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Black hole thermodynamics is usually carried out over a span of the real variables. In this study, we extend this process in the complex domain. Often resorting to the complex plane, we find that the Hawking-Page phase transition is the switch of the combination of two vortices of the heat flow, while the van der Waals-type phase transition is predicted by the local maximum winding number without lengthy conventional thermodynamic calculations. As a by-product, we also give the Riemann surface description of the black hole thermodynamics phase transition in the complex domain, so as to make the thermodynamics phase transition intuitive. This method is expected to be extended to various black hole thermodynamics systems and even to general thermodynamic statistical physical system., Comment: v1:6 pages, 3 figures; v2: add References and some comments; v3: title changed and clarifications added

In this study, we use complex analysis to investigate the phase transition of the black hole thermodynamic system. We find that the phase transition of a black hole is governed by a local winding number. By decomposing the obtained winding number, we endow the phase transition with concrete and vivid complex structure characteristics. There are three basic elements: (1) when the winding number is one, there is no phase transition, and the corresponding complex structure is the Riemann surface with one foliation; (2) the winding number is two, which corresponds to the second-order phase transition and has a Riemann surface with two foliations; (3) when the winding number is three, it means that the first-order phase transition will occur, accompanied by the second-order phase transition, which has a Riemann surface with three foliations. Especially, a black hole thermodynamic system with a triple point has a structure of the Riemann surface with five foliations. We can also predict the phase transition when the winding number is four. There will be two kinds of decompositions, implying that such a system will undergo a mixture of first-order and second-order phase transitions, or only the pure second-order phase transition respectively., v1:6 pages, 3 figures; v2: add References and some comments

A 16-GS/s single 1.2-V supply direct RF-sampling capacitive D/A converter for soft radio base-station applications is implemented in a 16-nm FinFET technology. Due to co-integration with a wideband on-chip matching network, the D/A converter (DAC) is capable of signal synthesis between 600 MHz and 8 GHz, with a measured peak RF output power of +5.6 dBm. The linearity required by cellular base-station applications is achieved by employing several analog-domain linearization techniques, yielding a two-tone intermodulation distortion of better than -70 dBc up to the first Nyquist frequency. When transmitting a true multiband signal covering an instantaneous bandwidth of 1 GHz, an adjacent channel leakage ratio (ACLR) of better than -69 dBc is achieved. Digital segment mismatch correction is shown to improve the linearity of a segmented DAC in deep digital backoff operation. A full-rate digital pre-distortion (DPD) method with efficient model identification is experimentally demonstrated. Using DPD, the design achieves an spurious free dynamic range (SFDR) of better than 72 dBc over the entire 0.6-6-GHz frequency range.

The integration of complex DSP functions together with the entire RF-frontend enables software-defined base-station radio and mandates the availability of power-efficient and compact data converters in advanced digital CMOS processes. In a self-calibrated RF-sampling ADC-system, a co-integrated digital non-linear post-equalizer (NLEQ) can substantially increase the receiver performance [1], but wideband ADC-calibration also requires a wideband, calibration-free and ultra-linear reference signal source to train the NLEQ-coefficients during a dedicated calibration period, for example at startup, or during maintenance intervals. Another function with renewed importance in complex transceiver systems is the analog built-in-self-test (aBIST) capability [2] to spot-check the performance of data converters and RF-frontend building blocks, for example during scheduled test intervals in the field. In this work, a single-bit calibration-DAC (CALDAC), capable of wideband reference signal and linear sinewave generation at fclk/4 for these applications is presented. With fclk $=16$ GHz and $4 \times-$ interpolation, the CALDAC effectively samples at 64GS/s and implements a semi-digital FIR-architecture [3] –[5].

Future multi-band software-defined-radio base-stations for digital beamforming and massive MIMO applications depend heavily on the availability of highly linear and compact data converters with good power efficiency, while at the same time offering multi-GHz signal-bandwidth at sampling rates well in excess of 10GS/s. Wideband RF-sampling D/A-converters have traditionally been implemented in current-steering architectures, mostly with extensive calibration infrastructure [1] –[3]. The transistor stack required to achieve the necessary static and dynamic output impedance for the code-steered current sources leads to limited supply voltage scalability, while the capacitive self-loading by the current-source array makes true wideband matching at the RF-output inherently difficult. Capacitive digital-to-analog converters (C-DAC) have been widely used as RF DAC or switched-capacitor power amplifiers. Up to now digital transmitters have used C-DACs with inherent mixing functionality in polar or IQ systems for synthesis of high-power RF signals of moderate bandwidth of up to 160MHz [4] –[6]. This work uses a capacitive DAC as a direct RF-sampling DAC with moderate output power level for direct signal synthesis over a bandwidth from 0.5GHz up to at least 8GHz.