OPTIMAL DESIGN OF SPECTRUM CONSTRAINED SIGNAL SETS WITH CORRELATION ANALYSIS Tao Wang, Weiyong Yan * Dept of Electrical and Computer Engineering Curtin University of Technology GPO Box U 1987, WA 6840, Australia Zhuquan Zang † WATRI 35 Stirling Hwy, Crawley WA 6009 Australia ABSTRACT This paper is concerned with the design of an optimal set of analog signals with prescribed magnitude spectrum and quadratic phase structure such that the maximum cross-correlation is minimized. An analytic expression for the maximum cross-correlation between two signals is derived through mathematical analysis. The optimal set of signals with the lowest maximum cross-correlation is explic- itly characterized under certain conditions. 1. INTRODUCTION The design of signal sets with prescribed spectral properties and low values of correlation is an important element of modern mul- tidimensional signaling and multiuser communications system de- sign [3, 6, 9]. Such a design also plays a vital role in many other areas of signal processing such as multi-target detection in radar and sonar systems [1,3,7]. Considerable effort has been devoted to synthesizing signal sets with low values of cross-correlation at all lags and low values of auto-correlation at nonzero lags [2, 8]. Suc- cessful design of signal sets with these characteristics is desirable because of the needs for increasing the number of simultaneous access users and the reduction of inter-symbol interference and co- channel interference in Code Division Multiple Access (CDMA) communication systems [5, 9]. For target detection in radar and sonar systems, the signal-to-noise ratio (SNR) can be improved significantly by the proper choice of signal sets [1]. In order to maximize the SNR at the output of the receiver, one effective ap- proach is to shape the transmitted signal to be the inverse backscat- tering spectrum [1], which has an approximately rectangular shape in both time and frequency domains. In [3], a specific set of sig- nals, which have unit energy, a constant passband magnitude and quadratic phase structure in frequency domain, has been investi- gated. The reason for designing such signals is that the shapes of the complex envelopes of the signals are approximately rectangu- lar in both time and frequency domain, and the magnitude of the cross-correlation function of the signal set can be less than that of other signal sets such as Gold sequences or Kasami sequences [8]. The signal sets characterized in [3] have the property that the co- efficients of their quadratic phase function lie on an ellipse which is uniquely determined by the bandwidth and time duration of the signals. As such, for a given signal set with specific bandwidth and ∗ The work of this author was partially supported by a research grant from the Australian Research Council † The work of this author was partially supported by a research grant from Curtin University of Technology time duration, the cross-correlation magnitude becomes a major criterion for measuring the performance and quality of the signal set. In practice, it is always desirable to make the maximum cross- correlation of a signal set as small as possible. In [3], a very simple upper bound for the maximum cross-correlation between two sig- nals was established, although no method for designing an optimal signal set was given. In [4], it was further shown that the upper bound can be minimized simply by selecting the coefficients in such a way that they are equally spaced along the horizontal axis of an ellipse. In addition, a new and tighter upper bound on the maximum cross-correlation magnitude between two signals was derived. However, this bound is almost as hard to compute as the original maximum cross-correlation. In this paper, the problem of computing and minimizing the maximum cross-correlation magnitude is investigated. It is shown that under certain conditions on the design parameters, the maxi- mum cross-correlation magnitude itself can be expressed in closed form. In other words, an analytic expression for the maximum cross-correlation can be mathematically established which allows an efficient design method to be devised for designing a set of optimal signals with the prescribed spectrum and correlation prop- erties. 2. PRELIMINARIES 2.1. Signal set design problem The signal set design problem is to find a set of signals which possess prescribed properties in both time and frequency domains. Mathematically, the signal set design problem [3] can be stated as follows. Design a set of signals s = {s i (t); i =1, 2, ··· ,N } de- fined over  − T 2 , T 2  with their corresponding Fourier transforms S = {Si (f ); i =1, 2, ··· ,N } satisfying the following proper- ties:  + T 2 − T 2 |si (t)| 2 dt = 1; i =1, 2, ··· ,N (1) |R i,j (τ )|≤ δ; − T ≤ τ ≤ T ; i = j (2) |S i (f )| =  α(f ); |f |≤ W ; εi (f ); |f | >W ; i =1, 2, ··· ,N (3) where R i,j (τ ) is the cross-correlation function between signals s i (t) and s j (t) defined by R i,j (τ )=  + T 2 − T 2 s i (t)s ∗ j (t − τ )dt; − T ≤ τ ≤ T (4) III - 949 0-7803-8874-7/05/$20.00 ©2005 IEEE ICASSP 2005 ➠ ➡