928 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 2, FEBRUARY 2010 Adaptive Design of OFDM Radar Signal With Improved Wideband Ambiguity Function Satyabrata Sen and Arye Nehorai, Fellow, IEEE Abstract—We propose an adaptive technique to design the spectrum of an orthogonal frequency division multiplexing (OFDM) waveform to improve the radar’s wideband ambiguity function (WAF). The adaptive OFDM signal yields a better auto-correlation function (ACF) that results into an improved delay (range) resolution for the radar system. First, we develop a mutlicarrier OFDM signal model and the corresponding WAF at the output of the matched filter, emphasizing that the received signal depends on the scattering parameters of the target. Then, we devise an optimization procedure to select the OFDM waveform such that the volume of the corresponding WAF best approximates the volume of a desired ambiguity function. We demonstrate the improvement in the resulting ambiguity function, along with the associated ACF, through numerical examples. We find that the optimization algorithm puts more signal energy at subcarriers in which the target response is weaker. Index Terms—Adaptive waveform design, multifrequency scattering, OFDM radar, wideband ambiguity function. I. INTRODUCTION In this correspondence, we consider a mutlifrequency radar that employs an orthogonal frequency division multiplexing (OFDM) signal [1], and we compute its wideband ambiguity function (WAF) [2], [3] including the effects of the target response on the received signal. The motivation for employing multiple frequencies is that the different scattering centers of a target resonate differently at each frequency, and this also allows us to demonstrate the effects of target response on the WAF. Moreover, the use of a mutlicarrier OFDM signal improves the delay-resolution by a factor equal to the number of subcarriers [4], [5, Ch. 11]. In addition, we propose an algorithm to design the spectrum of the transmitting OFDM signal adaptively in order to improve its ambiguity profile. A. Background The advantage of multicarrier radar signaling has been well estab- lished in various applications, such as remote sensing of clouds and precipitation [6], detection of landmines [7], interpretation of an urban scene [8], etc. One of the ways to accomplish simultaneous use of sev- eral subcarriers is the OFDM signaling scheme, which employs mul- tiple orthogonal signals in the time domain [9]. Although OFDM has been elaborately studied and commercialized in the digital communi- cation field [10], it has not so widely been studied by the radar com- munity apart from a few recent efforts [11]–[14]. The ambiguity function for radar was originally introduced by Ville [15]; however, it is generally referred to as Woodward’s ambiguity Manuscript received May 05, 2009; accepted August 23, 2009. First pub- lished September 18, 2009; current version published January 13, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Sergiy A. Vorobyov. This work was supported by the De- partment of Defense under the Air Force Office of Scientific Research MURI Grant FA9550-05-1-0443, and ONR Grant N000140810849. The authors are with the Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130 USA (e-mail: ssen3@ese.wustl.edu; nehorai@ese.wustl.edu). Color versions of one or more of the figures in this correspondence are avail- able online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2009.2032456 function because of his popular work [16], [17]. According to Wood- ward, an ambiguity function is defined as a two-dimensional correla- tion between the transmitted narrowband signal and its time-delayed (related to target range) and frequency-shifted (related to target ve- locity) received version. Several literature interpret the ambiguity func- tion as a matched filter response [18, Ch. 4], [19, Ch. 11], [20, Ch. 5], whereas a few others as a two-dimensional point-spread function [21], [22]. But in general, these formulations of the ambiguity function ei- ther do not include a scattering coefficient of the target in the received signal model, or they assume identical values for the scattering coeffi- cients corresponding to different directions and/or frequencies. In our work, we follow a similar formulation of the ambiguity function but only after including the effect of different target responses at different frequencies. Additionally, Woodward’s version of the ambiguity function does not hold for large-bandwidth signals, such as OFDM signals. Target movements result in either expansion or compression in time for the wideband transmitted signal, and this effect can no longer be approx- imated by a simple “shift” in frequency. Therefore, in this work we follow the wideband ambiguity function (WAF) introduced by Kelley- Wishner [2] and Speiser [3]. Different properties of the WAF, similar to those of its narrowband counterpart, can be found in [23]–[26]. B. Outline In Section II we describe the parametric models of the transmitted and received signals; then we compute their WAF. We emphasize that the received signal and hence the corresponding WAF at the output of the matched filter depend on the scattering parameters of the target. In Section III we propose an optimization algorithm to compute an adap- tive OFDM waveform such that the volume of the corresponding WAF best approximates that of the desired ambiguity function. Our numer- ical examples, presented in Section IV, demonstrate the advantage of such an adaptive waveform design. Section V contains the conclusions and highlights of possible future work. II. SIGNAL MODEL AND WIDEBAND AMBIGUITY FUNCTION In this section, we first introduce the transmitted and received signal models of an OFDM signaling system. Along with the delay and Doppler effects, the received signal model also incorporates the scattering coefficients of the target at multiple frequencies. Then, we compute the expressions of WAF for a single pulse and a coherent pulse train. A. Signal Model We consider a monostatic radar employing an OFDM signaling system [9] with active subcarriers, a bandwidth of Hz, and pulse duration of seconds. Let contain the complex weights transmitted over different subcarriers, satisfying . Then the complex envelope of a single pulse can be represented as (1) and denotes the subcarrier spacing. Let be the carrier frequency of operation, the transmitted signal is given by (2) where represents the th subcarrier frequency. 1053-587X/$26.00 © 2010 IEEE Authorized licensed use limited to: WASHINGTON UNIVERSITY LIBRARIES. Downloaded on January 21, 2010 at 11:34 from IEEE Xplore. Restrictions apply.