This paper describes the architecture and the IC implementation of a direct digital frequency synthesizer (DDFS) that is based on an angle rotation algorithm (similar to CORDIC). It is shown that the architecture can be implemented as a multiplierless, feedforward, and easily pipelineable datapath. A prototype IC has been designed, fabricated in 1.0-/spl mu/m CMOS, and tested. The IC produces 16-b sine and cosine outputs with a spurious-free dynamic range of more than 100 dBc. A 36-b frequency control word gives a tuning resolution of 0.0015 Hz at a 100-MHz sampling rate.

A 100-MHz, 16-b, direct digital frequency synthesizer with a 100-dBc spurious-free dynamic range

This paper presents a high-speed method for computing elementary functions using parallel table lookups and multi-operand addition. Increasing the number of tables and inputs to the multi-operand adder significantly reduces the amount of memory required. Symmetry and leading zeros in the table coefficients are used to reduce the amount of memory even further. This method has a closed-form solution for the table entries and can be applied to any differentiable function. For 24-bit operands, it requires two to three orders of magnitude less memory than conventional table lookups.

The Symmetric Table Addition Method for Accurate Function Approximation

Direct digital frequency synthesizers (DDS or DDFS) are widely used in modern communications and measurement devices. Their advantages are small size and power consumption together with excellent frequency stability, high frequency resolution, and short switching times. The difficulties are rather low output frequencies (500 MHz at the present state of the art) and a large set of the spurious signals very often above the -80 dB level. One source of spurious signals in DDS is the use of smaller number, W, of the most significant bits (MSB) applied for the output sine wave reconstruction from all R bits stored in the accumulator. The result is a phase modulation of the output signal. The problem was first solved in a rather complicated way with the result that the level of the largest spurious signal is about -6 W dB below the carrier with an increase of 3.9 dB in some instances. A simpler solution of the problem of spurious signal level due to the phase truncation in DDS was found earlier. However, no attention was paid to the validity of the corrections suggested. In this paper we will be concerned with this problem and investigate the validity and correctness of these generally cited results and provide a simple way for finding positions, levels, and numbers of these spurious signals generated by truncation to W bits of the phase information stored in the DDS accumulator memory of R bits (W<R).

Spurious signals in direct digital frequency synthesizers due to the phase truncation

This is a rigorous mathematical theory of the operation of the flying-adder (FA) frequency synthesizer (also called direct digital period synthesizer). The paper consists of two parts: Part I presents a detailed mathematical model of the FA synthesizer, capturing the relationships between the properties of the FA's output and internal signals and the FA's parameters. The counting of the rising edges in the FA's multiplexer's output establishes a discrete-time index that is used to analytically derive the fundamental discrete-time periods of all FA's signals. The continuous-time intervals between the rising edges are calculated and used to derive the fundamental continuous-time periods of the signals from the corresponding discrete-time ones. It is shown that the FA behaves differently within different ranges of the frequency word, and the practically useful range is identified. The FA's output average frequency, along with its maximum and minimum values, is analytically derived by calculating the number of cycles in the output signal within a fundamental continuous-time period of it. The relationship between the average and the fundamental output frequencies is also established, indicating the potential frequencies and density of output spurious frequency components. Part II of the paper characterizes the timing structure of the output signal, providing analytical expressions of the pulses' locations, analytical strict bounds of the timing irregularities, and exact analytical expressions of several standard jitter metrics. Spectral properties of the output waveform are presented, including the dominance of the frequency component at the average frequency, and analytical expressions of the dc value and average power of the output signal are derived. The FA has been implemented in a Xilinx Spartan-3E field-programmable gate array, and spectral measurements are presented, confirming the theoretical results. Extensive MATLAB simulation has also been used to generate numerous examples illustrating the developed theory.

Theory of Flying-Adder Frequency Synthesizers—Part I: Modeling, Signals' Periods and Output Average Frequency

This is a rigorous mathematical theory of the operation of the flying-adder (FA) frequency synthesizer (also called direct digital period synthesizer). The paper consists of two parts. Part I presents a detailed mathematical model of the FA synthesizer, capturing the relationships between the properties of the FA's output and internal signals and the FA's parameters. The counting of the rising edges in the FA's multiplexer's output establishes a discrete-time index that is used to analytically derive the fundamental discrete-time periods of all the FA's signals. The continuous-time intervals between the rising edges are calculated and used to derive the fundamental continuous-time periods of the signals from the corresponding discrete-time ones. It is shown that the FA behaves differently within different ranges of the frequency word, and the practically useful range is identified. The FA's output average frequency, along with its maximum and minimum values, is analytically derived by calculating the number of cycles in the output signal within a fundamental continuous-time period of it. The relationship between the average and the fundamental output frequencies is also established, indicating the potential frequencies and density of output spurious frequency components. Part II of the paper characterizes the timing structure of the output signal, providing analytical expressions of the pulses' locations, analytical strict bounds of the timing irregularities, and exact analytical expressions of several standard jitter metrics. Spectral properties of the output waveform are presented, including the dominance of the frequency component at the average frequency, and analytical expressions of the dc value and average power of the output signal are derived. The FA has been implemented in a Xilinx Spartan-3E field-programmable gate array, and spectral measurements are presented, confirming the theoretical results. Extensive MATLAB simulation has also been used to generate numerous examples, illustrating the developed theory.

Theory of Flying-Adder Frequency Synthesizers—Part II: Time- and Frequency-Domain Properties of the Output Signal

The table-lookup method is a convenient and very flexible way to generate high-quality sinusoidal test signals for measurements in psychoacoustics, speech perception, etc. Exact spectra of such signals, which are of great interest in experiments, are derived, and the relation between spectral shape, table length, and sine frequency is shown.

S. Mehrgardt

Papers

Noise spectra of digital sine-generators using the table-lookup method