Waveform selection and adaptation

Array radar resource management Alexander Charlish* and Fotios Katsilieris*

3.2 Task management

3.2.3.3 Waveform selection and adaptation

model in the filter is matched to the current target manoeuvre. Consequently, the revisit interval time decreases as the target manoeuvres and increases as the target follows predictable motion. From the benchmark problems, the combination of IMM and multiple hypothesis tracking [9, Section 14.8] emerged as the tracking methodology that resulted in lowest radar loading for each target track [37,38].

likelihood probability density functions are assumed to be (or approximated as) Gaussian, denotedpðxkjZ^kÞ Nðxk; ^x_k;P_kjkÞandpðzkjxkÞ Nðzk;H_kx_k;R_kðqkÞÞ respectively, whereH_k is the measurement matrix that relates states to measure- ments, andR_kðqkÞis the measurement error covariance when waveformq_kis used.

The accuracy of the delay and Doppler measurement z_k¼ft;vg can be derived from the narrowband ambiguity function¹:

fðt;vÞ ¼ ð₁

1s_qk tþt 2

s_qk tt 2

expðj2pvtÞdt (3.20) wheretis the time delay, vis the Doppler shift,sqk is the complex envelope of waveformq_kand * denotes the complex conjugate.

The Fisher information matrix describes the information that the observations z_k carry on the state of interestx_k. The Fisher information matrix is related to the ambiguity function [45, Section 10.2]:

J¼c

@²fðt;vÞ

@t²

^t¼t_v¼v⁰₀^; @²fðt;vÞ

@t@v ^t¼t_v¼v⁰₀^;

@²fðt;vÞ

@t@v ^t¼t_v¼v^0;

0

@²fðt;vÞ

@v^{2 t¼t}_v¼v^0;

0

2 66 66 66 4

3 77 77 77 5

(3.21)

wherecis a normalization constant, see [45, Section 10.2] or [43],t₀is the target time delay andv0 is the target Doppler shift.

This Fisher information matrix determines the Crame´r–Rao Lower Bound (CRLB) on the estimation oftandvusing the waveform with complex envelope sqk. It can be argued that for measurements with high SNR, the CRLB can serve as an approximation to the measurement error covarianceR_kðqkÞ.

Given a waveform library Qcomprising multiple waveforms, the best wave- form to schedule for the next time step can be based on track performance criteria.

As it is desired to minimize the error in the track state estimate, a logical choice is to select a waveform that minimizes the trace or determinant of the expected error covarianceP_kjkafter a measurement has been generated with the candidate wave- form at the next time step [46]:

q_k ¼arg min

qk2Q tr P_kjkðq_kÞ

(3.22) This criterion is used for scheduling waveforms in [47,48], where it is shown scheduling waveforms in this manner can significantly reduce tracking errors.

Waveforms can also be selected based on information theoretic criteria [49], motivated by the assumption that the radar must maximize information production in order to minimize the uncertainty in the surveillance picture. The mutual

1Equation (3.20) is denoted as ‘time-frequency autocorrelation function’ in [45, Sec. 10.1] but as

‘ambiguity function’ in [43,46].

information between the expected measurement generated using a candidate waveform and the target state can be used for waveform selection. The mutual information Iðxk;z_kÞ, is the reduction in entropy of x_k due to the expected mea- surementz_kðqkÞgenerated using candidate waveformq_k:

Iðxk;z_kðqkÞÞ ¼HðxkÞ HðxkjzkðqkÞÞ (3.23) where HðÞis the entropy of a random variable. Assuming that the target state is represented by a Gaussian, as with a Kalman filter and its variants, the mutual information is:

Iðxk;z_kðqkÞÞ ¼1

2 ln jIþP_kjk1H^T_kR¹_k ðqkÞHkj

(3.24) where I is an identity matrix of proper dimensions. Consequently, waveform selection can be performed based on mutual information according to:

q_k¼arg max

qk2Q½Iðxk;z_kðqkÞÞ (3.25)

Alternative information theoretic criteria have also been proposed, such as the alpha-divergence between the prior and posterior estimates [50] and the Kullback–

Leibler divergence, which is a single case of the alpha-divergence. In addition, it has been shown that using the conditional entropy, mutual information or Kullback–Leibler divergence for sensor management purposes has the same sensor selection results, for more details see the discussion [51].

Information theoretic criteria are valuable for waveform selection as they act as a surrogate function, in that regardless of higher level objectives, it is always desirable to maximize the information content of the measurement with respect to already acquired information in the tracker. However, care should be taken [52]

when using information theoretic criteria for radar management problems. The assumption that the radar wants to maximize information may not always be valid, or at least, the value of different information sources may vary significantly, as also shown in [53].

Ideally, a waveform library should be small but well designed [54]. A small but effective library is motivated due to the possible computation explosion of online waveform design, especially when waveforms are scheduled non-myopically, that is, considering possible waveform sequences extending over a time horizon into the future. It has been shown that including non-linear frequency modulation (FM) waveforms [46,55] or fractional Fourier transformed waveforms [47,54] in the library can enhance tracking performance, especially in clutter.

In addition to selecting or adapting the intra-pulse modulation to optimize tracking performance, the PRF can be selected based on the target. This can be performed using a number of rules. First, a PRF should be selected such that the target is not eclipsed or in a clutter notch, based on the estimated target kine- matic parameters. Also, as track information is already available, active track updates can be performed using an ambiguous PRF set, which reduces the total number of PRFs used in the dwell [56–58].