Binary frequency shift keying for continuous waveform radar

Binary frequency shift keying for continuous waveform radar Nadav Levanon, Life Fellow, IEEE, Itzik Cohen Dept. of Electrical Engineering Systems, Tel Aviv University, Tel Aviv 6997801, Israel. Astract A new inary frequency shift keying (BFSK) waveform is suggested for continuous wave (CW) radar. It provides ideal periodic autocorrelation (PAC) when processed y a matched filter, and perfect periodic cross-correlation (PCC) when processed y a mismatched filter. Ideal PAC implies uniform sideloe level, whose ratio to the PAC peak equals the inverse of the code length. Perfect PCC implies zero sideloes. BFSK is relatively spectrum efficient. Design details and processing issues are discussed. Index Terms CW radar, continuous waveform, BFSK, Legendre, m-sequence, periodic correlation, radar. I. INTRODUCTION An experimental istatic CW radar, descried recently [1], employed a two-valued, periodic, phase shift keying (PSK), in which non-antipodal phases [2] and proper families of inary sequences (e.g., Legendre or m-sequences [3]), resulted in perfect periodic autocorrelation (PPAC). When processed y a matched filter, PPAC implies periodic range response with zero sideloes (SL), namely the periodic autocorrelation of the sequence s[n], s N 1 (1) * mod R n s n k s k N k0 is of the form: R n E δ nmod N (2) s namely periodic peaks of height E. The main drawack of the two-valued PSK waveform is its poor spectral efficiency - road spectrum and slowly decaying spectral sideloes. Spectral sideloes interfere with spectral neighors and require higher sampling rate in order not to alter the waveform s correlation properties. To reduce spectral sideloes the radar descried in [1] replaced the rectangular it shape y a Gaussian windowed sinc (GWS) shape [4]. However, using GWS it representation resulted in variale amplitude waveform, which required a transmitter with linear power amplifier (LPA). The radar descried in [1] used a low power (one Watt) linear amplifier, which is readily availale. The prolems with PSK prompted a search for a constant amplitude and spectrally clean CW waveform that will yield ideal or perfect periodic range response. The issue of spectrally clean pulse waveforms is well referenced [4-6]. Some of the approaches used for pulses can e adopted to CW, ut modifications are needed. The approach suggested in the present paper results in a waveform similar to the one used in [6], which is asically inary FSK (BFSK). Since [6] deals with a pulse waveform it needed a special treatment of the pulse s rise and fall time, in order to reduce their contriution to spectral sideloe. The periodic CW case has the advantage of no rise-time and no fall-time (Fig. 1). This and other differences are listed in Tale I. The concept of a coded periodic CW waveform and its processing is presented in Fig. 1. The top suplot represents the transmitted waveform containing an endless numer of periods, each period is BPSK or BFSK-coded y a common inary sequence of N elements (its). The 2 nd suplot descries a reference waveform, digitally stored in the receiver. It contains P periods of a matched or mismatched reference. Fig. 1. Indefinite periodic CW coded waveform (top) and three finite P-period references with different amplitude weightings. 1

Tale I. Differences etween pulse and CW coded waveforms Property Pulse CW Effect Rise time and fall time Exist Do not exist Cause higher spectral sideloes Mismatched filter Perfect delay response (= Zero delay sideloes) Can e longer than the pulse Unattainale (unless a complementary pair) Of equal length to the signal s period Attainale A longer mismatched filter yields lower delay sideloes Requires specific coding of signal and reference That reference has uniform inter-period weight. In the 3 rd suplot the reference is smoothly amplitude weighted y a Hamming window constructed y NP elements. In the ottom suplot the amplitude weighting is a stepwise P elements Hamming (simpler to implement). Amplitude weighting of the reference reduces Doppler sideloes. The penalty is SNR loss and a wider Doppler mainloe [7]. The resulted periodic delay-doppler response is slightly etter when the weight window is smooth rather than stepped. The waveform proposed here is BFSK. It was selected ecause of four properties: (a) Constant amplitude; () Simple two-valued modulation; (c) Cleaner spectrum than BPSK; (d) With proper coding it can produce ideal or perfect periodic delay response. Ideal periodic cross-correlation (IPCC) can e otained with a matched reference, meaning that the periodic autocorrelation of the waveform s[n] is two-valued: E, nmod N 0 Rs n F, elsewhere (3) where typically F 1. Perfect periodic crosscorrelation (PPCC) requires a unique mismatched reference waveforms, where the periodic crosscorrelation of the waveform s[n] with the mismatched filter h[n] would have zero sideloes: N 1 Rsh, n sn k h* k modn E δn modn (4) k0 Using a mismatched filter would entail signal-tonoise ratio (SNR) loss. In the mismatched periodic case (4) s and h are periodic and have the same period N. The different correlations (a-periodic and periodic) can e further demonstrated y noting the corresponding MATLAB scripts: The a-periodic correlation is given y R = xcorr(s,h), while the periodic correlation is given y R = ifft(fft(s).* conj(fft(h)), recalling that in the periodic case the s and h sequences must e of the same length. The last two lines of the MATLAB script in Appendix A, demonstrate ideal and perfect periodic cross-correlations. A waveform has three parameters that can e modulated or keyed (individually or comined): Amplitude, phase and frequency. Periodic amplitude on-off keying for a CW laser range finder was field tested and descried in [8]. Periodic BPSK for a istatic CW radar was field tested and descried in [1]. The present paper completes the trio y descriing CW-BFSK waveforms. It should e made clear at the eginning that in the proposed BFSK coding the delay resolution is a function of the duration of the frequency step, namely the width of the code element. There is no similarity what so ever etween our BFSK coding and the relatively slow BFSK switching used to implement the so called two frequency measurement [9, 10]. In the latter the purpose of the FSK is to repeatedly transmit two close frequencies next to each other and the duration of the frequency step has no earing on the range resolution. It is also important to clarify that the proposed BFSK waveform is not viale to stretch processing [11, 12], hence the sampling rate should match the waveform andwidth, and will e independent of the expected targets delays. The following sections will descrie the construction of the BFSK waveform; the mismatched reference needed to move from ideal to perfect periodic delay response; the resulted correlations and spectrums; and discuss issues of parameter tolerances and their effect on the range response. Before going into the details of the waveform, Fig. 2 demonstrates the spectrums of BPSK and BFSK waveforms, oth with a rectangular it shape. The frequency scale is normalized y multiplying the frequency variale y the code element duration, t. The waveforms used code length of 103 elements and the CPI contained 32 code periods. In the BPSK coded waveform the 1 st null appears at f t = 1 and the remaining nulls appear at the following integers. The delay resolution (first null of the delay response) of the 2

BPSK waveform is at of [1]. t, as shown in Fig. 5 The transformation will e demonstrated using a 23 element Legendre-ased inary phase code n 1 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 1 1 (6) Applying the transformation in (5) to the sequence in (6) yields t f 1 0 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 1 1 0 0 0 0 (7) 2 n When processed y a matched filter, a constant amplitude waveform, frequency-coded y (7), produces a two-level ideal periodic auto-correlation, where the sideloe level is 1/N relative to the mainloe peak (Fig. 3). However, the same signal will produce perfect periodic cross-correlation with a reference signal having the same frequency coding, ut also amplitude on-off coding, identical to the sequence in (7) (Top suplot of Fig. 4). Fig. 2. Spectrum comparison of BPSK and BFSK coding. In the BFSK coded waveform, descried in the next section, the spectrum mainloe is narrower and the first spectral null appears at 3/4. The remaining nulls are spaced y 1/2. The narrower BFSK spectral mainloe hints that the correlation mainloe of the BFSK will e wider than that of the BPSK, which is indeed the case. In general Fig. 2 demonstrates the much faster decay of spectral sideloes in the BFSK waveform case. II. DESIGN OF THE BFSK WAVEFORM The waveform design is ased on inary m- sequences [3], availale at lengths k N 2 1, k 2,3,4,..., or on inary Legendre sequences availale at all lengths Q if Q is an odd prime that satisfies Q 4k 1, k 1, 2,3,.... Appendix A suggests a MATLAB function that generates inary Legendre sequences. Note that Q contains aout half the prime numers, those that cannot e descried as the sum of two squares. For example the primes 103 and 107 are suitale, while the primes 109 (= 10 2 + 3 2 ) or 113 (= 8 2 + 7 2 ) are not. The transformation from the original inary phase n 0,, n1,2,..., N to the normalized elements BFSK elements n f 0, 1 t, n 1,2,..., N is given [13] y the following expression ( is XOR or sum modulo 2): 1 1 2t f n n n 1 mod N (5) Fig. 3. One period of the BFSK code, Legendre 23: Amplitude (top). Frequency (middle). Periodic autocorrelation (ottom). Fig. 4. One period of the BFSK code, Legendre 23: Amplitude of the BFSK reference (top). Frequency of signal and reference (middle). Periodic cross-correlation (ottom). 3

That result is related to the fact that the crosscorrelation etween a unipolar version {1, 0} of a Legendre or m-sequence code and its inary version {+1, 1} is perfect with a peak equal to the numer of 1 s in the code and uniform off-peak sideloe level which is identically zero [14, 8]. Since the numer of 0 s in a Legendre sequence of length N is (N+1)/2 or (N-1)/2, turning the amplitude of the reference off during each 0 code element, implies using only half the received signal s energy, namely an SNR loss of approximately 3 db. From (7) we learn that the FSK s frequency step is f = 1/(2 t ). The physical meaning of that relationship says that during a 0 it the IF f c, on which the receiver performs the synchronous sampling, completes n c (0) cycles per it, where nc 0 t fc, while during a 1 it it should complete exactly one half cycle more, as shown elow, 1 nc 1 t fc f t fc 2t 1 1 (8) t fc nc 0 2 2 The result in (8) is independent of the it duration or the delay resolution. From the periodic cross-correlation (Fig 4, ottom suplot), we see that the null is at a delay = 2 t, which can e referred to as the delay resolution. We can therefore relate f and t to the range resolution R as follows: 1 C R f, t 2 R C (9) where C is the velocity of propagation. For example, range resolution of R = 3m will require frequency step f = 50 MHz and it duration t = 10 ns. If the IF is 200 MHz, the numer of IF cycles during a 0 it is nc 0 t fc 2. If the next it is 1 then the numer of cycles in it will e nc 1 t fc f 2.5, a difference of half a cycle, as predicted in (6). Returning to the example aove, the specification requires that for the next 10 ns, the intermediate frequency should e exactly 250 MHz. The physical meaning of this specification is interesting and rather harsh. It says, for example, that in order to get range resolution of 3 m the it duration should e t = 10 ns. It also says that at the transmitted frequency, or at the receiver s intermediate frequency, the frequency jump etween different code its must e exactly 50 MHz; Fig. 5 shows how the periodic cross-correlation deviates from a perfect one (ottom suplot of Fig. 4), if the frequency step is 0.2% off the nominal value (e.g. 50.1 MHz instead of 50 MHz), or 0.5% (e.g. 50.25 MHz instead of 50 MHz). Fig. 5. Periodic cross-correlation with frequency step errors of 0.2% and 0.5%. A demonstration of the expected real signal is given in Figs. 6 and 7. The intermediate frequency (IF) where the received signal is synchronously sampled, was selected in such a way that the low and high BFSK frequencies are (Fig. 6): ft 0.5 flow 50MHz 8 t 10 f ft 1 100MHz HIGH 8 t 10 f 50MHz LOW is an aritrary choice used to enale simple demonstration of the half cycle difference etween 1 and 0. As a result of this selection the real signal intersects the its oundaries at two fixed values (Fig. 7). Because of this specific selection, during its # 1, 2 and 3, the signal completes one cycle per it, while during its # 4 and 5 it completes ½ cycle per it. The strict requirement on the accuracy of the frequency step raises concern regarding implementation of the frequency modulation through voltage input to a voltage controlled oscillator (VCO). 4

while the recurrent Doppler ridge at 1 T r dropped to a level of -30 db. That trend is expected to continue as the code length increases. Fig. 6. Frequency evolution of 4 periods of FSK 11 (Total of 44 its), f LOW = 0.5/t Fig. 8. Normalized periodic delay-doppler response. Matched FSK, Legendre 1019, P = 32. Hamming weighted reference. Fig. 7. Two periods (22 its) of the real signal, whose frequency is shown in Fig. 6 III. DELAY-DOPPLER RESPONSE Ideal periodic delay-doppler response of the BFSK waveform (Fig. 8) is demonstrated using a long code (Legendre, 1019 elements). The numer of code periods coherently processed is P = 32. Ideal response is otained when the processor uses a reference waveform identical to the transmitted signal, except for Hamming weighting. The expected pedestal at zero Doppler should have a level of 20 log(1/1019) = -60.16 db. Fig. 8 zooms on a limited delay axis 20t. The floor of the db scale was set to -65 db to enale oservation of the SL pedestal at zero Doppler. The main features of Fig. 8 are: (a) Mainloe that reaches a null at 2t ; () Typical Hamming sideloes (< - 40 db) along the Doppler axis; (c) A recurrent Doppler sideloe, with height of -26 db, at 1 T r, namely at PT r 32, where Tr N t is the code repetition period; (d) A range sideloe pedestal of -60 db at zero-doppler, which is acceptale for most applications. Increasing the Legendre code length to 2011 reduced the SL pedestal at zero-doppler to -66 db ( 20 log(2011) ), Fig. 9. Normalized periodic delay-doppler response. Mismatched BFSK, Legendre 1019, P = 32. Hamming weighted reference. The normalized perfect periodic delay-doppler response (Fig. 9) is demonstrated using the same long code (1019 element), and the same numer of code periods coherently processed (P = 32). The mismatched reference signal has on-off amplitude modulation. It was also Hamming-weighted in order to reduce Doppler sideloes. In Fig. 9 the delay axis extends to a full period, namely 1019t. The normalization is reflected y a peak value of 0 db. Without normalization the entire function, including the peak at the origin, will e lower y 4.5 db, reflecting the loss due to the on-off modulation and the Hamming weighting of the mismatched reference waveform. The main features of Fig. 9 are: Zero sideloes on the delay axis; Typical Hamming sideloes (< - 40 db) on the Doppler axis; Doppler recurrent ridge at PT 32 1 T with a typical level - 20 db. r r 5

IV. DELAY-DOPPLER PROCESSING CONSIDERATIONS The following two facts have important implications on the delay-doppler processing of the BFSK-coded signal: (a) The BFSK s frequency step of f = 1/(2 t ) implies that during a it the signal s phase accumulates (linearly) +/2 or /2 radians; () In an m-sequence or Legendre sequence of length N, the numer of 1 s is (N+1)/2 or (N-1)/2, namely one more or one less than the numer of 0 s. phase-wise periodicity can e made equal to one code period if the frequency switching is not symmetrical aout the reference frequency. If the numer of 1 s is larger than the numer of 0 s the normalized frequency around the reference frequency (marked as ft 0 ) should switch etween the two values: 1 1 ft 1 4 N (10) Namely, the reference frequency is offset from the center. Fig. 10. Frequency (ottom) and phase (top) evolution during 5 periods of BFSK 11 (total of 55 its). Centered reference frequency. The frequency and phase evolution of the complex envelope of the signal, presented in Fig. 10, is otained using a third fact: (c) The reference unmodulated sinewave, used in the coherent synchronous detection in the receiver, is set exactly at the middle etween the two BFSK frequencies. In Fig. 10 the signal is BFSK-coded using Legendre 11 and the numer of periods displayed is P = 5. The fact that the reference frequency is exactly centered is seen in the ottom suplot where the normalized frequency f t is switching etween and +1/4. The resulted phase evolution is seen in the top suplot of Fig. 10. We note clearly the linear accumulation of +/2 or /2 radians during each it. Because of fact () mentioned aove, there is a phase accumulation of +/2 per each code period lasting 11 its. With the three settings (a, and c), from the phase point of view, the true periodicity of the waveform is 4 code periods. Only after 4 periods (44 its) the phase, modulo 2, returns to its initial value. Normally, after matched filtering with one code period, the processor performs FFT on P complex samples taken at an equal delay, from each one of the coherently processed P consecutive code periods. Phase-wise periodicity that is different from the code periodicity can create a false fixed Doppler shift (of 1 4T r in this case). To prevent that, the Fig. 11. Frequency (ottom) and phase (top) evolution during 5 periods of BFSK 11 (Total of 55 its). Offset reference frequency. In this case the frequency and phase evolution, for code length 11, are shown in Fig. 11. Equation (10) tells us that the frequency offset decreases with the code length N. In a monostatic CW radar, where the transmitted signal is availale also to the receiver, the two FSK frequencies f H, f L, relative to the reference frequency f C, can e simply preset as indicated in (11) and the carrier at f C should e used in the synchronous detection. N1 N1 fh fc, fl fc (11) 4N t 4N t In a i-static or multi-static radar system, a synchronization mechanism is required at the receiver that will estimate and lock to f C, just as it needs to estimate the exact it width t or the code T Nt. period r V. SEPARABILITY An important property of coded waveforms is the richness of waveforms elonging to the same family. The variaility is mostly due to different code lengths. This property is important in multi-static radar scenes, where several transmitters operate simultaneously, emitting different waveforms. In order to enale a receiver to choose any signal with minimum interference from the others, the signals 6

should e separale. One simple measure of separaility is the a-periodic cross-correlation etween P periods of BFSK coded waveform of length N 1, and P periods of BFSK coded waveform of length N 2 (same t and same intensity). ecause the signal s parameters allow little tolerance. However, since the VCO characteristic is tapped at only two points, staility of the frequency vs. voltage curve is more important than the linearity of that curve. Another useful property of the BFSKcoded waveform is the good separaility etween two codes of close ut different period durations, when a large numer of periods are processed coherently. REFERENCES Fig. 12 A-periodic cross-correlation etween: (pink) P (=128) periods of BFSK-code of length 1019 and P periods of BFSK-code of length 1031; (lack) a-periodic auto-correlation of P periods of BFSK-code of length 1019. A-periodic cross-correlation is used ecause the signals period durations are different. Such a crosscorrelation, for the case N 1 = 1019, N 2 = 1031 and P = 128, is shown in Fig. 12 (pink plot). For comparison the a-periodic auto-correlation of the signal N 1 = 1019, P = 128 is also plotted (lack). The drawing shows difference of 45 db etween the auto-correlation peak and the highest value of the cross-correlation. That difference increases with an increase of the numer of periods processed coherently. Note that ecause we plotted the a- periodic (rather than the periodic) auto-correlation of the waveform with length 1019, the sideloes are not uniform at a level of -60 db (= 20 log(1019)), ut fluctuate around that value. VI. CONCLUSIONS A new inary frequency-coded, constant amplitude, periodic waveform was proposed for CW radars. Depending on whether the reference is matched or mismatched the output exhiits either ideal or perfect periodic range response, respectively. The waveform is mainly ased on Legendre inary sequences, hence the availale lengths include aout half the prime numers. When processed y a mismatched reference the output sustains an SNR loss of aout 3 db. The BFSK waveform exhiits etter spectral efficiency than a inary phase-coded (BPSK) waveform. Being frequency-coded waveform, it can e generated y a VCO, ut the VCO needs to e of high quality [1] Cohen, I., Elster, R. and Levanon, N., Good practical continuous waveform for active istatic radar, IET Radar Sonar Navigation, vol. 10, no. 4, pp. 798-806,2016. [2] Golom, S. W., Two-valued sequences with perfect periodic autocorrelation, IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 2, pp. 383-386, 1992. [3] Golom, S. W., Shift Register Sequences (revised ed.), Aegean Park Press, 1982. [4] Chen, R., and Cantrell, B., Highly andlimited radar signals, IEEE Radar Conf., 2002, Long Beach, CA, USA, pp. 220-226. [5] Faust, H. H., Connolly, B., Fireston, T. M., Chen, R. C., Cantrell, B. H. and Mokole, E. L., A spectrally clean transmitting system for solidstate phased-array radars, IEEE Radar Conf., 2004, Philadelphia, PA, USA, pp. 140-144. [6] Taylor, J. W. and Blinchikoff, H. J., Quadriphase code a radar pulse compression signal with unique characteristics, IEEE Trans. On Aerosp. Electron. Syst., vol. 24, no. 2, pp. 156-170, 1988. [7] Harris, F. J., On the use of windows for harmonic analysis with the discrete Fourier transform, Proc. IEEE, vol. 66, no. 1, pp. 51-83, 1978. [8] Levanon, N. Cohen, I., Arel, N. and Zadok, A., Non-coherent pulse compression aperiodic and periodic waveforms, IET Radar Sonar Navigation, vol. 10, no. 1, pp. 216-224, 2016. [9] Artis, J. P. and Kemkemian, S., Low cost millimeter wave radars in the automotive field, CIE International Conf. on Radar, 2006, Shanghai, PRC. [10] Qiu, L., Huang, Z., Zhang, S., Jing, C., Li, C. and Li, S., Multifrequency phase difference of arrival range measurement: Principles, implementation and evaluation, Int l J. Distriuted Sensor Networks, 2015, ID 715307. [11] Keel, B. M. and Baden, J. M., Advanced pulse compression waveform modulations and Techniques, Ch. 2 in Malvin, W. L. and Scheer, J. A., editors, Principles of Modern Radar 7

Advanced Techniques, Edison, NJ, USA, Scitech, 2013. [12] Yeh, L., Wong, K. T. and Mir, H. S., Viale/inviale polynomial-phase modulations for stretch processing, IEEE Trans. Aerospace and Electronic Syst., vol. 48, no.1, pp. 923-926, 2013. [13] Levanon, N. and Levanon, U., Two-valued frequency-coded waveforms with favorale periodic autocorrelation, IEEE Trans. Aerosp. Electron. Syst., vol 42, no.1, pp. 237-248, 2006. [14] Takeuchi, N., Sugimoto, N., Baa, H. and Sakurai, K., Random modulation CW lidar, Applied Optics, vol. 22, no. 9, pp. 1382-1386, 1983. APPENDIX A A MATLAB function for generating Legendre sequences function [ p_code ] = inary_legendre_sequence( nn ) % Creates a inary Legendre sequence of length nn % nn must e an odd prime that satisfies nn=4k-1, k is an integer if isprime(nn)==0 disp('not a prime') return end if rem((nn+3)/4,1)==0 disp('not a suitale prime') return else p_code=ones(1,nn); p_code(mod((1:nn-1).^2,nn)+1)=0; end A MATLAB Demonstration of ideal and perfect periodic correlations: s=2*p_code-1; % conversion to i-polar sequence r_ideal=ifft(fft(s).*conj(fft(s))); r_perfect= ifft(fft(s).*conj(fft(p_code))); Authors ios Dr. Nadav Levanon is Emeritus Professor of Electrical Engineering at Tel-Aviv University, Israel, where he has een a senior faculty memer since 1970. In addition to over 70 journal papers and 11 US patents, he authored two ooks Radar Principles (Wiley, 1988) and Radar Signals (Wiley, 2004). He is an IET fellow and IEEE Life Fellow. His 1998 IEEE Fellow citation is for Contriutions to radar signal analysis and detection". He is the 2016 recipient of the IEEE Dennis J. Picard Medal for Radar Technologies and Applications, cited for Contriutions to radar signal design and analysis, pulse compression, and signal processing. Itzik (Izchak) Cohen was orn in Israel in 1981. He received his B.Sc. in physics in 2010 (magna cum laude) and M.Sc. in electrical engineering in 2016, oth from Tel-Aviv University, Israel. Between 2000 and 2007 he served in the Israeli army. From 2007 to 2015 he has een with Elit Syetems, Israel. Currently he is a Ph.D. student in the department of Electrical Engineering Systems, Tel-Aviv University, Israel. Mr. Cohen is a recipient of the Weinstein prize for a scientific pulication in the field of signal processing (2016) and the Weinstein prize for excellence in studies (2017). 8