Overlapped modulation structure Overlapped modulation structure Overlapped harmonic structure Overlapped harmonic structure Spurious spectral components due to finite word length effects Spurious spectral components due to finite word length effects numerical frequency r = phase ACC register length ( frequency resolution ) m = memory (LUT) address length ( phase resolution ) n = data width ( amplitude resolution) r = 11, m = 7, without AQ, D = 150 r =10, m = 10 n = 12, D = 32 P = 2 r / gcd(D, 2 r ) = 32 ( sample period ) N o. of the AQ-spurs: (P/4) - 1 = 7 E = 2 r-m / gcd(D, 2 r-m ) = 8 ( period of the phase error ) N o. of the PT-spurs: E - 1 = 7 r = 11, m = 7, with AQ: n = 12, D = 150 N o. of the AQ-spurs: (P/4) - 1 = 255 Same as above, i.e. there are no PT - spurs because E = 1 at the specific D tuning word r =10, with PT: m = 7, n = 12, D = 32 Prolog to an Internet link... An example of interaction between PT and AQ : zero-phase sine, 8K FFT, BH7 (7-term Blackman-Harris) window, without and with AQ Online interactive ( Mathcad ) simulations and movie clips : Online interactive ( Mathcad ) simulations and movie clips : ACRONYMS DDS: Direct Digital Synthesis NCO: Numerically Controlled Oscillator ACC: accumulator LUT: Look-Up Table DAC: Digital-to-Analog Converter AIF: Anti-Imaging Filter reg: register gcd: greatest common divisor PT: Phase Truncation AQ: Amplitude Quantization * FFT: Fast Fourier Transform FPGA: Field Programmable Gate Array DDC: Digital Down Converter DUC: Digital Up Converter SDR: Software Defined Radio ARB: arbitrary harmonically related cos components with -5dB level change (and 8K FFT, BH7 window) There is a simple procedure for evaluating the spur structures due to numerical distortion in a single-tone DDS. * There is a simple procedure for evaluating the spur structures due to numerical distortion in a single-tone DDS. * Determining the location and the level of spurs is a good starting point for the NCO design or selection ( e.g. parameterization of an NCO core in FPGA ). * In digital communication systems there are several applications that do not convert the numerical samples into an analog signal : digital radios and modems, software-defined radios, digital down/up converters. * Interactive simulation helps to discover the composed spectral maps for specific applications and tuning ranges ( spurs vary rather irregularly with tuning ), proves to be a usable tool for multi-tone ( DDS-based ARB waveform generators ). Interactive simulation helps to discover the composed spectral maps for specific applications and tuning ranges ( spurs vary rather irregularly with tuning ), proves to be a usable tool for multi-tone ( DDS-based ARB waveform generators ). Spectral purity is one of the most important specifications to the signal synthesis. * An example of multi-tone : PT - spurs :AQ - spurs : ( D is all two in hexa, E = )( P = )