Recursive Decimation/ Interpolation for ML Chirp Parameter Estimation PAUL M. BAGGENSTOSS Naval Undersea Warfare Center We present a fast implementation of the optimal linear processor for “chirp” signals with O(N 2 ) computational load for data length N. The method is technically an approximation since it has an interpolation step but we show that the errors are negligible. In simulations it performs up to 20 times faster than the existing method using radix-2 fast Fourier transform (FFT) and has significantly lower threshold signal-to-noise ratio (SNR) against noise and interference compared with a leading fast method. Manuscript received October 26, 2012; revised February 7, 2013, April 22, 2013; released for publication May 13, 2013. IEEE Log No. T-AES/50/1/944800. DOI. No. 10.1109/TAES.2013.120677. Refereeing of this contribution was handled by K. T. Wong. Author’s address: NavalUndersea Warfare Center, 1176 Howell St., Newport, RI 02841. E-mail: (p.m.baggenstoss@ieee.org). 0018-9251/14/$26.00 C  2014 IEEE I. INTRODUCTION A. Background and Prior Work The linear frequency modulated (LFM) waveform or “chirp” is extensively used in SONAR and RADAR signal processing and time-series analysis. A well-studied problem is to estimate the frequency and frequency-rate (slope) of a chirp in noise [1–4]. The simplest and most intuitive approach is to correlate the data with a model signal, varying the parameters until the highest correlation is obtained [2, 5]. In the presence of white Gaussian noise, this is the maximum likelihood (ML) processor which achieves the Cramer-Rao lower bound (CRLB) in the presence of noise [6] and has superior signal-to-noise ratio (SNR) threshold effect [7–9]. It has, however, a high computational load of O(N 3 ), where N is the data length in samples, which can be reduced to O(N 2 log 2 N) with the fast Fourier transform (FFT). Because of the high computational load of ML and the need to search over the entire parameter space, the problem has been the subject of many and varied papers. These can be grouped into the following general categories. 1) Specialized fast algorithms for chirp estimation, such as the fast chirp transform [10], or fractional Fourier transform [11], but these also achieve O(N 2 log N). 2) Generalizations of the short-time Fourier transforms (STFT) for polynomial-phase signals [12]. This approach is computationally demanding and, like the STFT, uses a shorter transform than the signal length, giving it a signal-to-noise disadvantage. 3) Algorithms that integrate trajectories in Wigner or related time-frequency (TF) distributions [13–17]. These methods are versatile and of important theoretical value but they require calculating the TF distribution which adds significant processing. 4) A set of similar algorithms called discrete chirp Fourier transform (DCFT) [18, 19], discrete quadratic phase transform (DQPT) [20], and polynomial time-frequency transform (PTFT) [21, 22]. Despite impressive claims of computational reduction, the algorithms require integer slope values. Since useful chirp signals use fractional values of l between –1/2 and 1/2, this transform that uses integer l is of little value for our purposes. 5) Nonlinear algorithms that attain fast performance by initially searching only over the one-dimensional space of frequency rate [3, 5, 8, 23–29]. These algorithms are O(N log 2 N), can attain the CRLB estimation error variance at higher SNR, and can be extended to higher order polynomial-phase signals. However, because they are based on quadratic functions of the data or are based on phase information, they require a higher SNR against noise and interference to work properly. 6) Various ML-based methods including approximate ML approaches [30] with a threshold performance penalty and a modified ML method that has better smoothness for easier optimization [9]. IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 50, NO. 1 JANUARY 2014 445