Impact of Imperfect Channel Estimation on MIMO Two Hop Relay Network with Beamforming K.G.A Madushan Thilina∗ , Nandana Rajatheva† Information and Communication Technologies Field of Study, School of Engineering and Technology, Asian Institute of Technology, Klong Luang, Pathumthani 12120, Thailand
[email protected]∗,
[email protected]†
Abstract—In multiple-input multiple-output relay networks, the imperfect channel state information (CSI) at the transmitters and receivers degrades the system performance. In this paper, we quantify the detrimental effect of imperfect CSI on the performance of such a network with beamforming. The sourceto-relay and the relay-to-destination transmissions take the forms of maximum ratio transmission (MRT) and maximum ratio combining (MRC), respectively. The MRT weight vector at the source is computed based on the outdated CSI, and the MRC weight vector at the destination is computed based on CSI with Gaussian errors. The cumulative distribution function, the probability density function and the moment generating function of the end-to-end signal to noise ratio are derived in closed forms. The outage probability and the average symbol error rate are also derived. The analytical and Monte Carlo simulation results are presented to determine the detrimental effect of imperfect CSI on the system performance and to validate our results.
I. I NTRODUCTION The system performance in wireless channels degrades significantly due to the signal fading. Spatial diversity can be used effectively to combat signal fading. In particular, transmit/receive diversity combining for the two-hop multipleinput multiple-output (MIMO) amplify-and-forward (AF) relay networks systems can improve the system performance significantly [1]–[4]. The combination of maximal ratio transmission (MRT) and maximal ratio combining (MRC) are used widely as a simple and efficient beamforming technique for two-hop MIMO AF relay networks [1]–[3], [5], [6]. In [2], the performance of dual-hop MIMO AF variable-gain relaying with MRT/MRC beamforming is studied over independent and identically distributed (i.i.d.) Rayleigh fading. The outage probability, average symbol error rate (SER) and high SNR analysis of these performance metrics are derived in closed-forms. In [5] authors analyze the performance of two-hop beamforming by using a single-antenna semi-blind (fixed-gain) relay over the i.i.d. Nakagami-m fading. Closed-form expressions for the outage probability, moment generating function (MGF), and generalized moments of the end-to-end signal-to-noise ratio (SNR) are derived. Further, in [6], the average SER of two-hop MIMO AF variable-gain relaying over Nakagami-m fading is analyzed. In [3], the effect of multiple-antennas at the source on the outage probability is investigated by considering MRT for the source-to-relay (S → R) transmission in two-hop AF relaying. In all the above mentioned studies, the channel state
information (CSI) at the relay and the destination is assumed to be perfect. In a practical implementation of MRT/MRC beamforming, the imperfect CSI may be used for computing the MRT and MRC weight vectors at the source and the relay, respectively. The weighting errors of MRT and MRC have a significant detrimental effect on the system performance. In [7], the effect of outdated CSI due to feedback delays on the performance of MRT for S → R in two-hop AF relaying is studied. Closedform expressions for the outage probability, probability density function (PDF) and average SER are derived. Although this analysis considers multiple-antenna source, both the relay and the destination are single-antenna terminals. Thus, the singleinput single-output relay-to-destination (R → D) channel governs the system diversity order, despite having a multipleinput single-output S → R channel. The diversity order of the system in [7] equal to one, irrespective of perfect or imperfect CSI fed to the source for computing the MRT weight vector. In this paper, the effect of imperfect CSI on the performance of two-hop MIMO variable-gain AF relay networks is investigate. In our analysis, the source and the destination are equipped with multiple-antennas whereas the relay is equipped with a single-antenna [2], [5], [6]. The S → R and R → D transmissions take the forms of MRT and MRC, respectively. The MRT weight vector at the source is computed based on the outdated CSI due to feedback delays. On the other hand, the MRC weight vector at the relay is computed based on the CSI with channel estimation errors. The cumulative distribution function (CDF), the PDF and the MGF of the endto-end SNR are derived in closed-forms and used to obtain the average SER, which is valid for a wide range of modulation schemes. Numerical and Monte-Carlo simulation results are also provided to quantify the detrimental effect of imperfect CSI on the performance of MRT/MRC beamforming for the two-hop MIMO AF relay networks. The rest of this paper is organized as follows: In section II, the system and the channel model of two-hop AF beamforming relay Network is described. Section III presents the performance analysis. Section IV provides the numerical and simulation results. Section V concludes the paper. II. SYSTEM MODEL We consider a twohop AF relay network, where the source and destination are equipped with Nt and Nr antennas
MRT
MRC
h 1,1
S
... .
where n2 (t) is the AWGN vector at the destination with a variance σ22 INr , and h2 (t) is the R → D channel gain vector. The post preprocessing received signal at the destination after applying MRC can then be written as follows:
h 2,1
h 1,2
R
h 2,2
.. ..
D
yeq (t) = w2 (t)yD (t),
h 2,
h 1, S-R Feedback Channel
Fig. 1.
channel estimation error
System Model
(see Fig. 1). The single-antenna relay uses variable-gain AF protocol. The source employs MRT whereas the destination employs MRC. The weight vectors for the MRT and MRC are computed based on the imperfect CSI. The source-todestination (S → D) transmission takes place in two orthogonal time-slots [8], [9]. The S → D direct channel is assumed to be unavailable due to heavy shadowing. The S → R and R → D channels undergo i.i.d. Rayleigh fading. Under this system and channel model, the end-to-end SNR γeq can be derived as follows: During the first time-slot, the received signal at the relay is given by p yR (t) = P1 w1 (t|Td )h1 (t)x(t) + n1 (t), (1)
where x(t) and n1 (t) are the transmitted signal and zero mean additive white Gaussian (AWGN) noise at the relay, satisfying E(|x(t)|2 ) = 1 and E(|n1 (t)|2 ) = σ12 , respectively. P1 is the transmitted power at the source. The weight vector w1 (t|Td ) is computed based on the outdated CSI and given by [10] w1 (t|Td ) =
hH 1 (t − Td ) , |h1 (t − Td )|
(2)
where h1 (t − Td ) is the delayed channel gain vector by a time delay Td of the perfect estimate h1 (t) of S → R channels. The relationship between h1 (t − Td ) and h1 (t) is given by a commonly used outdated CSI model as follows [10]–[12]: h1 (t) = ρd h1 (t − Td ) + ed (t),
(3)
where ρd = J0 (2πfd Td ) is the normalized correlation coefficient between the h1,j (t) and h1,j (t − Td ) , j = 1...Nt . J0 (·) is the zeroth oder Bessel function of the first kind, fd is the Doppler frequency and ed (t) is the zero mean Gaussian error vector with variance (1 − ρ2d )INt . During the second time-slot, the relay amplifies yR (t) by a variable-gain G and forwards to the destination. Assuming that the relay perfectly estimates the channel gain w1 (t|Td )h1 (t), the G can be written as follows [7]–[9]: s P2 (4) , G= H P1 |w1 (t|Td )h1 (t − Td )|2 + Cσ12 where P2 is the transmitted power at the relay. Here, C = 1 stands for the channel-noise-assisted AF relays whereas C = 0 stands for channel-assisted AF relays [2]. Then the signal at the destination can be written as yD (t)
=
GyR h2 (t) + n2 (t),
(5)
(6)
where w2 (t) is the MRC weight vector at the destination. The R → D channel estimation is assumed to have Gaussian errors and is given by [13] ˆ H (t) h , (7) w2 (t) = 2 ˆ 2 (t)| |h ˆ 2 (t) is the imperfect channel estimate of h2 (t). In our where h ˆ 2 (t) are correlated each analysis, we assume that h2 (t) and h other by a normalized correlation coefficient ρe [13], [14]. By substituting (7) into (6) and after carrying out some trivial mathematical manipulations, the end-to-end SNR γeq can be derived as γ1 γ2 , (8) γeq = γ1 + γ2 + C where γ1 = |w1H (t|Td )h1 (t)|2 γ¯1 and γ2 = |w2H (t)h2 (t)|2 γ¯2 . i γ¯i = P for i=1,2 are the average SNRs of S → R and σi2 R → D channels. III. P ERFORMANCE
ANALYSIS
In this section, the CDF and the PDF of end-to-end SNR γeq are derived. The outage probability and the average SER, which is valid for a wide range of modulation schemes, are also derived. A. The CDF and the PDF of the end-to-end SNR The CDF of γeq in (8) can be written as a single-integral expression as follows [2]: Z ∞ fγ2 (ω + x)dω, (9) 1−Fγ1 x(ω+x+C) Fγeq (x)=1− ω 0
where Fγ1 (x) and fγ2 (x) are the CDF and the PDF of γ1 and γ2 , respectively. The PDF of γ1 for full-rate feedback without quantization errors can be written as [10] Nt − 1 Nt −1 n 1 X 2(N −n−1) |ρd | t fγ1 (x) = Nt γ¯1 n=0 (Nt − n − 1)! 2
(¯ γ1 (1 − |ρd | ))n xNt −n−1 e
×
− γ¯x
1
.
(10)
The corresponding CDf of γ1 can be derived as Fγ1 (x)
=
1−
N −n−1 t −1 NtX X n=0
i=0
Nt − 1 n
|ρd |2(Nt −n−1)
x
×
2
(1 − |ρd | )n
xi e− γ¯1 . i!¯ γ1i
(11)
The PDF of γ2 can be written as [13], [14] fγ2 (z) = ×
Nr −1 2 (1 − |ρe | )(Nr −1) − γ¯z X Nr − 1 e 2 k γ¯2 k=0 k 1 |ρe |2 z . (12) 2 γ ¯ (1−|ρ | ) 2 e k!
By substituting (11) and (12) into (9), the following integral expression for the CDF of γeq can be obtained. N −n−1 NX r −1 t −1 NtX X
Fγeq (x)= 1−
n=0
0
Nr − 1 k
+ γ1 )
(13)
By applying the Binomial expansion and then by using [15, Eq. (3.471.9)], the integral in (13) can be solved to yield a closed-form expression for the CDF of γeq as follows: N −n−1 NX r −1 t −1 NtX X
i k X X Nt − 1 = 1−2 n n=0 i=0 k=0 l=0 m=0 Nr − 1 i k 2(N −n−1) × |ρd | t l m k
Fγeq (x)
2 n
×
2k
(1 − |ρd | ) |ρe | i! k!
2 Nr −k−1
(1 − |ρe | )
) ( k+m+l+1 ) ( k+2i−m−l+1 2 γ1 γ2 2
( 2i+k+m−l+1 ) 2
× x
× Kk−m−l+1
(x + C)( 2
q
k+l−m+1 ) 2
x(x+C) γ 1γ2
1
e−x( γ 1
,
+ γ1 )
(14)
N −n−1 NX i k r −1 X t −1 NtX X X Nt − 1 fγeq (x)= 2 n n=0 i=0 k=0 l=0 m=0 Nr − 1 i k 2(N −n−1) 2k × |ρd | t |ρe | l m k 2
2
(1 − |ρd | )n (1 − |ρe | )Nr −k−1 xA−1 (x + C)B−1
× e
) ( k+2i−m−l+1 2
i! k! γ 1 −Dx
) ( k+m+l+1 2
γ2
[ (2x + C)Z Kν−1 (2Z) + Kν (2Z)
×(Dx2 − (A + B − DC − ν)x − C(A −
×
N −n−1 NX r −1 t −1 NtX X
= 1 − 2s
n=0
× ×
k=0
p=0
Nt − 1 Nr − 1 k + i 2(N −n−1) |ρd | s n k p 2
×
i=0
k+i X
2k
(1 − |ρd | )n |ρe |
2
(1 − |ρe | )Nr −k−1
( k+2i−p+1 ) 2
( k+p+1 ) 2
i! k! γ 1 γ2 √ λ π (2φ) Γ(ψ + λ) Γ(ψ − λ) (ω + φ)λ+ψ Γ(ψ + 21 ) 2 F1 (ψ
1 1 ω−φ + λ, λ + ; ψ + ; ), 2 2 ω+φ
(17)
1 1 where ψ = (i + k) , λ = (k − p + 1), ω = (s + γ1 + γ2 ) 2 and φ = √γ1γ2 . 2 F1 (a, b; c, d) is the Gauss Hypergeometric function [15, Eq. (9.14.2)].
C. Outage probability
2
where Kv (x) is the Modified Bessel function of the second kind of order ν [15, Eq. (8.407.1)]. The PDF of γeq can be derived by differentiating the CDF of γeq with respect to x by using [15, Eq. (8.486.12)] as
×
The MGF of the end-to-end SNR is an important statistic which can be used to obtain many other performance metrics [16]. The MGF of γeq can be derived by substituting into Mγeq (s) = Eγeq (exp(−sγeq )). Mγeq (s) can also be expressed in an alternative form as follows: Z ∞ (16) sFγeq (x)e−sx dx. Mγeq (s) =
Mγeq (s)
2
x(x+C) w (w + x + C)i (w + x)k e(− wγ 1 − γ 2 ) dw . i w
2
B. The MGF of the end-to-end SNR
By substituting (14) with C = 0 into (16) and solving the resulting integral by using [15, Eq. (6.621.3)], the MGF of γeq can be derived in closed-form as
2
1
1
k+l−m+1 , 2
0
(1 − |ρd | )n i! k! γ 1i γ 2k+1
|ρe |2k xi (1 − |ρe |2 )Nr −k−1 e−x( γ 1
× ∞
k=0
2(Nt −n−1)
|ρd |
×
Z
i=0
Nt − 1 n
q x(x+C) 2i+k+m−l+1 , B = where Z = γ1γ 2 , A = 2 1 1 D = γ + γ and ν = k − m − l + 1.
ν )) ], (15) 2
The outage probability is an important wireless system parameter and is defined as the probability that the instantaneous received SNR drops below a predetermined SNR threshold γth . Thus, the outage probability can be obtained by evaluating the CDF of γeq at γth as Fγeq (γth ) = P r(γeq < γth ).
(18)
D. Average symbol error rate In this section, the average SER is derived for the channelassisted AF relays for the sake of analytical tractability. The average SER of dual-hop relay networks with channel-assisted AF relays is a tight upper bound for that of with channel-noiseassisted relays [2]. The average SER, which is valid for a wide range of modulation schemes can be written as [17] p Ps = aEγ [Q 2bγ] , (19) where Q(·) is the Gaussian-Q function. a and b represent modulation specific constant. For example, the average bit error rate (BER) of binary phase shift keying (BPSK), and binary frequency shift keying (BFSK) with orthogonal signaling can be obtained by substituting (a, b) = (1, 1) and
0
10
Simulation Analytical
Nt = 3, Nr = 3
0.5
Simulation Analytical −1
d
e
ρ = 1, ρ =1
ρd = 0.79, ρe = 0.72
Outage Probability
Probability Density
10
ρ = 0.79, ρ = 0.62
0.4
0.3 ρd = 0.9037, ρe = 0.85 0.2
d
e
−2
10
ρ = 1, ρ =0.95 d
−3
e
ρd= 0.9037, ρe=0.85
10
ρd = 1, ρe = 0.95
ρ = 0.79, ρ =0.72 d
e
−4
0.1
10
ρ = 0.4720, ρ =0.43 d
e
ρ = 0.2906, ρ =0.27 d
0
e
−5
1
2
3
4
5 γth
6
7
8
9
10 −10
10
−5
0 5 10 Average SNR of First Hop
15
20
Fig. 2. The PDF of the end to end output SNR versus normalized correlation coefficient for Nt = 3, Nr = 3,γ 1 = 7dB and γ 2 = 1
Fig. 3. Outage probability vs. Average SNR at first hop for Nt = 3, Nr = 3 and γ 2 = 2γ 1 and C = 1
(a, b) = (1, 0.5) into (19). Moreover, the average SER of M ary pulse amplitude modulation (PAM) is given by (19) with (a, b) = (2(M − 1)/M, 3/(M 2 − 1)). By using integration by parts, (19) can be written in a single-integral form as r Z a b ∞ Fγ (x) −bx √ e dx . Ps = (20) 2 π 0 x
that when both ρd and ρe approach one, the PDF curve shifts to the right. The exact agreement between the analytical results and the Monte-Carlo simulations validates our analysis. Fig. 3 shows the impact of imperfect CSI on the outage probability for MIMO AF relaying with beamforming. The outage probability is plotted for several values of ρd and ρe . Further, the outage probability for the perfect system (ρd = ρe = 1) is also plotted to compare the performance degradation due to imperfect CSI. When ρd and ρe approach one (i.e., the weight vectors are computed with perfect CSI), the system achieves the full diversity order; Gd = min (Nt , Nr ). However, if either of the two weight vectors are computed based on the imperfect CSI, the system achieves no diversity gains, despite the multiple-antennas at the source and the destination. Thus, this figure reveals clearly that the outage performance degrades significantly if the weighting vectors at the source and the relay is computed based on the imperfect CSI. The Monte-Carlo simulations results matches exactly with the analytical results validating our analysis. Similarly, the Fig. 4 depicts the effect of imperfect CSI on the outage probability for several antenna configurations at the source and the relay. The outage probability for a single-antenna AF relaying is plotted as a benchmark to compare the performance gains of MIMO relaying. Despite a relative coding gain, no diversity benefits can be achieved by MRT/MRC beamforming for dual-hop relay networks if a time delay exists in the R → S feedback channel or channel estimation error on the R → D channels. Fig. 5 shows the impact of imperfect CSI on the performance of average BER of BPSK for a MIMO relay network with Nt = 3 and Nr = 4. The average BER is plotted for several ρd and ρe combinations. The BER curve for ρd = ρe = 1 corresponds to the system with perfect CSI at the source and the relay. On the other hand, the BER curve with ρd = ρe = 0 corresponds to the full-imperfect CSI case. This figure reveals that even a slight time delay on
Now, the average SER can be derived by substituting (14) with C = 0 into (20), and solving the resulting integral by using [15, Eq. (6.621.3)] as follows: r Nt −1 Nt −n−1 Nr −1 i X X X b X a PS = − a 2 π n=0 i=0 k=0
× × × ×
l=0
k X Nt − 1 Nr − 1 i k n k l m
m=0
|ρd |
2(Nt −n−1)
2
2k
(1 − |ρd | )n |ρe | ) ( k+2i−m−l+1 2
2
(1 − |ρe | )Nr −k−1
) ( k+m+l+1 2
i! k! γ 1 γ2 √ π (2β)ν Γ(µ + ν) Γ(µ − ν) (α + β)µ+ν Γ(µ + 12 ) 2 F1 (µ
1 1 α−β + ν, ν + ; µ + ; ), 2 2 α+β
(21)
1 1 + γ2 ) where µ = (i+k + 32 ) , ν = (k −m−l +1), α = (b+ γ1 2 and β = √γ1γ2 .
IV. N UMERICAL A NALYSIS This section presents the numerical and the Monte-Carlo in simulation results study of the detrimental effect of imperfect CSI on the system performance and to validate our analytical results. In Fig. 2, the probability density function of the end-to-end SNR is plotted for several combinations of ρd and ρe . It shows
0
0
10
10
Simulation Analytical
Simulation Analytical −1
10
−1
Nt =4, Nr =4
−2
10
ρ1=0, ρ2 = 0
−2
10
Nt =4, Nr =3
BER
Outage Probability
10
Nt =3, Nr =3 −3
10
Nt =3, Nr =2
−4
10
Nt =2, Nr =2 −4
ρ = 0.95, ρ = 1 1
Nt = 1, Nr = 1
10
ρ1 = 0.5, ρ2 = 0.5
−3
10
ρ = 1, ρ = 0.95
−5
1
10
ρd = 0.9037 ρe = 0.85
2
ρ = 1, ρ =1 1
−5
10 −10
2
2
−6
−5
0 5 10 Average SNR of First Hop
15
10 −10
20
−5
0
5 10 Average SNR of First Hop
15
20
Fig. 4. Outage probability vs. Average SNR at first hop for ρd = 0.9037, ρe = 0.85, γ 2 = 2γ 1 and C = 1
Fig. 5. C=0
the R → S feedback channel or channel estimation error at the R → D channels has a significant detrimental effect on the average BER. In fact, our system achieves full diversity order (Gd = min (Nt , Nr ) = 3) if the CSI is perfect in either hop. However, no diversity advantages can be obtained if the CSI is imperfect. This fact is revealed by the BER curves corresponding to (ρ1 = 1, ρ2 = 0.95) and (ρ1 = 0.95, ρ2 = 1), where one channel has no CSI imperfections, and the other hop has a slight CSI imperfection. Further, the BER curves correspond to (ρ1 = 1, ρ2 = 0.95) and (ρ1 = 0.95, ρ2 = 1) achieve signigicant coding gains over the full-outdated CSI case; ρd = ρe = 0. Monte-Carlo simulation results verify our BER analysis.
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V.
CONCLUSION
In this paper , we analyze the detrimental effect of imperfect CSI on the performance of the MIMO two-hop AF relay network with beamforming. Closed-from expressions for the CDF, the PDF and the MGF of the end-to-end SNR are derived. The outage probability and the average SER are also derived in closed-forms. Numerical and Monte-Carlo simulation results are presented to investigate the effect of imperfect CSI on the system performance and to verify our analytical developments. Numerical results show that the imperfect CSI at either hop has a significant detrimental effect on the system performance. Although MRT/MRC beamforming for dual-hop relaying achieves full diversity oder (Gd = min (N t, N r)) when the CSI is perfect in both hops, this diversity gain diminishes to one, if either hop’s MRT/MRC weight vector is calculated based on the outdated CSI. Our analysis may be used for the designing of the two-hop MIMO AF relay networks with beamforming. R EFERENCES [1] R. H. Y. Louie, Y. Li, H. A. Suraweera, and B. Vucetic, “Performance analysis of beamforming in two hop amplify and forward relay networks with antenna correlation,” IEEE Trans. Wireless Commum., vol. 8, no. 6, pp. 3132–3141, June 2009.
Average BER using BPSK for Nt = 3, Nr = 4, γ 2 = 0.5γ 1 and
