%% Digital Waveform Generation: Approximating a Sine Wave % % Real-Time direct digital synthesis of analog waveforms using % embedded processors and digital signal processors (DSPs) connected % to digital-to-analog converters (DACs) is becoming pervasive even % in the smallest systems. Developing waveforms for use in embedded % systems or laboratory instruments can be streamlined using the tight % integration of MATLAB and Simulink. You can develop and analyze the % waveform generation algorithm and its associated data at your desktop % before implementing it with Real-Time Workshop on target hardware. This % demonstration goes through some of the main steps needed to design and % evaluate a sine wave data table for use in digital waveform synthesis % applications in embedded systems and arbitrary waveform generation % instruments. (Open the model: ) % % When feasible, the most accurate way to digitally synthesize a sine % wave is to compute the full precision sin() function directly for each % time step, folding omega*t into the interval 0 to 2*pi. In real-time % systems, the computational burden is typically too large to permit % this approach. One popular way around this obstacle is to use a table % of values to approximate the behavior of the sin() function, either % from 0 to 2*pi, or even half wave or quarter wave data to leverage % symmetry. % % Tradeoffs to consider include algorithm efficiency, data ROM size % required, and accuracy/spectral purity of the implementation. Similar % analysis is needed when performing your own waveform designs. The % table data and look-up algorithm alone do not determine performance % in the field. Additional considerations such as the accuracy and % stability of the real-time clock, and digital to analog converter are % also needed in order to assess overall performance. The Signal % Processing Toolbox and the Signal Processing Blockset complement the % capabilities of MATLAB and Simulink for work in this area. % % The distortion analysis in this demo is based on principles presented % in "Digital Sine-Wave Synthesis Using the DSP56001/DSP56002", by % Andreas Chrysafis, Motorola Inc. 1988 % Copyright 1990-2004 The MathWorks, Inc. % $Revision: 1.1.4.1 $ $Date: 2004/04/01 16:23:03 $ %% Create a table in double precision floating point % The following commands make a 256 point sine wave and measure its % total harmonic distortion when sampled first on the points and then % by jumping with a delta of 2.5 points per step using linear % interpolation. For frequency-based applications, spectral purity can % be more important than absolute error in the table. % % The M-file is the core function in this demo. % It is used for calculating total harmonic distortion (THD) for digital % sine wave generation with or without interpolation. This THD algorithm % proceeds over an integral number of waves to achieve accurate results. % The number of wave cycles used is A. Since the step size 'delta' is % A/B and traversing A waves will hit all points in the table at least % one time, which is needed to accurately find the average THD across a % full cycle. % % The relationship used to calculate THD is: % % THD = (ET - EF) / ET % % where ET = total energy, and % EF = fundamental energy % % The energy difference between ET and EF is spurious energy. % N = 256; s = sin( 2*pi * (0:(N-1))/N)'; thd_ref_1 = ssinthd( s, 1, N, 1, 'direct' ) thd_ref_2p5 = ssinthd( s, 5/2, 2*N, 5, 'linear' ) %% Put the sine wave approximations in a model % You can put the sine wave designed above into a Simulink model and % see how it works as a direct lookup and with linear interpolation. This % model compares the output of the floating point tables to the sin() % function. As expected from the THD calculations, the linear % interpolation has a lower error than the direct table lookup in % comparison to the sin() function. % open_system('sldemo_tonegen'); sim('sldemo_tonegen'); subplot(2,1,1), plot(tonegenOut.time, tonegenOut.signals(1).values); grid title('Difference between direct look-up and reference signal'); subplot(2,1,2), plot(tonegenOut.time, tonegenOut.signals(2).values); grid title('Difference between interpolated look-up and reference signal'); %% Taking a closer look at waveform accuracy % Zooming in on the signals between 4.8 and 5.2 seconds of simulation % time (for example), you can see a different characteristic due to the % different algorithms used: close_system('sldemo_tonegen'); ax = get(gcf,'Children'); set(ax(2),'xlim',[4.8, 5.2]) set(ax(1),'xlim',[4.8, 5.2]) %% The same table, implemented in fixed point % Now convert the floating point table into a 24 bit fractional % number using 'nearest' rounding. The new table is tested for % total harmonic distortion in direct lookup mode at 1, 2, and 3 % points per step, then with fixed point linear interpolation. bits = 24; is = num2fixpt( s, sfrac(bits), [], 'Nearest', 'on'); thd_direct1 = ssinthd(is, 1, N, 1, 'direct') thd_direct2 = ssinthd(is, 2, N, 2, 'direct') thd_direct3 = ssinthd(is, 3, N, 3, 'direct') thd_linterp_2p5 = ssinthd(is, 5/2, 2*N, 5, 'fixptlinear') %% Compare results for different tables and methods % Choosing a table step rate of 8.25 points per step (33/4), jump through % the double precision and fixed point tables in both direct and linear % modes and compare distortion results: % thd_double_direct = ssinthd( s, 33/4, 4*N, 33, 'direct') thd_sfrac24_direct = ssinthd(is, 33/4, 4*N, 33, 'direct') thd_double_linear = ssinthd( s, 33/4, 4*N, 33, 'linear') thd_sfrac24_linear = ssinthd(is, 33/4, 4*N, 33, 'fixptlinear') %% Using preconfigured sine wave blocks % Simulink also includes a sine wave source block with continuous and % discrete modes, plus fixed point sin and cosine function blocks that % implement the function approximation with a linearly interpolated % lookup table that exploits the quarter wave symmetry of sine and cosine. % (Open the model: ) open_system('sldemo_tonegen_fixpt'); %% Survey of behavior for direct lookup and linear interpolation % The M-file % performs a full frequency sweep of the fixed point tables will let % us more thoroughly understand the behavior of this design. Total harmonic % distortion of the 24-bit fractional fixed point table is measured at % each step size, moving through it D points at a time, where D is a % number from 1 to N/2, incrementing by 0.25 points. N is 256 points in % this example, the 1, 2, 2.5, and 3 cases were done above. Frequency is % discrete and therefore a function of the sample rate. % % Notice the modes of the distortion behavior in the plot, they match with % common sense: when retrieving from the table precisely at a point, % the error is smallest; linear interpolation has a smaller error than % direct lookup in between points. What wasn't apparent from using % common sense was that the error is relatively constant for each of % the modes up to the Nyquist frequency. figure('color',[1,1,1]) tic, sldemo_sweeptable_thd(24, 256), toc %% Next steps % To take this demonstration further, try different table precision and % element counts to see the effect of each. You can investigate different % implementation options for waveform synthesis algorithms using % automatic code generation available from the Real-Time Workshop and % production code generation using Real-Time Workshop Embedded Coder. % Embedded Target products offer direct connections to a variety of real-time % processors and DSPs, including connection back to the Simulink diagram % while the target is running in real-time. The Signal Processing Toolbox % and Signal Processing Blockset offer prepackaged capabilities for % designing and implementing a wide variety of sample-based and frame-based % signal processing systems with MATLAB and Simulink. close_system('sldemo_tonegen_fixpt')