Unimodular (i.e., constant modulus) sequences with good autocorrelation properties are useful in several areas, including communications and radar. The integrated sidelobe level (ISL) of the correlation function is often used to express the goodness of the correlation properties of a given sequence. In this paper, we present several cyclic algorithms for the local minimization of ISL-related metrics. These cyclic algorithms can be initialized with a good existing sequence such as a Golomb sequence, a Frank sequence, or even a (pseudo)random sequence. To illustrate the performance of the proposed algorithms, we present a number of examples, including the design of sequences that have virtually zero autocorrelation sidelobes in a specified lag interval and of long sequences that could hardly be handled by means of other algorithms previously suggested in the literature.

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New Algorithms for Designing Unimodular Sequences With Good Correlation Properties

Active sensing applications such as radar, sonar and medical imaging, demand proper designs of the probing waveform. A well-synthesized waveform can significantly increase the system performance in terms of signal-to-interference ratio, spectrum containment, beampattern matching, target parameter estimation and so on. The focus of this work is on designing probing waveforms using computational algorithms. 
We first investigate designing waveforms with good correlation properties, which are widely useful in applications including range compression, channel estimation and spread spectrum. We consider both the design of a single sequence and that of a set of sequences, the former with only auto-correlations and the latter with auto- and cross-correlations. The proposed algorithms leverage FFT (fast Fourier transform) operations and can efficiently generate long sequences that were previously difficult to synthesize. We present a new derivation of the lower bound for sequence correlations that arises from the proposed algorithm framework. We show that such a lower bound can be closely approached by the newly designed sequences. 
A two-dimensional extension of the time-delay correlation function is the ambiguity function (AF) that involves a Doppler frequency shift. We give an overview of AF properties and discuss how to minimize AF sidelobes in a discrete formation. 
Besides good correlation properties, we also consider the stopband constraint that is required in the scenario of avoiding reserved frequency bands or strong electronic jammer. We present an algorithm that accounts for both correlation and stopband constraints. 
We finally consider transmit beampattern synthesis, particularly in the wideband case. We establish the relationship between a desired beampattern and underlying waveforms by using the Fourier transform. We highlight the increased design freedom resulting from the waveform diversity of a MIMO (multi-input multi-output) system.

Waveform Design for Active Sensing Systems: A Computational Approach

A merit factor based on the sequence autocorrelation function, whose minimization leads to the reduction in the Cramer-Rao lower bound (CRLB) for the variance of "two-sided" intersymbol interference (ISI) channel estimation is introduced. Pairs of binary pilot symbol sequences (a preamble and a postamble) for channel estimation are jointly designed to minimize this merit factor. Given that the number of channel taps is L and the length of a pilot symbol sequence is (N+L-1), where N/spl ges/L, we distinguish between the case when N is even and the case when it is odd. For even N, we show that complementary sequences not only minimize the merit factor, but also the CRLB. For a subset of odd N we construct almost-complementary periodic sequence pairs that minimize the merit factor. The optimal pilot symbol block signaling requires alternating between two (in most cases) different binary sequences that form the merit-minimizing pair.

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Complementary sequences for ISI channel estimation

Time-discrete signals with good autocorrelation properties are used in various applications in communications engineering. From an implementation point of view, usually sequences with a small phase alphabet and a maximal energy efficiency, i.e., a uniform envelope, are most favorable. For this reason, known methods as well as several new construction techniques of binary and quadriphase sequences with optimal or best known autocorrelation properties are discussed in this correspondence. In many cases, the achievable correlation properties can be improved significantly if a single zero element per sequence is accepted. These "almost" binary and "almost" quadriphase sequences are considered as well. The optimality criteria used include the maximum absolute sidelobe and the merit factor of the periodic and the odd-periodic autocorrelation function, respectively. For sequence lengths of up to 44 in the binary case and up to 32 in the quadriphase case, the best known parameters obtained by computer search are compared with the constructed results.

Binary and quadriphase sequences with optimal autocorrelation properties: a survey

This article's main goal is to show how to design the receive filter (including its length) and the transmit code sequence via the optimization of a number of relevant metrics considered separately or in combination. Several numerical studies are given to illustrate the performance of the presented designs. Compared with most of the previously published works on the subject, the present contribution is more coherent as well as more complete, and yet the approach taken here is generally simpler both conceptually and computationally. Finally, weremark on the fact that while the focus is on radar systems, for the sake of being specific, the theory and design methods presented should be useful to several other active sensing applications, such as sonar, nondestructive testing, seismic exploration, and biomedical imaging.

Transmit codes and receive filters for radar

A set of perfect correlation sequences is discussed. The sequences are binary with exception of a single zero element and possess a perfect odd-autocorrelation function. The construction uses an isomorphy to special conference matrices and gives sequences with all lengths M=p/sup a/+1, p odd prime, a=1,2,... . Applications in CW radar and in measuring mobile radio channels are described using the ambiguity function of multiple odd-perfect sequences. >

Odd-perfect, almost binary correlation sequences

Signals of two or more dimensions with ideal impulse-like periodic autocorrelation functions are used in higher-dimensional signal processing or in radar systems. Four synthesizing methods for perfect ternary arrays are presented. Based on known perfect binary and ternary sequences and arrays, many new perfect ternary arrays with increasing number of elements are constructed. New strong conditions for the existence of perfect arrays are developed. By combining these conditions with an advanced computer search, new ternary arrays are found. These new ternary arrays can be used as starting arrays to construct additional perfect ternary sequences and arrays. >

Perfect ternary arrays

Perfect binary arrays

A pair of binary sequences is called complementary if the sum of their aperiodic autocorrelation functions is zero except for zero shift Such sequences are known for relatively few lengths This correspondence therefore introduces pairs of binary odd-periodic complementary sequences, which can be constructed using q-ary m-sequences for many more lengths

Hans Dieter Lüke

Papers

Binary and quadriphase sequences with optimal autocorrelation properties: a survey

Odd-perfect, almost binary correlation sequences

Perfect ternary arrays

Perfect binary arrays

Binary odd-periodic complementary sequences