CHAPTER-1 INTRODUCTION. Radar is an integral part of any modern weapon systems. ability to work in all weather environments at long ranges is

Transcription

1 CHAPTER-1 INTRODUCTION Radar is an integral part of any modern weapon systems. Its ability to work in all weather environments at long ranges is incomparable with any other existing sensors. Use of wideband microwave technology, and advanced radar signal processing techniques have greatly enhanced the detection probability and resolution characteristics of the modern radar systems. The capability of wideband/high-resolution radar in target detection, recognition and analyzing the backscattering media, have increased the role of the radar for defence and many areas of civil applications. Most of the civil applications are concentrated on remote sensing, investigation of natural resources, ground mapping and high-resolution imaging of objects; whereas, the applications of military systems include intelligence, surveillance, navigation, detection, recognition, guidance of weapons, battle field surveillance, antiaircraft fire control etc. [1-2]. The high-resolution requirements of ever changing real time situations demand constant upgradation in the design and development of High-Resolution Radar (HRR) waveforms. This thesis is an attempt to give a comprehensive approach of designing high-resolution radar waveforms, which would ensure improved detection probability and high-resolution in multiuser as well as in multitarget environments.

2 2 Basically, the performance of a radar can be categorized into two aspects, one is the target detection, and the other is the estimation of target parameters like the range, velocity, acceleration, size etc. Both are typical tasks and may be treated as simple in single target environment, and/or when the target is at a reasonable distance. However, the target detection and parameter estimation become difficult practical problems of interest, when the target is small and/or is at a longer distance. In addition, a third practical aspect also comes into picture, when the radar performance characteristics are discussed, namely the resolution. In general, the resolution of a radar can be defined in terms of its capability to distinguish a desired target in a multitarget environment. Further, the detection and estimation problems become quite challenging, when the target is located in a multiple target scenario, where interference from several targets need to be addressed first. Resolution thus becomes a significant parameter of interest in all the discussions related to modern high performance radars. The real test for the capability of a good radar therefore involves the detection and parameter estimation aspects, which may have to be performed for several targets simultaneously, in a multitarget environment [3-4]. It is therefore necessary to deal with the different aspects of resolution, and identify the resolution capabilities of a radar. A radar, which can successfully detect the target and accurately measure its range, velocity etc. in a multitarget environment, can be considered as high-resolution radar.

3 3 The resolution capabilities of the radar can be addressed in different ways. interest are Two of the most prominent types of resolution of (i) the range resolution and (ii) the doppler resolution. The range resolution of a pulse radar is described in terms of the transmitted pulse width and can be expressed as ctp/2, where c is the velocity of light and Tp is the width of the pulse. However, this simple definition of range resolution is predominantly applicable for basic pulse radars, transmitting unmodulated (constant frequency) pulses [5]. The width of the transmitted pulse (single pulse) is also a functional parameter in expressing the doppler resolution of a pulse radar, as 1/Tp Hz. Accordingly, the velocity resolution can be expressed as /2Tp m/sec [4-5]. It can therefore be seen that the range and doppler resolutions have conflicting requirements. A higher range resolution needs a short pulse width, and a good doppler resolution requires a large pulse width. Shortening the pulse width improves the range resolution capabilities, but at the cost of decreased doppler resolution or decreased velocity resolution and vice versa. As the pulse width increases (Tp ), the pulse radar tends to become CW radar, which has an excellent doppler/velocity resolution but no range resolution [6]. In addition to these two important resolutions in terms of range and doppler, some more aspects of resolutions, like angular resolution in azimuth and elevation, cross-range resolution etc. also exist [5].

4 4 Once the three significant performance features of radar are identified, the next focus of attention will be on the most powerful equation that governs the performance characteristics of typical pulse radar, namely the radar range equation, which can be expressed as [7] E G A R 4 t t e max = 2 4π k BT0F E n N 0 min σ (1.1) where Rmax is the maximum radar range, Et = PtTp, is the energy contained in the transmitted waveform, Gt is gain of the antenna, Ae is antenna effective aperture area, is target cross section, Fn is receiver noise figure, E is received signal energy, N0 is noise power per unit bandwidth, Pt is radar transmitter power, k B is Boltzmann s constant and T0 is the standard temperature of 290 K [7]. This radar equation does not take into account the various losses and other factors/efficiencies that affect the radar performance. The derivation of this equation is based on a rectangular pulse transmission and the above expression is a more convenient way for extracting the related information. It can be applied to any waveform that the radar can transmit, provided the receiver is designed on the basis of matched filter detection [7]. The radar range equation can be rewritten in various forms, depending on the utility and application. In particular, the range equation can also be expressed in terms of signal-to-noise ratio (S/N) as P G A R 4 t t e max = 2 4π k BT0BFn S min σ N (1.2)

5 5 where B is the receiver bandwidth and (S/N)min is the minimum acceptable signal-to-noise ratio [8]. The maximum range of the radar can thus be calculated in terms of the minimum signal-to-noise ratio required, which in turn is evaluated from the required probability of detection (Pd), and probability of false alarm (Pfa). It is meaningful to bring into focus here, the empirical formula developed by Albersheim [5, 8], which expresses the relationship between S/N, Pd and Pfa as S/N = ln(0.62/pfa ) ln(0.62/pfa) ln(pd/1-pd)+ 1.7 ln(pd/1-pd) (1.3) The above equation reveals that the required minimum signal-tonoise ratio can be calculated on the basis of specified values of detection and false alarm probabilities on a single pulse basis. More specific formulations for the optimized performance of the radar reveal the significance and implementation of the radar receiver in matched filter form [9]. For the detection of the target and extraction of the desired information from the echo signal, the realization of the matched filter characteristic leads to signal-to-noise ratio criterion, which in turn establishes the aspect of maximizing the signal-to-noise ratio at the output of matched filter; i. e., the peak signal to average noise power ratio has to be maximized to achieve the desired target detection capability [9]. The radar receiver design based on matched filter characteristics therefore becomes a significant feature to analyze the performance capabilities of the radar. Derivation of the matched filter characteristic shows that the maximum S/N at the output of the

6 6 matched filter can be expressed as S/N = 2E/N0 [7-9]. The radar detection capability thus becomes a function of the energy associated with the received signal only and does not depend upon the shape, time duration or bandwidth of the received signal waveform [5, 9]. Thus, for better detection capability, radar needs a high peak signal to mean noise ratio, which can be achieved through the use of a signal having high energy content. To achieve high energy content in transmitted signal, either the peak transmitted power may be increased for a given pulse width or an elongated pulse length may be used for a given peak power [9]. As most of the radar transmitters are operated near peak power conditions [10], for good detection, the energy content can only be increased when a long duration pulse is transmitted; whereas high range resolution radar needs short pulses. Due to these divergent needs of long pulse for detection and short pulse for range resolution, the early radars had the limitations in achieving both the functions simultaneously [10] Literature Survey In the development of the modern radar theory, Woodward s [11] studies assume a tremendous significance in this context. Woodward s investigations and his monograph presentations were perhaps the leading significant contributions in the analysis and design of radar waveforms and the resulting developments in advanced radars. His presentations suggested that, a wide pulse can be transmitted to achieve the energy required for detection; however, after meeting the

7 7 detection requirements, the desired range resolution conditions could be achieved by modulating/coding the transmitted pulse using frequency or phase modulation, to have a bandwidth greater than that of an unmodulated/uncoded pulse of the same duration. The received echo is processed to yield a narrow compressed pulse, which depends on the signal bandwidth, and not on the duration of transmitted pulse [7, 9, 11]. This process led to the development of another significant technological development in the design of radar waveforms, which is popularly known as Pulse Compression Technique [5, 9, 12-13]. The two significant design objectives of the high performance radars, namely the high detection capability (requiring high energy content) and high range resolution (requiring short pulses/wide bandwidth) can successfully be met by the use of pulse compression techniques [5]. The pulse compression features allow radar designers to specify pulse width and range resolution independently. Pulse compression is the method, which permits transmission of long duration modulated pulse (with increased bandwidth B) and compresses the received echo to a short duration pulse (with width = 1/B). Compression of long duration coded pulse of width Tp (B 1/Tp), to short duration pulse of width T, is achieved by matched filter processing [5, 7-8, 12, 14]. Pulse compression technique allows improvement in detection performance that arises out of a large pulse, simultaneously providing the better range resolution due to a shorter pulse. The pulse

8 8 compression waveform has a time-bandwidth product BTp 1, in contrast with the unity time-bandwidth product of unmodulated pulse waveform. The ratio of the wide pulse width Tp to the compressed pulse width T (T = 1/B), is the pulse compression ratio, which will be same as the BTp product. Typical compression ratios are in between 100 to 300, though they could be as low as 10 or as high as 10 5 [6]. The major merits of pulse compression techniques include Increased average power at the radar transmitter, without increasing peak power or pulse repetition frequency (PRF) of the radar. Better resolving capability of system in doppler (velocity) due to wide pulse. Reduction in vulnerability to interfering signals, which are different from the coded transmitted waveforms. Capability to minimize clutter echoes in case of sea, land, rain and other meteorological particles. Improvement in range measurement accuracy and minimum detection range of the radar [7, 9]. In view of the different advantages offered by the pulse compression waveforms, many pulse compression techniques have been investigated, and the corresponding waveform design and their performance characteristics have been reported in the literature [8, 15-34]. Different types of modulations can be used for achieving the pulse compression; two of the most significant and popular modulation

9 9 schemes are frequency modulation and phase modulation. Accordingly, the existing pulse compression methods can be broadly categorized and listed as frequency coding techniques and phase coding techniques; the former includes Linear Frequency Modulation (LFM), stepped LFM, Non-linear FM (NLFM), discrete frequency shift (time-frequency coding) waveforms, and the latter includes biphase (Barker codes, compound Barker codes etc.) [8, 16-18] and polyphase codes (Frank codes, P1, P2, P3, P4 codes etc.) [8, 19-30]. Radar waveforms designed with above coding techniques require complex receiving systems, which are more complicated than that of simple pulse radar [9]. Thus the analysis of the radar signals or waveforms, coupled with the design and development of the matched filter receiver, dictate and characterize the important features required for the high-resolution radar [35-36]. Further, a signal reflected from a moving target is affected by doppler shift, which changes the carrier frequency. The special mathematical function, which is used for the study of the matched filter output, when the received signal is time shifted (delay and doppler shifted ( is known as the time frequency autocorrelation function or Ambiguity Function (AF) or uncertainty function, which is the best analytical tool for the investigation of the above aspects [34]. Ville [37] seems to be the first researcher, who conceptually introduced the time-frequency response function in complex domain, whereas Woodward [11] studied its importance and investigated its

10 10 utility in radar applications. Later, Siebert [38] carried out extensive studies on the analytical aspects of the uncertainty function, Wilcox [39] worked on the synthesis aspects of the ambiguity function and Klauder [40] presented the design features of radar signals estimating the properties of the time-frequency response function [9]. In general, the output of the matched filter for a moving target case is represented by the complex function (, ), where represents the time delay, and represents the doppler shift. (, ) is designated as the ambiguity function [4, 34], which can be mathematically represented as χ(τ,ν) u(t) u (t τ).exp(j2πνt) dt * (1.4) where u(t) represents the transmitted signal, (, ) being the complex envelope of the matched filter output. In radar terminology, the output of the matched filter response function is conveniently represented in three slightly different mathematical forms, namely (, ), (, ) and (, ) 2, all of them being commonly designated as ambiguity functions. Rihaczek [4] designated (, ) as uncertainty function and (, ) 2 as ambiguity function. Skolnik [6], Cook and Bernfeld [9], Nathanson [15], Sinsky and Wang [41] used (, ) 2 as the ambiguity function and Levanon [34] represented (, ) as the ambiguity function itself. In principle, the AF has a three dimensional representation and the plot of the AF (, ) 2 is often characterized as the ambiguity

11 11 diagram (AD) [15], and the plot of (, ) 2 above ( plane is designated as the ambiguity surface [4]. In the graphical representation of the AF of different signals, Levanon [34] observed that sidelobe suppression will be better seen if (, ) 2 plots are used, and use of log scale may apparently result in enhanced sidelobe presence, and therefore recommended (, ) representation for the ambiguity function. As the major focus of the present analysis and investigations is on evaluation of sidelobes relative to the mainlobe, Levanon s recommendation of (, ) representation of the AF is implemented in the present thesis. In (1.4), τ = 0, and ν = 0, represent the origin of the ambiguity diagram, which is centered on the actual location of the desired target in the delay-doppler plane [15]. The study of the matched filter output as a function of both delay and doppler-frequency is important in terms of resolution, sidelobe behavior, ambiguities in range and doppler, measurement accuracy and clutter rejection characteristics of the waveform. An ideal ambiguity function for target detection and resolution would consist of a single peak of infinitesimal thickness at the origin and zero for all other points away from the origin. Unfortunately, fundamental properties of the ambiguity function do not allow the existence of such impulse like ambiguity function [6]. Two major restrictions are that the height of the ambiguity function can nowhere be greater than that at the origin, and the volume under the ambiguity

12 12 surface is constant. Therefore, the height of the peak at the origin and volume enclosed by the ambiguity function are fixed [34]. The properties of the AF also reveal that its maximum value is constant and is represented as (, ) 2 = (2E) 2. This property has also been discussed under volume invariance category as radar uncertainty principle or law of conservation of ambiguity [4, 9]. The same condition has also been treated as the most important constraint of the AF by Cook and Bernfeld [9]; while stating that all signals and waveforms are equally good or bad as long as there is no prior knowledge of the specific application or radar environment. In view of the above restrictions, the logical choice of ambiguity function of radar waveform is to have a narrow central spike at the origin, with the remaining energy spread into the low-level pedestal region surrounding the spike. The resolving capability of the radar in range and doppler will be guaranteed by narrowness of the central spike, and low level sidelobes or pedestal will ensure no range and doppler ambiguities. The width of the central spike of such ambiguity function in delay and doppler is of the order of 1/B and 1/Tp, respectively. Ambiguity surface of this type is commonly known as thumbtack ambiguity function [8, 34]. Selection of the desired waveform depends on location of the clutter or competing targets in the delay-doppler plane of the radar environment. In other words, before selecting the waveform (waveforms) for given radar application, the ambiguity function of the

13 13 selected waveform must be tested against the radar environment. Thus, the ambiguity function helps the waveform designers in selection of proper waveform, so that the selected waveform matches with the specific radar application [15, 34]. The foregoing discussions reveal the conceptual studies and developments associated with the performance characteristics of the radar, from the point of view of analysis and design of the radar waveforms and the associated matched filter features. The concepts discussed, the parameters introduced along with the necessary mathematical tools so far, will lay the foundations for analysis, design and development of waveforms, which are the prime objectives of the present work. The most powerful architectural feature in the analysis and design of radar waveforms is the utility of pulse compression technique that can achieve high-resolution and better detection performance simultaneously. Several researchers [8, 15-30, 42-47] have contributed in a significant manner for the development of different pulse compression waveforms that led to the opening and/or establishment of different areas of research in the pulse compression domain. In this category, perhaps Linear Frequency Modulation (LFM) can be treated as the first building block, which later led to the development of Nonlinear FM (NLFM). However, both of them have been classified under analog pulse compression schemes.

14 14 Further, these LFM, NLFM modulations can also be used either within a single pulse (burst of N subpulses) or on a pulse-to-pulse basis. The former technique is represented as intra-pulse modulation and the latter scheme is designated as inter-pulse modulation. In addition, FM may be achieved through a continuous (analog) process or a discrete process [6, 15]. Inspite of their popularity and potential applications, both the LFM and NLFM waveforms suffer from some drawbacks. Although LFM waveforms are recommended for wideband (where swept bandwidth is in order of hundreds of MHz to 1 GHz) radar systems, implementation of the associated digital signal processing scheme is difficult because of non-availability of high quality analog to digital converters (ADC) at the required high sampling rates [5]. Other significant limitations in LFM waveforms are the poor doppler resolution and ambiguous range measurement due to range doppler coupling [5, 13, 15]. The output of the LFM matched filter displays objectionably large range sidelobes (the first sidelobe is approximately 13.2 db), which can be made low at the cost of sacrificing the signal-to-noise ratio [34, 48 * ]. On the other hand, NLFM waveforms are sensitive to doppler frequency shift and are not doppler tolerant. Further, the major limitations in implementation of NLFM are - (i) system complexity, (ii) limited development of NLFM generating devices, and (iii) stringent phase control requirements [5-7]. In principle, the LFM and NLFM waveforms may be categorized as * Author s publication

15 15 the first major type of classification of pulse compression waveforms. Accordingly, the phase coded waveforms may be designated as the second major class of pulse compression waveforms [5]. Phase coding technique is an innovative pulse compression technique, where the effects of pulse compression are achieved by using discrete phase changes rather than frequency variations. Phase coding/modulation schemes are also equally popular as the frequency modulation method for achieving the pulse compression. Thus, the resulting pulse compression technique can be classified as phase coding technique, which can be implemented on intra-pulse basis or inter-pulse basis [15], which in turn led to the development of several discrete phase coded pulse compression techniques. Contributions of Barker [16], Turyn [17-18], Friese [19-20] Borwein and Ferguson [21], Frank et al. [22-23, 26], Chu [25], Lewis and Kretschmer [27-30], are some of the most significant works in the area of discretely phased or discrete phase coded pulse compression waveforms/sequences. Their investigations using discrete phase coded waveforms have been categorized broadly into biphase and polyphase codes. In phase-coded waveforms, the frequency remains constant but the phase of each subpulse is switched between certain predetermined M values at periodic intervals. That is, each pulse of length Tp can be considered as a contiguous set of N subpulses of duration T = Tp/N, and the phase of each subpulse is chosen either 0 or (M = 2, binary) or any other values of phases between 0 to 2 given by 2 /M, M being

16 16 an integer greater than 2 (polyphase). In principle, the phase coding techniques are designated on the basis of number of predetermined values of phases that have to be changed, in the transmitted pulse; M=2 resulting in binary phase coding, and M 2 accounting for polyphase coding. In case of phase coding, the pulse compression ratio becomes N, the number of subpulses (N = Tp/T), which is approximately equal to BTp, where B 1/T. The output of the matched filter will have a narrow peak (mainlobe), which is N times larger than that of long received pulse, and width of the peak response will be T. The portions of the matched filter output, other than the peak response, which are extended over Tp to Tp, are referred as timesidelobes. The number of subpulses N, in a transmitted pulse is also called the length of the sequence, and the individual subpulses are designated as chips [5] or subpulses [8] or bits [34]. The phase coded pulse compression waveforms categorized under biphase codes include - Barker Codes, Maximal Length Sequences, and Minimum Peak Sidelobe Codes etc. whereas, Frank Codes, P- Codes etc. are categorized as polyphase codes. Barker codes [16] belong to that set of binary phase codes having equal sidelobes, the peak sidelobe (PSL) being 1/N for a Barker code of length N. One of the major limitations of Barker codes is that they are available up to length of N=13 only; many variants of Barker codes are not available. Turyn [17-18] showed that no odd Barker code sequences can exist for N 13, and Barker codes of even length are available up to N = 4 only. He has

17 17 also shown that no such codes exist for even length, in the range 4 N The longest length of 13 in Barker code offers a lower compression ratio, which may not be acceptable for many practical applications. Larger values of N (>13) are possible through the use of maximal length pseudorandom sequences or Maximal Length Sequences (m-sequences) [5, 8-9, 15]. Implementation of maximum length sequence of periodic type has been suggested for high-resolution radar applications [9]. However, when N is large, the peak sidelobe of the truncated maximal length sequence approaches 1 N, which is higher compared to the sidelobe level of Barker codes. It can therefore be seen that the Barker code of length 13 produces a peak sidelobe level (PSLL) of 22.3 db (20 log (1/N)), whereas the length of the m-sequences required for producing the sidelobe level of that order is about 255 (sidelobe level 25.9 db). As the biphase Barker codes offer a PSL value of 1 or PSLL of 1/N, (maximum N being 13), studies on the existence of larger sequence length codes with minimum PSLL have also been carried out [5, 15]. The resulting codes are known as Minimum Peak Sidelobe Codes, and utilization of exhaustive search techniques showed that minimum peak sidelobe codes exist up to a length of N = 70 [5]. Minimum peak sidelobe codes are also binary phase sequences having the lowest PSLL for a given length. It can also be seen that all Barker codes are also minimum peak sidelobe codes but converse is not true.

18 18 The PSL of minimum peak sidelobe codes of 7, 11 and 13 is 1.0, indicating that they are on par with the Barker codes. Information available in the literature [5, 15, 34] on minimum peak sidelobe codes, indicates that the peak side lobe is 2 for N 28, 3 for 29 N 48 and N = 51, and peak sidelobe of 4 for N = 50 and 52 N 70. This peak sidelobe variation shows that there is a marginal improvement in peak sidelobe as the code length N increases. Even for N = 70, the PSL is 4 and PSLL is 24.9 db [5]. All these studies indicate that the best peak sidelobe is 1/N for Barker codes up to 13, and reduction in peak sidelobe level is possible only when - (i) one can go for m-sequences to a length of the order of N = 255 (PSL of 13 or PSLL of 25.9 db), or N = 511 (PSL of 19 or PSLL of 28.6 db) or N = 1023 (PSL of 29 or PSLL of 30.9 db) [15], (ii) minimum peak sidelobe codes are used with a higher length of N = 51 (PSL of 3 or PSLL of 24.6 db). The main advantage of biphase codes is the ease of implementation, but the limitations are that these codes are not known for large length of N and their doppler tolerance is limited. Research reports indicate that polyphase codes can produce lower sidelobe levels than the binary phase codes and have appreciable doppler tolerance when the doppler frequencies are smaller [5]. Number of researchers Friese, Frank et al., Heimiller, Chu, Lewis and Kretschmer [16-30] investigated different types of polyphase codes and showed that they possess good autocorrelation and cross-correlation properties.

19 19 Friese [19-20] established the existence of uniform polyphase sequences up to length 36, targeting the Barker condition, i.e. the magnitude of all autocorrelation sidelobes is less than or equal to one. Stochastic optimization technique was used to obtain these sequences with properly selected starting vectors. Borwein and Ferguson [21] further extended the list of known polyphase sequences that satisfy the Barker condition, up to length 63, by using two different optimization algorithms applying stochastic and calculus techniques. Frank et al. [22-23] established the correlation properties of the polyphase codes with M number of phases and code length N, where N=M 2. He also verified the properties of the polyphase codes for M up to 8, and showed that the peak mainlobe to peak sidelobe level ratio will be substantially better than that of biphase codes for M greater than 5. These codes are popularly known as the Frank Codes, and it has been suggested that the doppler shift effect of these codes can be similar to those of LFM pulse compression codes. In principle, their main limitation is that they are applied only for codes of perfect square length (N=M 2 ). Frank codes can be treated as an approximate set of stepped LFM versions of polyphase codes. Lewis and Kretschmer [27-30], introduced 4 types of P-Codes, namely P1, P2, P3 and P4 codes, which may be treated as variants of Frank type polyphase codes. It is claimed that the P-codes are more tolerant to receiver band limiting before pulse compression than the Frank codes [8, 22-23, 34]. Various sets of polyphase codes have been

20 20 proposed and listed in the literature [8, 34]. In a broad sense, it can be stated that all the Frank codes and P-codes are derived/related versions of LFM signals. The P1 and P2 sequences are permutations of the Frank code, and are applicable only for N = M 2 ; P3 and P4 codes are derived from the linear FM waveforms, and are applicable for any length N [34]. It has been shown that the P3 and P4 codes have larger zero-doppler peak sidelobes than the other codes, but degrade less as the doppler frequency increases. Yang and Sarkar [49] presented a new set of polyphase codes, derived from the step approximation of the phase curve of the hyperbolic frequency modulation and reported that these codes degrade less with the increase of doppler frequency. It is stated that such polyphase codes have the desired property of doppler tolerance; however, they have relatively larger sidelobe levels under zero doppler conditions. The studies on various types of polyphase codes reveal that the polyphase codes produce lower sidelobe levels than that of binary phase codes, and are tolerant to doppler frequency shifts. A typical Frank polyphase code with N = 25 (for M = 5) has a PSLL of 23.9 db, whereas the nearest length m-sequence biphase code of length N = 31 results in a PSLL of 17.8 db only [8]. This clearly indicates the advantage of polyphase coding over biphase coding. The major advantage of phase coded waveforms is less vulnerability to repeater jamming than LFM; however, the

21 21 implementation of phase coded pulses is more complex. The principal limitation of polyphase coding is that - in case of higher sequence length, these codes become more sensitive to doppler shift than that of the shorter ones. Therefore, these waveforms are suitable for low speed target applications with smaller time-bandwidth product [13]. Rihaczek s [50] studies also indicate that the phase shift coding techniques are not desirable in high-resolution radar applications, which demand larger time-bandwidth product (higher compression ratio), as they need exceedingly large length, which in turn leads to higher system complexity. The inherent limitations of the LFM and NLFM waveforms, as well as binary and polyphase coding techniques listed above, emphasize the need for further investigations to find alternate techniques, and corresponding sets of coded waveforms that can lead to better performance, in terms of higher compression ratios and better sidelobe suppression characteristics, especially for high-resolution radar systems. It would be even more interesting to study the characteristic features of such codes from the required correlation properties point of view as well as from the optimal length (for a desirable sidelobe) point of view. The focus of attention thus narrows down to the study of the design techniques of frequency coded waveforms or frequency hop codes. Frequency hop coding is another innovative and powerful pulse compression method, which uses Discrete Frequency Coded Waveforms

22 22 (DFCWs) or Frequency Hopped Waveforms (FHWs) to realize better range resolution and detection probabilities in multiuser and multitarget environments. Frequency hopping technique can be designated as the third major class of pulse compression waveforms, in which the transmitting frequency is switched discretely between a set of predetermined values to achieve a large overall bandwidth B, whereas each subpulse utilizes smaller bandwidth, which is equal to B/N [6]. In this context, the studies carried out by John P. Costas [51] deserve a special mention in the classification and analysis of frequency hop codes. His innovations and investigations led to the classification of the most popular set of waveforms designated as Costas arrays or Costas sequences or Costas codes or Costas waveforms. Costas studies laid the foundation for the design and development of Frequency Coded Waveforms (FCWs), having desirable sidelobe levels and near ideal ambiguity function. The special features of the Costas waveforms, their construction and characteristics have been discussed in detail in the literature, and several researchers have extended these studies for achieving the desired delay-doppler resolutions and detection capabilities in different radar applications [52-76]. Costas [51] suggested an approach of designing detection waveforms, which exhibit low-level pedestal in the delay-doppler plane, with careful selection of frequency hopping patterns. Such patterns are represented in the form of N N arrays (where columns represent N

23 23 time-slots of each duration T, and rows represent N discrete frequencies equally separated by Δf ). The synthesis procedure of Costas can lead to a good or near ideal thumbtack ambiguity function with controlled delay-doppler sidelobe levels. He illustrated that these codes produce ambiguity function with sidelobes of the order of 1/N times of the mainlobe in all regions in the delay-doppler plane away from the origin, whereas, near the mainlobe, these sidelobes rise to a factor of 2/N [51]. For any given value of N, the total number of different NxN arrays are N! but all of them do not satisfy Costas condition. Costas investigated all such possible codes for N 12 which satisfy Costas condition, using Difference Triangle Method [51]. To satisfy the Costas condition, all the elements in a row of difference triangle must be different from each other [34]. John Robbins found all possible Costas arrays for N = 13 [52]. Golomb and Taylor [52-53] extended the pioneering work of Costas, and illustrated the possibilities of several array constructions for N 360 (for 271 values of N only, with gaps in between), using various array construction methods like Golomb, Welch, Lempel and several of their variants. Their findings indicate the presence of some Costas arrays for a given value of N; and all possible Costas array up to N = 13. Costas arrays can also have some special features which include Costas Arrays with Nonattacking Queen Sequences, Honeycomb Array Realizations and Nonattacking Kings Configurations. Silverman et al. [76] further extended the studies of Costas array constructions for N up to 17, by using a probabilistic model. As mentioned by Levanon [34], no Costas arrays

24 24 are known to exist for N = 32 and 33, and the challenge posed by Silverman et al. to find such an array for N = 32 and N = 33 is quite intriguing [76]! The findings of Beard et al. [77-79] are remarkable, and need a special mention at this juncture, in the sense that their search methods have been successfully used for generation of all possible Costas arrays up to N = 27. The construction features of all the Costas arrays listed above can be classified broadly into two categories one, involving Algebraic Construction Methods used by Golomb et al., and Popovic etc. [52-54, 59], and the second, Exhaustive Search Method, which has been used for the computation of Costas codes of different lengths N [77-80]. Costas arrays exhibit the best possible autocorrelation properties. However, these codes are not suitable in multiuser radar systems, as they do not exhibit good cross-correlation properties. Titlebaum et al. [57-58, 63] examined the cross-correlation properties of algebraically constructed Costas arrays, for a specific construction and a given N. They also established a cross-correlation upper bound for pairs of Welch- Costas arrays. The research works of Moreno, Golomb and Rod Gow reveal certain interesting properties of Costas arrays like - shifting property [60], periodicity property [61], and regularity property [71]. Possibilities of generation of Costas arrays for orders up to N = 200 have also been discusssed by Beard [80]. A neat compilation of all important aspects

25 25 detailing the methods of construction, algorithms development and associated results has been presented by Konstantinos [81] in his review paper, which is an open access article. Other significant research contributions on the status of Costas arrays and their constructions have been due to Golomb et al. [82-83]. All these research works clearly illustrate the entry, existence and evolution of radar waveforms design, based on frequency hopping codes to achieve Costas arrays. The research work is still going on in these areas related to frequency hopping techniques and the field is wide open for discussions; both algebraic construction methods and exhaustive search methods are being explored to develop and configure the properties of Costas arrays of higher order. All these methods primarily contribute to check the realization of Costas arrays for any given value of N. In most of such Costas array constructions, little attention is given to the cross-correlation properties of such arrays. A good radar signal design for high performance radar applications demands good autocorrelation property - to have low sidelobe levels, and good cross-correlation property or minimal interference, which is essential for multiuser radar systems. It would therefore be interesting to investigate the properties of frequency coded waveforms, not necessarily from Costas point of view alone, but with attractive autocorrelation and cross-correlation properties points of view, which are essential for modern radar applications.

26 26 Accordingly, the first investigation in this thesis is focused on the design of the sets of orthogonal frequency coded waveforms, which have desired autocorrelation and cross-correlation properties. An innovative approach, for selecting the appropriate order of frequency hop codes, to achieve the desired autocorrelation and crosscorrelation properties, is the Optimization Technique. By using a suitable global optimization algorithm, frequency coded waveforms can be designed to have good autocorrelation and cross-correlation properties as suggested by H. Deng [84]. Deng [84] proposed a novel statistical approach of designing high-resolution radar waveforms by optimizing discrete frequency coded waveforms sets. DFCW sets are optimized using Simulated Annealing (SA) Algorithm [85], for the lengths N = 32 and N = 128. It has been shown that the peak levels of autocorrelation sidelobes and crosscorrelation sidelobes will be about L/N, where L is the number of sequences in a set (set-size), N being the sequence length. He illustrated the applicability of the SA algorithm, for the design of frequency coded waveforms for netted radar applications. Deng also established the near optimum realizability of impulse function like autocorrelation property with nearly zero level cross-correlation function using SA algorithm and optimization procedure. Deng s statistical approach suggested utility of SA algorithm; however it will be worthwhile to study the effects of implementation of algorithms like Threshold Accepting (TA) Algorithm and Hamming Scan

27 27 (HS) Algorithm etc. The TA algorithm [86] has been described as a general purpose statistical optimization algorithm, and is a modification of SA algorithm with superior performance characteristics. However, it has slow convergence property. On the other hand, HS algorithm has been shown to have faster convergence rate [87-88], but it has the demerit that it can get stuck at the realization of local minimum point. To overcome these limitations, Singh et al. [89] proposed a New Hybrid Algorithm (HA) and studied its applicability for polyphase waveforms. An attempt is made in this present research work, to investigate the effects of the new hybrid algorithm for the realization of discrete frequency coded waveforms, perhaps for the first time. It is therefore proposed to study the implementation of the HA for realizing an optimal solution for design of DFCW codes, combining the design simplicity of TA algorithm, and superior convergence property of HS algorithm, for high-resolution radar applications. It can be expected that such a hybridized algorithm approach may lead to achieve better autocorrelation and cross-correlation sidelobe levels than that of SA alone. Further, the adaptability of such an algorithm, for optimization of DFCW sequence length for larger value of N (up to 300), can also be established. The waveform design studies reveal that no waveform is optimum in general [4]. The DFCW sequences designed on the basis of sidelobe correlation characteristics and optimization level features are really appropriate for multi radar environment with low target density.

28 28 Special design aspects are to be taken into account, in developing the DFCW sequences for dense target environments. In high-resolution radar applications, especially in dense target environments, when the detection of small target in the neighbourhood of large target is more important, the weak echo signal of the desired small target may be masked by the stronger sidelobes of large target (self-clutter). In such cases, it is desirable to study the design and development of radar signals that can produce large sidelobe-free area around the mainlobe in the ambiguity function [90-91]. In other words, the waveform design has to be explored to ensure and to establish a desirable sidelobe-free region, by pushing the sidelobes away from the mainlobe, beyond the acceptable region in which the desired targets may exist. The resulting property can be designated as sidelobe pushing property, which becomes an important input feature for the design of radar waveforms [92]. When sidelobe pushing property is employed, the weaker wanted target may become free from the sidelobes of the stronger echoes of the nearby targets. In the design of DFCW sequences for dense and multi radar environments, the sidelobe pushing property of the sequences thus becomes equally significant, in addition to the realization of the desired auto and cross-correlation properties. It can also be mentioned here that some researchers [4, 9, 90-91] suggested the uniform pulse train type of radar signals for achieving the desired AF, for detection and resolution of targets in dense target

29 29 environments. Their approach suggested that the inter-pulse period must be selected in such a way that entire delay spread as well as expected doppler shifts of the entire target space must exist within the unambiguous central region of the AF of the uniform pulse train. However, their recommendations, to take care of doppler shifts in many situations, may not yield practical values of carrier frequencies [9]. Although the uniform pulse train based waveform design did not suggest the practical solution for many radar applications, it indirectly leads to the significant aspect of maintaining desired clear area around the mainlobe, which is an essential feature of a pushing sequence property. The design of radar signals with desired sidelobe pushing property thus becomes yet another classification from the author s point of view. Focusing the attention on designing the waveform that can give a desired clear area around the mainlobe demands a comprehensive study and exploitation of its AF properties. A successful signal design should lead to a thumbtack AF of the type shown in Fig. 1.1, which has a clear area around the central mainlobe, the region in which no sidelobes are to be present. The limits of this region can be identified as rt and r/t, where T is pulse width of contiguous subpulses (of DFCW) and r is the measure of the clear region between mainlobe and consecutive sidelobes, denoted as the pushing power. When r = 1, Fig. 1.1 can be considered as equivalent to that of the AF of uniform pulse train [4, 90]. For taking into account the pushing

30 30 sequences, the r parameter becomes an integer greater than 1, as suggested by Chang and Bell [92]. Fig. 1.1 Thumbtack ambiguity function with a clear area around the central spike. As shown in Fig.1.1, the AF can be divided into two regions - (i) a clear area or primary region (consisting only of mainlobe), (ii) sidelobe area or secondary region, and the research should aim at increasing area of the primary region as much as possible. In this context, the innovative techniques of Chang and Bell [92] become quite significant for the construction of DFCW sequences with sidelobe pushing property. They illustrated the notion of pushing sequences using Lempel T4 construction and Lee Code Words. Further, their detailed investigations reveal two distinct methods of construction of DFCW pushing sequences, from a given pushing sequence one based on symmetry property, and other based on frequency omission property. In

31 31 particular, their illustrations indicated that the symmetry property (which has been designated as group D4 dihedral symmetry property) has got rotational symmetry, in the sense that it permits the construction of additional DFCW pushing sequences from the original sequence of specified length N, by rotation. Chang et al. used these techniques effectively to construct more Costas sequences of Lempel T4 type, and Lee code words with pushing properties. The definition of the pushing sequence property [92] implies that all available Costas sequences like Welch, Golomb, Lempel (and their variants), non-attacking Queen Sequences, Honeycomb Arrays and non-attacking Kings Configurations etc. can not be categorized as pushing sequences. Chang and Bell proved that Costas arrays using Lempel T4 constructions, generate ambiguity function with sidelobefree area (with r = 1) around the mainlobe, and improve the delaydoppler resolution. Such sequences are named as pushing Costas sequences [92]. It would be of interest, to study and prove whether any other type of Costas sequence (other than T4) can also have pushing property. An attempt is made to investigate this pushing aspect with reference to Costas sequences having non-attacking Kings property, in the present thesis. The properties of the pushing sequences are useful for dense target environments; however, the properties required for multiuser radar systems are the good cross-correlation properties of the waveforms. The present day requirements of the HRR systems point

32 32 out the design and development of radar waveforms for multiuser and dense target environments. Investigations are therefore required for identifying radar sequences or waveforms that possess not alone pushing sequence property, but also desirable cross-correlation properties. Accordingly, the cross-correlation properties of nonattacking Kings sequences need to be evaluated. Chang and Bell [92] described construction of radar sequences for enhanced delay-doppler resolution using Lee code words, which do not possess the Costas property. They established the pushing characteristics of the Lee code words. The attractive feature of these codes is that one can design the waveforms for the clear area around the mainlobe, if the Lee code word length is N = 2r 2 + 2r + 1, and the Lee metric between each pair of code words is equal to (2r + 1), where the pushing power r is an integer. However, no significance has been attached for the calculation of cross-correlation properties of the Lee code words and their suitability for multiuser radar systems. This aspect has been given significant importance in the present research work. The detection and high-resolution in dense target environments need two significant inputs one in terms of pushing properties and other in terms of cross-correlation properties. It would be worthwhile, to investigate whether any of the time-frequency hop codes, suggested in the literature as having good cross-correlation properties, can have pushing sequence features also. This is a kind of a mirror image talk to

33 33 the discussions presented in the earlier paragraph, and investigations of Titlebaum et al. [93-98] assume a great significance in the present aspect. Titlebaum [93] developed set of frequency coded waveforms using the Theory of Linear Congruences that possess good crosscorrelation properties. The codes may thus be designated as LC (Linear Congruence) codes, which have been suggested for use in multiuser radar systems. The LC codes have the characteristic features that they can be designed for any code length N, where N is a prime number. Further, Titlebaum suggested that the upper bound for the crosscorrelation function can reach 2/N for large values of N, and there exists a tradeoff between autocorrelation properties and crosscorrelation properties. It is the interest of the present author, to investigate the properties of these LC codes, and check whether they have appreciable pushing property or not. The literature survey details presented above, indicate the explorability of waveform design features, for evaluation of good cross-correlation properties for codes having pushing sequence characteristics and vice versa. Another significant feature that can be brought into analysis is capability of modifications of the existing pushing sequences. Such waveform designs are more suitable for dense target and multiuser radar environments. Of equal importance is the detection and resolution of a small/weak target in the neighbourhood of strong target, especially for military applications and war zones, which demand an