Feasibility of P 3 - Cognitive waveform design for spectral coexistence

Cognitive waveform design for spectral coexistence

3.6 Appendix

3.6.2.1 Feasibility of P 3

Problem P3 is strictly feasible due to the strict feasibility of P. In fact, let us assume that cs is a strictly feasible solution of P, i.e., c^yscs¼1, c^ysRIcs<EI

and jc^ysc0j² ðReðc^ysc0ÞÞ²>de. Then there are u1;u2;. . .;uN1 such that

5In fact, takinga1¼a3¼a4¼0 anda2¼1, thena1Rþa2Iþa3RIþa4C0:¼I 0.

U¼½cs;u1;u2;. . .;uN1is a unitary matrix [67], and for any 0<h<1 the matrix Cs¼ ð1hÞUe1e^y₁U^yþ h

N1U I e1e^y₁

U^y 0 (3.47)

withe1¼ ½1;0;. . .;0^T is a strictly feasible solution of the SDP problem EQPR.

In fact,

trðCsÞ ¼1 (3.48)

trðCsRIÞ ¼ ð1hÞc^ysRIcsþ h N1

n¼1

u^ynRIun (3.49)

trðCsC0Þ ¼ ð1hÞc^ysC0csþ h N1

n¼1u^ynC0un (3.50)

which highlight that ifhis suitable chosen, thenCs is a strictly feasible solution, i.e., trðCsÞ ¼1; trðCsR1Þ>E_Iand trðCsC0Þ>de.

3.6.3 Waveform design algorithm for signal-dependent scenario In this section, it is described the polynomial time procedure employed to obtain an optimized solution to the non-convex optimization problem P in the presence of signal-dependent interference. In this case, the design problem is given by

P₁ maxc;w

jaTj² w^yc² w^yM cð Þw s:t: kck²¼1

c^yRIcE_I kcc0k²e 8>

(3.51)

Notice that problemP1 is a non-convex optimization problem, as the objective function is a non-convex function and the constraintkck² ¼1 defines a non-convex set. Therefore, following the guidelines in [57], the goal is to derive optimized solutions toP1 via a alternating maximization procedure. The idea is to iteratively improve the SINR, controlling, at the same time, the total amount of energy injected in the licensed bandwidth, as well as radar waveform features. Specifically, given w^ðn1Þ, an admissible radar codec^ðnÞat stepnimproving the SINR corresponding to the receive filterw^ðn1Þand the transmitted signalc^ðn1Þis searched. Wheneverc^ðnÞ is found, the filterw^ðnÞthat improves the SINR corresponding to the radar codec^ðnÞ and the receive filterw^ðn1Þis searched, and so on. Otherwise stated,w^ðnÞis used as starting point at stepnþ1. To trigger the procedure, the optimal receive filterw^ð0Þ, for an admissible code c^ð0Þ, is considered. Notice that the proposed optimization procedure requires a condition to stop the iterations; to this end, an iteration gain constraint can be forced, namelyjSINR^ðnÞSINR^ðn1Þj z; where z0 is the desired precision.

From an analytical point of view,w^ðnÞ can be computed solving the optimization problem

P^ðnÞ_w max

jaTj²w^yc^ðnÞ² w^yMðc^ðnÞÞw (

(3.52) whose optimal solution, for any fixedc^ðnÞ, is given by

w^ðⁿ^Þ¼ Mðc^ðnÞÞ¹c^ðnÞ

c^ðnÞyMðc^ðnÞÞ¹c^ðnÞ (3.53)

On the other hand,c^ðnÞ is an optimal solution to the following non-convex optimization problem

P^ð_cⁿ^Þ maxc

jaTj²jw^ðⁿ¹^Þycj² w^ðn1ÞyMðcÞw^ðn1Þ s:t: kck²¼1

c^yRIcEI

kcc0k²e 8>

(3.54)

It is possible to show that problem P^ðnÞ_c is a hidden-convex optimization problem. Precisely, its optimal solution can be computed in polynomial time (resorting to the rank-one matrix decomposition theorem [55, Theorem 2.3]), starting from an optimal solution to the SDP problem

maxC;t trðQCÞ s:t: trM1ðw^ðⁿ^ÞÞC

¼1 trð Þ ¼C t

trðRICÞ tEI

trðC0SÞ tde

C≽0 t0 8>

(3.55)

witht an auxiliary variable,C0 ¼c0c^y₀,C2H^N,Q¼w^ðn1Þw^ðn1Þy,M1ðw^ðnÞÞ ¼ PN1

k¼Nþ1;k6¼0 bk J^T_k w^ðn1Þ w^ðn1Þy Jk þ w^ðn1ÞyM w ^ðn1ÞI and de ¼ ð1e=2Þ². Algorithms 2 describes the procedure leading to an optimal solution P^ðnÞ_c .

Algorithm 2:Algorithm for Radar Code Optimization Input:M1ðw^ðnÞÞ;Q;RI;c0;de;E_I.

Output:An optimal solutionc^ðⁿ^Þ toP^ðnÞ_c .

1: solve SDPP₂finding an optimal solutionðC;tÞand the optimal valuev; 2: letC:¼C=t;

3: ifrankðCÞ ¼1then

4: perform an eigen-decompositionC¼cðcÞ^yand getc 5: else

6: apply the rank-one decomposition theorem [55, Theorem 2.3] to the set of matricesðC;QvM1ðw^ðⁿ^ÞÞ;c0c^y0;I;RIÞand getc;

7: end

8: outputc^ðnÞ:¼ce^jargðc^y^c⁰^Þand terminate.

Finally, Algorithm 3 summarizes the devised alternating optimization procedure. To trigger the recursion, an initial radar codec^ð⁰^Þ from which obtaining the optimal filterw^ð⁰^Þ is required; a natural choice isc^ð⁰^Þ¼c0.

Algorithm 3:Algorithm for Transmit-Receive System Design in the Presence of Signal-Dependent Interference

Input:f gbk ,M,RI,E_I,e,c0. Output:A solutionðc;wÞto P.

1: setn¼0,c^ðⁿ^Þ¼c0, w^ðnÞ¼ Mðc^ðⁿ^ÞÞ¹c^ðⁿ^Þ

c^ðⁿ^ÞyMðc^ðⁿ^ÞÞ¹c^ðⁿ^Þ;

and the value of the SINR for the pairðc^ðnÞ;w^ðnÞÞ; 2: do

3: n¼nþ1;

4: construct the matrixM1ðw^ðnÞÞ;

5: solve problem P^ð_cⁿ^Þfinding an optimal radar codec^ðnÞ, through the use of Algorithm 2;

6: construct the matrixMðc^ðnÞÞ;

7: solve problem P^ð_wⁿ^Þfinding an optimal receive filter w^ðⁿ^Þ¼ Mðc^ðnÞÞ¹c^ðnÞ

c^ðnÞyMðc^ðnÞÞ¹c^ðnÞ;

and the value of the SINR for the pairc^ðnÞ;w^ðnÞ

; 8: let SINR⁽ⁿ⁾¼SINR;

9: untiljSINR^ðnÞSINR^ðn1Þj z 10: outputc¼c^ðnÞandw¼w^ðnÞ.

As to the computational complexity of Algorithm 3, it is linear with respect to the number of iterationsN, whereas in each iteration, it includes the computation of the inverse ofScc^ðⁿ^Þ

þRindand the complexity effort of Algorithm 2. The former is in the order of OðN³Þ. The latter is connected with the complexity of SDP solution, i.e.,OðN^4:5Þ.

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Dalam dokumen Novel Radar Techniques and Applications, Volume 2.pdf (Halaman 135-144)