Computational design of optimal waveforms for MIMO radar via multi-dimensional iterative spectral approximation

Designing Unimodular Waveform with Low Range Sidelobes and Stopband for Cognitive Radar via Relaxed Alternating Projection

“…where denotes the phase of the th element, (⋅) indicates the matrix or vector transpose, C represents the complex field, and is the number of samples. As discussed in [13,14,17], the correlation vector (x) could be regarded as the output of matched filter, and each entry of (x) has (x) = ∑ − =1 x( )x( + ) * = * − (x), and (⋅) * denotes the complex conjugate. Since even-number (e.g., 2 ) is more suitable for implementing FFT, we usually pad "0" to the tail of (x) to make (x) ∈ C 2 ×1 for the convenience of FFT.…”

“…Note that low range sidelobes of waveform could improve target detection when range compression is applied in receiver [12][13][14]. Generally, low range sidelobes could be obtained by minimizing the ISL [6,8] as follows:…”

“…Based on discussion above, spectral power in stopbands could be expressed as min x ‖PC x‖ 2 . Referring to [13], this objective function could be further rewritten as follows:…”

“…Solving these problems efficiently is challengeable because of its nondeterministic polynomialtime hard property. Alternating projection (AP) method as a cyclic mechanism was introduced for these problems [20,21], and our previous works have also demonstrated its performance [13,14]. Particularly, let denote the collection of waveforms which possesses constant envelope, that is, |x( )| = 1, = 1, 2, .…”

“…The LBFGS algorithm was also utilized to design unimodular waveform with low range sidelobes [10]. Other approaches, such as iterative method [11], alternating projection [12][13][14], and heuristic search algorithm [15], have also been presented. However, the initialization of these algorithms under unimodular constraint almost always incurs local stagnation and loss of robustness.…”