CODED WAVEFORM DESIGN FOR RADAR

Abstract

Good signals for radar can be designed as coded waveforms more easily. Let the sequence which represents this coded waveform be a-(aQ, a^ ..<, a^ ), Then the aperiodic autocorrelation of this sequence is defined as N-l~k r(k) .Z^ a. a.+k, k»0,1,..., N-l For good range resolution, r(0) should be large and r(k ^0) should be ideally zero. Such sequences would also be useful in seismic exploration. This ideal behaviour is not easy to achieve. Therefore one looks for sequences for which the merit factor o N-l 0 F = r^(o) / 2 E rzO) j-l or the discrimination D - r(p)/max |r(k)j k>£0 are large. If {ai} is chosen from aDinary alphabet [+ lj the value of Fcannot exceed 2e2, Golay [59] and sequ~lces with Fgreater than 12.32... are not known at large lengths, Golay [60]. The optimal value of F for agiven length is often poorer than the bound above, Lindner [i23j, Golay [60]. Therefore in this thesis ternary sequences with [o, +i] as alphabet have been considered, and values of Fas high as ii 20 are obtained. Moreover the merit factors obtained are much better than those achievable by binary sequences of corresponding length. This proves the advantage derived by using ternary sequences. However ternary sequences have a disadvantage in comparison with the binary sequences in that the energy efficiency E . r(o)/N | a I2 N I ' max1 is suboptimal ( less than 1 ) for the former. To include this disadvantage into the comparison so that it is very fair a new quality factor Q » FxE, is defined. Even on the basis of Q ternary sequences do better than binary sequences. The ternary sequences referred to above were obtained using the concepts of terminal admissibility, Moharir 04^148], and odd shift orthonormality or skew-symmetry, Golay [56,59] conjunctively and together with some other concepts. Thus the search is restricted to some majority-optimal sequences only, majority-optimal sequence being one for which r(k) is ideal for majority of the k values, Moharir [148J. The notion of complementary or cooperative sequences is also an important one in the field of radar, a pair of sequences is said to be cooperative if their autocorrelations rxM and r2(k) add to r(k) which is zero for kfi 0, Golay [55] and Barker cooperative if r(k) = h- 1 for k^ 0. In this thesis this concept has been extended to achieve double cooperation. That is, in addition to the above, the sequences have the iii property that where one sequence has nonzero elements, the other has zeroes and vice versa. The cooperative sequences have to be transmitted as two different sequences of equal length by frequency modulation thus effectively reducing the energy efficiency to 0.5. Moreover as the channel adds frequency - dependent distortion the individual autocorrela tions cannot be coherently added. In case of doubly cooperative Pair of ternary sequences, only one sequence is to be trans mitted at a fixed carrier frequency with phase modulation. This resultant sequence has an ideal energy efficiency and does not suffer due to frequency dependent distortion added by the channel, The return signal is to be correlated with the individual ternary sequences and the results added to get the advantage of cooperation. The cooperative merit factors (merit factors for r(k) =^(10+ r2(k)) obtained by this procedure are very good and keep pace with the length of the sequence. Noting that the scheme requires transmitting only one four phase sequence, these values of Fcould be legitimately compared with the best possible values of F for binary sequences, Golay [59], to appreciate the advantage the proposed scheme has achieved at the cost of slightly more complex processing of the return signal. We have discovered that there could exist Barker sequences with Barker segments in them. Such sequences are called Barker towers. Each of the Barker segments is separately correlated

with the received waveform. The delay at which all the cross-correlations show a peak is taken as the two - way travel time to the target. This notion of coincidence detection would reduce the effects of additive noise and the probability of error. For simultaneous range and Doppler resolution the requirements are not merely on the autocorrelation but on the ambiguity surface of the transmitted signal a(t). The ambiguity surface is defined as ? °° -jwt 2 I© ( T," )| - |/ a( t-T/2 ) a* (t+r/2)e ' dt| and should be as impulse - like as possible. The design procedures available in the literature, Sussman |216], Wolf, Lee and Suyo, [245] to design coded waveforms having good ambiguity surfaces lead to sequences with poor energy effi ciency. Therefore the optimum ambiguity surface should be sought subject to the constraint that the energy efficiency should not fall below a prescribed threshold. This problem has been solved in the present thesis with the help of interior point penalty function approach, Fiacco and McCormick [48]. Alter natively a convex combination of the volume under the ambiguity surface excluding a small region around the origin and the negative of energy efficiency has been minimized. For this purpose sequences of length N normalized to have unit energy are considered. Trigonometric notation has been used so that the sequence is described in terms of N-l angles. There are many V such descriptions, their number grows very rapidly as N increases. Notions from graph theory have been borrowed to work out the optimum description in terms of N-l angles so that the expressions for energy efficiency and its derivatives with respect to these angles have manageable complexity. As the energy efficiency is given increasing weightage, the reduction in the ambiguity volume away from the origin decreases, Thus the contribution of this thesis is to invoke energy effi ciency as an important design criterion and to suggest ways in which it can be included in the design procedure. The ambiguity surface has large number of local maxima. There was no efficient procedure available till recently to find out the largest of these, excluding the one at the origin. Therefore the notion of discrimination introduced in the con text of the aperiodic autocorrelation r(k) could not be rea listically extended to define ambiguity discrimination. But now an algorithm called tunnelling algorithm, Levy et al [119], to obtain global maximum has become available. Using one of the tricks in that algorithm the maximum at the origin can be suppressed and the global maximum of the modified ambiguity surface can be obtained, which will correspond to the largest sidelobe in the ambiguity surface. The ratio of the peak at the origin and the largest sidelobe will give us the ambiguity discrimination. Then the sequences with large ambiguity dis crimination can be searched for by recursive techniques.

Instead of aperiodic autocorrelation and ambiguity surface, their periodic (or cyclic) counterparts can be defined. It was known, Moharir [149] that the periodic autocorrelation cf the Chinese product of sequences is a Chinese product of the individual periodic autocorrelations. It has now been proved in this thesis that the cyclic ambi guity function of the Chinese product cf sequences is the Chinese product of their individual cyclic ambiguity func tions, suitably permuted along the Doppler axis. With this theorem the possibilities of designing large sequences with good cyclic ambiguity properties in terms of smaller sequences with such properties open up. Energy efficiency properties of a class of cyclic sequences designed on the basis of autoregressive moving average models, Rutter and Grant [179], are systematically studied in this thesis to suggest tnat energy efficiency is an important design consideration. The notion of autocorrelation r(k) has been generalized to that of monogenic signature M(k). A function f(aQ, a-^ a2,...) is said to be monogenic if it is expressible as f(ao, all a2,...) = S g± (aQ, a2, a£,. .. ) where gi+l^ao'al'a2",#^ = gi(a1,a2,a3f..,) where g0(ao,a1,a2,...) is called the gene.

It is easily seen that r(k) . aQ ak+ ^ a^ +a2 ak+2+ ... is merely amonogenic function with a0 ak as agene. Therefore r(k), k • 0,1,2,... can be replaced by M(k), k = o 1 2 which are different monogenic functions. A sequence is said to have ideal monogenic signature if M(k) is large positive value for k = 0 and is zero for k / 0. Then merit factor and dis crimination can be defined for monogenic signatures. Sequences with good monogenic signatures with certain choices of genes have been discussed in this thesis. Availability of such sequen ces leads to the notion of monogenic function radar wherein the role of autocorrelation is replaced by monogenic signature. The notions of cooperation and Sarker towers can also be deve loped for monogenic signatures in variety of ways. In summary, the contributions of this thesis lie in powerfully indicating the advantages of nonbinary (particularly ternary and four-phase) sequences over the binary sequences, in putting the energy efficiency considerations in proper perspec tive and focus in introducing newer types of processing such as those required in monogenic function radar and in using doubly cooperative sequences and Barker towers of various types and in extending the notion of cooperation so as to remove some of the practical difficulties in its application.