  Quadrature compressive sampling (QuadCS) is a sub-Nyquist sampling scheme for
acquiring in-phase and quadrature (I/Q) components in radar. In this scheme,
the received intermediate frequency (IF) signals are expressed as a linear
combination of time-delayed and scaled replicas of the transmitted waveforms.
For sparse IF signals on discrete grids of time-delay space, the QuadCS can
efficiently reconstruct the I/Q components from sub-Nyquist samples. In
practice, the signals are characterized by a set of unknown time-delay
parameters in a continuous space. Then conventional sparse signal
reconstruction will deteriorate the QuadCS reconstruction performance. This
paper focuses on the reconstruction of the I/Q components with continuous delay
parameters. A parametric spectrum-matched dictionary is defined, which sparsely
describes the IF signals in the frequency domain by delay parameters and gain
coefficients, and the QuadCS system is reexamined under the new dictionary.
With the inherent structure of the QuadCS system, it is found that the
estimation of delay parameters can be decoupled from that of sparse gain
coefficients, yielding a beamspace direction-of-arrival (DOA) estimation
formulation with a time-varying beamforming matrix. Then an interpolated
beamspace DOA method is developed to perform the DOA estimation. An optimal
interpolated array is established and sufficient conditions to guarantee the
successful estimation of the delay parameters are derived. With the estimated
delays, the gain coefficients can be conveniently determined by solving a
linear least-squares problem. Extensive simulations demonstrate the superior
performance of the proposed algorithm in reconstructing the sparse signals with
continuous delay parameters.
