Mathematical models of noise characteristics of high-speed digital-to-analog converters of radar signal generators

A mathematical model of the power spectral density of phase noise of high-speed digital-to-analog converters in one side frequency band of tuning is obtained. Calculated using the developed mathematical model of the power spectral density of phase noise of high-speed digital-to-analog converters from Analog Devices in the efficiency mode in the NRZ baseband showed a small error in comparison with the experimental characteristics. It is shown that in the 2 × NRZ oversampling mode, the phase noise level of the high-speed DAC is 7-8 dBc/Hz lower than in the main NRZ operation mode.


Introduction
Digital-to-analog converters (DACs) [1][2][3] have recently been widely used in radio engineering systems to generate high-frequency radio signals. They have the following advantages: the ability to form a radio signal at once at a high frequency with the required types of modulation, high accuracy of the synthesized frequency, digital control of the frequency and phase of the output signal, very high resolution in frequency and phase, high frequency tuning speed [4,5]. Digital-to-analog converters are an integral part of direct digital synthesizers (DDS) [6][7][8][9][10], which are used to create reference signals in radio systems [11].
An important characteristic that determines the quality of the generated radio signals is the power spectral density (PSD) of phase noise in one sideband. This value is characterized by the ratio of the noise power at the frequency F of one side band in the frequency band 1 Hz to the power of the carrier signal [12].
PSD models of phase noise based on power functions are used for the theoretical analysis of the noise characteristics of radio devices [8]. The use of these models significantly simplifies the design of signal generators and other devices, allowing you to do without complex experimental studies. For digital computing synthesizers, similar models are proposed and considered in [13,14].
Since the noise characteristics of the DAC are mainly determined by the noise of the DAC, we will take the model of the characteristics of the DAC as the basis for the PSD model of phase noise in a single sideband of the DAC. The aim of the work is to develop a mathematical model of the power spectral density of phase noise in one sideband of digital-to-analog converters in various operating modes.

The model of the power spectral density of phase noise in a single sideband of digital-toanalog converters
For a mathematical model of the power spectral density of phase noise in one sideband of the DAC, we use the model for the DAC [13], based on power functions of the form where coefficients k1, k2, k3, k4 determine the level PSD 1/F 2 phase noise, 1/F phase noise, the natural noise component of the input circuits and the natural noise component of the load resistance, respectively, F -offset from the carrier frequency, The expression is determined by the PSD value of the phase noise in dBc/Hz, defined for the lowest output frequency , for which there are experimental spectral characteristics.
Value k2 determines the level of white frequency noise 1/F 2 , which is defined for the minimum offset frequency F = 10 Hz.
The coefficients for the natural noise components are determined for the offset frequencies and the k4 coefficient is determined for the maximum output frequency of the synthesizer for The noise level of the DAC quantization is determined by the number of bits N, for N≥14 quantization noise is significantly less than additive noise and can be ignored.
To create a model of the noise characteristics of digital-to-analog converters, we use the experimental noise characteristics of a digital-to-analog converter AD9164 [1] shown in figure 1. This DAC can operate in various modes: normal operation mode (non-return-to-zero (NRZ)), 2хNRZ -The DAC operates at twice the clock speed if new data samples are captured in the DAC core on both the leading and trailing edges, radiofrequency (RF) or mixmode -in this case, each clock pulse of the NRZ mode is represented by two different-polar pulses with a duration equal to half of the period. In this case, the noise characteristics of the chip are given taking into account the noise of the clock generator.  (1). Therefore, to calculate the approximation coefficients ki, we will determine the noise characteristics of the DAC itself     For other frequencies, the simulation results shown in figure 4 were also satisfactory.  . A comparison of the results of modeling noise characteristics with experimental characteristics in the NRZ mode is shown in figure 5. As you can see, the greatest error is observed at offset frequencies of more than 300 kHz. This is probably due to the error in the approximation of the initial experimental characteristics. We will conduct a study of the noise characteristics of the DAC in the mode 2xNRZ. In this mode, the signal samples are made on the leading and trailing edges of the clock generator pulses, which is equivalent to doubling the clock frequency. The noise characteristics for the clock frequency fCLK=4000 MHz shown in the datasheet of the AD9164 integrated DAC correspond to this mode and are shown in figure 6.
The dependencies for the modes NRZ and 2xNRZ constructed using the model (7)   For a low output frequency of 70 MHz, the characteristics for the NRZ and 2xNRZ modes up to 10 kilohertz differ from each other by 6 dBc/Hz, with a larger offset, both dependencies are limited to thermal noise at -171 dBc/Hz. For a frequency of 900 MHz, the dependencies for any detuning differ by 6 dBc/Hz, and with an increase in the output frequency to 3900 MHz, the DAC transmission coefficient and phase noise in the NRZ mode begin to affect significantly (by 20 dBc/Hz) increase.
This does not happen in 2xNRZ, since at twice the clock frequency, the effect of the DAC transmission coefficient on natural noise is insignificant.
The phase noise of the reference generator has a great influence on the formation of high-frequency signals. Taking into account the noise of the REF in accordance with (6), the obtained noise characteristics of the AD9164 are shown in figure 8.
As can be seen, the proposed PSD model of DAC phase noise approximates the experimental dependences with sufficient accuracy. A small outlier in the experimental characteristics is due to the PSD of the phase noise of the reference generator.

Conclusion
Thus, the proposed mathematical model of the power spectral density of phase noise in one sideband of digital-to-analog converters corresponds to the experimental characteristics with a sufficiently high accuracy and allows us to theoretically study the noise characteristics of high-speed DACs at any output frequencies.