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We present a turbo (iterative) equalization receiver with fixed-lag nonlinear Kalman filtering for coded data transmission over doubly-selective channels. The proposed receiver exploits the complex exponential basis expansion model (CEBEM) for the overall channel variations, and an autoregressive (AR) model for the BEM coefficients. We extend an existing turbo equalization approach based on symbol-wise AR modeling of channels to channels based on BEM's. In the receiver an adaptive equalizer using nonlinear Kalman filters with delay is coupled with a soft-input soft-output (SISO) decoder to iteratively perform equalization and decoding. The adaptive equalizer jointly optimizes the estimates of the BEM coefficients and data symbols, thereby automatically accounting for correlation between data symbols and channel tap gains. An extrinsic information transfer (EXIT) chart analysis of the proposed approach is also presented. Simulation examples demonstrate that our CE-BEM-based approach significantly outperforms the existing symbol-wise AR model-based turbo equalizer.