f = 5 is equal to the delay resolution of a B =12. 5 is shown in Fig. 1. Using M 5

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1 Orthogonal rain of Modified Costas Pulses Nadav Levanon and Eli Mozeson Dept. of Electrical Engineering Systems, el Aviv University P.O. Box 394 el Aviv 6998 Israel Astract wo recent results are comined to create a radar signal with improved performances. he signal is created initially from a coherent train of N identical modified Costas pulses. An orthogonal set of N phase codes is then overlayed on the N pulses. I. INRODUCION he timeandwidth product (BW) of a Costas pulse [] is determined solely y its duration and the numer of its elements M. Since the delay resolution of such a signal is inversely related to the BW, improving delay resolution requires increasing M. In recent papers [2,3] we show how the andwidth of a modified Costas pulse can e increased without increasing M, and yet avoiding grating loes. he Costas signal, its modification and any other pulse compression signal usually exhiit sideloes in their autocorrelation function (ACF). In another recent paper [4] we show that in a coherent train of N identical pulses (any type), the ACF sideloes can e completely removed from most of the single pulse duration, where y most we mean ( N ) N of. he sideloe removal is achieved y overlaying an orthogonal phase coding on the N pulses. In the present paper oth techniques are comined to yield an orthogonal train of modified Costas pulses. he new signal achieves oth high BW, hence high delay resolution; low ACF sideloes in the remaining N part of that is not identically zero; and low recurrent loes at multiples of the pulse repetition interval B t Sections II and III descrie riefly the modified Costas pulse and the orthogonal overlay. Section IV gives an example of an 8 pulse signal. In section V it is compared to an orthogonal train of LFM pulses with the same BW. Finally, in section VI we show that despite the diversity etween the pulses in the coherent train, creating filters matched to nonzero Doppler shifts can still e implemented effectively using discrete Fourier transform (DF), as is usually done when the pulses are identical. II. MODIFIED COSAS PULSE A modified Costas pulse differs from a conventional Costas pulse [] y increasing the sucarriers spacing f eyond the nominal spacing f t, where t is the transmission duration of each sucarrier. Normaly, when f > t, the autocorrelation function (ACF) exhiits grating loes at delay multiples of f. As shown in [2,3], replacing the fixed frequency during t y linearfm with frequency deviation B, can nullify the grating loes, when one of several particular relationships exist etween t f and t B. he advantage of the modified signal is the increased andwidth, hence improved delay resolution, without an increase in the numer of elements in the Costas array. A modified Costas pulse with M elements (its) achieves the same pulse compression as a conventional Costas with M t f elements and equal total pulse width. For example, the delay resolution of a modified Costas pulse with M 8 and t f 5 is equal to the delay resolution of a conventional Costas pulse, of the same duration, ut with M 8 elements. f * t f t he frequency evolution of a modified Costas pulse with 8, t f and t B 2. 5 is shown in Fig.. Using M 5 up and down LFM slopes [3] minimizes the overlap etween neighoring sucarriers, hence minimizes autocorrelation sideloes Fig.. Frequency evolution of a modified Costas pulse, f t 5, Bt 2.5 t / t III. ORHOGONAL OVERLAY Match processing a coherent train of N identical pulses, of any kind, yields the same delay resolution as a single pulse. As a matter of fact, over the delay span τ, where is the

2 pulse duration, there is no difference etween the ACF of a single pulse and that of a train of pulses. Using a train of identical pulses improves the Doppler resolution, which now drops from to ( N r ), where r is the pulse repetition interval. However, as shown in [4], when the pulses are not identical, ut overlaid y an orthogonal phasecoded set, the ACF of the train can e improved in two ways: (a) the ACF sideloes within most of the pulse duration can e reduced to zero, and () the recurrent ACF loes (at multiples of the repetition interval) are drastically reduced. Frequency ime Fig. 2. Binary orthogonal overlay on 8 Costas pulses Fig. 2 presents an example of overlaying the 8 rows of a inary orthonormal 8x8 matrix on N 8 costas pulses, each pulse is constructed from M 8 elements (its). he and symols indicate the overlayed inary phase coding of the its in each pulse. It makes no difference if during the its the frequency remains constant (conventional Costas) or shifts linearly (modified Costas). Orthogonal overlay works as well for any other pulse signal, not necessarily devided into its, ut aritrarily sliced into P slices. If the signal is constructed from M its, it can still e sliced into P slices, and it is not required that M P. An NyP matrix A is said to e orthogonal when the dot product etween any two columns of A is zero (A A is diagonal). Note that orthogonal NyP matrices exist only for P N. An important case is where all elements in the matrix have the same asolute value (normalized to unity) and differ only in their phase denoted y φ n,p (in this case we will refer to the matrix A as an orthonormal phase coding matrix). For an orthonormal phase coding NyP matrix A{a n,p }{exp(jφ n,p )} the signal maintains its envelope power properties and we can write that A ANI where I is a PyP identity matrix. In the example given in this paper M N P 8. Note that the n th Costas pulse in Fig. 2 was overlaid y the n th row of A {exp(jφ n,p )}, where φ n,p is the inary phase matrix in (). An example of a polyphase orthonormal matrix is given in (2). Its rows are all the cyclic shifts of a P4 signal of length 8. Amplitude Autocorre lation [db] t / t Fig. 3. Real envelope of 8 Costas pulses τ / t Fig. 4. AF and ACF (zoom on a repetition period) ϕ inary π () ϕ P4 4 π 8 4 (2)

3 IV. ORHOGONAL RAIN OF MODIFIED COSAS PULSES Autocorrelation [db] τ / t Fig. 5. AF and ACF (zoom on a single pulse) In our example the pulse repetion period is r 3 24t. he relatively large duty cycle was selected to simplify the drawings. he real envelope (amplitude) is plotted in Fig. 3. Because the numer of slices P is equal to the numer of its M, the slice duration t s equals the it duration t s t. It was shown in [4] that due to the orthonormal overlay, the ACF sideloes are identically zero for ts τ r. In our example this implies zero sideloes for t τ 6t. At τ r 6t the st recurrent loe egins. While not cancelled, the diversity introduced through the overlay has reduced the recurrent loe peak to approximately 2 db. he amiguity function (AF) and the ACF, plotted in Fig. 4, extend in delay as far as the end of the st recurrent loe. hey demonstrate the sideloe and recurrent loe ehaviour outlined aove. he AF (Fig. 4, top) shows the first Doppler null (of the zerodelay cut) at ν ( N r ). his (and the entire shape of the zerodelay cut) is a universal property of a pulse train in which the real amplitudes of the different pulses are identical. While not shown, the AF will exhiit a recurrent Doppler peak at τ, ν. he AF recurrent Doppler peak value will r e only slightly less than one. Fig. 5 zooms in on the pulse duration of oth the AF (top) and ACF (ottom). Note that for t τ 8t the ACF sideloes are indeed zero, ut at higher Doppler they slowly uild up. Fig. 6 zooms in even further, and shows only the first it. Only here the ACF sideloes are not zero, ut decrease toward zero as Doppler increases. From the ACF in Fig. 6, we note that the mainloe width ( st null) ocurs at τ t st null 4, implying a signal andwidth of BW τ st null 4 t. Indeed, since M 8, t f 5, t B 2.5, we get from Fig. Autocorrelation [db] τ / t ( f f ) ( M ) t f t B 4. 5 t (3) he total BW of one pulse is therefore approximately ( f f ) M t ( f f ) (4) Fig. 6. AF and ACF (zoom on a single it)

4 S( f ) f * t Fig.. Spectrum of 8 modified Costas pulses with inary orthogonal overlay he frequency evolution, as indicated in Fig., suggests less time spent at the edge frequencies than at center frequencies. In a conventional Costas signal, the frequency distriution is more uniform. he one sided spectrum of the complex envelope of our signal (Fig. ) shows the tapering eginning at f 2 t. V. COMPARISON WIH OROGONAL RAIN OF LFM PULSES An expected question is how different are the performances of the signal discussed aove, with a train of LFM pulses, overlaid with the same orthogonal phase coding. Using M 8 its, setting t f 6 and t B 6 and maintaining the same slope polarity in all 8 its, will create a single LFM pulse with ( f f ) 38. hen N 8 such pulses with the same duty cycle r 3 are overlayed with the same inary code, creating the signal to e compared with. he resulted AF and ACF (zoom on a single it) appear in Fig. 8. Comparing it with Fig. 6 indicates significantly lower level of nearsideloes (within the first it) when the train was constructed from modified Costas pulses. Outside the first it the ACF sideloes are inherently zero in oth signals. VI. MACHED FILERS FOR HIGHER DOPPLER SHIFS he zerodelay cut of the amiguity function is identical to the zerodelay cut of the AF of any signal with the same real amplitude (no matter what other frequency or phase modulation is used, including none). As is often done in a coherent train of identical pulses, filters matched to higher Doppler shifts can e implemented y performing DF on the N pulsecompression outputs. his is equivalent to adding interpulse phase steps to the reference signal. Fig. 9 shows the interpulse phase steps that should e added to the reference signal in order for it to Phase / π Fig. 8. AF and ACF (zoom on a single it) of a train of LFM pulses with orthogonal overlay 3/4 /2 /4 /4 /2 3/4 Amplitude t / r t / r Fig. 9. Interpulse phase compensation for a reference signal matched to ν ( ) N r ν N r. Fig. shows the delaydoppler response of a correlation receiver for the orthogonal train of modified Costas pulses, which uses a reference signal with the added interpulse phase steps. he peak has moved in Doppler to ν ( N r ). he new match a return signal with Doppler shift of ( )

5 of 64. he added orthogonal phasecoded pulse diversity limited the nonzero ACF sideloes to only /8 of the pulse duration, with the peak of those sideloes at approximately 28 db. he diversity attenuated the recurrent loes to a level of approximately 2 db. We also showed that despite the pulse diversity the signal lends iteslf to simple Doppler processing, commonly used in more simple coherent trains of identical pulses. A similar orthogonal train of LFM pulses (with the same andwidth) exhiited higher nearsideloes. Fig.. DelayDoppler response with a reference signal matched to ν ( N r ) delaydoppler response looks like a copy of the amiguity function, shifted in Doppler y ( N r ). he Doppler axis in Fig. extends as far as the end of the first Doppler recurrent loe, whose center is spaced from r the center of the main Doppler loe. Because the zerodelay cut of the AF is independent of any phase and/or frequency modulation, it soars aove the otherwise low sideloes pedestal, achieved thanks to the frequency and phase modulation. he only mean to reduce Doppler sideloes on the zerodelay cut is to add amplitude modulation. One possiility is to introduce interpulse amplitude steps according to one of the many well known windows (Hann, Hamming, etc.). However, interpulse amplitude weighting will violate the orthogonality etween the different pulses in the train. Without orthogonality the property of zero ACF sideloes eyond the first it ( slice), will e lost. Fig. showes some attenuation of the first Doppler recurrent loe relative to main Doppler loe. If the ratio r were consideraly larger than 3, that attenuation would have een much smaller. Since LFM pulses exhiit etter Doppler tolerance than the modified Costas pulses, in their delay Doppler response the Doppler recurrent loe will e less attenuated, ut slightly shifted in delay. In addition to the improved amiguity function, the new signal offers advanges in the contex of coexistance with other radars, and reduced proaility of intercept. LFM pulses have only two asic permutations positive and negative frequency slope. Eight element Costas array can e produced in 444 different permutations (6 element array in 24 permutations). Furthermore, the 8 different pulses (different due to the overlay) can e arranged in 8! 432 differrent orders. here can also e several different inary overlays, each one providing its N! different orders. hus, a signal with a given set of parameters can still e produced in milions different permutations. Each permutation will e detected properly only y its own matched filter. he cross amiguity etween different permutations of the same general signal (using the same sucarrier frequencies) is likely to exhiit a low pedestal shape. REFERENCES [] J. P. Costas, A study of a class of detection waveforms having nearly ideal rangedoppler amiguity properties, Proc.IEEE, vol. 2, pp , August 984. [2] N. Levanon, and E. Mozeson, Nullifying ACF grating loes in steppedfrequency train of LFM pulses, IEEE rans. Aerospace and Electronic Systems, vol. 39, pp. 6943, April 23. [3] N. Levanon, and E. Mozeson, Modified Costas Signal, Sumitted to the IEEE rans. Aerospace and Electronic Systems. [4] E. Mozeson, and N. Levanon, Removing autocorrelation sideloes y overlaying orthogonal coding on any train of identical pulses, IEEE rans. Aerospace and Electronic Systems, vol. 39, pp , April 23. VII. CONCLUSIONS Our example of 8 modified Costas pulses, of 8 elements each, yielded pulse compression of 32. Conventional 8 element Costas pulses would have yielded pulse compression