Rational Function Interpolation of Electromagnetic Transfer Functions of High-Speed Interconnect Systems from Discrete Time-Domain and Frequency-Domain Data

We present new methodologies to generate rational function approximations of broadband electromagnetic responses of linear and passive networks of high-speed interconnects, and to construct SPICE-compatible, equivalent circuit representations of the generated rational functions. These new methodologies are driven by the desire to improve the computational efficiency of the rational function fitting process, and to ensure enhanced accuracy of the generated rational function interpolation and its equivalent circuit representation. Toward this goal, we propose two new methodologies for rational function approximation of high-speed interconnect network responses. The first one relies on the use of both time-domain and frequency-domain data, obtained either through measurement or numerical simulation, to generate a rational function representation that extrapolates the input, early-time transient response data to late-time response while at the same time providing a means to both interpolate and extrapolate the used frequency-domain data. The aforementioned hybrid methodology can be considered as a generalization of the frequency-domain rational function fitting utilizing frequency-domain response data only, and the time-domain rational function fitting utilizing transient response data only. In this context, a guideline is proposed for estimating the order of the rational function approximation from transient data. The availability of such an estimate expedites the time-domain rational function fitting process. The second approach relies on the extraction of the delay associated with causal electromagnetic responses of interconnect systems to provide for a more stable rational function process utilizing a lower-order rational function interpolation. A distinctive feature of the proposed methodology is its utilization of scattering parameters. For both methodologies, the approach of fitting the electromagnetic network matrix one element at a time is applied. It is shown that, with regard to the computational cost of the rational function fitting process, such an element-by-element rational function fitting is more advantageous than full matrix fitting for systems with a large number of ports. Despite the disadvantage that different sets of poles are used in the rational function of different elements in the network matrix, such an approach provides for improved accuracy in the fitting of network matrices of systems characterized by both strongly coupled and weakly coupled ports. Finally, in order to provide a means for enforcing passivity in the adopted element-by-element rational function fitting approach, the methodology for passivity enforcement via quadratic programming is modified appropriately for this purpose and demonstrated in the context of element-by-element rational function fitting of the admittance matrix of an electromagnetic multiport.

Phase noise and spur reduction in an array of direct digital synthesizers

Many applications, including communications, test and measurement, and radar, require the generation of signals with a high degree of spectral purity. One method for producing tunable, low-noise source signals is to combine the outputs of multiple direct digital synthesizers (DDSs) arranged in a parallel configuration. In such an approach, if all noise is uncorrelated across channels, the noise will decrease relative to the combined signal power, resulting in a reduction of sideband noise and an increase in SNR. However, in any real array, the broadband noise and spurious components will be correlated to some degree, limiting the gains achieved by parallelization. This thesis examines the potential performance benefits that may arise from using an array of DDSs, with a focus on several types of common DDS errors, including phase noise, phase truncation spurs, quantization noise spurs, and quantizer nonlinearity spurs. Measurements to determine the level of correlation among DDS channels were made on a custom 14-channel DDS testbed. The investigation of the phase noise of a DDS array indicates that the contribution to the phase noise from the DACs can be decreased to a desired level by using a large enough number of channels. In such a system, the phase noise qualities of the source clock and the system cost and complexity will be the main limitations on the phase noise of the DDS array. The study of phase truncation spurs suggests that, at least in our system, the phase truncation spurs are uncorrelated, contrary to the theoretical prediction. We believe this decorrelation is due to the existence of an unidentified mechanism in our DDS array that is unaccounted for in our current operational DDS model. This mechanism, likely due to some timing element in the FPGA, causes some randomness in the relative phases of the truncation spurs from channel to channel each time the DDS array is powered up. This randomness decorrelates the phase truncation spurs, opening the potential for SFDR gain from using a DDS array. The analysis of the correlation of quantization noise spurs in an array of DDSs shows that the total quantization noise power of each DDS channel is uncorrelated for nearly all values of DAC output bits. This suggests that a near N gain in SQNR is possible for an N-channel array of DDSs. This gain will be most apparent for low-bit DACs in which quantization noise is notably higher than the thermal noise contribution. Lastly, the measurements of the correlation of quantizer nonlinearity spurs demonstrate that the second and third harmonics are highly correlated across channels for all frequencies tested. This means that there is no benefit to using an array of DDSs for the problems of in-band quantizer nonlinearities. As a result, alternate methods of harmonic spur management must be employed.