Equiripple MIMO Beampattern Synthesis Using Chebyshev Approximation

This letter presents a method for synthesizing equiripple MIMO transmit beampatterns using Chebyshev approximation. The MIMO beampattern is represented as a non-negative real-valued trigonometric polynomial where the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell ^{\text{th}}$</tex-math></inline-formula> order polynomial coefficient is the sum of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell ^{\text{th}}$</tex-math></inline-formula> order diagonal of the waveform correlation matrix. The optimal coefficients for a given equiripple beampattern design is then posed as a Chebyshev approximation problem which is efficiently solved using the Parks-McClellan algorithm from optimal Finite Impulse Response (FIR) filter design theory. The unique advantage of this synthesis method is that it provides a closed form method to generating MIMO correlation matrices that realize the desired equiripple beampattern. This correspondingly facilitates the design of waveform sets that closely approximate those correlation matrices. This method is demonstrated via two illustrative design examples; the first using traditional partial signal correlation methods and the second using transmit beamspace processing. Both examples realize equiripple beampatterns using constant envelope and spectrally compact waveform sets.