Analysis of Digitally Generated Sinusoids with Application to A / D and D / A Converter Testing

has been planning for the influx of digital and Coding, " Bell System Tech. Abstract-Several aspects of the design of digital frequency synthesizers are considered, and a detailed analysis of the spectral purity of digitally synthesized frequencies is given. These results are then applied to a specific application: the testing of A/D and D/A converters. F REQUENCY synthesis, which is the process of deriving one frequency from another reference frequency, is best known as a means for obtaining many synchronous sinusoids or clocks. Such signals are required, for example, in communication receivers and digital communications systems. A less well known application of frequency synthesis is to the testing of A/D and D/A converters (codecs) where digitally derived sinusoids are useful. While the work reported here is motivated by the latter application, the results are also applicable to the former. There are several methods of frequency s y n t h e s i ~ , ~ ~ ~-~ , ~ but this paper concentrates on digital frequency synthesis1,2 in which the samples of a sinusoid are generated digitally and a D/A converter is used to generate the desired sinusoid. In applications of the type first mentioned above the generated sinusoid is the primary objective, while in the second, the objective is to test the operation of the D/A converter itself. Section 2.0 considers several aspects of the design of digital frequency synthesizers, including an analysis of the spectral properties of the quantization noise. In Section 3.0 we apply these results to the testing of A/D and D/A converters. Appendix A supplies some mathematical details. The set of frequencies which can be obtained from a digital frequency synthesizer is considered in Section 2.1, and in Section 2.2 the problem of obtaining the best approximation to a desired frequency with a given synthesizer complexity is solved. Section 2.3 studies several types of symmetries which can be exploited to reduce the memory requirements for stored sinusoids (recursive generation techniques can also be used1). Finally, in Section 2.4 the spectrum of the synthesized signal is derived and formulas are given for the SNR. Assume that a sinusoid A sin (k f d t i-0) with amplitude A , frequency f d and phase 0 is desired. We can generate such a sinusoid by storing digital encodings of its samples x (k) = A sin (2TfdkT + 0) (2.1) in a read-only-memory (ROM) and presenting …