a) 4 repeats of a 7-bit Barker code c) spectrum b) autocovariance 0 5 10 20 25 Time (ms) Lag (ms) Frequency (Hz) -1000 1000 20 40 60 -60 -40 -20 Figure 1. a) Schematic Representation of a repeat sequence code. The transmitted waveform consists of a seven bit Barker code repeated four times. b) The corresponding autocorrelation function. Values are small except at even multiples of the subcode length. c) The frequency spectrum of the code. The spectrum of an uncoded sinusoidal pulse of the same duration would consist of a single peak of width equal to the width of any individual peak in the above spectrum. Improvement of Doppler Estimation through Repeat-Sequence Coding J. A. Smith and R. Pinkel Scripps Institution of Oceanography, La Jolla, CA 92093-0213 Abstract Repeat sequence coding offers a robust method for improving the precision of velocity estimates from incoherent Doppler sounders, while retaining the simplicity of the complex covariance estimation technique. This method involves transmitting a number of repeats of a broad band "subcode." The Doppler shift is estimated from the complex autocovariance value at a time lag equal to the subcode duration. The repeat sequence code is an extension of the simple pulse trains developed in the early days of radar. By transmitting codes, rather than discrete pulses, greater average transmitted power is achieved. The performance is enhanced roughly in proportion to the usable bandwidth. Recent field work demonstrates performance improvements in both open ocean and shallow water applications. 1. Introduction Pulse to pulse incoherent sonar was first considered for oceanographic research by Emmanuel and Mandics. 1 Development of prototype systems followed. 2,3 Performance bounds on simple Doppler systems can be calculated using methods derived for atmospheric acoustic and radar research. 4,5 Theriault 6 gives a simple approximate relation for the incoherent sonar ∆V ∆R = K 1 c 2 f -1 P -1/2 (1) where ∆V is the rms velocity imprecision ∆R is the effective range resolution ≈ cT/2 T is the duration of the transmitted pulse c is the speed of sound f is the acoustic frequency P is the number of independent incoherent averages used in forming the velocity estimate (e.g., the number of transmissions). K 1 is a constant. With f in Hz, K 1 is ideally (8π) -1 . Velocity precision increases with acoustic operating frequency. However, both acoustic attenuation and background ambient noise increase with increasing frequency in the sea, at frequencies above 90 kHz. These factors give rise to a tradeoff between the maximum range achievable with a given system and the expected range-velocity precision. The only way to improve on this is to increase the information content of the returning echo. Here a simple coding scheme is investigated: repeat sequence coding. While the method is sub-optimal from a signal processing viewpoint, it is easy to implement and the performance improvement survives signal distortion by the real ocean. Repeat sequence codes are produced by taking broad-band "subcodes" and repeating them sequentially M times. A variety of codes have been developed with the objective of having minimal self- correlation, except at zero lag. We consider here codes generated by reversing the sign of the carrier frequency of the sonar at controlled intervals. The sonar output signal can be thought of as a sinusoid which is multiplied by either 1 or -1, depending on the dictates of the code. The seven-bit Barker code is shown in Fig. 1 as an example. 7 For a digitally synthesized subcode of bandwidth τ -1 and duration Lτ, the time bandwidth product is the number of "bits" in the subcode, L. The time bandwidth product of the overall transmission is ML. If the subcode is properly chosen, the autocorrelation of the overall transmission is nearly zero, except at lags t = nLτ, n = 0 to M-1 (Fig. 1b). The number of peaks in the code spectrum is roughly proportional to the number of bits, L, in the subcode (Fig. 1c). Repeat sequence codes are contemporary analogs of the simple "pulse train" sequences developed in the early days of radar. 5,7 A pulse train is a sequence of M sinusoidal pulses, each of length τ, transmitted at intervals of Lτ. The pulse train has an autocorrelation which is exactly zero except at lags NLτ, N=1 to M-1. These pulse trains represent an ideal limit of the repeat sequence codes discussed here. However, the average power transmitted in a pulse train is low relative to the peak power. Repeat sequence codes approximate the pulse train ideal, but the average power transmitted is a factor of L greater than the pulse train, with little loss in velocity precision. This can extend the usable range of a practical sounder. Repeat sequence codes also generalize the pulse-pair approach investigated by Edwards and others. 5,8 Here we consider the option of repeating the pulse or sequence more than just once. This enables the overall transmission length to be specified independent of the maximum unambiguous velocity, which is determined by the interpulse spacing, Lτ. This generalization is useful in adapting systems to different environmental conditions. 2. Performance Assessment Methods for estimating a Cramer-Rao lower bound for precision are well known. 4,5,6,9 Here we take a different approach, based on the sample error inherent to covariance estimation from real data. This approach is in some ways more appealing, since it applies directly to the technique employed. The results explain part of the discrepency between observed error levels and the Cramer-Rao lower bound estimate. a) Sample Error Consider first a filtered, discretely sampled echo, from an uncoded transmission of duration Mτ, where τ is the sample interval. We conceptually divide the echo into "range cells" of length cτ/2. The echo from each cell typically arises from sums over many scatterers. Regardless of the actual distribution of scatterer cross-sections, the central limit theorem implies that the complex returns from these cells are nearly normally distributed. For spatially homogeneous cells with average echo