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Communication at millimeter wave (mmWave) frequencies is defining a new era of wireless communication. The mmWave band offers higher bandwidth communication channels versus those presently used in commercial wireless systems. The applications of mmWave are immense: wireless local and personal area networks in the unlicensed band, 5G cellular systems, not to mention vehicular area networks, ad hoc networks, and wearables. Signal processing is critical for enabling the next generation of mmWave communication. Due to the use of large antenna arrays at the transmitter and receiver, combined with radio frequency and mixed signal power constraints, new multiple-input multiple-output (MIMO) communication signal processing techniques are needed. Because of the wide bandwidths, low complexity transceiver algorithms become important. There are opportunities to exploit techniques like compressed sensing for channel estimation and beamforming. This article provides an overview of signal processing challenges in mmWave wireless systems, with an emphasis on those faced by using MIMO communication at higher carrier frequencies., Comment: Submitted to IEEE Journal of Selected Topics in Signal Processing