Basis functions of the Discrete Hermite transform (DHT) can be formed as the set of eigenvectors of a symmetric tridiagonal matrix which commutes with the centered discrete Fourier transform matrix. In this paper, we consider the optimization of the associated time-axis scaling factor, which is of crucial importance in the DHT-based applications. The parameter optimization is performed to enhance the signal representation by matching the smallest possible number of basis functions with the time-domain signal waveform, therefore minimizing the number of Hermite coefficients with significant or non-zero values. Such highly concentrated signal representation is particularly amenable for signal compression, filtering, and denoising, while the coefficients of the optimized transform can be exploited as features in signal classification and other machine learning applications. The proposed parameter optimization approach is verified on numerical examples, including the experiments with QRS complexes, specific segments of electrocardiogram (ECG) signals being particularly important in biomedical applications.

离散 Hermite 变换 (DHT) 的基函数可以形成为对称三对角矩阵的一组特征向量，该矩阵与中心离散傅立叶变换矩阵交换。在本文中，我们考虑了相关时间轴比例因子的优化，这在基于 DHT 的应用程序中至关重要。执行参数优化以通过将最小可能数量的基函数与时域信号波形匹配来增强信号表示，从而最小化具有重要值或非零值的 Hermite 系数的数量。这种高度集中的信号表示特别适用于信号压缩、滤波和去噪，而优化变换的系数可以用作信号分类和其他机器学习应用中的特征。所提出的参数优化方法在数值示例中得到验证，包括 QRS 复合波实验，心电图 (ECG) 信号的特定片段在生物医学应用中尤为重要。