This paper investigates an adaptive equalizer with carrier frequency offset compensation for digital mobile radio transmission over several tens kbit/s. The performance of conventional adaptive equalizers for mobile radio severely degrades for receiving signals having carrier frequency offset which is caused by local oscillator frequency offset due to misadjustment, temperature dependence, and aging.Carrier frequency offset compensation for TDMA burst signaling requires both accurate and short-time measurements. However, both requirements have a tradeoff relationship and must be optimized. This paper proposes the carrier-acquisitionduringtraining (CAT) algorithm for transmissions allowing a training signal. First, two basic configurations and their equivalence are discussed. Next, computer simulations and real-time experiments with onechip DSP show the effectiveness of CAT by demonstrating training convergence, optimum parameter characteristics, and average BER performance under two-path selective fading with one-symbol delay difference and independent Rayleigh components.