Modified Costas Signal

Transcription

1 I. INTRODUCTION Modified Costas Signal NADAV LEVANON, Fellow, IEEE ELI MOZESON Tel Aviv University Israel A modification to the Costas signal is suggested. It involves an increase of the frequency separation f eyond the inverse of the supulse duration t,cominedwithaddinglinearfm (LFM) with andwidth B, in each supulse. Specific relationships etween f and B will prevent autocorrelation grating loes, that would normally appear when t f > 1. A Costas array [1] is an N N inary array, filled with N ones and N(N 1) zeros. There is exactly one 1 in each row and in each column. The two-dimensional discrete autocorrelation function (ACF) of the array should exhiit only 1 and 0 values, except at the origin where the autocorrelation value is N. Construction algorithms for Costas arrays appear in [2]. In a Costas signal the N rows of the array represent N frequencies spaced f apart, and the N columns represent N contiguous supulses ( its ), each of duration t.a1inthe(i,j) element of the array indicates that during the jth time interval the ith frequency is transmitted. A Costas signal dictates a specific relationship etween the frequency spacing and the it duration f = 1 t : (1) This crucial relationship results in orthogonality etween the different frequencies, when the integration time is equal to t. Due to the orthagonality, the ACF of the complex envelope of the Costas signal exhiits nulls at all multiples of t. It will also result in nulls at all ut N(N +1)+1 grid points of the amiguity function (AF), when the grid spacings are f and t. (In addition to the origin, where the AF value is 1, at each delay nt, n = 1, 2,:::, (N 1) there are, respectively, N jnj Doppler gird points with non-zero AF value. Namely, there are 2[ (N 1)] = N(N +1) grid points with AF value of exactly 1=N.) The pulse compression ratio of a Costas signal is N 2. Namely, the first ACF null is found at = T=N 2 = t =N, where T is the total pulse duration. Lower frequency spacing ( f < 1=t )degrades the delay resolution. The ACF mainloe widens and the ACF nulls disappear. Higher frequency spacing ( f > 1=t ) narrows the ACF mainloe (ecause the signal andwidth has increased) ut results in additional high ACF peaks within the delay interval t <<t, known as grating loes. We show later (see (7)) that when f = a=t, a>1 (2) Manuscript received May 4, 2003; revised Novemer 6, 2003 and April 27, 2004; released for pulication April 27, IEEE Log No. T-AES/40/3/ Refereeing of this contriution was handled y P. Lomardo. Authors address: Dept. of Electrical Engineering Systems, Tel Aviv University, POB 39040, Tel Aviv, 69978, Israel, (nadav@eng.tau.ac.il) /04/$17.00 c 2004 IEEE the grating loe(s) appear at g = g t, g =1,2,:::,ac (3) a where ac indicates the largest integer smaller than a. The AF and ACF of a Costas signal in which N =8 and f =5=t are shown in Figs. 1 and 2. The grating loes (four on each side of the mainloe) are clearly evident in oth figures. The frequency evolution of the signal is plotted in Fig. 3. The andwidth, normalized with respect to the 946 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 3 JULY 2004

2 Fig. 1. AF of Costas signal with N =8, f =5=t, (zoom on jj 2t = T=4). Fig. 2. ACF (db) of Costas signal with N =8, f =5=t. Fig. 3. Frequency evolution of Costas signal with N =8, f =5=t. entire signal duration T, isgiveny T (f max ) f min )+ 1t = N(N 1)t f + N = = 288: (4) This product is the time-andwidth product (TBW) of the signal, which is approximately its compression ratio. Note that the TBW (= 288), achieved ecause of the larger frequency spacing, is much higher than the TBW (= 64) of a regular Costas signal of length 8. The penalty is the grating loes. The next section shows how the grating loes can e nullified while maintaining the large frequency spacing, hence the large andwidth. II. ADDING INTRABIT LFM NULLIFIES THE GRATING LOBES The prolem of grating loes is encountered in a popular radar signal known as a train of stepped-frequency pulses. In that signal large overall andwidth is achieved gradually y frequency stepping unmodulated pulses separated in time. In stepped-frequency train of unmodulated pulses, LEVANON & MOZESON: MODIFIED COSTAS SIGNAL 947

3 Fig. 4. ACF of a train of 8 stepped-frequency pulses (T r =t =3, t f =5).Costasfrequencyorder: Top:t B =0. Bottom: t B =12:5. ACF grating loes appear when the product of the pulsewidth times the frequency step exceeds one. Replacing the fixed-frequency pulses with linear FM (LFM) pulses [3] was used to attenuate grating loes (as well as sideloes). In a recent paper [4] an analytic expression of the AF of such a stepped-frequency train of N LFM pulses was derived for delay within the individual pulse duration, assigned the symol t. The AF expression is valid as long as the pulse repetition interval T r oeys T r > 2t. The zero-doppler cut of the AF, which is the magnitude of the ACF, within the mainloe area (jj t ), turns out to e a product of two expressions µ µ µ R = R1 t R2, 1: (5) t The first is due to a single LFM pulse with andwidth B, µ µ µ R1 t = 1 t sinc TB t 1 t, 1: (6) The second term descries the grating loes, µ sin µn¼t f R2 t t = N sin µ¼t f t, t 1: (7) Clearly jr 2 (=t )j exhiits peaks (mainloe and grating loes) at loes t = g t f, g =0, 1, 2,:::,t fc, jj <t : t t t (8) TABLE I Examples of Valid Cases m n t f t B B= f Nullifying the grating loes is ased on requiring that the nulls of jr 1 (=t )j coincide with the grating loes of jr 2 (=t )j. In [4] that requirement was shown to imply the following relationships t f = 4m n 2m n t B = (4m n)2 2(2m n) (9) (10) where m and n are integers. Some examples, in which all the grating loes are nullified, are given in Tale I. The ACF expression in (5) is independent of the order in which the pulses are arranged in time. Hence, arranging the pulses in a Costas sequence maintains the nullifying. Equation (5) is also independent of the pulse repetition interval T r, as long as it is larger than twice the pulse duration, namely T r > 2t. An example of the ACF of a train of 8 separated stepped-frequency pulses, with and without LFM, is given in Fig. 4. The delay axis extends as far as the first recurrent loe, in order to show the significant difference etween the ACF mainloe area and the recurrent loe area. 948 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 3 JULY 2004

4 Fig. 5. Alignment of signal and reference pulses when calculating: (a) mainloe area of ACF, () first ACF recurrent loe area. Fig. 6. Alignment of signal and reference Costas supulses when calculating: (a) first it area of the ACF, () second it area. Fig. 5 demonstrates schematically the different alignments of the received and reference pulses yielding the mainloe area of the ACF (pair a), and yielding the first ACF recurrent loe (pair ). Regarding the ACF mainloe area we can write where R()= R up u p ()= Z t 0 NX R up u p (), 0 t (11) p=1 u p (t)u p (t + )dt, 0 t : (12) The exact expression of ACF outlined in (11) was given in (5), (6), and (7). When calculating the first recurrent loe area of the ACF, cross terms are involved NX R()= R up u p 1 (): (13) p=2 Because the complex envelopes u p (t) of different pulses have different center frequencies, the spectral overlap is relatively small, yielding ACF recurrent loes that are consideraly lower than the mainloe. In addition to showing the lower recurrent loe, Fig. 4 also shows the disappearance of the grating loes within the mainloe area (jj t ), when LFM with a specific t B product was added. III. CONVERTING THE TRAIN OF SEPARATED PULSES INTO CONTIGUOUS SUBPULSES Converting the train of separated pulses into contiguous supulses (T r = t ) changes the ACF expression. The new alignment of pulses (now supulses) is demonstrated in Fig. 6. The changes in the ACF over jj t (where the grating loes are found) are additional terms due to the cross correlation etween adjacent supulses, NX NX R()= R up u p ()+ R up u p 1 (), 0 t : p=1 p=2 (14) At higher delays (t < jj) only cross terms are found in the ACF. Note that the dominant contriution to the ACF in (14) is the first sum. Exactly the same sum was found when the pulses were separated, and its expression was given in (5) (7). That first sum may exhiit ACF grating loes (without LFM), or they may e nullified if the proper LFM is added. That sum is not affected y the polarity of the LFM slope of the individual supulses. The additional sum of cross terms does depend on the polarity of the LFM slope of the individual supulses. When the slope polarity is the same in all supulses, the cross terms involving adjacent its, and for positive delays <t, were found to e µ sin ¼t (k p k p 1 ) f + B t jr up u p 1 ()j = t t ¼t µ(k p k p 1 ) f + B t, t where k p is an integer and k p f is the carrier frequency of the pth supulse. 0 t (15) LEVANON & MOZESON: MODIFIED COSTAS SIGNAL 949

5 Fig. 7. ACF of train of 8 stepped-frequency pulses, Costas order: , (t f = 3). Top: T r =t =3, t B =0. Middle: T r =t =3, t B =13:5. Bottom: T r =t =1, t B =13:5. In order to check if the nullifying holds we should e interested in the value of (15) at the position of the grating loes, namely at = r= f, where r is an integer smaller than t f. Sincer is an integer and k p k p 1 is an integer, (15) can e reduced to µ r Ru p u p 1 f = µ ¼Br r sin f t f 1 ¼t (k p k p 1 ) f + B r t f B : (16) The numerator of (16) is zero if and only if (IFF) the argument of the sine function is an integer multiple of ¼. Using the values of t B and t f that were given in (9) and (10) gives the argument of the sine function (after division y ¼) as r n(r 1)r [r(2m n) 4m + n]=m(r 2)r 2 2 (17) which is always integer since r(r 1) is even for all r. Thus the numerator of (16) will e zero whenever t B and t f nullify the original grating loes. The only case where the entire fraction in (16) will not equal zero is when the denominator is also zero, resulting in sin(0)=0. Thus if (k p k p 1 )+ B f µ r t f 1 =(k p k p 1 )+ 4m n 2 r (2m n) 1 4m m n(r 1) =(k p k p 1 )+m(r 2) = 0 (18) 2 for some r, m, n, and(k p k p 1 ), the corresponding rth grating loe will not e nullified. For the Costas sequence , out of all the cases in Tale I, only three will loose some or all the nulls. For example in the case m =3,n =3,k 4 k 3 = 6 3=3,ther = 1 null will e lost. This example is demonstrated in Fig. 7, which plots the ACF of the first it. For the top part of Fig. 7 the pulses are separated, and there is no LFM. The two ACF grating loes, caused y using (t f = 3), are clearly evident. For the middle part, the pulses remained separated ut an appropriate LFM was added. The resulted nullifying of the grating loes is clearly evident. For the ottom part the spacing etween the pulses was closed, converting them into a single modified Costas pulse. The second grating loe at =t =2=3 remained nullified, however the null corresponding 950 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 3 JULY 2004

6 Fig. 8. Frequency evolution (top) and ACF (ottom) of 8 element modified Costas pulse, Costas frequency order: , t B =12:5, t f = 5, fixed slope polarity. Fig. 9. Frequency evolution (top) and ACF (ottom) of 8 element modified Costas pulse, Costas frequency order: , t B =12:5, t f = 5, alternating slope polarity. to the first grating loe was lost, as predicted in the preceding paragraph. Yet, while the original grating loe at =t =1=3 was 3:5 db (top suplot), the new value at that delay is approximately 28 db (ottom suplot). Comparing the middle and ottom suplots of Fig. 7 shows that the transition from separated pulses to contiguous supulses increased the ACF sideloes somewhat, ut kept them elow 22 db. We can conclude that loosing a null at a particular = r= f does not imply that the grating loe was restored. It only implies that the ACF value at that delay is no more identically zero. Because of the relatively weak contriution from the cross terms, the ACF value will remain very low (typically elow 20 db). This is true for the case of equal slope polarity (for which the analytic expression in (15) was found) as well as for the case of alternating slope polarity, for which we do not have an analytic expression. In Figs. 8 and 9 we present the frequency evolution and the resulted ACF for modified Costas pulses of length 8, when the slope polarity is fixed (Fig. 8) or alternating (Fig. 9). An appealing intuitive rule for deciding which of the supulses should have an inverted frequency slope, says that supulses at adjacent frequency slots (not necessarily adjacent time slots) should have opposite frequency slopes, ecause the spectral overlap etween such pairs is the largest. This is the alternatingruleusedinfig.9. LEVANON & MOZESON: MODIFIED COSTAS SIGNAL 951

7 Fig. 10. AF of modified Costas signal with N =8, t f =5, t B =12:5, zoom on jj 2t = T=4. Fig. 11. AF of classical Costas signal with N =8, t f =1, t B = 0, zoom on jj 2t = T=4. The peak-to-peak frequency deviation, multiplied y the total pulse duration T, isgiveny T(f max f min )= T t [(N 1)t f + t B] = N[(N 1)t f + t B]: (19) This product is approximately the TBW of the signal, and also its time compression ratio. For the example giveninfigs.8or9(inwhichn = 8) the aove equation yields T(f max f min ) = 380. A conventional Costas signal needs 19 or 20 (¼ p 380) elements to get a similar compression ratio. So far we presented ACF plots of the modified Costas signal. In order to demonstrate how well the ACF nulls (or reduction) hold for higher Doppler shifts we present in Fig. 10 the partial AF (delay zoom on the first two its, out of eight) of the signal shown in Fig. 9. This plot should e compared with Fig. 1 (same t f utnolfm),andwithfig.11 which presents the AF of a classical 8 element Costas signal, in which t f =1. IV. CONCLUSIONS Adding LFM to the individual supulses of a Costas signal allows an increase in the frequency spacing f much eyond the 1=t spacing dictated in a conventional Costas signal. The resulted increase in andwidth improves the delay resolution, which 952 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 3 JULY 2004

8 otherwise would have required a consideraly longer Costas sequence (y a factor of p t f, ifthetotal signal duration is unchanged). Maintaining one of several possile relationships etween the frequency spacing and the LFM andwidth will prevent ACF grating loes, which would have appeared if LFM was not added. The suggested modified Costas signal could e useful in situations where adding LFM is easier than increasing the length of the Costas sequence. [2] Golom, S. W., and Taylor, H. (1984) Construction and properties of Costas arrays. Proceedings of the IEEE, 72, 9 (1984), [3] Raideau, D. J. (2002) Nonlinear synthetic wideand waveforms. In Proceedings of the IEEE Radar Conference, Los Angeles, CA, May 2002, [4] Levanon, N., and Mozeson, E. (2003) Nullifying ACF grating loes in stepped-frequency train of LFM pulses. IEEE Transactions on Aerospace and Electronic Systems, 39, 2 (Apr. 2003), REFERENCES [1] Costas, J. P. (1984) A study of a class of detection waveforms having nearly ideal range-doppler amiguity properties. Proceedings of the IEEE, 72, 8 (1984), Nadav Levanon (S 67 M 70 SM 83 F 98) received a B.Sc. and M.Sc. degrees from the Technion Israel Institute of Technology, in 1961 and 1965, and a Ph.D. from the University of Wisconsin Madison, in 1969, all in electrical engineering. He has een a faculty memer at Tel Aviv University since 1970, where he is a professor in the Department of Electrical Engineering Systems. He was chairman of the EE-Systems Department during He spent saatical years at the University of Wisconsin, The Johns Hopkins University Applied Physics La., and Qualcomm Inc., San Diego, CA. Dr. Levanon is a memer of the ION and AGU and a Fellow of the IEE. He is the author of the ook Radar Principles (Wiley, 1988) and coauthor of Radar Signals (Wiley, 2004). Eli Mozeson was orn in He received his B.Sc., M.Sc. and Ph.D. degrees from Tel Aviv University in 1992, 1999, and 2003, all in electrical engineering. Since 1992 he serves in the Israeli Air Force as an electronic engineer. Dr. Mozeson is coauthor of the ook Radar Signals (Wiley, 2004). LEVANON & MOZESON: MODIFIED COSTAS SIGNAL 953