The nonorthogonal basis generalized Fourier transform is used as orders extraction technique during machinery speed-up and slow-down tests due to nonstationary nature of vibration signals in these tests. The kernels of this transform have time-dependent frequency which is related to the operating speed of the machine. Since these kernels may belong to different groups or shafts, they are generally nonorthogonal. The actual amplitudes and phases of the orders can be found by solving the system of linear equations resulting from decomposition process which is proposed in this work as an improvement to the time variant discrete Fourier transform (TVDFT) method. The proposed scheme is proved to be efficient and the processing time is very small as compared to other schemes such as the Vold–Kalman order tracking (VKOT) method. The accuracy and efficiency of the proposed scheme are investigated using simulated vibration signal and also actual signals.