## It's a Gaussian World

A previous article suggests that most digital output waveforms follow a nearly Gaussian profile (Reference 1). Let's test that theory.

Figure 1 depicts the rising edge from a Texas Instruments DL100 44T LVDS (low-voltage-differential-signaling) driver. This driver exists on a custom test board with SMA output connectors. A coaxial cable feeds the test outputs directly to the 50O terminated inputs of a LeCroy SDA6000 oscilloscope. A small bit of residual ringing from the previous bit appears in the left side of the figure. Then, just before the signal begins its major ascent, you may see a tiny precursor, probably the result of crosstalk from the test equipment and predrivers inside the DL100 44T. The amplitude of these features amounts to only about 1% of the main signal's step size. Directly after the main edge, the signal overshoots a tiny amount and then rings back approximately 2%. This artifact is more likely due to the layout of the test board than anything related to the DL100 44T. With artifact amplitudes this low, spectral analysis of the waveform should clearly reveal whether the true spectral content of a DL100 44T driver follows the Gaussian theory.

Figure 1 plots the step response of the driver. Spectral calculations require the impulse response. The frequency response associated with that impulse response gives you a filter you can then apply to any square-edged digital signal, making the output look just like the step shapes in Figure 1.

To make the necessary spectral calculations, first convert the step waveform to an impulse response by computing, at each point, the slope of the signal:

x'n = (xn+1xn)/T

Where:

• T equals the sampling interval.

Next, truncate the signal to a finite-length region. Make the region wide enough to capture the main features of the signal in question but narrow enough to eliminate unrelated events before and after the main edge. Try using the whole region shown in Figure 1, from 0 to 5 nsec. Within this region, 2000 points are sampled at 2.5 psec each.

Simple truncation of a sampled-data sequence can induce Gibbs phenomena. To mitigate these unusual effects, multiply the truncated sequence by a Hamming window of length N=2000:

hn = 0.54 – 0.46 cos(2πn/N)

A Hamming window smoothly feathers the sequence to zero at both ends, reducing the impact of discontinuities at the endpoints. It accomplishes this task at the expense of a modest loss in frequency-domain resolution.

Finally, use an FFT to compute the frequency response of the windowed impulse response and normalize the result so that it starts at 0 dB. Figure 2 compares the result against a Gaussian template. It's a match. ### Reference

 Johnson, Howard, "Real signals," EDN, Oct 8, 2009, pg 13.