There are many misconceptions surrounding the relationship between risetime and bandwidth, in oscilloscopes. These misconceptions extend to time-domain reflectometer (TDR) instruments in the form of questions about spatial resolution of impedance profile plots. Things get even more confusing when thinking of impedance profile plots generated from instruments like the vector network analyzer (VNA). This paper will clarify the thinking about these concepts.
The Relationship Between Risetime and Bandwidth
In oscilloscopes, based on the stated bandwidth, there is an expectation on the risetime of the instrument, as measured when a very fast step is applied to the instrument. This expectation is expressed as a multiplier m, such that bandwidth · risetime = m, where m = 0.35 is most commonly used. It turns out that this multiplier is valid only under a special case. Therefore, it is important to understand the source of this expectation.
Various threshold crossing times are tabulated in table 1. For a single-pole system, the step response, while starting at zero, never actually reaches its final value, as the response is infinite in duration. Therefore, on the first row in table 1, one can see that it crosses the threshold at zero time, but never crosses the threshold at 1.
second and third rows are the most important when talking about risetime. Usually, risetime is specified as the time it takes the signal to traverse either the 10% and 90% thresholds (the 10–90 risetime), or the 20% and 80% thresholds (the 20–80) risetime, with the 10–90 risetime being the most important.
WavePulser 40iX Risetime
While the absolute risetime of the TDR is indicative of the frequency content in the TDR pulse, it is only the frequency content relative to the noise floor that is really important for the quality of the measurements made insofar as dynamic range is concerned.  Otherwise, it has no effect on the effective risetime implied by the s-parameters nor on the spatial resolution of the instrument. That being said, the WavePulser 40iX utilizes an impulsive stimulus, whose typical characteristics are shown in figure 2. The incident impulse is shown in figure 2a, and the spectral content of the impulse is shown in figure 2b, where it is seen to have an essentially flat content of -62dBm/GHz out to 40GHz. The wiggles in the response are due to the small impedance discontinuity of the launch, which is seen at around 300ps in figure 2a. It is important to note that the frequency content in figure 2b is actually the product of the pulser frequency content and the frequency response of the sampler, so the actual frequency content of the impulse is probably higher.
Because the WavePulser employs an impulsive stimulus, it has higher dynamic range at high frequency, but it must be integrated for comparison with a traditional TDR. The integrated step response is shown in figure 2c and zoomed in figure 2d. In figure 2c, it is seen to reach a nominal amplitude of 2pV·s.1 In figure 2d, markers
are placed at the 20% point and the 80% point. The 20% threshold is 0.4pV·s, which is reached at -2.685ps and the 80% threshold is 1.6pV·s, which is reached at 4.545ps for a 20–80 risetime measurement of 7.23ps. Multiplying this risetime by the 40GHz instrument end frequency gives 0.289, which is higher than for the single-pole system shown in the third row of table 1. This is because the response characteristic drops more sharply than a single-pole response, as seen in figure 2b; the response characteristic affects the multiplier.
Despite the fast risetime of the WavePulser, the time-domain plots generated by the instrument are calculated from the measured s-parameters in much the same manner as a VNA would generate them. In the case of the VNA, the DC point is extrapolated, which sometimes causes problems. The WavePulser directly measures the DC point, which is critical to proper time-domain analysis. Because the time-domain plots are generated from calibrated s-parameter measurements, the frequency response shape from which the time-domain waveforms are generated can be considered to have unity gain up to the end frequency and then ends abruptly.
Brick-wall Limited Systems
In high-end oscilloscopes (i.e., those with bandwidths exceeding 15GHz), the frequency response rolls off very quickly after the 3dB bandwidth is reached, which affects the multiplier that relates bandwidth to risetime. The most extreme situation is the brick-wall limited system. A brick-wall system is one that passes frequency content with unity gain right up to a given frequency and no content after that frequency. It turns out that a brick-wall system is exactly what one obtains when examining time-domain effects from s-parameters, whether measured using a VNA or TDR.
When considering s-parameters, the end frequency defines an effective sample rate for the time-domain waveforms.  Thus, for a given end frequency Fe, the effective sample rate is Fs = 2·Fe (i.e., the end frequency is considered the Nyquist rate) and the sample period is Ts = 1/Fs = 1/(2·Fe). A perfect system would have a step response that rises from 0 to 1 in exactly one sample. Thus, given s-parameters to 40GHz (an effective sample rate of 80GS/s), the step would rise in one sample period, or 12.5ps. One might think that this, therefore, is the effective risetime, but this is not the whole picture.
Using a normalized end frequency of 0.5, allows for the various threshold crossing times and risetimes to be calculated in samples, where the sample period is calculated from the end frequency as previously described. Such a normalized system is shown in figure 3, where figure 3a shows the frequency response, and figure 3b shows the step response.
These results are tabulated in table 2. There are several things to note:
The multiplier used in the 10–90 bandwidth–risetime relationship is 0.446, as opposed to the commonly stated 0.35 multiplier for single-pole systems.
The multiplier used in the 20–80 bandwidth–risetime relationship is 0.317, as opposed to the 0.221 multiplier in single-pole systems.
The WavePulser multiplier of 0.289 for the 20–80 risetime is somewhere in between the single-pole system and brick-wall system.
Both the 10–90 and 20–80 risetimes are less than one sample.
One might erroneously think that the brick-wall system provides the worst case multiplier to be used, but it does not. This is because of the phase response of the system. The phase of the brick-wall response is linear phase, while most responses, such as the single-pole system, are minimum phase. This is the topic of another discussion. 
Spatial Resolution and Propagation Velocity
The relationship between TDR risetime and spatial resolution is recommended by the IPC3 Test Methods Manual  for measuring the characteristics of lines on printed circuit boards by TDR. In the manual, the temporal resolution is defined as half the 10–90 risetime of the instrument. In order to convert this to physical length, the propagation velocity must be known.
Since this document was written, however, it is more common for controlled impedance traces to be constructed as either stripline, or covered microstrip with advanced laminates. A stripline example with an advanced laminate is provided in the table 3, which gives the WavePulser a 1× resolution of 0.870mm. For this example, a trace length in excess of 3.479mm is recommended for accurate impedance measurements.
As the IPC document points out, TDR effects other than spatial resolution must be considered in making impedance measurements, including ringing and other aberrations. Certainly, measurements that result from converting frequency-domain s-parameters to time-domain impedance profiles will exhibit these effects because of the nature of sin ¹xº/x interpolation. Fortunately, this can be mitigated within the WavePulser instrument by specifying a risetime to apply to the time-domain measurements.
To illustrate this, the open-source software SignalIntegrity  is used to simulate TDR waveforms applied to a 40Ω 1× and 4× structure with applied risetimes ranging between 0 and 20 ps, shown in figure 4. The 1× and 4× structure simulation is shown in figures 4a and 4b, respectively. It is important to understand that the risetimes applied in the simulations do not by themselves determine any spatial resolution. In other words, 0ps risetime applied to 40GHz s-parameters will have the spatial resolution of 40GHz s-parameters. In some sense, the risetime of the applied step in the simulation adds in quadrature with the risetime inherent to the s-parameters.
In order to see the effects of the incident step risetimes more clearly, zooms of the simulation waveforms are shown in figures 4c and 4d. In both of these plots, markers are placed at the correct measurement points; two vertical lines are placed at the beginning and ending time location of the 40Ω discontinuity, and a horizontal line marks the correct 40Ω impedance. In figure 4c, it is clear that the time boundaries of the measurement along with the actual impedance measured are incorrect, however the instrument has no problem time locating and, for the most part, measuring the discontinuity. The error is 2Ω with 0ps incident risetime. The instrument measures half the difference between 40Ω and the 50Ω discontinuity using 20ps incident risetime. While this error is not insignificant, it allows the instrument to identify the impedance change. It is the opinion of the author that the 20ps risetime setting measures the impedance properly within the spirit of temporal or spatial resolution for the 1× structure, given that the 4× structure is recommended to make the precise impedance measurement. This is further demonstrated by examining figure 4d, which is on the same scale as figure 4c. Here, it is seen that the time boundaries of the structure are properly measured with all incident risetimes, but the 20ps risetime setting shows the 40Ω impedance perfectly measured at 200ps with no aberrations present in the waveform.
Bandwidth and risetime are related in a manner that depends on the thresholds used to measure the risetime and in the shape of the magnitude and phase response of the system. Spatial resolution of TDR instruments depend on the risetime, but for instruments that compute time-domain impedances from s-parameters, the dependency is actually on the end frequency.
The WavePulser 40iX measures from true DC to 40 GHz, giving it an effective 10–90 risetime of 11.15ps, although the actual risetime is much faster. This provides the ability to resolve impedances that are 5.575ps in electrical length, which is approximately 1mm of resolution for microstrip on FR4, and < 1mm in more common situations.
Adjusting the incident risetime in the time-domain measurement results, such as the impedance profile traces, reduces ringing and aberrations and improves impedance measurements.