Daqarta
Data AcQuisition And Real-Time Analysis
Scope - Spectrum - Spectrogram - Signal Generator
Software for Windows
Science with your Sound Card!  The following is from the Daqarta Help system:

# Burst Shape

Macro: BurstShape

This controls the shape of the Rise and Fall portions of the burst. It sets the exponent of a cosine-power function used to create the shape, from Cos^0 to Cos^16 in half-power steps. Cos^0 is a special linear mode. Although Rise and Fall must use the same Shape exponent, they may use different widths to obtain, for example, a fast attack and slow decay. Each Stream allows its own independent Shape setting.

Although Cos^n is the standard notation, the tone is actually multiplied by

(0.5 - 0.5 * cos(theta))^n

where theta goes from 0 to 180 degrees on the rising portion of the burst, and from 180 to 0 on the falling portion. The exponent is thus applied after an ordinary cosine is shifted and scaled to the 0 to 1.0 range, rather than the normal -1.0 to +1.0 range. (Otherwise, odd exponents would flip the polarity of the result.)

The cosine of 0 degrees is 1.0, so at the start of the rise the above formula becomes 0.5 - 0.5 or 0, no matter what the exponent. Similarly, the cosine of 180 degrees is -1, so at the top of the rise the above formula becomes 1.0, again independent of exponent. The tone thus always rises smoothly from 0 to 100% of the Level value, but the selection of the exponent controls the steepness of this shape.

The Rise and Fall durations also control the steepness by setting the width of the rise/fall areas. Many investigators standardize on one exponent like the Cos^2 default used here, and specify only rise/fall durations.

If you set Cos^0, no cosine function is used. Instead, the rise and fall are simply linear ramps from 0 to 100%, over the specified Rise or Fall duration.

To see the effect of Shape and the interaction with rise/fall durations, perform the rise/fall experiment with different Shape exponents.

The Shape used here is really a "window function" applied to the stream signal, similar to the functions applied to Spectrum display data with the Spectrum Window option. The purpose of each is the same: to reduce the "spectral splatter" that would be introduced by a sudden onset or offset of a signal, either a signal being generated (as here) or one being analyzed, whose "onset" and "offset" are artifacts of a finite number of analysis samples.

To compare the cosine-power Shape window to the standard Spectrum Window functions, set the Rise and Fall durations to 512 samples, which is half the FFT analysis width, and set Lag and Duration to zero samples. In this condition, the Cos^1 Shape is identical to a Hann (Hanning) window applied to the same wave having Rise and Fall durations set to zero and Duration set to 1024 samples.

Since applying a window to a continuous wave reduces the signal energy seen by a subsequent FFT, Daqarta automatically applies the appropriate correction factor to the spectrum when the Spectrum Window option is active. In the case of the Hann window, this is a factor of exactly two. Thus, if you are comparing the spectrum of a windowed continuous wave to a shaped tone burst, you might want to cut the amplitude of the continuous wave in half for a direct comparison.

The Cos^0 and Cos^1 shapes have a special property: You can use these to create bursts on different streams or output channels that always sum to unity. If the Rise of one stream starts exactly when the Fall of the other starts, and both use Cos^0 or both use Cos^1, then the sum of the two channels will always be 100%. If both streams are producing the exact same wave, then the sum will appear continuous. This effect is used to advantage with the AltSine.GEN setup file for manual Generator calibration using the Match method.

The Cos^0.5 shape has a similar special use for summing two uncorrelated noise sources. In this case the sum of the squares of the two shape envelopes is always unity. This is used to advantage in the GausWhit.GEN setup file for comparing different noise distributions.

Note that instead of the cosine method discussed above, a more efficient formula is used for all half-power shape settings:

(sin(theta/2))^2*n

Macro Notes:

L.1.BurstShape=1.3 will set Left Stream 1 Burst Shape to 1.5, rounding to the nearest 0.5 step.

L.1.BurstShape=>1 will increment and L.1.BurstShape=>-1 will decrement Shape by 0.5. Note that only steps of +/-1 are accepted by the command, even though the actual change is +/-0.5. GO: