Cognitive waveform design for spectral coexistence
3.3 Signal-independent interference scenario
the overlaid telecommunication networks is emphasized, with respect to choosing EI;e
ð Þequal to the point B. In the last case, other radar features, such as low PSL and/or ISL, are privileged.
The interested reader may refer to Appendix 3.6.1 for all the analytical details concerning the study of the I/S achievable region.
shape and autocorrelation features. Hereafter, a radar whose baseband equivalent transmitted signal has a two-sided bandwidth of 810 kHz and a Nyquist sampling frequency are considered. In addition, the interference is composed of unlicensed narrowband continuous jammers, white interference and licensed coexisting tele- communication networks spectrally overlaid to the radar of interest. Specifically, the disturbance covariance matrix is modelled as:
M ¼s0IþXK
k¼1
sI;k
DfkRkI þXKJ
k¼1
sJ;kRJ;k (3.11)
where s0¼0 dB is the thermal noise level; K ¼7 is the number of licensed radiating systems; sI;k accounts for the energy of the kth coexisting tele- communication network operating on the normalized frequency band ½f2k;f1k (sI;k¼10 dB,k¼1;. . .;K); Dfk ¼f2kf1k is the bandwidth associated with the kth licensed system, for k¼1;. . .;K; KJ ¼2 is the number of active and unli- censed narrowband jammers; sJ;k,k¼1;. . .;KJ, accounts for the energy of the kth active jammer (sJ;1 dB¼50 dB, sJ;2 dB¼40 dB); RJ;k,k¼1;. . .;KJ, is the normalized disturbance covariance matrix of the kth active unlicensed jammer, defined asRJ;k¼rJ;kryJ;k;withrJ;kðnÞ ¼ej2pfJ;kn=fs, wherefJ;kdenotes the Doppler shift of thekth jammer (fJ;1=fs¼0:7 and fJ;2=fs¼0:75).
Concerning the overlaid licensed radiating systems, the baseband equivalent radar stop-bands given in (3.10) are considered. Furthermore, all the licensed/foreseen overlaid systems are assumed with the same relevance, namely, wk¼1 for k¼1;. . .;7. Based on the assumed stop-bands (3.10) and weightswk’s, the inter- ference energy constraint on the transmitted radar waveform can be now enforced.
Notice that, the matrixRI does not depend on the frequencies of the unlicensed jam- mers and is the only function of the spectral bands (3.10) and weights associated to the licensed networks. Finally, the LFM pulse in Section 3.2.3 is used as reference waveformc0; it results inN ¼162 samples due to the considered sampling frequency.
The I/S achievable region for the considered scenario is represented in Figure 3.2.
For comparison purpose also, the transmit sequence~c0and the receive filter~g0
devised according to the procedure in [39] are considered. Specifically, with reference to thesoft-power constrainttransmit waveform design technique [39], the code c0 is used as starting sequence, and d¼0:9, lT ¼105 and Rð2Þ¼RI. Moreover,b¼0:05,lT ¼105,Dn¼10 andRð2Þ¼M are used for the receive filter design. The interested reader may refer to [39] for additional details about Lindenfeld’s algorithms, the aforementioned parameters and their setting [39].
In Figure 3.4, the ESD, the squared modulus of the ACF, and the normalized SINR of the waveforms devised for three operative points (red dot, black star and cyan hexagram in Figure 3.2) are reported. Specifically, the points ðEI;eiÞ,i¼1;2;3, withEI ¼0:066,e1¼0:264,e2 ¼0:444,e3 ¼0:654 are con- sidered, whereðEI;e1Þcorresponds to the pointAin Figure 3.2, and the results are compared with those achievable by the signal/receiver pairð~c0;~g0Þ. In Figure 3.4(a), the ESD of the synthesized signals versus the normalized frequency, together with that of the reference code is shown. The stop-bands in which the licensed systems are located are shaded in light grey. The curves highlight the capability of Algorithm 1 to suitably control the amount of energy produced over the shared frequency bands.
−60
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−40
−30
−20
−10 0 10
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0 1
Normalized frequency
ESD [dB]
c0
(EI, ε) = (0.066, 0.654) (EI, ε) = (0.066, 0.444) (EI, ε) = (0.066, 0.264)
−150 −100 −50 0 50 100 150
−80
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−50
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−20
−10 0
Delay bin
Squared Modulus of the ACF [dB]
(c)
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 ε
SINR [dB] |αT|2
(b) (a)
c0
(EI, ε) = (0.066, 0.654) (EI, ε) = (0.066, 0.444) (EI, ε) = (0.066, 0.264)
Algorithm 1
c0 (Algorithm [39])
c0 (Algorithm [39])
c0 (Algorithm [39])
˜
˜
˜
Figure 3.4 (a) ESD; (b) Squared modulus of the ACF. Green curve: reference codec0; blue curve: Algorithm 1, EI ¼0:066,e¼0:264; magenta curve: Algorithm 1, EI ¼0:066,e¼0:444; black curve: Algorithm 1, EI ¼0:066,e¼0:654; cyan curve: Algorithm [39]; (c) blue curve:
Algorithm 1, Normalized SINR vse, EI ¼0:066; cyan curve:~c0, Algorithm [39]
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.