
1
n) Ψ(q Fξ (z) = 1 − zk . (50) n (s)
q ! s=−z n=1 n k=0
ΩX
From (45) and (50) it is clear that the final expression for the outage probability when the received desired signal at every antenna is arbitrarily distributed will be L−N N
1 1
(qn ) Pout = Ψ (s) . (51)
n (βPI )k q ! s=−1/βPI n=1 n ΩX
k=0
Note that (51) is in fact the same result obtained in (15) for interference-limited MRC systems for equal average power of the CCI signals by just changing in (15) the number of interferers L to L − N + 1. That is, OC diversity with equal average CCI signal powers for overloaded systems achieves the cancellation of N − 1 interferers, while still retaining the diversity improvement of MRC under interference. This fact was previously known to hold when the signals at every antenna from the desired user are i.i.d. [2], [7], but it is now proved to hold even for arbitrary distribution of the signals from the desired user at every receive antenna. In particular, when the distribution of the received power at every antenna from the desired user is i.n.d. Nakagami, from (51) we can write the outage probability as Pout =
L−N k=0
×
1 (βPI )k
N ΩX
n Psqnn mm n Γ (mn + qn ) , (mn + Psn /(βPi ))mn +qn Γ (mn ) qn ! n=1
(52)
while for the case of i.n.d. Ricean fading for the received signals from the desired user at every antenna we have Pout
=
L−N
Psqnn (1 + Kn )qn ! 1 (βPI )k (1 + Kn + Psn /(βPi ))qn +1 k=0 ΩX i=1 −Kn Psn /(βPi ) × exp (1 + Kn + Psn /(βPi )) l q n (1 + Kn )Kn 1 × . (l!)2 (qn − l)! 1 + Kn + Psn /(βPi ) N
l=0
(53)
It can be shown that if the received signals at every antenna from the desired user have the same average powers and parameter K (i.e., i.i.d. Ricean fading), (53) collapses to [7, eq. (8)]. Note that our results can be applied to macrodiversity combining systems, and are given in a complete analytical form and no simulations are needed. In particular, our results are exact, and don’t require numerical integration or hypergeometric functions (which are in fact infinite series that are typically truncated in calculations). IV. O UTAGE P ROBABILITY OF A N ULL -S TEERING B EAMFORMER A. System Model and Interference Statistics In this section we consider that the N antennas at the receive end are employed for interference cancellation. The optimum weights for a linear beamformer are also given by (42)-(43). However, there is a fundamental difference with respect to OC, as in this case, cs and ci are the array response vectors for the desired an the i-th interfering signals, respectively. Note that only the angle of arrival (AoA) of a signal is needed to calculate the array response [5], that is, only one parameter per interferer must be known at the receiver in this case. In contrast, for OC, N parameters per interferer (the channel gain at every antenna) must be known at the receiver to calculate the antenna weights. Instead of calculating the outage probability starting from the optimum antenna weights, which will be a function of the particular antenna array layout and the AoA of the different receive signals, we will use an alternative approach that will allow us to obtain closed form expressions for the outage probability. An ideal null-steering algorithm is assumed which is able to perfectly cancel the N −1 interferers with the highest powers while not having a detrimental effect on the desired signal. However, we consider that not all interferers can be canceled, that is, for L total interferers, if L > N − 1, then L − N + 1 interference signals remain after cancellation. In addition to interference, the desired signal at each antenna is corrupted by additive white Gaussian noise (AWGN). The signals at the receive end from the desired user is assumed to experience Nakagami, Rice or Rayleigh fading. On the other hand, we assume that there are L interfering users experiencing Rayleigh fading, and the average power P i for i = 1 . . . L of every interfering user at the N-element array is assumed to be arbitrary. Let the set {yk }L k=1 denote the instantaneous interference signal powers. The order statistics y k;L are obtained by arranging the values of the set in increasing order of magnitude y1:L ≤ y2:L ≤ . . . yL:L . The pdf of the order statistics is given by [22] [17]: fy1:L ≤y2:L ≤...yL:L (x1 , x2 . . . xL ) =
L
fσ(i) (xi ), (54)
σ∈SL i=1
where σ = {σ(1)σ(2) . . . σ(L)}, and S L denotes the set of the L! permutations of the integers {1, 2, . . . , L}. As CCI signals undergo Rayleigh fading, we have that f σ(i) (xi ) = M 1 −xi /Pσ(i) . Let Y = i=1 yi:L denote the remaining Pσ(i) e
8
interference after interference cancellation corresponding to the sum of the M = L − N + 1 weakest signal powers. The MGF of Y can be calculated as follows: M ∞ ΨY (s) = E exp s yi:L = fσ(L) (xL )dxL
i=1
∞
xL−1
σ∈SL
··· × fσ(M+1) (xM+1 )dxM+1 ∞ xM × esxM fσ(M) (xM )dxM xM −1 ∞ ∞ sx2 ··· × e fσ(2) (x2 )dx2 esx1 fσ(1) (x1 )dx1 . (55)
It is shown in [17] that the solution of (55) can be written as
θ 0 L−M
σ∈SL
l=1
M
θl
k=1
1 , θL−M+k − ks
(56)
L k−1 where θ 0 = i=1 1/Pi and θk = i=0 1/Pσ(L−i) . We can rewrite (56) in a more convenient form as ΨY (s) =
θ 0 L
M
l=1 θl k=1
σ∈SL
1 , 1 − αk s
(57)
θ 0
L
σ∈SL
Jα
l=1 θl i=1
ΨY (s) =
θ 0
L
σ∈SL
1 . (1 − αi s)nαi
(58)
nαi Jα
l=1 θl i=1 j=1
δij
1 , (59) (1 − αi s)nαi −j+1
where the δij are coefficients given by δij
= (−1)j−1 ×
L
σ∈SL
l=1 θl i=1 j=1
Jα
Ωα k=1,k=i qk nk αk αi , (αi − αk )nk +qk
nk + qk − 1 nk − 1
δij
y nαi −j (ni −
nα −j+1 e j)!αi i
−y/αi
.
(61)
The cdf of Y will be θ 0 FY (y) = 1 − L σ∈SL
×
nαi Jα
l=1 θl
δij e
−y/αi
i=1 j=1
1 y k . (62) k! αi
nαi −j k=0
B. Outage Probability Calculation The outage probability can be calculated using the same framework as that presented in Section II-C. In particular, we have to solve (19), where in this case X is the power of the desired signal undergoing Rice, Rayleigh or Nakagami fading, and Y is the remaining interference power after cancellation, which is distributed according to (61). The general expression obtained after solving yields Pout = FX (βσ ) +
σ∈SL
θ 0
L
nαi nαi −j k Jα
l=1 θl i=1 j=1 k=0 l=0
δij
2
The partial fraction decomposition in (58) yields
nαi Jα
θ 0
2
where αk = k/θL−M+k , for k = 1 . . . M . To calculate the pdf of Y we first achieve a partial fraction decomposition of (57). To do so, and in order that our results can be completely general, we first must assume that the coefficients α k may have an order of multiplicity higher than one 2 . Therefore, we divide the M coefficients α k into Jα groups with n αi elements each, where all the coefficients in a given group take the same Jα nαi = M , and (57) can be rewritten value αi . That is: i=1 as: ΨY (s) =
fY (y) =
0
x1
ΨY (s) =
shown that, by inverse Laplace transformation of (59), the pdf of Y can be written as
×
eσ /αi (−σ 2 )k−l GX (l, −1/(βαi ), βσ 2 ), l!(k − l)!αki β l
(63)
where the distribution function of X, F X (·), is given in Table I for the different types of fading of the desired signal considered here, and G X (n, s, Λ) is defined as in (22). Introducing in (22) the density functions f X (·) given in Table I leads to the following results when the desired signal experiences Nakagami fading: (m/Ps )m Γ (m + n, (m/Ps − s) Λ) , m+n (m/Ps − s) Γ(m) (64) and the following result under Ricean fading: GX (n, s, Λ) =
1+K GX (n, s, Λ) = 1 + K − sPs n Ps sKPs × exp 2(1 + K − sPs ) 1 + K − sPs ⎛ ⎞ 1 + K (1 + K)K × Q2n+1, 0 ⎝ 2 , 2 − s Λ⎠ , 1 + K − sPs Ps (65)
(60)
where Ωα is the set of (Jα − 1)-tuples such that Ωα = Jα (q1 , . . . qJα ) : qk ∈ N, qi = 0, k=1 qk = j − 1 . It can be 2 As a simple example, consider that we have a systems with L=4, M=3, and the powers are given by (P1 , P2 , P3 , P4 ) = (1/3, 1, 1, 1). It is easy to show that the coefficient will be (α1 , α2 , α3 ) = (1/2, 2/3, 1/2), i.e., we obtain a coefficient with value 1/2 and multiplicity 2.
where Q2n+1, 0 (a, b) is calculating using (29) with N = 1. When the desired user experiences Rayleigh fading we have
1 n!Psn exp −Λ − s GX (n, s, Λ) = (1 − sPs )n+1 Ps u
(66) n 1 1 × −s . −Λ u! Ps u=0 However, for the Rayleigh case, the use of the alternative approach for the outage probability calculation presented in
9
0
0
10
10
Ld=4 Ld=5 −1
−1
10
10
−2
−2
10
10
−3
Outage probability
Outage probability
−3
10
−4
10
−5
10
−6
10
IC
−4
10
MRC −5
10
−6
10
10 N=2,3,4,5
−7
−7
10
−8
10 −20
10 MRC IC −15
−8
−10
−5
0
5
10
15
20
Normalized Average SINR (dB)
Section II-C.3 will result in a much simpler expression for Pout . From (34) and conditioning on the remaining interference after cancellation Y, it is straightforward to show that Pout |Y =y = P (X < β(y + σ 2 )) = 1 − e−β(y+σ
2
)/Ps
. (67)
The unconditioned outage probability can be calculated by simply integrating (67) over all possible values of the random variable Y, that is: ∞ 2 Pout = 1 − e−βσ /Ps e−βy/Ps fY (y)dy. (68) 0
Note that the integral in (68) is in fact the MGF of Y with s = −β/Ps . Therefore, with the help of (57) the resulting outage probability when the desired signal experiences Rayleigh fading will be: 2
/Ps
σ∈SL
θ 0
M
l=1 θl
k=1
L
−15
−10
−5
0
5
10
15
20
Normalized Average SINR (dB)
Fig. 1. Outage probability comparison between MRC and IC for different number of receive antennas. SOI is Rayleigh distributed. L=6. Ld =3. INR=90 dB. R=60 dB.
Pout = 1 − e−βσ
10 −20
1 . (69) 1 + αk β/Ps
V. N UMERICAL R ESULTS In this section numerical results are presented for the outage probability of a multiuser wireless system employing N antennas at the receive end. The N antennas at reception are employed to provide interference cancellation (IC), Maximal Ratio Combining (MRC) or Optimum Combining (OC). All the results in the figures below have been obtained analytically by the direct computation of expressions (16), (17), (21)-(29), (52), (53) and (63)-(66). For MRC and OC, the computational complexity required to calculate our expressions of P out will be primarily determined by the complexity of the required summations over integer partitions. We demonstrate in Appendix II that this complexity is in general exponential. In particular, the complexity to calculate the coefficients A ij given in (6) grows exponentially with J, the number of different interfering powers. However, for the case when every interferer has a different average power (i.e. n i = 1 for all i) the
Fig. 2. Effect of number of dominant interferers on outage probability. SOI is Rayleigh distributed. L=6. N=5. R=60 dB. INR=90 dB.
complexity of these coefficients is linear. Also, the complexity of the MGF derivative in (13) and the coefficients B hl in (31) grow exponentially with the number of antennas. For IC the complexity of our expressions is primarily determined by a summation over a set of permutations, and so the expressions show in this case a factorial complexity with the number of interfering users. Note however that a practical system will generally have a small number of antennas and/or dominant interferers, for which this complexity is reasonable. In the results we show, we assume that the system is affected by L interferers undergoing Rayleigh fading. The signal of interest (SOI) is affected by Rice, Nakagami or Rayleigh fading. In some of the figures we show, we consider that there are Ld dominant interferers out of the L total interferers. Every dominant interferer is assumed to be received with an average power R dB higher than a non-dominant interferer. The figures shown in this section present results for the outage probability as a function of the average SINR per antenna normalized by β, which is the threshold SINR. The average SINR per antenna is defined as: N Ps 1/N SIN R = L n=1 n , 2 i=1 Pi + σ where Psn is the mean received power from the SOI at antenna n and Pi is the mean received power from the i-th interferer. In Fig. 1 we provide a comparison between MRC and IC when the number of antennas increases from 2 to 5. We consider L = 6 interferers with L d = 3 dominant interferes which have a power 60 dB higher then the non-dominant interferers. In addition the total interference to noise ratio (INR) is assumed to be 90 dB. For MRC the results show the typical performance improvement with the number of antennas, with an increase of the slope of the curves and diminishing returns as the antenna number increases. However, for IC the situation is very different: N antennas will cancel N − 1 interferers, so when N increases from 2 to 3 there is a performance improvement as an additional interferer is
10
0
0
10
10 INR=90dB INR=45dB
−1
10
10
−2
−2
10
10
MRC
−3
−4
Outage probability
Outage probability
−3
10
10
IC
−5
10
−6
10
−4
10
−5
10
N=2,3,4
−6
10
10
−7
−7
10
10
−8
10 −20
OC MRC IC
−1
−8
−15
−10
−5
0
5
10
15
10 −20
20
−10
0
Normalized Average SINR (dB)
10
20
30
40
Normalized Average SIR (dB)
Fig. 3. Effect of INR on outage probability. SOI is Rayleigh distributed.L=6. Ld =4. N=5. R=60 dB.
Fig. 5. Outage probability comparison for OC, MRC and IC for different number of receive antennas. SOI is Rayleigh distributed. L=4. No noise. Equal power interferers.
0
10
R=60dB R=35dB R=10dB
−1
0
10
OC MRC IC
10
−1
10 −2
10
−2
10 MRC
10
−3
IC
Outage probability
Outage probability
−3
−4
10
−5
10
10
−4
10
−5
10
−6
10
−6
10
N=2,3,4
−7
10
−7
10 −8
10 −20
−15
−10
−5
0
5
10
15
20 −8
Normalized Average SINR (dB)
10 −20
−10
0
10
20
30
40
Normalized Average SIR (dB)
Fig. 4. Effect of R on outage probability. SOI is Rayleigh distributed. L=6. Ld =4. N=5. INR=90 dB
canceled. However, there is a dramatic improvement when the number of antennas increases from 3 to 4. This is due to the fact that when N = 4 all 3 dominant interferers are canceled, and therefore the interference is only due to low power signals, resulting in an improvement of the outage probability of more than 4 orders of magnitude. In Fig. 2 we can see that when N is fixed to 5 antenna elements, the performance of IC improves dramatically when the number of dominant interferers decreases from 5 to 4, because in the latter case all these dominant interferers are canceled. MRC will be better than IC only if the average received SINR is very high, but that is not an effective design criterion from a system perspective. Also note that the impact of the number of dominant interferers on MRC is almost negligible as the total average interference remains constant for a given average SINR and SOI power, and MRC only improves the performance by improving the output SOI. In
Fig. 6. Outage probability comparison for OC, MRC and IC for different number of receive antennas. SOI is Rice distributed with K=7 dB. L=4. No noise. Equal power interferers.
addition, both Fig 1. and Fig. 2 show that even when the dominant interferers are not canceled, IC is better than MRC in the low SINR regime. Fig. 3 shows the effect of increasing the noise in relation to the total interference (i.e. decreasing the INR). The effect is, as expected, a much higher outage probability for IC, as the noise cannot be canceled, while the impact on MRC is nearly negligible due to the reason mentioned above. Fig. 4 shows the effect of decreasing the power of the dominant interferers with respect to the power of the nondominant interferers (i.e., decreasing R). It can be observed that the higher the relative power of dominant interferers, the better the performance of IC. Again, the effect on MRC is very small. Figs 5, 6 and 7 show a performance comparison of MRC, IC
11
0
0
10
10 OC MRC IC
−1
OC MRC −1
10
10
−2
−2
10
10
K=0 (Rayleigh)
−3
Outage probability
Outage probability
−3
10
−4
10
−5
10
10
−4
10
−5
10
N=2,3,4
−6
10
K=10dB
−6
10
K=15dB −7
10
−7
10
−8
10 −20
Constant SOI
−8
−15
−10
−5
0
5
10
15
20
Normalized Average SIR (dB)
10 −10
−8
−6
−4
−2
0
2
4
6
8
10
Normalized Average SIR (dB)
Fig. 7. Outage probability comparison for OC, MRC and IC for different number of receive antennas. SOI is Nakagami distributed with m=10. L=4. No noise. Equal power interferers.
Fig. 9. Effect of parameter K on outage probability. SOI is Rayleigh distributed in all antennas except one, which receives a RIcean distributed SOI. L=10. N=4. No noise. Equal power interferers.
0
10
OC MRC −1
10
−2
10
Outage probability
−3
10
−4
10
MRC, N=1 −5
10
−6
10
−7
10
D=40, 20, 0 dB −8
10 −20
−10
0
10
20
30
40
Normalized Average SIR (dB)
Fig. 8. Effect of D on outage probability. SOI is Nakagami distributed with m=1.5. L=10. N=4. No noise. Equal power interferers.
and OC for a SOI undergoing Rayleigh, Rice and Nakagami fading respectively. All interferers are assumed to be received with the same average power and the noise is neglected. The curves show that IC is only better for low average SIR, and the benefit of IC with respect to MRC for low average SIR increases as the channel of the SOI improves. This figures also show that OC is the best technique, because it achieves the same cancellation as IC while at the same time achieves the same diversity as MRC (note that the OC and MRC curves are parallel). Fig. 8 shows the effect of unbalanced average power from the desired user at the different antenna elements on MRC and OC. It is assumed that all antenna elements receive signals with Nakagami fading from the desired user, but one of the antennas receive an average power D dB higher than
the average power received at any other antenna. It can be seen that an unbalanced received power results in a worse performance for a given average SIR. Specifically, a higher average SIR is required to achieve the diversity gain as the power in the selected antenna increases with respect to the rest. This can be seen from the fact that a higher SIR is required to achieve a high slope in the curve. In the case of asymptotically high factor D, all the power is received in one antenna, and no diversity gain is achieved. Note also that OC has the same slope at every point as MRC, but a lower outage probability, due to interference cancellation. Fig. 9 also shows the effect of power imbalance from the desired user but from a different perspective. In this case all the receive antennas have the same power from scattered signals, that is, from NLOS components of the transmitted signal. In addition, one of the antennas has a LOS component which is received with a power K times higher than the NLOS component of that antenna and the rest of antennas. In other words, all antennas except one receive a Rayleigh-distributed signal, while one antenna receives a Ricean-distributed signal with parameter K and a total average power (1 + K) times higher than the power received in the rest of antennas. Our results show a performance improvement as the LOS component increases. In the limiting case (K = ∞) the performance is equivalent to a single receive antenna with a constant received signal from the desired user. VI. C ONCLUSION We provide general and easily computable closed-form expressions for the outage probability of wireless systems employing different array processing techniques. In particular, we present new expressions for the outage probability of MRC, OC and ideal null-steering beamforming (which provides interference cancellation, IC) in the presence of CCI and general signal fading. Our analysis of MRC allows an arbitrary number of antennas and interfering users. The
12
developed expressions for IC require the number of antennas to be equal or lower than the number of interferers. However, since we may consider interferers with arbitrary powers, we can assign a negligible power to some of the interferers, and therefore the comparative analysis presented here is suitable for the study of underloaded systems, i.e., systems with more antennas than dominant interferers, as well as for overloaded systems, when the number of dominant interferers is equal or higher than the number of antennas. Our results show that, for underloaded systems, the null steering beamformer is able to cancel all dominant interferers and its performance is better than MRC except for a very high average SINR. In contrast, for overloaded systems, or when the background noise is high with respect to the interference, a higher benefit is obtained if MRC is used at reception, except for the case in which the received average SINR is low. The SINR threshold below which IC performs better than MRC is a function of the channel of the desired user and the background noise. When the channel of the desired user improves, a higher benefit is obtained for low average SINR by using a beamformer. On the other hand, the background noise has a higher detrimental effect on the beamformer, as the noise cannot be canceled. For interference-limited system (negligible background noise), expressions have also been derived for the performance analysis of OC, and comparative results have been presented between this technique, MRC, and IC. As expected, OC achieves the lowest outage probability due to its ability to offer both interference cancellation and diversity gain simultaneously, but at a significantly higher implementation complexity. A comparative analysis has also been presented between OC and MRC when the desired user signals at every receive antenna are arbitrarily distributed for interference-limited overloaded systems with equal power interferers. Our results show that OC performs the same as MRC with N-1 interferers canceled. This fact was known to hold for i.i.d. signals from the desired user at every antenna, and has now been proved for the i.n.d. case. A PPENDIX I A N E XPRESSION FOR C ALCULATING THE O UTAGE P ROBABILITY The outage probability is defined as the event that the SINR falls below a predefined threshold β, that is X < β , (70) Pout = P Y + σ2
only take positive values, it is obvious that we will have: Pout |X=x 1, M > 1. (79)
S(1, M ) = M ∀ M > 0 ;
S(K, 1) = 1 ∀ K > 0
13
By computing (79) iteratively we can write: S(K, M ) = M +
M−1
(M − m) · Pm (K)
(80)
m=1
where Pm (K) =
i1 K−1
im−1
···
i1 =1 i2 =1
1.
(81)
im =1
From (79) it is clear that the complexity of S(K, M ) is linear when K = 1, and thus we canwrite S(1, M ) ∈ O(M ). For K > 1, taking into account that ni=1 ai ∈ O(nk+1 ) given that ai ∈ O(nk ), it is clear from (81) that P m (K) ∈ O(K m ). As a simple check of this result, the exact expressions of Pm (K) for m=1, 2 and 3 are: P 1 (K) = K − 1, P2 (K) = K(K − 1)/2 and P3 (K) = (K(K − 1)(2K − 1) + 3K(K − 1))/12, and these expressions show the derived complexity. Noticing that (80) has a summation of terms of exponential complexity with increasing exponent, we can finally write S(K, M ) ∈ O(K M−1 ). R EFERENCES [1] A.J. Goldsmith, Wireless Communications, Cambridge University Press, 2005. [2] M.K Simon, M.-S Alouini, Digital Communications over Fading Channels, Hoboken, N.J, Wiley 2005. [3] J.H. Winters, ”Optimum Combining in Digital Mobile Radio with Cochannel Interference”, IEEE Journal on Selected Areas in Comm, vol. 2, no. 4, pp. 528 - 539, July 1984 [4] L.C. Godara, ”Applications of Antenna Arrays to Mobile Communications, Part I: Performance Improvement, Feasibility, and System Considerations”, Proc. IEEE, vol. 85, no. 7, pp. 1031 -1060, July 1997 [5] R. Vaughan J.B. Andersen, Channels, Propagations and Antennas for Mobile Communications, The Institution of Electrical Engineers, London, U.K., 2003. [6] A. Shah and A.M. Haimovich, ”Performance Analysis of Maximal ratio Combining and Comparison with Optimum combining for Mobile Radio Communications with Co-channel Interference”, IEEE Trans. Veh. Technol., vol. 49, pp. 1454-1463, July 2000. [7] C. Chayawan and V.A Aalo, ”On the Outage Probability of Optimum Combining and Maximal Ratio Combining Schemes in an InterferenceLimited Rice Fading Channel,” IEEE Trans. Comm., vol. 50, no 4, pp. 532-535, April 2002. [8] V.A. Aalo and C. Chayawan, ”Outage probability of cellular radio systems using maximal ratio combining in Rayleigh fading channel with multiple interferers”, Electronics Letters, vol. 36, no. 15, July 2000. [9] X.W. Cui, Q.T. Zhang, Z.M. Feng; ”Outage performance for Maximal Ratio Combiner in the Presence of Unequal-Power Co-channel Interferers,” IEEE Communications Letters, vol. 8, no. 5, May 2004. [10] J.P. Pe˜na-Martin and J. M. Romero-Jerez, ”Outage Probability with MRC in presence of multiple interferers under Rayleigh fading channels”, Electronics Letters, vol. 40, no. 14, July 2004. [11] A. Shah and A.M. Haimovich, ”‘Performance Analysis of Optimum Combining in Wireless Communications with Rayleigh Fading and Cochannel Interference,”’ IEEE Trans. Commun., vol. 46, pp. 473479, Apr. 1998. [12] M. Kang, M.-S. Alouini and L.Yang, ”Outage Probability and Spectrum Efficiency of Cellular Mobile Radio Systems with Smart Antennas,” IEEE Trans. Commun., vol. 50, pp. 18711877, Dec. 2002. [13] Q.T Zhang and X.W. Cui, ”Outage Probability for Optimum Combining of Arbitrarily Faded Signals in the Presence of Correlated Rayleigh Interferers,” IEEE Trans. Veh. Technol., vol. 53, no. 4, pp. 1043-1051, July 2004. [14] M.O. Hasma, M.-S. Alouini A. Bastami and E.S Ebbini, ”Performance Analysis of Cellular Mobile Radio Systems with Succesive Co-channel Interference Cancellation,” IEEE Trans. Wireless Comm., vol. 2, no. 1, pp. 29-49, January 2003. [15] R. Mostafa, A. Annamalai and J.H. Reed, ”Performance Evaluation of Cellular Mobile Radio Systems with Interference Nulling of Dominant Interferers,” IEEE Trans. Comm., vol. 52, no. 2, pp. 326-335, February 2004.
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