Parameter estimation of random modulated chirp signals

Source or target tracking is an important problem in radar and sonar. If the target is moving with significant velocity, the received signal is nonstationary because of the time-varying delay. Usually the transmitted signal s(t) is narrowband with carrier frequency w, so that the delay does not distort the amplitude but affects only the carrier term. The signal received at the observation sensor is: x(t)=m(t)s(t-φ(t))+n(t)≈m(t)s(t)e(jw₀φ(t))+n(t) (1) where, n(t) is the zero-mean, stationary additive noise which models the sensor noise and other ocean interference. The transmitted signal s(t) undergoes random multiplicative transformations m(t) due to platform motion and vibration, ocean effects, and target scintillation. If the target or source is moving with constant acceleration, the time-varying delay is a second-order polynomial φ(t)=d_e+d _lt+d₂t², (2) with the coefficients related to its range, velocity, and acceleration. Hence, tracking the target is equivalent to estimating the coefficients of the phase polynomial φ(t). Such signals also arise in continuous phase modulation and passive array processing with moving sources. The process in (1) is nonstationary and has time-varying second and higher-order moments and cumulants. However, specific higher-order cumulants of it are (almost) periodic with the period being related to the polynomial coefficients. The Fourier Series coefficient of the time-varying correlation called the cyclic cumulant conveys information about the time-variation and can be exploited to estimate the time-varying delay via the coefficients