Circular motion analysis of time-varying bioimpedance

This paper presents a step forward towards the analysis of a linear periodically time-varying (PTV) bioimpedance ZPTV(jw, t), which is an important subclass of a linear time-varying (LTV) bioimpedance. Similarly to the Fourier coefficients of a periodic signal, a PTV impedance can be decomposed into frequency dependent impedance phasors, [Formula: see text], that are rotating with an angular speed of wr = 2πr/TZ. The vector length of these impedance phasors corresponds to the amplitude of the rth-order harmonic impedance |Zr( jw)| and the initial phase is given by Φr(w, t0) = [Symbol: see text]Zr( jw) + 2πrt0/TZ, with t0∈[0, T] being a time instant within the measurement time T. The impedance period TZ stands for the cycle length of the bio-system under investigation; for example, the elapsed time between two consecutive R-waves in the electrocardiogram or the breathing periodicity in case of the heart or lungs, respectively. First, it is demonstrated that the harmonic impedance phasor [Formula: see text], at a particular measured frequency k, can be represented by a rotating phasor, leading to the so-called circular motion analysis technique. Next, the two dimensional (2D) representation of the harmonic impedance phasors is then extended to a three-dimensional (3D) coordinate system by taking into account the frequency dependence. Finally, we introduce a new visualizing tool to summarize the frequency response behavior of ZPTV( jw, t) into a single 3D plot using the local Frenet-Serret frame. This novel 3D impedance representation is then compared with the 3D Nyquist representation of a PTV impedance. The concepts are illustrated through real measurements conducted on a PTV RC-circuit.