The Kramers-Kronig (KK) receiver has recently drawn a great deal of attention due to its capability of retrieving the phase information from the directly-detected intensity waveforms. However, a technical challenge associated with the implementation of this attractive receiver is the high sampling rate of digital signal processing (DSP). Since nonlinear operations (such as logarithm and exponential functions) included in the conventional KK algorithm broaden the signal spectrum significantly, digital up-sampling is required at the beginning of DSP. To solve this problem, we have recently proposed and demonstrated a new KK algorithm operable without the digital upsampling. In this upsampling-free algorithm, we removed the exponential function by expressing the complex signal in the Cartesian form (i.e., real plus imaginary) and replaced the logarithm function with its mathematical approximation. In this paper, we present the detailed comparison between the conventional and upsampling-free KK algorithms through numerical simulations. We investigate the performance of the KK receivers when we employ orthogonal frequency-division multiplexing and Nyquist subcarrier multiplexing signals formatted in 16-quadrature amplitude modulation (16-QAM) and 64-QAM. Also investigated in this paper is the performance limitation of the upsampling-free KK algorithm. Finally, we propose the schematic diagrams of hardware implementation for the two KK algorithms, and study their complexity as well as their power consumption. The results show that the upsampling-free KK algorithm reduces the complexity and power consumption by a factor of 7 to 10, in comparison with the conventional algorithm at the expense of small sensitivity penalties. For example, the sensitivity difference between the two algorithms is shown to be < 1 dB for the 16-QAM formatted signals.