Simulating Peak Shift (Jitter)

The most commonly used timing impairment is peak shift. The objective is to move the location of one or more peaks forward or backward relative to other peaks. At first glance, this may seem to be a trivial exercise solved by a simple cut-and-paste operation. That is, why not just move a selected segment of a waveform forward or backward in the record? Figure 28 illustrates this approach. An approximately 20 ns segment of a modulated data waveform is simply advanced in time by 1 ns, or one record point for a 1 GS/s clock. The peak shifts 1 ns, but this creates two abrupt transitions at the splice points. In this example, the splices are placed where the signal slopes are large to illustrate that simple shifting can introduce discontinuities resulting in an unrealistic test signal. You can attempt to hide the splice by selecting splice points at peak locations or where the slope of the signal is zero. But this simply masks the fact that the altered signal is not a realistic implementation of peak shift. Remember that our model of a disk waveform is a sum of transition responses where the effect of a single write transition extends beyond its local peak. Thus, in a realistic test signal you cannot alter one peak location without accounting for the effect on adjacent peaks.

Figure 28. The second positive peak is advanced by simply cutting a 20 ns slice and pasting it 1 ns earlier in time. This introduces discontinuities at the splice points and skirts the realities of the read-write process.

The solution is to simply re-convolve the transition response waveform with an altered data pattern. In particular, the data transition corresponding to the desired peak is simply advanced by one location to simulate a 1 ns shift in the location of the write-current transition. Recall that for our data pattern example, each bit interval is 20 ns long, consisting of 20 repeated 1's or 0's for a 1 ns clock period. An extra "1" replaces the last "0" from the preceding bit interval thus advancing the location of the positive-going transition by 1 ns. Figure 29 shows the original data waveform and two successive re-convolved data patterns where the positive-going transition was advanced and delayed by 1 ns. The peak shift is smoothly integrated into the waveform. Since our model of the transition pulse response was 256 ns long, we should expect that any transition should affect an interval tens of nano-seconds before and after the actual peak location. Note that this technique is limited to shifts that are integer multiples of the clock period. We will examine sub-period shifts later in this section.

Figure 29. A single transition in the data pattern was advanced and delayed by 1 ns. The two new data patterns were convolved with the transition response waveform. The two new waveforms are compared against the original waveform. Although the location of a single transition was altered, the two adjacent peaks are also affected. The scope was triggered by an AWG marker output located at the same point in each waveform.

Simulating Frequency Shift to Verify Clock Recovery Loops

A second common timing impairment is frequency shift. Frequency shifts are especially useful for verifying acquisition and tracking of clock recovery loops. The most direct method of modulating the clock rate of the AWG is via its external clock input, receiving modulation from an external signal generator. That is, arbitrarily shifting the frequency of a complex waveform that is already modulated by the data pattern is best performed by an external clock generator. However, you can directly implement simple shifts, such as frequency modulation of a sinusoid, using only the AWG. Figure 30 illustrates another simple case of frequency shift modulation of a data pattern using just the AWG.

Figure 30. The AWG's horizontal scaling function is used to expand the 9600-point waveform by 1%, making the new waveform 9696 points. If the same 1 GS/s clock is used, the new waveform is 1% slower than the original.

The objective is to use our basic data waveform but to alternate its rate by ±1%. We will use the AWG's horizontal scaling function to create two new waveforms--one that is 1% longer and one that is 1% shorter than the original. The original record length is 960 points, so we need to add and subtract 9.6 clock periods to the record length. Since fractional periods are not allowed, the simplest approach is to use the sequence editor to concatenate 10 copies of the original waveform and create a 9600-point waveform. Then we scale the new waveform to fit 9696 points and 9504 points to implement the frequency shift. Figure 30 shows the horizontal scaling function with a factor of 1.01 to expand the 9600-point waveform to 9696 points.

Figure 31. The 1% fast and 1% slow waveforms are concatenated to create a data pattern (Ch1) that toggles between two data rates.

Figure 31 shows the result as viewed on the TDS. We took advantage of the two independent marker outputs of the AWG to generate separate trigger pulses to signify the start of each direction of frequency shift. Use the marker triggers to synchronize the scope when examining the error signals in the clock recovery circuit. You can take advantage of the logic triggering capability in the TDS to overlay and examine the alternating patterns. In Figure 32, the TDS is set to trigger when either marker input goes high, so the scope captures the starting point of both patterns. The peaks diverge from the splice point since a constant frequency difference integrated over time causes a linearly increasing timing shift.

Figure 32. The TDS' logic trigger captures and overlays the slow and fast waveforms. The logical "OR" function is enabled so that the TDS will be triggered on the rising transition of either Ch3 or Ch4, which are connected to the AWG's Marker 1 and 2 outputs.

The AWG's two independent marker output signals, the lower two traces, were programmed to indicate the alternating rates; they signify each change of direction in frequency. Ch3 marks frequency decreasing by 1%; Ch4 marks frequency increasing by 1%.

"Beating the Spec" to Achieve Higher Resolution

The previous examples altered timing by changing the AWG's record in increments of the clock period. This limits the timing resolution to 1 ns for a 1 GS/s AWG clock. The next two examples illustrate techniques to achieve timing shifts with resolutions better than 1 ns, surpassing the specified timing resolution of the AWG. The first example demonstrates how to alter the transition pulse response equation to shift a waveform. The second example shows how to shift a selected segment of an existing waveform. Recall that the nominal location of the transition pulse peak was defined by the 128 ns offset in the pulse equation. Figure 33 shows the AWG's equation editor with the timing offset changed to 127.8 ns. This means that when our data pattern is convolved with this pulse response, each peak will arrive 200 ps earlier. If the offset is changed to 128.1 ns, then each peak in the convolved waveform will arrive 100 ps later than in the original waveform.

Figure 33. The timing offset in the transition response equation is changed from 128 ns to 127.8 ns. When convolved with the original data pattern, the resulting waveform is mathematically advanced by 200 ps.

Figure 34 demonstrates the result. The original 48-bit data pattern was re-convolved with four additional transition responses, having offsets of ­200, ­100, +100, and +200 ps relative to the original offset of 128.0 ns. The TDS displays the five resulting waveforms. The scope was triggered by an AWG marker output pulse stored at the same location in each waveform.

Figure 34. The TDS displays a single edge of the original data waveform and four other waveforms with 100 ps increments of timing offset in the transition response equation.

The waveforms are vertically expanded to isolate the same negative-going zero-crossing. Note that the horizontal scale is 250 ps per division. Ideally, the waveforms would be equally spaced in 100 ps horizontal increments, spanning a total of 400 ps. Instead, the increments are not uniform and the total span is about 450 ps. Why? The reason is that while the mathematical calculation of the convolved waveforms is precise, we are limited by the resolution and linearity of the AWG's 8-bit D/A converter. Thus, when using this technique, you need to examine the results, and you may need to adjust the timing offset empirically to achieve a specific output shift.

The final example addresses the task of a sub-nanosecond shift of a segment of an existing waveform. In other words, you may have captured an actual drive signal with the TDS and transferred it to the AWG. In this case, you do not have the option of applying various transition pulse response functions with the convolution function. We will use the technique introduced in the first example of this section. That is, we will select a segment of the waveform and shift it in time. Because we are shifting the segment by only a fraction of a clock period, we can ignore the discontinuities introduced at the splice points.

In summary, we will select a segment and scale it to ten times its original length. The expanded waveform is shifted by two points and then collapsed to its original length. The net effect is a shift of only 0.2 points, or 200 ps for a 1 ns clock period. In Figure 35, the AWG's vertical cursors isolate a forty point segment (points 503 to 542) of our 960-point basic data waveform. The AWG's horizontal scaling function is used to expand this segment by a factor of 10 to 400 points.

Figure 35. The AWG's horizontal scaling function will expand a 40-point segment between two positive peaks by a factor of 10 to 400 points.

After scaling, the AWG's shift function moves the expanded segment forward in time by 2 points (Figure 36). The two original leftmost points in the segment are lost, and we defined the two new rightmost point values to be copies from the original rightmost point value. The AWG's horizontal scaling function is then re-applied with a factor of 0.1, so that the expanded segment is reduced to its original size of 40 points.

Figure 36. The 40-point segment has been expanded to a 400-point segment. Then the AWG's horizontal shift function moves each data point forward in time (leftward) by 2.

Figure 37 illustrates this procedure. The filled points in the lower waveform represent the original waveform. The unfilled points represent the expanded version with ten times the points of the original. The expanded signal is shifted left by two points at the expanded scale, and the shifted waveform is scaled back to its original size.

Figure 37. The filled points on the lower waveform are interpolated to simulate a tenfold increase in the timing resolution. The expanded waveform is shifted left by two points at the expanded scale. When the expanded waveform is scaled back (decimated) to its original size, the waveform has been shifted left by two tenths of a point.

Figure 38 shows the results as captured by the TDS. The expand-shift-contract process was done twice--once to advance the segment by 200 ps and once to delay the segment by 200 ps. The TDS captures the two new waveforms as well as the original waveform. Note that the selected segment spanned two consecutive positive peaks. The TDS zoom function was used to expand the three waveforms around the beginning of the affected segment. The three waveforms overlay nicely before the affected area, but they diverge in time as they enter the 40-point (40 ns) segment. Although the ideal result of three equally spaced waveforms, 200 ps apart, is not achieved, we have transcended the AWG's resolution specification. The outcome is repeatable and meets the objective of improving on the apparent "limitation" of a 1 ns timing resolution for a 1 GS/s AWG clock.

Figure 38. The segment to the right of the positive peak (above the top of the screen) was advanced and delayed by 200 ps relative to a fixed marker output sync pulse.