Relating wideband DSO rise time to bandwidth: Lose the 0.35!
Ultrawideband real-time oscilloscopes exhibit maximally flat frequency response below the –3-dB point. Therefore, the old rules that relate rise time to frequency response no longer apply.
By Dennis Weller, Agilent Technologies -- EDN, December 12, 2002
Traditionally, oscilloscopes have exhibited a Gaussian frequency response. A Gaussian response results from the scope design's combining many circuit elements that have similar frequency responses. Analog oscilloscopes achieve their frequency response in this manner, thanks to chains of amplifiers from the input BNCs to the CRT display. (Analog oscilloscopes used the input signal to directly deflect the electron beam in a CRT. This architecture required amplifying the input signal by three orders of magnitude and driving the large capacitive load that the CRT deflection plates presented.) The properties of Gaussian-response oscilloscopes are fairly well-taught and -understood throughout the industry.
Less understood, though, is the flat response that is becoming common in modern, high-bandwidth digital oscilloscopes. A digital oscilloscope has a shorter chain of analog amplifiers to contribute to a Gaussian response and can use DSP techniques to optimize the response's accuracy. More important, a digital oscilloscope is subject to sampling-alias errors, which have no counterpart in analog scopes. (Sampling-alias errors occur when a signal contains frequencies greater than half the sample rate. Half the sample rate is called the Nyquist frequency.) Compared with a Gaussian response, a flat response reduces these errors, thus satisfying an important requirement in the design and operation of digital oscilloscopes.
Gaussian-response oscilloscopes
Figure 1 shows a typical Gaussian frequency response for a 1-GHz oscilloscope. A Gaussian response offers good pulse response without overshoot, regardless of the input signal's speed. Figure 2 shows the response to a fast step input of a 1-GHz Gaussian-response oscilloscope.
In a Gaussian-response oscilloscope, the oscilloscope's rise time is related to the oscilloscope's bandwidth by the familiar and commonly used formula, rise time=0.35/bandwidth. (Rise time is measured from the pulse's 10 to 90% amplitude points. Bandwidth is defined as the frequency at which the response is down 3 dB relative to dc. The theoretical relationship for a Gaussian system is rise time=0.339/bandwidth, but the industry has settled on 0.35/bandwidth as a practical formula.)
Another commonly used property of Gaussian systems is the overall system bandwidth, which is the rms value of the individual bandwidths. You can calculate it using the familiar relationship, system bandwidth=1/(1/BWPROBE2+1/BWOSCILLOSCOPE2)0.5. "System bandwidth" refers to the bandwidth you achieve with a combination of an oscilloscope probe and oscilloscope.
Oscilloscope probes are often designed to have sufficiently higher bandwidth than the oscilloscope bandwidth, so that the above formula is unnecessary for derating the system bandwidth.
Inversely, the measured rise time is commonly related to the system and signal rise time, using measured rise time=(RTSIGNAL2+RTSYSTEM2)0.5. You use this relationship to estimate the actual signal rise time when the oscilloscope's system rise time is not sufficiently faster than the signal's rise time.
Flat-response oscilloscopes
Figure 1 compares a flat response with a Gaussian response. Note that the flat response is much flatter below the –3-dB bandwidth but then rapidly drops off above the –3-dB bandwidth. You sometimes refer to this response shape as a maximally flat, or brick-wall, response.
Flat responses present a couple of advantages. First, the frequency contents of the signal below the –3-dB bandwidth are less attenuated, thus you can more accurately measure them. Second, the steeper roll-off helps reduce sampling alias errors.
In the time domain, applying a fast step input to a system that exhibits flat frequency response produces a pulse response with overshoot and ringing (Figure 2). Oscilloscope users often perceive overshoot and ringing as undesirable effects. However, ringing occurs only if the signal rise time is significantly faster than fastest rise time that the oscilloscope can accurately measure. In such cases, you need a higher bandwidth oscilloscope.
Unlike in Gaussian systems, the rms value of the subsystem parts does not determine the system bandwidth of a flat-response oscilloscope. The commonly used bandwidth and rise-time formulas for Gaussian-response oscilloscope systems do not apply to flat-response oscilloscope systems. Instead, you should rely on the oscilloscope vendor to specify the bandwidth of an oscilloscope/probe combination.
In the case of a flat-response oscilloscope, the rise time is related to the bandwidth as rise time=N/bandwidth, where N=0.4 to 0.5. The larger N is, the steeper or more of a "brick wall" the frequency response becomes. Oscilloscope specifications sometimes include this relationship, which can indicate the type of response the oscilloscope exhibits.
Measurement accuracy
To determine which type of frequency response offers the best measurement accuracy, you need to consider the maximum signal frequency and the oscilloscope's sampling-alias errors. The example in Figure 1 shows that a flat response offers less signal attenuation below the –3-dB bandwidth (1 GHz in the example) than does a Gaussian response. It stands to reason, then, that for signals with frequencies primarily below the –3-dB bandwidth, a flat-response oscilloscope offers better measurement accuracy than a Gaussian-response oscilloscope. (This statement implies that the phase response in the passband must be linear.)
For example, compare the measurement of the rise time of a digital signal with a 700-psec rise time using scopes that have both types of response. You can determine the signal's maximum frequency component from the rise time as, maximum signal frequency=0.5/rise time.
The maximum signal frequency is defined so that any system (including an oscilloscope) that can accurately measure components at frequencies up to and including the maximum signal frequency accurately reproduces the signal (Reference 1).
Using this relationship, a signal with a 700-psec rise time primarily contains frequencies of less than 714 MHz. From Figure 1, you can see that at less than 714 MHz, a flat-response scope attenuates less than does a unit with a Gaussian response. Indeed, a flat-response oscilloscope measures the rise time of this 700-psec edge more accurately than does a Gaussian-response oscilloscope (Figure 3). The flat-response oscilloscope measures the rise time with 3% error, whereas a Gaussian response oscilloscope exhibits a 9% error.
As the signal rise time decreases (faster edges), a Gaussian-response system eventually surpasses a flat-response system's rise-time-measurement accuracy. This situation occurs because, as the rise time decreases, the frequency content of the signal increases above the –3-dB bandwidth. In this region, the flat-response oscilloscope has less amplitude response than does the Gaussian-response oscilloscope.
Figure 4 illustrates the 1-GHz-bandwidth oscilloscope's rise-time-measurement error for various signal rise times. Note that the rise-time measurement error is already 15% at the point at which the Gaussian-oscilloscope measurement becomes more accurate than the flat-response oscilloscope. Thus, for less-than-15%-error measurements of signal rise times, a flat-response oscilloscope is superior to a Gaussian-response oscilloscope of equal bandwidth. This result is counterintuitive, given that, with an ideal (fast) step input, a Gaussian-response oscilloscope has a faster rise time than does a flat-response oscilloscope. Remember, though, that a scope's rise-time specification does not by itself indicate how accurately the scope can measure rise times; you must also consider the character of the scope's response.
Sampling-alias errors
Digital oscilloscopes use repetitive and real-time sampling methods. A repetitive sampling oscilloscope samples the signal over many repetitions of the signal and is not subject to sampling-alias errors. A real-time oscilloscope samples and captures the signal in one pass, or occurrence, of the signal. This discussion applies to the more common real-time sampling oscilloscopes, which offer many benefits over repetitive-sampling oscilloscopes.
For a digital real-time oscilloscope to accurately measure a signal, the signal must not have significant frequency content above the Nyquist frequency, which is half the sampling frequency. Frequency content above the Nyquist frequency folds back into the frequency-domain region below the Nyquist frequency. In the time domain, this error manifests itself as a pulse response with "wobbling" edges (Figure 5). These wobbling edges result in inconsistent rise times and time-delay measurements.
For the example in Figure 1, the sample rate is 4 GHz, so the Nyquist frequency is 2 GHz. A Gaussian-response oscilloscope allows the sampling of significant frequency content beyond 2 GHz and produces sampling-alias errors for signals with significant frequency content above 2 GHz. A flat-response oscilloscope, however, significantly attenuates all frequency content above 2 GHz, essentially eliminating alias errors.
To avoid alias errors, the oscilloscope must have sufficient sample rate. A Gaussian-response oscilloscope may need a sample rate as much as six times the oscilloscope's bandwidth, though four times is more typical. On the other hand, a flat-response oscilloscope with a sharp filter may need to sample at only 2.5 times the oscilloscope's bandwidth to avoid alias errors.
How much bandwidth?
Table 1 lists the steps to estimate the necessary oscilloscope bandwidth to make accurate measurements. You first determine the maximum signal frequency, FMAX, based on the signal's rise time. (Depending on the application, additional bandwidth may be necessary to reproduce noise or jitter beyond the maximum signal frequency.) Next, you determine whether the scope has a Gaussian or a flat response. Then, based on the accuracy you need, you multiply FMAX by the appropriate multiplier to determine the required oscilloscope bandwidth. Finally, ensure that the oscilloscope has sufficient sample rate to achieve the required bandwidth without introducing alias errors.
For example, measuring a 100-psec-rise-time (20 to 80%) signal with a flat-response oscilloscope to an accuracy of 10% requires (0.4/100 psec)·1.2=4.8-GHz bandwidth and a minimum sample rate of 4.8 GHz·2.5=12G samples/sec.
This procedure is only a tool to estimate the necessary bandwidth. It is prudent to verify actual rise-time accuracy with measurements, because frequency response varies among oscilloscope models.
For less-than-15%-error measurements of digital-signal rise times, a flat-response oscilloscope offers better accuracy than a Gaussian-response oscilloscope of equal bandwidth. Another benefit of a flat-response oscilloscope is that such scopes typically have brick-wall filters that reduce or prevent alias errors.
The signal's rise time, not its repetition rate primarily determines the necessary oscilloscope bandwidth. For accurate measurements, pick an oscilloscope that has the flattest possible frequency response up to the maximum signal frequency, determined by 0.5/rise time (10 to 90%). In the case of a modern flat-response oscilloscope, an oscilloscope bandwidth that is 1.4 times the maximum signal frequency usually suffices for accurate rise-time measurements.
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