Interpretable Optimization-Inspired Deep Network for Off-Grid Frequency Estimation

The accuracy of on-grid frequency estimation methods suffers from the quantization error of discrete grids. In this article, a deep unfolded network for off-grid frequency estimation is proposed, dubbed OGFreq. In the OGFreq, there exist two kinds of variables. One is the batch-oriented dictionary for frequency-domain transform, and the other one is the instance-specific on-grid frequency and off-grid bias. As the dictionary is required to be universally applicable among all observed signals, network layers are designed and network weights are updated to approximate the transform bases in a data-driven way. Besides, instance-specific on-grid frequencies and off-grid biases are solved by unfolding the iterative soft-threshold algorithm (ISTA). In addition, the instance-specific hyperparameters for sparsity in ISTA are obtained by an encoder-decoder soft-threshold (EDS) module with the attention mechanism. In this way, the dictionary, on-grid frequency, and off-grid bias are learned in a unified data-driven framework. Numerical experiments show that the OGFreq obtains 4% lower false negative rate (FNR) when the SNR is 20 dB. Moreover, the computational complexity is one order of magnitude lower than the iteration-based off-grid frequency estimation methods. Finally, the robustness of the OGFreq is discussed when extended to the impulse noise and damped signals.