Matrix-Wigner Distribution

Abstract

In order to achieve time–frequency superresolution in comparison to the conventional Wigner distribution (WD), this study generalizes the well-known τ-Wigner distribution (τ-WD) with only one parameter τ to the multiple-parameter matrix-Wigner distribution (M-WD) with the parameter matrix M. According to operator theory, we construct Heisenberg’s inequalities on the uncertainty product in M-WD domains and formulate two kinds of attainable lower bounds dependent on M. We solve the problem of lower bound minimization and obtain the optimality condition of M, under which the M-WD achieves superior time–frequency resolution. It turns out that the M-WD breaks through the limitation of the τ-WD and gives birth to some novel distributions other than the WD that could generate the highest time–frequency resolution. As an example, the two-dimensional linear frequency-modulated signal is carried out to demonstrate the time–frequency concentration superiority of the M-WD over the short-time Fourier transform and wavelet transform.

Keywords

• lower bound
• <i>τ</i>-Wigner distribution
• time–frequency resolution
• uncertainty product
• Wigner distribution