By definition, ( ,g g h q is precisely the average of r, ) g g h q defined in (3.10) dq( , , )r over the distribution of the estimation errors of the R-D channels, that is,
rewrite each summand in (4.2) in the form of the expectation of a ratio of two quadratic forms in the error vector e. Starting from (4.2) together with further manipulations, we have (see Appendix D for the detailed derivations)
( )
fm = éêëg12
(
1-f r2( 1( )m ))
hr*,1gL 2(
1-f r2( L( )m ))
h*r L, ùúûT. (4.7) Now, each summand in (4.3) is the expectation of a Rayleigh ratio of the complex-valued Gaussian random vector e. To facilitate analysis, let us consider the augmented real-valued random vector associated with e, namely,Re( ) the aid of (4.8), we go on to rewrite each summand in (4.3) in terms of the real-valued Gaussian random vector x. The result, as shown in the next proposition, will allow us to derive an analytic formula for g g h q( , , ) . r
and 2L-dimensional Gaussian random vector with zero mean and variance s . To find a e2
closed-form expression for g g h q( , , ) based on (4.9), we need the following lemma. r
is derived in the following theorem.
Theorem 4.3: Let g g h q( , , ) be defined in (4.9). It follows that r
where
To the best of our knowledge, the double-integral in (4.20) does not admit further closed-form expressions. In the simulation section, it will be verified that the derived integral formula (4.20) is in close agreement with the corresponding simulation outcome. It can be seen that ( ,g g h q in the form (4.20) is a very complicated r, ) function of the beamforming coefficients g 's, and direct maximization of ( , , )i g g h q r
based on (4.20) is thus intractable. In the following section, we propose a method that can facilitate an analytic design of g 's. i
4.2 Approximation for g g h q( , , ) r
As noted before, the double integral form of ( ,g g h q in (4.20) makes the r, ) beamforming design problem formidable to tackle. To resolve this difficulty, we then turn to the expression of ( ,g g h q given in (4.9), which is a sum of the expected r, ) value of the ratio of quadratic functions in the random vector x. To ease subsequent analysis, we propose to adopt the commonly used approximation technique (see, e.g., [30]); more specifically, each summand in (4.9) is approximated as the ratio of the respective means of the numerator and denominator:
Through further rearrangement, the approximation in (4.23) admits a simple form as shown in the next lemma.
Lemma 4.4: Let x ( ,0 se2I2L). Then we have respectively, (4.10), (4.12), (4.14), and (4.15).
[Proof]: See Appendix. □
Based on (4.23) and (4.24), it directly follows that can be further rearranged into a ratio of two standard quadratic forms in terms of the beamforming weights gi’s; hence, the expression for ( ,g g h q can be further r, ) simplified. This is done in the next lemma.
Lemma 4.5: The approximation of g g h q( , , ) in (4.25) can be expressed as r
( )
4.3 Design of Beamforming Weights
With the aid of (4.26), the problem of beamforming design for SNR enhancement can be formulated as
where P is the total available power budget. The optimization problem (4.30), d unfortunately, is not convex and the optimal solution is difficult to find. Toward analytical tractability and complexity reduction, the following theorem further derives a lower bound for the objective function in (4. 30). As will be shown later, by conducting maximization of the derived lower bound, an analytic suboptimal solution can then be obtained.
With the aid of Theorem 4.6, we propose to obtain a suboptimal beamformer based on maximization of the lower bound derived in (4.31):
2
The solution to (4.32) is precisely the dominant eigenvector of Y W [26], -1 normalized so that the two-norm equal to P . d
4.4 Simulation Results
This section uses numerical simulations to illustrate the effectiveness of the proposed scheme. In the cooperative beamforming system, the channel gains of all S-R and R-D links are i.i.d random variables generated from (0,1). The crossover probability pi of each BSC is drawn from U(0.05, 0.1), the uniform random variable distributed over the interval [0.05, 0.1 . The quantization threshold is set according to the rule in [10, ] p-4779]. The total available power of transmit beamforming is Pd =1. In each Monte-Carlo run, a sequence of T =5000 BPSK source symbols is generated.
A. Validation of the Derived Conditional Average SNR (4.20)
To validate the formula of the derived conditional average SNR g g h q( , , ) in (4.20), r
we consider a network of L =4 relays, and the proposed suboptimal beamforming scheme (4.32) is employed at each relay. With R-D link channel estimation error variance se2 =0.01, Figure 4 compares the values of ( ,g g h q obtained based on r, ) the integral form (4.20) and corresponding simulated outcome at different average R-D receive SNR, defined to be gd Pd rs2/sw2 =sw-2 [8]. Also, with R-D receive SNR fixed to be g =d 15 dB, Figure 5 compares the theoretical and simulated values of
( , , )r
g g h q at different channel estimation error variance s . In both figures, the e2 simulated results are obtained by averaging over 100 realizations of the R-D link channel estimates. The figures show that the theoretical solutions computed based on (4.20) are indeed very close to the experimental results.
B. BER Performances
For fixed average S-R SNR g =s 13 dB, Figure 6 compares the BER curves of the proposed beamforming scheme (4.32) with the solutions in (3.17) and [10] at various average R-D SNR. The number of relays is L =6, and the R-D channel estimation error of each link is drawn according to e i (0, 0.05). As can be seen from the figure, the proposed scheme outperforms the method in [10]; the result is not unexpected since the solution in [10] is designed under the idealized assumption that the one-bit message is received at the destination without errors and that the CSI of the R-D links is perfect. Also, compared with the solution in (3.17), the proposed beamformer (4.32) yields improved performance since the effect of R-D channel estimation error is taken account of in the design. With fixed gs =13 dB, and gd =20 dB, Figure 7 further illustrates the BER of the three methods when the variance s of R-D channel e2 estimation error varies from 0 to 1, i.e. 0£se2 £ (the number of relay is 1 L =6).
The figure shows that, compared with the solutions in (3.17) and [10], the proposed scheme is indeed more robust against the increase in the channel estimation error variance. It is somewhat unexpected to see that the beamformer in (3.17), which takes into account only the effect of the one-bit S-R-link SNR transmission error, performs worse than [10] when se2 ³0.08. A plausible rationale behind this phenomenon is that, while the beamformer in [10] admits a simple closed-form expression (see (37) in [10]), the computation of the solution in (3.17) calls for solving a generalized eigenvalue problem. The involved eigen-decomposition increases the algorithmic complexity, and renders the resultant solution more vulnerable to system parameter mismatch. The proposed beamforming scheme, which further takes account of the effect of R-D link channel estimation errors, does improve the robustness against model uncertainty.
Figure 4. Values of the conditional average SNR g g h q
(
, r,)
obtained via the integral solution (4.20) and simulations for different R-D SNR (channel estimation error variance is set to be 0.01).
5 10 15 20
9 10 11 12 13 14 15 16 17 18 19 20
SNRrd(dB)
Conditional Average SNR (dB)
theoretical result simulation result
Figure 5. Values of the conditional average SNR g g h q
(
, r,)
obtained via the integral solution (4.20) and simulations for different channel estimation error variance (R-D SNR is set to be 15 dB).0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 15.8
16 16.2 16.4 16.6 16.8 17 17.2
Estimation Noise Variance
Conditional Average SNR (dB)
theoretical result simulation result
Figure 6. BER performance of the proposed beamformer (4.32) and the two solution in [10] and (3.17) for different R-D SNR.
0 2 4 6 8 10 12 14 16 18 20
10-8 10-7 10-6 10-5 10-4 10-3 10-2
average R-D SNR
BER
solution in [10]
solution in (3.17) proposed solution
Figure 7. BER performance of the proposed beamformer (4.32) and the two solutions in [10] and (3.17) for different channel estimation error
variance.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
estimation error power
BER
solution in [10]
solution in (3.17) proposed solution
Conclusion
Low-overhead cooperative beamforming design under mismatched inter-node CSI has been an important research problem in the study of relay-based wireless communications. In this these we investigate this problem by further taking into account the effects of imperfect transmission of quantized S-R link SNR and R-D link channel estimation errors. As in previous study, the transmission link of the one-bit S-R link SNR is modeled as a BSC with a non-zero crossover probability. In the first part of this thesis, we assume R-D link channel estimation is perfect. Given a set of received quantized SNR message, we derive the closed-form formula of the expected conditional SNR, averaged over the bit-flipping distributions of BSCs. While beamforming design via direct maximization of this SNR metric is formidable, we further derive an tractable SNR lower bound. By conducting maximization of this lower bound, a suboptimal beamformer can be obtained as the solution to a generalized eigenvalue problem. In the second part of this thesis, the assumption of perfect R-D channel estimation is relaxed, and the channel estimation errors are modeled as i.i.d. Gaussian random variables.
Given the received quantized S-R link SNR and a set of R-D link channel estimates, we further derive the exact formula of the expected conditional SNR averaged over both the distributions of the bit-flipping effect and R-D link channel estimation errors. Since the SNR metric thus obtained is difficult to analyze, we resort to certain approximation techniques to derive a tractable SNR lower bound. Still, through maximization of this lower bound a suboptimal beamformer can be obtained by solving a generalized eigenvalue problem. Computer simulations evidences the performance advantages of the proposed schemes.
Appendix A
Proof of Theorem 3.1
To derive (3.6), we shall find an explicit expression for
( ) q=q. It then follows immediately that
candidate q’s. Therefore, b can be accordingly expressed as 1
1 where (b) follows after some straightforward manipulations. The term b stands for 2 the event that q differs from the true q in two bits. Given q , there are totally
2( ) 2L ( 1)/ 2
S q =C =L L- possible candidate q’s in this case. By repeating the above arguments, it can be directly verified that
( ) ( )
Equation (3.6) follows since
0
Appendix B
Proof of Theorem 3.2
All we have to do is to rewrite the multiple summations in (3.6) as a single summation in the form of (3.10), and then to provide an explicit relation between m in (3.10) and the multiple indices l k, , ,1 kl involved in (3.6). For ease of discussion, main procedures for deriving (3.10) can be summarized as follows: (1) exhaustively list all the C0L +C1L ++CLL terms in (3.6) in the increasing order of l; (2) particularly, in b (l l ³1), the C terms in the l-fold multiple summations are listed in the lL following way: starting from k = , exhaustively list all involved terms indexed by 1 1 this k in the remaining summations, and then proceed to 1 k = , and so forth. Based 1 2 on such procedures, for given l k, , ,1 kl, the corresponding m given in (3.13) can then be obtained by induction and some straightforward manipulations.
The first term in (3.10) (indexed by m =1 ) is simply b , thus 0
Then it follows immediately that
Towards this end, we recall that, for a fixed l, thus totally l flipped bits, the indices
1 2 l
deduced that there are totally
2 2
. By induction, it can be concluded that there are totally l
1
1 1 With (A.5), the desired index m is thus
1
Appendix C
Proof of Theorem 3.3
By the Cauchy-Schwartz inequality, we have
Appendix D
Derivations of Equation (4.3)
Since hr =hr + =e éêëhr,1+e1, , hr L, +eLùúûT and ei (0, )se2 , "i, based on straightforward rearrangement (A.9) can be expressed as
( ) ( )
Appendix E
Proof of Proposition 4.1
We will first show that e b b eH m mH +2 Reéêëh b b eHr m mH ùúû+ b hHmr 2 =x A xT 1,m +aT1,mx
( )
where (e) holds since( )
m
hp m
Z is a real-valued matrix. By following similar procedures
as in the derivation of (A.12), the term 2 Re( )
Appendix F
Proof of Theorem 4.3
Recall that each summand of g g h q( , r, ) is expressed in (4.16). To prove the theorem, we first rewrite 1 2
1
where (f) follows from [32] and
( )
where (g) holds for the equality and some straightforward manipulation. With (A.16) and (A.17), (A.15) then admits the form
( )
where (h) holds based on equalities in (A.16) and (A.17), together with some basic manipulation. Hence, it suffices to derive 2 1 21
1 1 1 1
Appendix G
Proof of Lemma 4.4
Since x ( ,0se2I2L), it follows that
( ) ( )
1, 1, 1, 1, 1,
, , , ,
( ) 2 2
2 1, 1, 1, 1,
,
T T T
m m m m m
i
e L m m e m m
E d E d
tr s d s tr d
é + + ù = é ù+
ê ú ê ú
ë û ë û
= + = +
r r
e g h q x A x a x e g h q x A x
I A A
(A.23) where (i) holds due to [33, p-414]. Similarly, we have
Ee g h q, r, éêëx A xT 2,m +aT2,mx+d2,mùúû =se2tr
(
A2,m)
+d2,m. (A.24) The result follows from (A. 23) and (A.24). □Appendix H
Proof of Lemma 4.5
To prove (4.4), first we rewrite (4.3) as
( )
Appendix I
Proof of Theorem 4.6
By Cauchy-Schwartz inequality, we obtain
1 1 1
Appendix J
Extension of the First Study: Multi-bit SNR Quantization
J.1 Analyses
Assume that B bits (B ³1) are used at each relay for SNR quantization. Assume also that, at the ith relay, the B bits are BPSK modulated, and are then transmitted over a BSC with crossover probability pi, 1£ £i L. Denote by q the jth received bit i j,
in which the ith row consists of the received B-bit message of ith S-R link SNR. For a
given Q, let us collect all
, ,
which consists of the locations of all different entries between Q and Q. Based on (A.30) and (A.31), the conditional average SNR gdq( , , )g h Qr in the multiple-bit case
expressed as a single sum of Rayleigh quotients, which is shown as
and
0 L L i
i i
M C t
=
=
å
, where t is defined in (A.34).J.2 Simulation Results
For the two-bit case, i.e., B = 2, computer simulation is conducted to compare the BER performances of the proposed method and the solution in [10]. In the simulation setup, the number of relay nodes is L =4, the average SNR of the S-R link is set to be
s 20
g = dB, and the crossover probability pi of each BSC follows the uniform distribution over the interval [0.05, 0.1]. The simulated BER with respect to different R-D link SNR is then plotted in Figure A.1, shown on the next page. It can be seen from the figure that, as expected, the proposed method outperforms the solution in [18].
Figure 8. BER results for two methods when 2-bit SNR quantization is adopted at each relay.
0 5 10 15 20 25 30
10-7 10-6 10-5 10-4 10-3 10-2 10-1
average R-D SNR
BER
solution in [10]
solution in (3.17)
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Publication List
A. International Journal Papers
[1] Jwo-Yuh Wu, Chung-Hsuan Hu, Tsang-Yi Wang, and Sheng-Ho Tsai “Design of low-overhead cooperative beamforming under imperfect quantized SNR of source-to-relay links,” IEEE Trans. Signal Processing, revised, 2012.
[2] Jwo-Yuh Wu, Chung-Hsuan Hu, Tsang-Yi Wang, Wen-Hsuan Lee, “Low-overhead cooperative beamforming for information relaying in wireless sensor networks under mismatched inter-node link CSI,” IEEE Trans. Wireless Communications, submitted, 2012.
B. International Conference Papers
[3] Jwo-Yuh Wu, Chung-Hsuan Hu, and Tsang-Yi Wang, “Low-overhead cooperative beamforming under imperfect quantized SNR of source-to-relay links,” the Seventh IEEE Sensor Array and Multichannel Processing Workshop (SAM),
2012, Hoboken, USA.