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We consider downlink precoding in a frequency-selective multi-user massive MIMO system with highly efficient but non-linear power amplifiers at the base station (BS). A low-complexity precoding algorithm is proposed, which generates constant-envelope (CE) transmit signals for each BS antenna. To avoid large variations in the phase angle transmitted from each antenna, the difference of the phase angles transmitted in consecutive channel uses is limited to [-&#x03B1;&#x03C0;, &#x03B1;&#x03C0;] for a fixed 0 &lt;; &#x03B1; &#x2264; 1. With independent Rayleigh distributed channel gains, numerical studies reveal that the extra total transmit power required under the time variation constraint when compared to the case of no time variation constraint (i.e., &#x03B1; = 1), is small (&lt;; 2 dB) for many practical values of &#x03B1;. Further, irrespective of the value of &#x03B1;, the required total transmit power to achieve a fixed per-user information rate decreases by roughly 3 dB with every doubling in the number of BS antennas.

Constant-Envelope Precoding With Time-Variation Constraint on the Transmitted Phase Angles