ESTIMATION OF PULSE PARAMETERS BY CONVOLUTION Y.T. Chan', B.H. Lee', R. Inkol2, F. Chan' oDepartment ofElectrical and Computer Engineering Royal Military College of Canada, Kingston, Ontario, Canada email: {chan-yt, lee-h, chan-f}grmc.ca 2Defence Research and Development Canada, Ottawa, Ontario, Canada email: robert.inkolgdrdc-rddc.gc.ca Abstract The time-of-arrival (TOA) estimation ofpulse signals is widely used in radar, sonar and other sensor systems for geo-locating targets. This paper presents a new estimator for the TOA and width (W) of a pulse. Three auto-convolutions of a full and partial pulse determine the location of three convolution peaks. These peak locations are given by linear equations that contain as unknowns TOA and W. Solving these equations yields estimates of the pulse parameters. Simulation results show the performance of the estimator in noise and are compared with results obtained using an existing estimator. Keywords. Time-of-arrival estimation; pulse width estimation. 1. Introduction Assuming a direct propagation path, the propagation delay of a signal is proportional to the distance between a signal source and the receiver. Sensors, such as radar and sonar, make use of this relationship to geo-locate targets by measuring the time-of-arrival (TOA) of received signals. TOA estimation is also important in communications systems where it is often needed for recovering symbol timing information. Depending on the amount of available information on the signal s(t), there are three cases of TOA estimation: (A) Known s(t). The received signal is s(t - D), where D is the TOA. An estimate of D in noise is given by the time shift where the cross-correlation of s(t) and s(t - D) peaks. Examples are found in active radar, sonar and digital communications systems, where the transmitted signal s(t) is available as a reference. (B) Partially known s(t). There is prior knowledge that the signal is a pulse, or a pulsed sinusoid. The unknowns are the pulse width, its rise and fall characteristics, and the phase and frequency of the sinusoid. Examples are found in passive radar and sonar, and in frequency hopping communications signals. (C) Unknown s(t). The signal is completely random. Examples include an explosion detected by an acoustic sensor, or an underwater transient noise source. Due to the importance of radar, most TOA estimation references [1-6] deal with case (A). However, [4], which estimates TOA by measuring the time when the rising edge of a pulse exceeds a fixed fraction of the pulse amplitude, and [5], which determines the TOA as the instant when the slope of the 1-4244-0038-4 2006 IEEE CCECE/CCGEI, Ottawa, May 2006 leading edge of the pulse is a maximum, are applicable to case (B). Another approach, which also uses only the rising edge, [6] involves examining the ratios of successive amplitude samples to decide on TOA. Since it does not require a full pulse, [6] is applicable to case (B), and possibly (C) as well. There is a delay in TOA estimation for [4], in comparison with [5] and [6], since the pulse amplitude must be measured. There is no published work for case (C). Suppose a signal is detected in a data segment of length L. If it is assumed that the minimum signal duration for detection is at least L14 and that the TOA has a uniform distribution between 0 and L14, an obvious coarse estimate of TOA is 3L/8 and the error variance is 3L2/64. From the hierarchy of information availability, it is clear that an estimator for case (C) can include cases (A) and (B), and one for case (B) can include case (A). The reverse is not true. Note also that an estimator that utilizes all of the pulse information will give a smaller error than one that does not. For case (A), the cross-correlation or matched filter estimator is optimal [1], and others, for example [4-6], will yield sub- optimal results. There is no standard definition for the TOA of a signal. For case (A), the TOA is the centroid of the signal and for (B) and [4,5,6] it is a point on the rising edge of the pulse. For most applications, the exact definition is not important as long as it is consistent within a system. For example, source localization schemes often take the time-difference-of-arrival, (i.e., difference in TOA) at two spatially separated sensors, to determine a hyperbolic line of position for a signal source. The TOA estimator presented in Section 2 is for case (B), and specifically for a baseband pulse or a pulse with a sinusoidal carrier. With the presence of a pulse in a data segment, the estimator first takes an auto-convolution of the segment to find the location t of the convolution peak. The time t then becomes a reference for two additional auto- convolutions. The results are used to solve linear equations and obtain the TOA and pulse width. Note that if the pulse has a carrier, the envelope for TOA estimation can be extracted via a Hilbert transform. Section 3 contains the simulation results and compares the new estimator with [6] which checks the ratios of rising edge values for TOA estimation. The conclusions are given in Section 4. 17