In this section, models of time-varying delays which are not linearly or quadratically time variant are briefly considered.
The time-varying range R(t) can be expanded in Taylor series, around t0, with Lagrange residual term:
(7.370)
where R(n)(·) denotes the nth-order derivative of R(·), and if t > t0 and if t < t0. This expression can be substituted into (c07-mdis-0044) to get an n-order algebric equation in D(t). One of the roots of this equation is the time-varying delay D(t) for the given R(t) (see Section 7.4 for the case n = 2 and ).
Alternatively, by following the approach in (Kelly 1961) and (Kelly and Wishner 1965), the time-varying delay D(t) can be expanded in Taylor series, around t0, with Lagrange residual term:
(7.371)
where if t > t0 and if t < t0.
This approach, when is embedded in additive white Gaussian noise (AWGN), allows to address the joint detection-estimation problem by the generalized ambiguity function (Kelly 1961; Kelly and Wishner 1965).
A rotating reflecting object gives rise to a periodically time-variant delay in the received signal. If the object is also moving with constant relative radial speed, then the delay contains also a linearly time-variant term (Chen et al. 2006):
(7.372)
The effect on the received signal is called micro-Doppler. In (Chen et al. 2006), with reference to the complex-envelope received-signal
(7.373)
the “Doppler frequency” is defined as
(7.374)
and the micro-Doppler effect is characterized by time-frequency analysis.
Let x0(t) be the complex-envelope of a transmitted signal and let
(7.375)
the complex signal reflected under the narrow-band condition by an oscillating scatterer whose effect is to produce a sinusoidally time-varying time-scale factor. The same mathematical model is obtained if the carrier frequency is not constant but is sinusoidally time-varying due to imperfection of the transmitting oscillator.
The (conjugate) autocorrelation function of x(t) is
(7.376)
where
(7.377)
Let a 2πfcΔ and ωm 2πfm. By using the expansion of a complex exponential in terms of Bessel functions (NIST 2010, eqs. 10.12.1-3) (with z = au and θ = 2πfmt + ϕ) we have
where Jn(z) is the Bessel function of the first kind of order n (NIST 2010, eqs. 10.2.2, 10.9.1, 10.9.2). The function of two variables ξ(u, t) can be evaluated along the diagonal u = t leading to
(7.379)
The second-order lag product of z(t) is given by
(7.380)
The function r1(t) is periodic in t. From (7.378) with u = τ we have
(7.381)
Due to the function
(7.382)
the signal x(t) turns out to be GACS with nonlinear lag-dependent cycle frequencies with complicate analytical expression.
A simulation experiment is carried out to estimate the cyclic correlogram of x(t). In the experiment, x0(t) is a PAM signal with raised cosine pulse with excess bandwidth η = 0.25, stationary white binary modulating sequence, and symbol period Tp = 64Ts, where Ts is the sampling period. Furthermore, fc = 0.125/Ts, s = 0.99, fm = 0.0233/Ts, Δ = 0.005, and ϕ = 0.
In Figure 7.19, (top) graph and (bottom) “checkerboard” plot of the magnitude of the cyclic correlogram of x(t), estimated by 213 samples as a function of αTs and τ/Ts, are reported.
18.217.211.92