PORTRAIT AND HILBERT TRANSFORM METHODS FOR EVALUATING CONTINUOUS RELATIVE PHASE BETWEEN LOWER-LIMB JOINTS IN THE ELDERLY DURING WALKING

In order to calculate the continuous relative phase (CRP) between joints, the portrait method based on the joint angle and angular velocity and the Hilbert transform method based on the analytical signal have been widely used. However, there are few comparisons of these methods. Therefore, the aim of this study is to quantitatively compare these methods by calculating the CRP in the lower-limb joints of the elderly during level free walking. Eighteen elderly female adults ([Formula: see text] year-old, [Formula: see text][Formula: see text]cm, [Formula: see text][Formula: see text]kg) wearing a Helen Hayes full-body marker set walked 10[Formula: see text]m on level ground at a self-selected velocity. The angles of the hip, knee, and ankle were measured. To calculate the CRP using the portrait method, the angular velocities were measured. Then, the phases between the angle and the angular velocity were calculated. To calculate the CRP using the Hilbert transform method, analytical signals were acquired. Then, the phases between the real and imaginary parts were calculated. A CRP was calculated as the difference between the phase in the proximal joint and the phase in the distal joint. To evaluate the similarity in the shape between the portrait and Hilbert transform methods, the cross-correlation was calculated. Bland–Altman plot analyses were performed to assess the agreement between these methods. For the root mean squares (RMSs) and standard deviations (SDs), a paired [Formula: see text]-test and the Pearson correlation between methods were evaluated. There were similarities in the in-phase or out-of-phase features and in the RMS and SD between the methods. Additionally, a higher cross-correlation and agreement between them were found. These results indicated the similarity between the portrait and Hilbert transform methods for the calculation of the CRP. Therefore, either method can be used to evaluate joint coordination.

Certain Properties of Generalized M-Series under Generalized Fractional Integral Operators

The aim of this study is to introduce new (presumed) generalized fractional integral operators involving I -function as a kernel. In addition, two theorems have been developed under these operators that provide an image formula for this generalized M -series and also to study the different properties of the generalized M -series. The corresponding assertions in terms of Euler and Laplace transform methods are presented. Due to the general nature of the I -function and the generalized M -series, a number of results involving special functions can be achieved only by making appropriate values for the parameters.