# Digital Waveform Generation: Approximating a Sine Wave

This example shows 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.

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 Simulink® Coder™ on target hardware.

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 DSP System Toolbox™ complement the capabilities of MATLAB and Simulink for work in this area.

Another popular way to approximate the behavior of the sine wave is to use the CORDIC approximation method. CORDIC is an acronym for COordinate Rotation DIgital Computer. The Givens rotation-based CORDIC algorithm is one of the most hardware-efficient algorithms because it requires only shift-add iterative operations.

The distortion analysis in this example is based on principles presented in "Digital Sine-Wave Synthesis Using the DSP56001/DSP56002", by Andreas Chrysafis, Motorola® Inc. 1988

### 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. Similar computations are done by replacing the sine values with CORDIC sine approximation. For frequency-based applications, spectral purity can be more important than absolute error in the table.

The file ssinthd.m is the core function in this example. 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; angle = 2*pi * (0:(N-1))/N; s = sin( angle )'; thd_ref_1 = ssinthd( s, 1, N, 1, 'direct' ) thd_ref_2p5 = ssinthd( s, 5/2, 2*N, 5, 'linear' ) cs = cordicsin( angle, 50 )'; thd_ref_1c = ssinthd(cs, 1, N, 1, 'direct' ) thd_ref_2p5c = ssinthd(cs, 5/2, 2*N, 5, 'linear' )

thd_ref_1 = 4.7198e-32 thd_ref_2p5 = 1.4176e-09 thd_ref_1c = 1.2544e-30 thd_ref_2p5c = 1.4176e-09

### 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, with linear interpolation, and with CORDIC approximation. 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. The CORDIC approximation shows a lower error margin when compared to the linear interpolation method. This margin depends on the number of iterations when computing the CORDIC sin approximation. You can typically achieve greater accuracy by increasing the number of iterations (corresponding to a longer computation time). The CORDIC approximation eliminates the need for explicit multipliers. It is used when multipliers are less efficient or non-existent in hardware.

open_system('sldemo_tonegen'); set_param('sldemo_tonegen', 'StopFcn',''); out = sim('sldemo_tonegen'); currentFig = figure('Color',[1,1,1]); subplot(3,1,1), plot(out.tonegenOut.time, out.tonegenOut.signals(1).values); grid title({'Difference between direct look-up', 'and reference signal'}); subplot(3,1,2), plot(out.tonegenOut.time, out.tonegenOut.signals(2).values); grid title({'Difference between interpolated look-up', 'and reference signal'}); subplot(3,1,3), plot(out.tonegenOut.time, out.tonegenOut.signals(3).values); grid title({'Difference between CORDIC sine', '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:

ax = get(currentFig,'Children'); set(ax(3),'xlim',[4.8, 5.2]) 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')

thd_direct1 = 2.6423e-15 thd_direct2 = 2.8660e-15 thd_direct3 = 2.6423e-15 thd_linterp_2p5 = 1.4175e-09

### 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')

thd_double_direct = 4.7061e-05 thd_sfrac24_direct = 4.7061e-05 thd_double_linear = 7.9741e-10 thd_sfrac24_linear = 8.1751e-10

### 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. Example sldemo_tonegen_fixpt uses a sampled sine wave source as the reference signal and compares it with a lookup table with or without interpolation, and with CORDIC sine approximation in fixed point data types.

Open the sldemo_tonegen_fixpt model

open_system('sldemo_tonegen_fixpt'); set_param('sldemo_tonegen_fixpt', 'StopFcn',''); out = sim('sldemo_tonegen_fixpt'); figure('Color',[1,1,1]); subplot(3,1,1), plot(out.tonegenOut.time, out.tonegenOut.signals(1).values); grid title({'Difference between direct look-up', 'and reference signal'}); subplot(3,1,2), plot(out.tonegenOut.time, out.tonegenOut.signals(2).values); grid title({'Difference between interpolated look-up', 'and reference signal'}); subplot(3,1,3), plot(out.tonegenOut.time, out.tonegenOut.signals(3).values); grid title({'Difference between CORDIC sine', 'and reference signal'});

### Use of Sine Function with a Clock Input

The model also compares the sine wave source reference with the sin() function whose input angle in radians is time based (computed using a clock). This section tests the assumption that the clock input would return repeatable results from the sin() function for period 2*pi. The following plot shows that the sin() function accumulates error when its input is time based. This also shows that a sampled sine wave source is more accurate to use as a waveform generator.

subplot(1,1,1), plot(out.tonegenOut.time, out.tonegenOut.signals(4).values); grid title({'Difference between time based sin()', 'and reference signal'});

### Survey of Behavior for Direct Lookup and Linear Interpolation

The file sldemo_sweeptable_thd.m performs a full frequency sweep of the fixed point tables and 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 is not intuitive is 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
```

Elapsed time is 1.156165 seconds.

### Next Steps

To take this example further, try different table precision and element counts to see the effect of each. Using CORDIC approximation, try different numbers of iterations to see the effects on accuracy and computation time. You can investigate different implementation options for waveform synthesis algorithms using automatic code generation available from the Simulink Coder and production code generation using 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 DSP System Toolbox offer prepackaged capabilities for designing and implementing a wide variety of sample-based and frame-based signal processing systems with MATLAB and Simulink.

bdclose('sldemo_tonegen'); bdclose('sldemo_tonegen_fixpt')