Doubly-selective channel estimation represents a key element in ensuring communication reliability in wireless systems. Due to the impact of multi-path propagation and Doppler interference in dynamic environments, doubly-selective channel estimation becomes challenging. Conventional channel estimation schemes encounter performance degradation in high mobility scenarios due to the usage of limited training pilots. Recently, deep learning (DL) has been utilized for doubly-selective channel estimation, where convolutional neural network (CNN) networks are employed in the frame-by-frame (FBF) channel estimation. However, CNN-based estimators require high complexity, making them impractical in real-case scenarios. For this purpose, we overcome this issue by proposing an optimized and robust bi-directional recurrent neural network (Bi-RNN) based channel estimator to accurately estimate the doubly-selective channel, especially in high mobility scenarios. The proposed estimator is based on performing end-to-end interpolation using gated recurrent unit (GRU) unit. Extensive numerical experiments demonstrate that the developed Bi-GRU estimator significantly outperforms the recently proposed CNN-based estimators in different mobility scenarios, while substantially reducing the overall computational complexity.

This paper proposes a model-driven deep learning (MDDL)-based channel estimation and feedback scheme for wideband millimeter-wave (mmWave) massive hybrid multiple-input multiple-output (MIMO) systems, where the angle-delay domain channels' sparsity is exploited for reducing the overhead. Firstly, we consider the uplink channel estimation for time-division duplexing systems. To reduce the uplink pilot overhead for estimating the high-dimensional channels from a limited number of radio frequency (RF) chains at the base station (BS), we propose to jointly train the phase shift network and the channel estimator as an auto-encoder. Particularly, by exploiting the channels' structured sparsity from an a priori model and learning the integrated trainable parameters from the data samples, the proposed multiple-measurement-vectors learned approximate message passing (MMV-LAMP) network with the devised redundant dictionary can jointly recover multiple subcarriers' channels with significantly enhanced performance. Moreover, we consider the downlink channel estimation and feedback for frequency-division duplexing systems. Similarly, the pilots at the BS and channel estimator at the users can be jointly trained as an encoder and a decoder, respectively. Besides, to further reduce the channel feedback overhead, only the received pilots on part of the subcarriers are fed back to the BS, which can exploit the MMV-LAMP network to reconstruct the spatial-frequency channel matrix. Numerical results show that the proposed MDDL-based channel estimation and feedback scheme outperforms the state-of-the-art approaches.