In this paper, we examine the energy efficiency (EE) of a base station (BS) with multiple antennas. We use a state-of-the-art power consumption (PC) model that captures the passive and active parts of the transceiver circuitry, including the effects of radiated power, signal processing, and passive consumption. The paper treats the transmit power, bandwidth, and number of antennas as the optimization variables. We provide novel closed-form solutions for the optimal ratios of power per unit bandwidth and power per transmit antenna, and discover a new relationship in which the radiated power equals the total transceiver power at the EE-optimal operating point. A central finding is that the EE-optimal signal-to-noise ratio (SNR) collapses to a universal numerical constant of approximately 5.93 dB, independent of channel and hardware parameters. We present an algorithm that jointly optimizes the three design variables to achieve maximum EE under practical constraints, and provide analytical insight into whether maximum power or maximum bandwidth is optimal and how many antennas a BS should utilize. We further extend the optimization framework to incorporate quality-of-service (QoS) requirements and three advanced sleep modes of varying depth: absolute sleep, deep sleep, and idle mode. We characterize the optimal hardware configuration for each mode and determine when the rush-to-sleep strategy, which transmits briefly at the EE-optimal active configuration and sleeps the rest of the time, is optimal. Incorporating wake-up transition delays, we reveal how latency constraints and sleep-mode-specific transition times jointly dictate the optimal sleep mode for data packets with absolute deadlines. Together, these results indicate that energy-efficient operation requires treating transmission and sleep as a single coupled optimization.

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