Multiple-Order Singularity Expansion Method

Abstract

Physical systems and signals are often characterized by complex functions of frequency in the harmonic-domain. The extension of such functions to the complex frequency plane has been a topic of growing interest as it was shown that specific complex frequencies could be used to describe both ordinary and exceptional physical properties. In particular, expansions and factorized forms of the harmonic-domain functions in terms of their poles and zeros under multiple physical considerations have been used. In this work, we start from a general property of continuity and differentiability of the complex functions to derive the multiple-order singularity expansion method. We rigorously derive the common singularity and zero expansion and factorization expressions, and generalize them to the case of singularities of arbitrary order, whilst deducing the behaviour of these complex frequencies from the simple hypothesis that we are dealing with physically realistic signals.

Figures (6)

• Figure 1: General representation of an LTIS. (a) In the temporal domain, the system is characterized by its impulse response $h(t)$ which can be used to obtain the output $y(t)$ after a convolution with the input $x(t)$. (b) In the harmonic domain, the LTIS is described by its transfer function $H(\omega)$. It is a filter acting in the frequency domain on the input $X(\omega)$ to generate an output $Y(\omega)$. In this case, $H(\omega)$ is a low-pass filter which partially removes the noise present at higher frequencies, as it can be seen by comparing the temporal input $x(t)$ to the temporal output $y(t)$. The output $Y(\omega)$ is the product of $H(\omega)$ and $X(\omega)$. The harmonic domain functions $X(\omega)$, $H(\omega)$ and $Y(\omega)$ are obtained by bilateral Laplace transform of the temporal signals $x(t)$, $h(t)$ and $y(t)$. The temporal signals can then be recovered using an inverse Laplace transform.
• Figure 2: (a) Modulus of the transfer function $H(\omega)$ defined in Eq. (\ref{['eq:H_example']}), as well as its resonant terms $Res(p_\ell)/(\omega - p_\ell)$ associated with the poles $p_l$ and the modulus of the sum of these resonant terms. The frequencies range from $1.040 \times 10^16$ Hz to $1.093 \times 10^16$ Hz, which is equivalent to ultraviolet wavelengths between $172.9$ nm and $181.3$ nm. (b) Log-amplitude of $H(\omega)$ in the complex frequency plane, in a limited complex frequency window. The poles in (a) are highlighted in (b) as red points. The sum over the poles gives an approximation of the shape of $H(\omega)$ on the real axis (the minimum and maximum frequencies), but with the background term of Eq. (\ref{['eq:spe']}) omitted, the reconstructed red curve poorly matches the exact expression.
• Figure 3: (a) Phase diagram in the complex frequency plane of the transfer function $H(\omega)$ of an LTIS defined in Eq. (\ref{['eq:H_example_2']}) as $H(\omega) = 5 + \frac{2 + 0.1i}{\omega - (2 + 0.3i)} + \frac{4 + 0.1i}{\omega - (5 + 1.4i)} + \frac{6 + 0.1i}{\omega - (6.5 + 0.7i)}$. (b) Phase diagram of the transfer function $H(\omega)e^{i\omega\tau_0}$, with $\tau_0=\pi/6$, of the same LTIS with a distinct time origin. Shifting the time-origin of the input $X(\omega)$ in (a) is tantamount to considering $H(\omega)e^{i\omega\tau_0}$ for the transfer function in (b). The distribution of the poles (red points) and zeros (blue points) remains the same, only the phase is affected.
• Figure 4: Any physically realistic input $X(\omega)$, output $Y(\omega)$ or transfer function $H(\omega)$ can be described using either MOSEM or the SZF. The two expressions are equivalent.
• Figure 5: Complex Bode diagram of the function $H$ defined in Eq. (\ref{['eq:H_example']}). (a),(b) Log-amplitude of $H$ for a complex frequency window in (a) and its symmetric window with respect to the imaginary axis in (b). (c),(d) The phase of $H$ in the same frequency window (c) and its symmetric window (d). The amplitude is symmetric with respect to the imaginary axis and results in a symmetric distribution of the poles (red points). The phase is antisymmetric, as shown with the red and blue arrows indicating a clockwise $2\pi$ phase-shift around the poles with a positive and negative real part respectively.