IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 1, JANUARY 2006 51 A Subspace Theory for Differential Chaos-Shift Keying Arnt-Børre Salberg, Member, IEEE, and Alfred Hanssen, Member, IEEE Abstract—In this brief, we introduce a subspace theory for differential chaos-shift keying (DCSK) systems. We show that DCSK systems operate by transmission of chaotic signals residing in a low-dimensional subspace. The subspace formalism of DCSK schemes leads to the derivation of useful subspace detectors that can be applied to decode the DCSK signal for various types of channels. Closed form expressions for the bit error rate is derived for an -ary FM-version of DCSK, under the assumption of orthogonal subspace generating vectors. Numerical simulations demonstrate that the proposed subspace detector in general outperforms the conventional correlation detector for DCSK. Index Terms—Differential chaos shift keying (DCSK), subspace detectors, subspace theory. I. INTRODUCTION D IFFERENTIAL chaos-shift keying (DCSK) [1]–[3] has been proposed to overcome the principal phase-synchro- nization problem of chaotic communications schemes. DCSK and frequency-modulated DCSK (FM-DCSK) are based on the transmitted reference principle [4], [5], which was proposed in the early days of spread spectrum (see e.g. [6]). Transmitted reference systems accomplish detection of the (unpredictable) wideband carrier by transmitting two versions of the carrier, each version in a separate channel. In DCSK (and FM-DCSK) the two channels are formed by time division. In the recent years, the DCSK communications principle has been extended from simple binary communications [7], and more sophisticated detectors which exploit the repetitive nature of the chaotic signal have been proposed [7]. Moreover, mul- tipath performance of binary DCSK systems using a standard correlation detector was characterized in [8], [9]. In [9] and [10] it was shown that by combining the chaotic waveform with Walsh functions, binary DCSK may be mod- eled by means of two orthogonal basis functions, where each basis function represents a given bit. The chaotic signals repre- senting bit “0” and bit “1” are then orthogonal to each other for all chaotic sample functions, i.e., the cross correlation between the two chaotic signals is equal to zero. In fact, we show that the chaotic signal representing a given symbol is a subspace signal in the sense that it is restricted to lie in a subspace determined by the symbol that is being trans- mitted, and hence, each symbol is associated with an entire sub- Manuscript received September 13, 2004; revised February 18, 2005. This paper was recommended by Associate Editor F. C. M. Lau. A.-B. Salberg is with the Institute of Marine Research, NO-9294 Tromsø, Norway (e-mail: arntbs@imr.no). A. Hanssen is with the Department of Physics, University of Tromsø, NO-9037 Tromsø, Norway (email: alfred@phys.uit.no). Digital Object Identifier 10.1109/TCSII.2005.854590 Fig. 1. Selected values of the DCSK signals. (a) DCSK. (b) FM-DCSK. space of chaotic waveforms rather than a single, fixed wave- form. The subspace formalism of the transmitted chaotic wave- form leads to useful geometric insight, and suggests the use of subspace detectors [11], [12] at the receiver. We show that the subspace detector is the resulting detector in the case of a multi- path channel, and that it is independent of the channel parameter vector. Moreover, the subspace detector is also the natural de- tector to use in more general subspace communicating systems [12]. II. SUBSPACE FORMALISM FOR DCSK The basic idea of the binary DCSK modulation technique is that every information bit to be transmitted is represented by two chaotic sequences in succession, a reference sequence, and an information sequence. To increase the efficiency, instead of transmitting only one information bearing signal after the refer- ence signal, bits will be transmitted using the same refer- ence sequence [7]. Let be a -dimensional vector of the form . Using Kronecker products, the discrete-time DCSK sequence , repre- senting symbol , may be expressed as (1) where is a vector whose elements constitute the chaotic sequence, is the transmitted symbol at time instant , is an subspace matrix which is built from the vector , is the unity matrix, and . Equation (1) reveals an important geometrical property about DCSK. Since is a tall matrix, the transmitted symbol vector will be a subspace signal since it is restricted to lie in the subspace spanned by the columns of . Furthermore, this subspace, denoted by , is completely determined by the subspace generating vector . If the vectors are chosen to be orthonormal (which can be accomplished if only subspace generating vec- tors are used), then , where 1057-7130/$20.00 © 2006 IEEE