Leonardo Cardoso - Vandermonde Frequency Division Multiplexing for Cognitive Radio Networks

Transcript

1. Vandermonde Frequency Division Multiplexing for Cognitive Radio Networks Leonardo S. Cardoso leonardo.cardoso@supelec.fr Alcatel-Lucent - Sup´ elec Chair in Flexibe Radio L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 1 / 26
2. Outline 1 Scenario 2 Related Work 3 System Model 4 VFDM 5 VFDM’s Performance 6 Further Work L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 2 / 26
3. Scenario L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 3 / 26
4. Scenario Spectrum Sharing Problem Primary Network Secondary Network 2 networks sharing the same band (legacy and cognitive). L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 4 / 26
5. Scenario Spectrum Sharing Problem Primary Network Secondary Network 2 networks sharing the same band (legacy and cognitive). L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 5 / 26
6. Scenario Spectrum Sharing Problem Primary Network Secondary Network h(11) h(12) h(21) h(22) 2 networks sharing the same band (legacy and cognitive). L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 6 / 26
7. Scenario Assumptions Primary system does not need to be aware of the existence of a secondary one; Secondary base station does not know the primary’s message; Primary terminal and base station know perfectly h(11); Secondary terminal and base station know their local channels (h(21), h(22) and h(12)); Primary system employs OFDM transmission; Channels are frequency selective. L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 7 / 26
8. Scenario Our Contribution: VFDM (Vandermonde Frequency Division Multiplexing): A linear Vandermonde-based precoder that generates zero interference on the primary network by exploiting the frequency selective nature of the channel. L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 8 / 26
9. Related Work L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 9 / 26
10. Related Work On Vandermonde: Scaglione et. al. [1] ◮ Main differences: ⋆ Use of Vandermonde filter together with a Lagrange spreading code; ⋆ Interference cancelation exploits the orthogonality between the spreading code and the filter; ⋆ The channel realization is not taken into account. On cognitive radio: [2, 3, 4, 5] ◮ works available are based on unrealistic assumptions: ⋆ the secondary transmitter has some knowledge of the primary’s message; ⋆ both transmitters know all the channels perfectly. L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 10 / 26
11. System Model L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 11 / 26
12. System Model Received signals at RX1 and RX2, respectivelly: y1 = F T (h(11))x1 + T (h(21))x2 + n1 y2 = F T (h(22))x2 + T (h(12))x1 + n2 (1) ◮ T (h(ij)) is a N ×(N +L) Toeplitz with vector h(ij) ∼ NC (0, σij /(L+1)); ◮ nk ∼ NC (0, IN ) is an AWGN noise; ◮ TX1 signal is precoded with A; ◮ TX1 signal: x1 = AFHs1 ◮ TX2 signal: x2 = Vs2 ◮ Power restriction: tr(E[xkxH k ]) ≤ (N + L)Pk L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 12 / 26
13. VFDM L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 13 / 26
14. VFDM Our objective is to find a V such that: T (h(21))Vs2 = 0, ∀s2. (2) One solution to (2) is given by the Vandermonde matrix: V =        1 · · · 1 a1 · · · aL a2 1 · · · a2 L . . . ... . . . aN+L−1 1 · · · aN+L−1 L        (3) where {al , . . . , aL } are the roots of S(21)(z) = L i=0 h(21) i zL−i . Eq. (1) can be rewritten as: y1 = H(11) diag s1 + ν1 (4) y2 = H2s2 + H(12) diag s1 + ν2 (5) L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 14 / 26
15. VFDM VFDM Precoder Conditioning −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 1 (a) Roots of the polynomial of h(ij) −15 −10 −5 0 5 10 15 20 25 30 −20 −15 −10 −5 0 5 10 (b) Zoom out of the roots Roots of a Gaussian polynomial tend to fall on the unit circle [6]; Some roots fall outside and deteriorate the conditioning of V; L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 15 / 26
16. VFDM VFDM Precoder Conditioning: Solution Find a new precoder, that possess better conditioning, but conserving the same properties of V. Ideas: ◮ Force an orthogonalization of V using Gram-Schmidt process: E = gs(V) ◮ Generate a new precoder matrix based on the SVD of the channel [?]: Let T (h(21)) = UΛDH , then E = dN | · · · | d(N+L)−1 | dN+L , where D = [ d1 | d2 | · · · | dN+L ]. L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 16 / 26
17. VFDM’s Performance L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 17 / 26
18. Rate Optimization The primary network’s capacity is maximized by the classical water filling (WF) algorithm, due to the OFDM modulation and N parallel channels; We consider that the interfering signals from the primary network are seen as noise on the secondary network. Eq. (5) becomes: y2 = H2s2 + η where η is NC(0, Sη), S1 = E[s1sH 1 ] and Sη = H(12) diag S1H(12) diag H + IN L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 18 / 26
19. Case 1: Equal power allocation The secondary user’s rate is given by: Req 2 = 1 N log IN + (N + L)P2 tr(VH V) S−1/2 η H2HH 2 S−H/2 η (6) The number of degrees of freedom in V is sensitive to |al |; Poor performance is expected. L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 19 / 26
20. Case 2: Optimal power allocation Requires perfect knowledge of the interference plus noise covariance Sη at the secondary base station; The optimization problem is: maximize 1 N log IN + S−1/2 η H2S2HH 2 S−H/2 η subject to tr(VHVS2) ≤ (N + L)P2 (7) After manipulation it becomes a classical WF. L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 20 / 26
21. Numerical Examples: Secondary Rate vs. SNR Vandermonde only case 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 SNR [dB] Rate 2 [bps/Hz] waterfilling+greedy σ 12 =0 equal power N=64 L=16 σ 12 =0.01 σ 12 =0.1 σ 12 =1 VFDM offers significant gains; L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 21 / 26
22. Numerical Examples: Secondary Rate vs. SNR Orthogonalized Vandermonde case 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 VFDM performance N = 64 L = 16 SNR [dB] Rate [bps] R1 (WF) R2 unnormalized Vandermonde (WF), cross = 0 R2 Gram−Schmidt V (WF), cross = 0 R2 SVD (WF), cross = 0 Orthogonalizing Vandermonde offers further gains; Gram-Schmidt and SVD schemes offer the same performance. L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 22 / 26
23. Numerical examples: Imposing target rate to primary user 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Target rate R1 * [bps/Hz] R1 SNR=10dB SNR=5dB N=64 L=16 SNR=15dB R2 with waterfilling R2 with equal power R2 R1 , R2 [bps/Hz] Example for a band of 20Mhz (802.11a): For a target of 54Mbps at primary, secondary has 2.83Mbps; For a target of 36Mbps at primary, secondary has 7.60Mbps. L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 23 / 26
24. Further Work L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 24 / 26
25. Further Work What is the impact of non-ideal channel estimations? ◮ Up to what extent can the interference generated by the secondary system be limited? ◮ What will be the impact on the secondary network’s rate? Extension to multi-user scenarios: ◮ Target scenarios; ◮ Coordination among secondary base stations; ◮ Systemic aspects. Feasibility: ◮ Is VFDM feasible? ◮ Is it worth implementing? ◮ How could it be constructed? L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 25 / 26
26. Bibliography A. Scaglione, GB Giannakis, and S. Barbarossa. Lagrange/Vandermonde MUI eliminating user codes forquasi-synchronous CDMA in unknown multipath. IEEE Trans. on Signal Process., 48(7):2057–2073, 2000. N. Devroye, P. Mitran, and V. Tarokh. Achievable rates in cognitive radio channels. IEEE Trans. on Inform. Theory, 52(5):1813–1827, 2006. A. Jovicic and P. Viswanath. Cognitive radio: An information-theoretic perspective. cs/0604107, April 2006. I. Maric, RD Yates, and G. Kramer. Capacity of Interference Channels With Partial Transmitter Cooperation. Information Theory, IEEE Transactions on, 53(10):3536–3548, 2007. I. Maric, A. Goldsmith, G. Kramer, and S. Shamai. On the capacity of interference channels with partially-cognitive transmitter. In ISIT’2007, 2007. I. A. Ibragimov and O. Zeitouni. On roots of random polynomials. Trans. American Math. Soc. 349, 2427-2441, 1997. L.S. Cardoso () VFDM for Mobile Flexible Networks A-L - Sup´ elec Chair 26 / 26