OPTIMAL WAVEFORM PRECODER DESIGN FOR NARROWBAND MIMO RADAR SYSTEMS Luke A Balzan and Langford B White School of Electrical and Electronic Engineering The University of Adelaide North Terrace Adelaide, SA, 5005 e: {lbalzan,lwhite}@eleceng.adelaide.edu.au ABSTRACT This paper addresses the problem of coded waveform design for Multiple Input - Multiple Output (MIMO) radar systems. The paper proposes a signal model based on the assumption of a small array aperture relative to the signal wavelength and a single target in far field. It is argued that near maximum likelihood performance can be obtained by choosing the transmitted waveforms on different anten- nas to be orthogonal. The Cram´ er-Rao Bound (CRB) for estimat- ing the target distance, radial velocity and wavenumber is derived. The paper presents a convex optimisation procedure to yield an op- timal energy distribution across the transmitted signals. Simulations demonstrate a reduction in CRB in the order of 25% compared to the use of a random unitary code. The paper concludes by indicating a natural generalisation of the method to optimal design for MIMO tracking radars. Index Terms— MIMO, radar, tracking, optimisation, diversity 1. INTRODUCTION The past few years has seen a significant rise in interest in the con- cept of multiple input - multiple output (MIMO) radar. Consider- able research has demonstrated that the ideas behind a MIMO radar system offer a range of improvements in radar system performance, particularly for detection, estimation, and now tracking. This current research has focused on the consideration of receiver design [1] and transmitter design [2], where the imposed orthogonality of transmit- ter codes allowed for the simple derivation of a maximum likelihood (ML) model for MIMO radar, and the associated Fisher Information (FI). This was developed with a view to finding optimal transmitter codes that minimise estimation variance and tracking error, follow- ing the ideas first proposed by Kershaw and Evans in [3]. At the same time, there has been a growing bulk of research covering all aspects of MIMO radar, often with differing underlying models. Models for MIMO radar as proposed in [4]–[5] make use of large antenna arrays with widely spaced elements, thus capitalising on spatial diversity to improve target estimation. The model pro- posed in this current research is more similar to that in [6], where closer inter-element spacing is employed, and transmitter and re- ceiver arrays are co-located. An assumption of limited array aperture relative to the signal wavelength is a key one made in our model. The assumption of co-location of transmit and receive arrays may be generalised. A key motivation of this current research has been to find optimal waveforms for transmission, with the FI playing an important role in this task. The research in [7]–[8] similarly uses the FI in selecting waveforms, though differs significantly from this current research, where the approach is intimately linked with and motivated by tracking. The application of convex optimisation in areas of signal pro- cessing has also accumulated considerable research interest in recent years. Fast and efficient methods, such as interior point algorithms, have been developed and can be applied to a range of problems in signal processing [9, 10]. Convex optimisation techniques have been considered for the task of optimal waveform design in this current research, and are discussed in greater detail in section 3. The remainder of this paper is organised as follows: the pro- posed model for MIMO radar is described in section 2, with the ML estimator derived in 2.1. Sections 2.2 and 2.3 discuss the Cram´ er- Rao Bound (CRB) and necessary waveform constraints respectively. The optimisation problem, where waveform codes are chosen to minimise the CRB, is introduced in section 3, with an example given in section 4. Section 5 proposes an extension of the optimisation task to the tracking case, and outlines future research. Finally, section 6 concludes the paper. 2. SIGNAL MODEL The proposed radar system consists of N transmitting and N re- ceiving antennas. For the purpose of this paper and for simplicity, the antennas are considered to be arranged in a linear, equispaced array with close ( λ 2 ) inter-element spacing, and both transmitting and receiving antennas are co-located. Other configurations, such as bistatic and multistatic, also can also be described to fit the proposed system, though are not discussed here. Successive temporal blocks of M pulses will be transmitted from each antenna, each pulse of length T seconds. The transmitted signals have the form s n,k (t)= M-1  m=0 Xn,m(k) gn(t − (k ¯ M + m)T ) (1) for n =0,...,N − 1 where Xn,m(k) is the complex code trans- mitted on pulse m by antenna n on temporal block k, ¯ MT is the block-pulse repetition interval (all M pulses in a block are repeated every PRI ¯ MT ), and gn is the pulse shaping function on transmitter n. The pulse shaping function is assumed to be real and twice dif- ferentiable, is supported on [0,T ], and has gn(0) = gn(T )=0. We also assume that gn =1 for all n. A single point target is considered at a far-field location r metres from the reference antenna (at the origin), travelling at radial veloc- ity ˙ r, and with wavenumber ρ. The (two-way) delay and Doppler parameters of the target relative to receiving antenna j and transmit- 517 978-1-4244-2710-9/09/$25.00 c  2009 IEEE