Ambiguity properties of quadratic congruential coding

The ambiguity characteristics of multiple access frequency hop codes based on standard quadratic congruences are investigated in the light of results obtained for codes based on Costas arrays and extended quadratic congruences. While the autoambiguity properties are found to be very similar to those of Costa codes, i.e. nearly ideal, the cross-ambiguity properties of quadratic congruential codes are much better. These results are valid across the whole class of code sets considered, but they are obtained at some expense in the pulse compression characteristics of the codes. A uniform upper bound is placed on the entire cross-ambiguity function surface, and bounds are placed on the amplitude of spurious peaks in the autoambiguity function. These bounds depend on the time-bandwidth product and code length exclusively and lead naturally to a discussion of the design tradeoffs for these two parameters. Examples of typical autoambiguity and cross-ambiguity functions are given to illustrate the performance of quadratic congruential coding with respect to Costas coding     

Time-frequency hop codes based upon extended quadratic congruences

Time-frequency hop codes are developed that can be used for coherent multiuser echolocation and asynchronous spread spectrum communication systems. They represent a compromise between Costas codes, which have nearly ideal autoambiguity but not so good cross-ambiguity properties, and linear congruential codes, which have nearly ideal cross-ambiguity but unattractive autoambiguity properties. Extended quadratic congruential code words are shown to have reasonably good autoambiguity and cross-ambiguity properties across the whole class of code sets considered. A uniform upper bound is placed on the entire cross-ambiguity function surface, and bounds are placed on the position and amplitude of spurious peaks in the autoambiguity function. These bounds depend on the time/bandwidth product and code length exclusively and lead naturally to a discussion of the design tradeoffs for these two parameters. Examples of typical autoambiguity and cross-ambiguity functions are given to illustrate the performance of the new codes     

Costas array generation and search methodology

Costas arrays are permutation matrices that provide sequencing schemes for frequency hop in FSK waveforms. Such frequency-shift keying (FSK) waveforms can be designed to have nearly ideal ambiguity function properties in both the time and frequency directions: the Costas property permits at most one coincident tone in autocorrelations in both time and frequency. Costas arrays are found by number-theoretic generators and their extensions, and by exhaustive search methods. Two new extensions of number-theoretic methods are introduced here that find two new Costas arrays. All Costas arrays for orders 24, 25, and 26 are disclosed here, including previously unknown examples.     

Cross-correlation properties of algebraically constructed Costasarrays

The problem of determining the cross-correlation properties of signals based on algebraically constructed Costas arrays is addressed by examining the discrete cross-correlation of the algebraically constructed Costas arrays for a given construction and dimension. Finding two arrays that minimally correlate implies that the signals based on these arrays also minimally correlate. The properties of finite fields are reviewed, and the major algebraic constructions for Costas arrays are presented, i.e. the Welch construction and the Golomb construction. The discrete cross-correlation properties of the Costas arrays are derived for arrays of the same dimension derived from the same construction. The use of Costas arrays in the signal design problem is discussed, and examples are given to show the cross-correlation of the signals based on the algebraically constructed arrays     

Frequency hop multiple access codes based upon the theory of cubiccongruences

The need for families of frequency-hop codes which have mutually small auto-ambiguity and cross-ambiguity functions is discussed. Current coding methods are reviewed. A new family of frequency-hop codes based upon the number-theoretic concept of cubic congruences is introduced. It is shown that for about 50% of the prime numbers, families of full codes exist which have at most two coincidences for any time-frequency shift in their auto-ambiguity functions and at most three coincidences in the set of mutual cross-ambiguity functions     

Periodic ambiguity functions of EQC-based TFHC

The time-frequency hop codes based upon extended quadratic congruences (TFHC-EQC) were developed previously (1988, 1991). The known results about the ambiguity functions of the TFHC-EQC are extended from aperiodic cases to periodic cases. Precisely, the following new results are proved: (1) The tight upper bound for the periodic auto-ambiguity function of the TFHC-EQC is 6; (2) the periodic cross-ambiguity upper bound of the TFHC-EQC is at least (N-1)/2; (3) there exists a subset TFHC-EQC of volume (N-1)/2 such that its periodic cross-ambiguity is tightly upper bounded by 8     

Costas arrays with small number of cross-coincidences

A simple method is described for obtaining two or more Costas arrays with a maximum of one mutual bit or cross-coincidence, assuming that the vertical components of their relative shifts are appropriately bounded. The result can be useful for designing multiple Costas waveforms with small cross-ambiguities, if the potential Doppler shifts are restricted to a small portion of the waveform bandwidth     

Multiple Access Frequency Hopping Patterns with Low Ambiguity

A technique is discussed for the design of multiple access frequency hopping patterns which will allow a number of users to communicate reliably over a limited bandwidth in the presence of Doppler and timing asynchronies. The patterns are based on a subset of the Reed-Solomon codewords. They possess nearly flat ambiguity and cross-ambiguity functions and thus could be useful as discrete radar signals. In addition the structure of the patterns permits a straightforward Implementation.     

Ambiguity Functions for Quadratic Phase-Coded Pulse Trains

The matched filter ambiguity function is presented for a burst waveform composed of repeated subbursts, each one of which consists of N pulses in which the phase is varied quadratically from pulse to pulse. The resulting ambiguity function exhibits small residual ambiguities along the delay axis separated by the reciprocal of the pulse repetition frequency (PRF). A cross-ambiguity function is derived which reduces these ambiguities to zero amplitude. A third cross-ambiguity function is presented for a receiver matched to a generalized Hamming weighted repeated quadratic burst. The location in the delay/Doppler plane of the waveform ambiguities for these waveforms is compared with that of an uncoded pulse burst.     

Family of multicarrier bi-phase radar signals represented by ternary arrays

A K times L ternary array, comprised of the elements {0, 1, - 1}, with some unique features, represents a multicarrier radar signal with favorable autocorrelation and ambiguity functions. Constructing such an array using Galois fields is described. As in a Costas binary array, only one frequency is transmitted at any time slot, but in our array the same frequency is repeated in several time slots, yielding a signal with considerably larger pulse compression than a Costas signal that uses the same number of frequencies     

Generalized Response of a Linear FM Pulse Compression Matched Filter

A pulse compression matched filter is analyzed so that the response may be computed when the pulse width, FM rate, and center frequency simultaneously differ from design conditions. Unilateral and bilateral time domain amplitude weighting for sidelobe reduction is included. A general cross-ambiguity function is defined to include these effects and some basic computed results are presented for the peak envelope response with various degrees of Hamming weighting. Computer evaluation of this cross-ambiguity function allows one to choose a combination of mismatches for signal design trade-off between resolution and detection performance. Since no restrictions are placed upon the mismatch parameters, this analysis may also be used to evaluate the filter discrimination against various interfering signals.     

The hit array: an analysis formalism for multiple access frequencyhop coding

A formalism is presented for the analysis of general frequency hop waveforms, such as those suitable for use in coherent active radar and sonar echolocation systems as well as multiple-access spread-spectrum communications. This formalism is based on the concept of coincidence, or `hit', between two frequency hopping patterns. The collection of all possible hits, together with their locations, is recorded in time-frequency space, which produces the high array associated with the two patterns considered. If the code length is sufficiently small with respect to the time-bandwidth product chosen, the hit array can be viewed as a digital representation of the corresponding ambiguity function. Salient properties of the hit array formalism are derived, including simple relationships between hit arrays resulting from basic symmetry-preserving transformations. These properties make it possible to predict the performance of a given set of frequency hop waveforms directly from the associated set of frequency hopping patterns     

Efficient wideband signal parameter estimation using a radon-ambiguity transform slice

A novel efficient technique based on a single slice Radon-ambiguity transform (RAT) for time-delay and time-scale estimation is proposed. The proposed approach combines the narrowband cross-ambiguity function (NBCAF), the wideband cross-ambiguity function (WBCAF), and a single slice RAT to estimate multiple target parameters in noisy environments. The square modulus of Gaussian-enveloped linear frequency modulated (GLFM) signals has high-energy centrality in the ambiguity plane. Its peaks in the NBCAF fall along nearly straight lines whose slopes depend on the Doppler rates of the moving targets. These lines could be effectively detected by computing the entire Radon transform of the NBCAF for all possible angles; however, it is a computationally intensive procedure. It is shown that without calculating the entire RAT, it is possible to estimate target parameters using only a single slice of the RAT, i.e., using an appropriate projection of the NBCAF. It is demonstrated that the proposed method can successfully separate overlapping targets efficiently. The efficiency is achieved due to fast Fourier transform (FFT)-bascd processing, use of a single slice of RAT, and the use of only one-dimensional (1-D) searches.     

Staggered Costas Signals

A radar signal, based on coherent processing of a train of staggered Costas bursts, is suggested and investigated. The selection of sequences of each burst is based on a minimum number of collocation of their individual ambiguity function sidelobe peaks. The resulting ambiguity function combines qualities of both "thumbtack" and "bed of nails" signals. Comparison with linear-FM, V-FM, and complementary phase coded (CPC) signals is given, as well as comparison with hybrid signals consisting of both phase and frequency coding.     

Cochannel Interference Analysis between Spaceborne and Terrestrial Radars

An analysis of cochannel interference between spaceborne and terrestrial radars is presented. A generic modeling methodology is applied, which, it is felt, will represent the worst-case situation. In order to account for Doppler shifts, auto-and cross-ambiguity functions ions have been developed for determining matched filter outputs. The results support the ability of such radar systems to coexist relatively harmoniously utilizing currently accepted assignment procedures.     

Group-Complementary Array Coding for Radar Clutter Rejection

A radar waveform design technique which utilizes Lagrange's method of multipliers to control temporal sidelobes and to reduce Doppler sidelobes is described. This classical method of constrained optimization is applied to the problem of synthesizing a radar wave-form where mismatch loss is the objective function to be minimized. The associated constraints are taken from expressions for the composite temporal sidelobes of the cross-correlation response and the peak correlation response where sets of code words are used to modulate a series of radar pulses. The resulting code sets and receiver reference sets are called group-complementary and produce a trench parallel to or on the range axis of the cross-ambiguity surface.     

Performance Degradation of MTI Circuits Due to Amplitude Limiting

The loss in output signal-to-noise ratio (SNR) due to amplitude limiting is obtained for a radar circuit consisting of a bandpass limiter, coherent demodulator, matched filter, and moving-target-indicator (MTI) filter. The circuit is used in scanning MTI radars. The tandem connection of the limiter and coherent demodulator is a model for the saturation of the intermediate-frequency (IF) demodulator of an MTI radar. Results on special functions are used to obtain simple formulas for the loss in output SNR relative to a linear IF demodulator when the input SNR is less than -15 dB and the number of hits per 3-dB beamwidth exceeds 15.     

Costas编码信号的循环谱特征分析与参数估计

研究了准确反映Costas编码信号特点的关键参数集合。基于循环谱相关密度估计方法,分析了Costas编码的循环谱特征及其与辐射源侍号参数的对应关系,提出了一种没有先验知识条件下Costas编码截获信号特征提取方法。通过对典型的Costas编码信号参数提取归一化均方根误差的计算,验证了该算法的有效性。     

基于FPGA的数字Costas环的设计与实现

通过对数字Costas环路原理的分析，提出了新的环路鉴相方式，且环路参数可根据锁定情况及时修改，确保锁相环路稳定工作。详细阐述了环路各部件的参数及电路设计。ChipScopePro实时观测FPGA内部信号和Matlab仿真结果表明．该Costas环路具有十分优良的性能。     

Sequences and arrays with perfect periodic correlation

Properties and methods for synthesizing sequences with perfect periodic autocorrelation functions and good energy efficiency are discussed. The construction is extended to two-dimensional perfect arrays. The construction methods used are based mainly on a search in the frequency domain and on a multiplication theorem for periodic sequences and arrays