Block Circulant Decomposition of Cross-Correlation Matrix for Transmit MIMO Beamforming

Abstract

This paper comprises the transmit probing signal design under the trade-off between good target discrimination (low cross-correlation beam pattern) and beam pattern design (desired auto-correlation beam pattern) in a Multiple-Input-Multiple-Output (MIMO) configuration. The quartic optimization problem, with an finite alphabet constraint on the probing signal by using Quadrature Phase Shift Keying (QPSK) in a multiplexed antenna system, is solved by a Fourier series approximation of the desired beam pattern by exploiting a block circulant property of the transmit signal matrix. The analytical evaluation of the mean square error between an ideal and the proposed cross-correlation beam pattern is −35 dB.

Closed Form Discrete Unimodular MIMO Waveform Design Using Block Circulant Decomposition

“…Therefore, the RCS of the κ-th target, α κ ∈ C, is a statistical parameter. The receive steering vector a Rκ ∈ C N R ×1 and the transmit steering vector a T κ ∈ C N T ×1 for the κ-th target, defined below, contain the MIMO channel information under the far-field and plane wave assumptions [20], [22],…”

“…A commonly used design metric is the Signal-to-Interference-to-Noise Ratio (SINR) [6]- [10]. Another metric is the design of a desired radiation pattern as a function of the transmitted waveform [3], [4], [11]- [20]. However, in addition to the interest in radiation pattern design, the cross-correlation properties of the transmitted signals are also considered [4], [11], [18], [20].…”

“…Another metric is the design of a desired radiation pattern as a function of the transmitted waveform [3], [4], [11]- [20]. However, in addition to the interest in radiation pattern design, the cross-correlation properties of the transmitted signals are also considered [4], [11], [18], [20]. The cross-correlation properties of the transmitted signal is of significant interest in a virtual MIMO configuration, since signals with good cross-correlation properties, e.g., orthogonal signals, enable unique discrimination of the transmitted signals at the receiver and therefore facilitate the construction of a virtually filled array at the receiver [1], [2].…”

“…However, [15] does not consider cross-correlation properties and unimodular signal design constraint in the design. Further, authors in [15] introduce a Toeplitz structure to the signal covariance matrix, whereas in this paper a much simpler block circulant approach similar to [20] is proposed. This paper builds on the work [20] and extends the quadrature phase two channel sequence design of [20] to the generic case of unimodular sequences with multiple phase stages.…”

“…• Extension of the block circulant decomposition approach [20] to multi-phase unimodular waveform design • Derivation of a closed-form solution to an optimization problem which addresses the transmit beam pattern design and minimizes the transmit signal cross-correlation under finite alphabet and uniform antenna constraints • Algorithm capable of real time implementation…”