Statistical Signal Processing in Engineering

20Multichannel Time of Delay Estimation

Time of delay (ToD) estimation is a very common problem in many applications that employ wavefield propagation such as radar, remote sensing, and geophysics (Section 18.5) where the information of interest is mapped onto ToDs. Features of the wavefields call for measurement redundancy from a multichannel system where there are signals that are collected as s₁(t), s₂(t), …, s_M(t), and this leads to massive volumes of data for estimating the ToD metric. The uniform linear array in Chapter 19 represents a special case of a multichannel system where the source(s) is narrowband and the only parameter(s) that can be estimated is the DoA(s). Here the narrowband constraint is removed while still preserving the multidimensionality of the signals in order to have the needed redundancy for estimating the ToD variation vs. measuring channels (e.g., sensors) with known geometrical positions.

The multichannel concept is illustrated in Figure 20.1 from seismic exploration (see Section 18.5 or [82] ): the wavefield generated at the surface propagates in the subsurface and is backscattered at depth discontinuities, and measured with sensors at the surface. The wavefield is collected by a regularly spaced 2D array of sensors to give the measured wavefields s(x, y, t) in space (x, y) and time t in the figure. Note that in Figure 20.1 the 3D volume s(x,  ...