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Digital signal processing has completely changed the way optical communication systems work during recent years. In combination with coherent demodulation, it enables compensation of optical distortions that seemed impossible only a few years ago. However, at high bit rates, this comes at the price of complex processing circuits and high power consumption. In order to translate theoretic concepts into economically viable products, careful design of the digital signal processing algorithms is needed. In this paper, we give an overview of digital equalization algorithms for coherent receivers and derive expressions for their complexity. We compare single-carrier and multicarrier approaches, and investigate blind equalizer adaptation as well as training-symbol-based algorithms. We examine tradeoffs between parameters like sampling rate and tracking speed that are important for algorithm design and practical implementation.