This work proposes a novel low-complexity digital backpropagation (DBP) method, with the goal of optimizing the trade-off between backpropagation accuracy and complexity. The method combines a split step Fourier method (SSFM)-like structure with a simplified logarithmic perturbation method to obtain a high accuracy with a small number of DBP steps. Subband processing and asymmetric steps with optimized splitting ratio are also employed to further reduce the number of steps required to achieve a prescribed performance. The first part of the manuscript is dedicated to the derivation of a simplified logarithmic-perturbation model for the propagation of signal in an optical fiber, which serves for the development of the proposed coupled-band enhanced split step Fourier method (CB-ESSFM) and for the analytical calculation of the model coefficients. Next, the manuscript presents a DSP algorithm for the implementation of DBP based on a discrete-time version of the model and an overlap-and-save processing strategy. Practical approaches for the optimization of the coefficients used in the algorithm and of the splitting ratio of the asymmetric steps are also discussed. A detailed analysis of the computational complexity is presented. Finally, the performance and complexity of the proposed DBP method are investigated through simulations. In a five-channel 100 GHz-spaced wavelength division multiplexing system over a 15x80 km single-mode-fiber link, the proposed CB-ESSFM achieves a gain of about 1dB over simple dispersion compensation with only 15 steps (corresponding to 681 real multiplications per 2D symbol), with an improvement of 0.9 dB over conventional SSFM and almost 0.4dB over our previously proposed ESSFM. Significant gains and improvements are obtained also at lower complexity. A similar analysis is performed also for longer links, confirming the good performance of the proposed method.