Cognitive waveform design for spectral coexistence
3.4 Signal-dependent interference scenario
In fact, for eache, the energy transmitted in the stop-bands is lower than the allowed level, thus ensuring the coexistence with the other transmitting systems.
Significantly, for the considered simulation setup, the transmit signal ~c0 is capable of ensuring an even greater suppression of the interference at the stop- bands than the devised codes. Nevertheless, this behaviour is quite expected as the signal design technique of [39] only focuses on the coexistence problem. Otherwise stated, the figure of merit is represented by the interference reduction, and no additional constraints are forced neither on the shape of the sought waveform (whose autoambiguity properties are unpredictable) nor on the SINR at the receiver side. On the contrary, the aim of Algorithm 1 is to maximize the attainable SINR, providing at the same time a control over the total amount of interference produced at certain frequencies as well as on the resulting signal shape.
In addition, it can be observed that increasing the similarity parameter e, smarter and smarter distributions of the useful energy are achieved. Indeed, a progressive reduction of the radar emission in correspondence of the shared fre- quencies as well as an enhancement of the unlicensed jammer rejection capabilities is highlighted. As a result, higher and higher SINR values can be achieved. This is actually shown in Figure 3.4(c), where the SINR normalized tojaTj2 is plotted versuse, assumingEI ¼0:066. As expected, a proper choice of the design para- meters enables good interference rejection properties as well as high SINR values.
It is worth pointing out that, starting frome¼0:72, with reference to Algorithm 1, the maximum normalized SINR of the system is achieved.
In Figure 3.4(b), a performance analysis in terms of autocorrelation properties of the designed waveforms is provided. Better SINR values, spectral compatibility, and interference rejection are traded-off for worse and worse range resolutions and/
or ISLs/PSLs. It can also be observed that the waveform devised through [39]
exhibits worse range-sidelobe profiles than those associated with Algorithm 1, reflecting the fact that the algorithm in [39] does not directly control the auto- ambiguity properties of the sought waveform. Nevertheless, the smoother behaviour of the signals synthesized according to Algorithm 1 agrees with the design criterion P. In fact, the optimization problem itself involves a compromise between the desire of lowering the transmitted energy in the stop-bands as well as in corre- spondence of the jammer central frequencies, and the need of keeping under control the ambiguity features of the sought signals.
whereJk,k¼ 1;. . .;ðN1Þ, denotes the shift matrices1[49],f gbk k6¼0 repre- sents the mean scatterer powers of the adjacent range cells andM is the signal- independent contribution as in Section 3.3. Notice that the prediction of the actual scattering scenario can be accomplished by means of the cognitive framework, i.e., exploiting a dynamic environmental database including a geographical information system, previous scans, meteorological data, some theoretical (or possibly empirical) e.m. reflectivity and spectral clutter models [56,57].
In Appendix 3.6.3, it is shown that in the presence of signal-dependent inter- ference, Problem P boils down to a non-convex quadratically constrained fractional quartic problem that is, in general, very hard to solve [58,59]. Hence, an alternating optimization procedure for the transmit signal and the receive filter design that monotonically improves the SINR is devised. Each iteration of the algorithm, whose convergence is analytically proved, requires the solution of both a convex and a hidden convex optimization problem. In detail, exploiting the Charnes–Cooper [60] transfor- mation and a specific rank-one matrix decomposition procedure, it is shown that the optimal code for a fixed filter can be obtained based on the optimal solution to a specific SDP problem. The resulting computational complexity of the devised algo- rithm is linear with the number of iterations and polynomial with receive filter length.
Figure 3.5 depicts the block diagram of the optimization procedure adopted to handle Problem P when the radar system operates in signal-dependent interference.
Let us now assess the performance of the waveform design technique described in Appendix 3.6.3, i.e., Algorithm 3. The same simulation setup as in Section 3.3 is considered, but with K ¼2 licensed radiating systems (W1¼½0:05;0:08 and W2¼½0:4;0:435), a duration of 148 ms for the reference chirp waveform (resulting in N¼120 samples) and sJ;1¼25 dB,sJ;2¼30 dB.
Besides, as to the signal-dependent interference, a uniform clutter environment with bk¼8 dB, k¼ 1;. . .;ðN1Þ is assumed and a signal-to-noise power ratio jaTj2=js0j ¼10 dB is supposed.
In Figure 3.6(a), the SINR behaviour versus the number of iterations is provided, for the operative pointsðEI;eiÞ,i¼1;2;3, withEI ¼0:0017,e1 ¼0:1,e2 ¼0:15 and2e3¼0:3. As expected, increasinge, the optimized SINR value improves as the
1Jkði;mÞ ¼1 ifim¼kandJkði;mÞ ¼0 otherwise,ðl;mÞ 2 f1;. . .;Ng2,k¼ 1;. . .;ðN1Þ.
2As starting sequencecð0Þ to the iterative procedure, the one corresponding to the boundary point
EI¼0:0017,e0¼0:089 is considered.
Radar transmitter optimization fixed the receiver
Radar receiver optimization fixed the transmitter
c(n) w(n)
Figure 3.5 Block scheme of the considered transmit-receive optimization procedure in the presence of signal-dependent interference
feasible set of the optimization problem becomes larger and larger (with performance gains up to approximately 1.6 dB). In Figure 3.6(b), with reference to the same operative points of Figure 3.6(a), the ESD of the synthesized signals versus the normalized frequency is reported, together with that of the reference codec0. The stop-bands in which the licensed systems are located are shaded in light grey.
(b) –50 –40 –30 –20 –10 10
0
ESD [dB]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized frequency
(Ec(0)I, ε) = (0.0017, 0.3) c0
(EI, ε) = (0.0017, 0.15) (EI, ε) = (0.0017, 0.1) 6.6
6.8 7 7.2 7.4 7.6 7.8 8 8.2
(a)
SINR [dB]
5 10 15 20 25 30 35 40 45 50
#Iteration (EI, ε) = (0.0017, 0.3) (EI, ε) = (0.0017, 0.15) (EI, ε) = (0.0017, 0.1)
Figure 3.6 (a) SINR; (b) ESD (stop-bands shaded in light grey); brown curve:
reference codec0; blue curve: starting sequencec(0); red curve:
Algorithm 3, EI ¼0:0017,e¼0:1; magenta curve: Algorithm 3, EI ¼0:0017,e¼0:15; black curve: Algorithm 3, EI ¼0:0017,e¼0:3
The curves show that Algorithm 3 is capable to suitably control the amount of energy produced over the shared frequency bands. In addition, increasing the similarity parametere, namely increasing the available degrees of freedom, smarter and smarter distributions of the useful energy are achieved. In fact, as in Figure 3.4(a), both a lower and lower radar radiated energy in correspondence of the shared frequencies and improved jammer rejection capabilities are experienced.
Finally, in Table 3.1, the ISL and the PSL for the cross-correlation functions (CCFs) of the radar codes and receive filters, corresponding to the operative point ðEI;e3Þ ¼ ð0:017;0:3Þ, for different values of the iteration number (n¼1;10;30;50) are provided. The values in the table reflect the capability of the considered joint transmit-receive optimization procedure to iteratively achieve better and better signal-dependent disturbance suppression levels.