Chapter 13 focused on the optimization of the waveform covariance matrix R for transmit beampattern design. While designing R is an important step, the final goal is the waveform design and we might therefore think of designing directly the probing signals by optimizing a given performance measure with respect to the matrix X of the signal waveforms. However, compared with optimizing the same performance measure with respect to the covariance matrix R of the transmitted waveforms, optimizing directly with respect to X is a more complicated problem. This is so because X has more unknowns than R and the dependence of various performance measures on X is more intricate than the dependence on R (as R is a quadratic function of X).

In this chapter we consider the following problem: with R obtained in a previous (optimization) stage, we want to determine a signal waveform matrix X whose covariance matrix is equal or close to R, and which also satisfies some practically motivated constraints (such as constant-modulus or low PAR constraints). We present a cyclic algorithm for the synthesis of such an X. We also consider the case where the synthesized waveforms are required to have good auto- and cross-correlation properties in time. Several numerical examples are provided to demonstrate the effectiveness of the proposed methodology.

Problem formulation

Consider an active sensing system equipped with M transmit antennas. Let the columns of X ∈ ℂN×M be the transmitted waveforms, where N denotes the number of samples in each waveform.