Discussion on using DSP filtering technology on oscilloscope

　　Introduction
　　All current high-speed real-time digital oscilloscopes use some form of digital signal processing (DSP). Some engineers are concerned that using software to filter the acquired data waveform may not be consistent with the actual signal. However, the raw waveform captured by the oscilloscope does not necessarily represent the actual input signal. The "raw" waveform data captured by the oscilloscope includes distortion results caused by the oscilloscope's front-end hardware filters. In an ideal situation, a real-time oscilloscope would have an infinitely fast sampling rate, perfectly flat frequency response, linear phase response, no background noise, and high bandwidth. However, in a real environment, oscilloscopes have hardware limitations that produce errors. DSP filtering techniques can ultimately correct hardware-induced errors to a certain extent, improve measurement accuracy, and enhance display quality.
　　There are five DSP filtering techniques commonly used in current high-performance real-time oscilloscopes:


　　Each filter feature can be implemented using a finite impulse response (FIR) software filter. This article describes the uses of different DSP filters and the associated advantages and disadvantages. This article does not provide information about the actual software to implement various DSP filters.
 　　DSP filtering techniques for waveform reconstruction
　　Waveform reconstruction filtering is used to "interpolate" mathematical operation points between two actual data sampling points. The inserted data points can improve the waveform measurement accuracy under faster time base and make the waveform closer to reality. Equivalent/repeated sampling is also a waveform reconstruction technology realized by inserting points, but its

The application scenarios are limited and only work for strictly repetitive waveforms; equivalent sampling cannot be used for applications where the signal changes in real time. A complete waveform capture must be completed in one acquisition, so the only option is to reconstruct the waveform by software.
　　The simplest waveform reconstruction uses a linear interpolation filter. Although this type of filter will improve measurement resolution, accuracy, and display quality, a more accurate interpolation technique is sin(x)/x waveform interpolation filtering, which is a symmetrical filter. Figure 1 shows an example of a 3GHz sine wave using linear interpolation (blue curve at the top) and sin(x)/x interpolation (yellow curve at the bottom). With linear interpolation, we can clearly see that this oscilloscope, which uses 20 Gsamples per second, has a sample interval of 50 ps.

Figure 1: Linear interpolation and sinusoidal interpolation

　　Although sin(x)/x interpolation filtering is a more accurate way to represent the input signal, there are some issues to note. First, in order for the sin(x)/x interpolation filter to be absolutely accurate, the oscilloscope's sampling rate must be guaranteed to be able to handle any frequency components below the Nyquist frequency (fN). The Nyquist frequency is defined as ? of the sampling frequency (fS). For an oscilloscope that can sample at 20 GSa/s, the Nyquist frequency is 10 GHz. To provide maximum bandwidth while ensuring that frequencies above 10 GHz are completely filtered out, in theory, the oscilloscope must have a hardware "brick wall filter" at or below 10 GHz. Unfortunately, brick wall filters are physically impossible to implement in hardware. The red curve (top) in Figure 2 shows the characteristics of a brick wall filter, where all frequencies below the Nyquist frequency are completely passed and all frequencies above the Nyquist frequency are completely filtered out.

Figure 2: Frequency Response of Various Hardware Filters

　　In the past, oscilloscopes with lower bandwidths generally had a Gaussian-type roll-off characteristic, as shown by the green curve (bottom) in Figure 2. If you use this Gaussian-type slow roll-off filter for very fast signals, the frequency components above the Nyquist frequency (represented by the shaded area in this figure) will be aliased due to the large number of signals above the –3dB bandwidth. If the fundamental frequency of the object under test is close to or exceeds the Nyquist frequency, aliasing will make the displayed periodic waveform look like there is no trigger, and the measurement error of the waveform will increase exponentially. When the fundamental frequency of the input signal is lower than the Nyquist frequency, but the signal harmonics are higher than the Nyquist frequency, you may see a waveform with "wobbling" edges on the oscilloscope display. For this reason, Agilent has traditionally limited the bandwidth of real-time oscilloscopes with lower bandwidths and Gaussian roll-off characteristics to ? of the sampling rate, that is, ? of the Nyquist frequency, in order to filter out harmonic components higher than Nyquist.
　　For some real-time oscilloscopes with higher bandwidths between 2 GHz and 6 GHz, the hardware roll-off characteristics begin to approach the theoretical brick wall filter. This is a desirable characteristic in most oscilloscope measurements. This type of hardware filter is called a high-order maximum flat filter, as shown in the blue curve (middle) in Figure 2. Through this type of hardware filter, most in-band frequencies are transmitted with minimal attenuation, while most out-of-band frequencies are significantly attenuated. At higher order maximally flat responses, the oscilloscope bandwidth begins to approach the Nyquist limit. Agilent recommends that for oscilloscopes with higher order maximally flat responses, the oscilloscope bandwidth should be limited to no more than 0.4 times the sampling rate. In other words, for waveform reconstruction techniques using sin(x)/x filtering to be effective and accurate, the bandwidth of an oscilloscope sampling at 20 GSa/s should not exceed 8 GHz.
　　What are the disadvantages of using sin(x)/x software interpolation filters in an oscilloscope? If the input signal is band-limited early on, or if the oscilloscope hardware appropriately limits the sampled frequency components above the Nyquist frequency, then the problem can be minimized. However, if the input signal has significant high frequency components that exceed the system bandwidth, one of the problems with sin(x)/x filtering techniques is that software-generated undershoot and overshoot may appear on the reconstructed waveform. This effect is essentially a Gibbs phenomenon. The software-generated overshoot is often hidden by the overshoot inherent in the actual input signal and the overshoot generated by the oscilloscope's hardware filtering techniques. Oscilloscope users often question the effectiveness of sin(x)/x filtering techniques because undershoot is usually not physically present in the signal. However, when measuring out-of-band signals, software-induced errors such as undershoot may pale in comparison to errors introduced by uncorrected hardware.

　　Keep in mind that measuring out-of-band signals means you are trying to capture signals with frequency content that exceeds the specified bandwidth capability of the oscilloscope, so the measured results may include significant error components due to hardware limitations. For example, if you are trying to measure an input signal with an edge rate of 20 ps (10% - 90%), a 6 GHz oscilloscope will produce a measurement result of approximately 70 ps (10% - 90%), a 250% measurement error. Although the undershoot and overshoot caused by software filtering may be visually disturbing, these phenomena are only a small source of error compared to the overshoot and often overlooked edge rate measurement errors caused by hardware.
　　To reduce the software-induced undershoot, oscilloscope designers can use sin(x)/x interpolation filtering techniques without correcting the phase of the acquired out-of-band waveform. The result is that the filtered waveform has large overshoot and small undershoot. Although this may be more comfortable, the accuracy of amplitude and edge rate measurements will deteriorate. Therefore, for the measurement of fast rising and falling edges, the measurement results are most accurate when using DSP filtering techniques with linear phase correction. (Phase correction filtering techniques are discussed in more detail later in this article.)
　　The best approach is to try to ignore the undershoot and view this “wiggle” at the beginning of a fast edge pulse as a reflection of the correct D

The undershoot can also be seen as a sign that SP filtering is the technique that most accurately represents the overall characteristics of out-of-band signals. You can also view the undershoot as a sign that you should use a higher bandwidth real-time oscilloscope, or use a high bandwidth sampling oscilloscope such as the Agilent 86100C. If resampling is not possible, and a suitable high bandwidth real-time oscilloscope is not available, you may have to accept that real-time measurements are the best that can be achieved with current real-time sampling and filtering techniques.
　　As mentioned previously, sin(x)/x DSP filtering can significantly improve measurement resolution and accuracy to well above the real-time sampling interval (1/sampling rate). With the Agilent 20 GSa/s 54855A oscilloscope, delta time measurement accuracy can be improved to +/-7 ps (peak) when using sin(x)/x filtering in a single-shot acquisition. In some cases, the use of sin(x)/x filtering can compromise throughput, in other words, the filter causes the oscilloscope display to update too slowly. However, any disadvantages are outweighed by the enhanced accuracy of using sin(x)/x filtering.
　　All major real-time oscilloscope vendors now allow users to decide whether to use sin(x)/x filtering. This mode of operation is the default option on Agilent oscilloscopes, but users can select other options.
　　Amplitude Flattening Filtering
　　Amplitude flattening filters are used to correct for non-flat frequency responses in the oscilloscope hardware. Ideally, an oscilloscope should have a perfectly flat hardware response up to the natural bandwidth roll-off point of the oscilloscope, as shown by the curve in Figure 2. This means that if you measure a sine wave with constant amplitude but varying frequency, you should always measure the same amplitude until you get close to the roll-off frequency. Unfortunately, the flatness of the frequency response tends to deteriorate as you approach the bandwidth limit of the oscilloscope. Typically, the hardware itself causes the signal to attenuate at certain frequencies and amplify at certain frequencies. In fact, oscilloscope designers often intentionally introduce amplification near the bandwidth limit of the oscilloscope hardware to compensate for the frequency-dependent amplitude attenuation and push the oscilloscope to a higher bandwidth frequency response. The
　　red curve (top) in Figure 3 shows the typical hardware frequency response of an Agilent 54855A real-time 6 GHz oscilloscope. As can be seen, the hardware response of this oscilloscope meets the –3dB hardware simulated bandwidth standard of 6 GHz, but the response also shows a peak of about +1dB at about 3.5 GHz and a peak of nearly +2dB at about 5.5 GHz. Currently no oscilloscope manufacturer specifies the flatness of an oscilloscope’s frequency response. The only frequency domain specification specified for an oscilloscope is the –3dB bandwidth point. Even if an oscilloscope has +6dB peaking, which equates to a 60% amplitude error at some in-band frequency, as long as the –3dB point is above the specified bandwidth, the oscilloscope is considered to meet the specification. But just as attenuation at higher frequencies degrades measurement accuracy, so does amplitude amplification.

Figure 3: Amplitude Flattening Filter Response

　　The blue curve (bottom) in Figure 3 shows the corrected amplitude frequency response of the 54855A when using the amplitude flattening filtering technique. With this DSP/software filter, the oscilloscope's corrected frequency response deviation is typically less than +/- 0.5 dB before approaching 6 GHz bandwidth. This FIR filter is always present and cannot be removed. It is always working when the oscilloscope is sampling at the maximum sampling rate to correct the hardware filtering error. The combination of software and hardware filters provides higher measurement accuracy than that produced by hardware filters alone. [page]
　　Phase Correction Filtering Technology

Figure 4: In-Phase Harmonics

　　High-speed digital signals consist of multiple frequency components, including fundamental and harmonics. Ideally, the fundamental and harmonics of a digital signal are strictly in phase, with no phase difference or delay between the frequency components, as shown in Figure 4. Unfortunately, the oscilloscope's hardware introduces phase shifts in the high-order components of high-speed signals, which can only be eliminated by significantly increasing the instrument's analog bandwidth or using phase correction DSP filtering technology. Figure 5 shows an example of a fifth harmonic (green curve) with a time delay relative to the fundamental and third harmonics. The result is a distorted waveform display on the oscilloscope display. Without phase correction techniques, this distortion usually appears as excessive overshoot in the waveform display, with reduced edge rates. High-speed digital designers often ignore the overshoot component of the distortion, thinking that the measured overshoot actually appears on the measured input signal. However, this may not be the case and may actually be a measurement error caused by insufficient hardware capabilities.

Figure 5: Delayed 5th Harmonic

　　The red curve in Figure 6 shows the typical frequency-dependent phase error caused by the 54855A hardware at higher input frequencies. The blue curve in this figure shows the corrected phase response using the phase correction DSP/software filtering technique. It can be seen that this software filter corrects the phase error far beyond the bandwidth specification of the instrument.

Figure 6: Corrected and Uncorrected Phase Response

　　Figure 7 is a simulation of a fast edge signal with and without phase correction for a 6GHz hardware system based on a high-order maximum flat response. In the phase correction waveform (left/red curve), it can be noticed that there are undershoots and overshoots on the waveform, which are not actually present. This measurement indicates that the measured signal exceeds the frequency of the oscilloscope’s –3dB bandwidth and the oscilloscope uses a linear phase system response. The blue waveform on the right is the result of the oscilloscope measurement without phase correction. It can be seen that although there is no undershoot, the overshoot is very high. In the phase correction waveform (left/red curve), the overshoot errors at the top and bottom are improved overall. And most importantly, with phase correction technology, the timing measurements of in-band or out-of-band signals, such as rise time and fall time, are much more accurate. In the Agilent 54855A oscilloscope, the phase correction filter cannot be removed to ensure that the hardware phase error is corrected.

Figure 7: With and without phase correction

Pulse response at

　　Noise Reduction Filtering
　　As you might expect, noise reduction filtering reduces the effects of the oscilloscope's noise floor. Oscilloscopes are wideband instruments, and the higher the bandwidth, the higher the noise floor. This hardware-induced error is inevitable in wideband instruments. With the Agilent 54855A oscilloscope, you can select noise reduction filters to improve measurement accuracy by setting bandwidth limits over a wide range.

Figure 8: Without noise reduction filters, the measured noise floor is 2.8 mV RMS 

　　Figure 8 is an example of a 1 GHz sine wave captured with a 6-GHz bandwidth 54855A oscilloscope without noise reduction filtering. Using infinite persistence display mode, we see the noise caused by the oscilloscope's hardware noise floor on this captured sine wave, which is approximately 2.8 mV RMS after accumulating 1000 acquisitions. The upper/yellow curve is the input signal zoomed in to nearly full scale at 100 mV/div. The lower/green curve shows the peak portion of the waveform zoomed in 10 times. [page]

Figure 9: Noise reduction filter parameters set to 2 GHz, measured noise floor of 1.6 mV RMSFigure

　　9 shows the same 1 GHz sine wave, but now with a 2 GHz bandwidth noise reduction filter. After accumulating 1000 acquisitions, we see that the system noise floor has been reduced by nearly half. Here, the upper/yellow curve still shows the input signal amplified at 100 mV/div, and the lower/yellow curve shows the peak portion of the waveform zoomed in 10 times, so we can more clearly see that the oscilloscope noise floor has been greatly reduced after using noise reduction filtering. When
　　testing lower bandwidth signals or signals with relatively slow edge rates, the use of noise reduction filtering techniques will generally enhance the accuracy of amplitude measurements and time-related measurements. For example, when measuring jitter, the largest but often overlooked component of jitter measurement error is the jitter/timing error caused by vertical noise. There is a direct relationship between vertical noise and time-related measurement errors, which is a function of the signal slew rate. Although it is difficult to explain this technique intuitively, it is true that reducing the measurement system bandwidth will actually improve the accuracy of jitter measurements when measuring in-band signals. Enabling noise reduction filtering automatically reduces the jitter caused by the instrument's noise floor. Since increasing bandwidth and reducing the noise floor are in conflict, in the Agilent 54855A oscilloscope, we give the user the option of using noise reduction filtering.
　　Bandwidth Enhancement Filtering Technology
　　Bandwidth enhancement filtering technology, sometimes also called "bandwidth enhancement technology," is probably the least intuitive DSP filtering technology. This technology is currently used in some high-bandwidth real-time oscilloscopes. Once the hardware has attenuated the signal, how can the bandwidth of the system be increased? The answer is simple, use software to amplify the signal. Once the digitized signal is separated into various sine wave frequency components, the software can be used to selectively "amplify" individual frequency components, and the attenuated frequency components can be attenuated. The oscilloscope's -3dB point frequency response point is raised to a higher frequency using software filtering methods, as shown in Figure 10. The red curve (bottom) in this figure shows a typical hardware frequency response. The green curve (top) shows the bandwidth enhancement filter, while the blue curve (middle) shows the improved system bandwidth response, where the bandwidth has been "pushed" to a higher frequency. In addition to increasing bandwidth, this particular filter also creates a steeper roll-off characteristic for the oscilloscope, helping to reduce high frequency noise and helping to eliminate aliasing when testing out-of-band input signals.

Figure 10: Bandwidth Enhancement Filtering

　　There is also a big drawback here. As we mentioned, an oscilloscope is a wideband instrument, and the instrument's noise floor can significantly degrade the measurement results. Bandwidth enhancement filtering also amplifies the instrument's noise floor. Therefore, the signal-to-noise ratio is affected when using the bandwidth enhancement function of the oscilloscope's FIR DSP filter.
　　Although bandwidth enhancement filtering is a fairly new feature in some of today's higher bandwidth real-time oscilloscopes, it is not a new technology in the test and measurement industry. Agilent has been using bandwidth enhancement technology in network analyzers and spectrum analyzers for many years. In fact, Agilent has long used this technology in 20GHz sampling oscilloscopes to emulate faster edge rates when making TDR measurements. This technique is called "normalization" in current sampling oscilloscopes with TDR measurement capabilities.

Figure 11: Rise time measured without bandwidth enhancement

　　Figure 11 is a measurement of the bandwidth using a 6GHz oscilloscope.
 

Example of an external signal. The input signal has a rise time of approximately 50 ps (10% - 90%). However, since the rise time specification of the oscilloscope hardware is 70 ps, ​​our measurement result is 74 ps. By using the 7 GHz bandwidth enhancement filtering technique, we can now make a more accurate measurement of 66 ps, as shown in Figure 12. However, you can see that the baseline noise at the top and bottom of this waveform has increased. In standard 6 GHz bandwidth mode, the oscilloscope's noise floor is measured at approximately 3 mV RMS at a 100mV/div setting. When using 7 GHz bandwidth enhancement filtering, the noise floor increases to approximately 6 mV RMS.

Figure 12: Rise time measured using 7-GHz bandwidth enhancement

　　Another advantage of using bandwidth enhancement DSP filtering on the Agilent 54855A oscilloscope is that an 8 GHz active high-impedance probe can be used to achieve system bandwidths up to 7 GHz for measurement.
　　Summary
　　Many engineers today generally trust hardware filtering techniques and are skeptical of DSP filtering techniques because the latter are software-based. We have explained in this application note that the purpose of using DSP filtering on oscilloscope waveforms is to correct hardware filtering errors. Software filtering should not be viewed as an unrealistic processing method, but rather as a data restoration method. It is important to understand whether DSP filtering technology has any side effects, and if so, what they are. Over the years, we have used software to correct hardware filtering errors.

Hardware errors in the oscilloscope, including gain/offset calibration and deskew delays between channels. Software can also be used to correct for more complex frequency-dependent hardware error sources when using DSP filtering techniques.
　　Some of the filter features discussed in this application note have little or no side effects, such as amplitude flattening and phase correction filtering techniques. For this reason, these specific filter features are not user selectable and are used as the default mode of operation when the Agilent 54855A oscilloscope is sampling at the maximum sampling rate (20GSa/s). Because we believe that sin(x)/x waveform reconstruction filtering will improve measurement accuracy and display quality, this specific filter feature is also used as the default operating mode of the oscilloscope, but the user can easily disable this feature. The main side effect of using sin(x)/x filtering is a reduction in oscilloscope response rate.
　　Other features of the oscilloscope FIR DSP filter, including noise reduction and bandwidth enhancement filtering, have significant effects on bandwidth and noise floor. For this reason, neither of these filter features are the default oscilloscope operating mode and must be enabled by the user to use them.
　　Once you understand the problems inherent in certain types of filtering, you can confidently use DSP filtering techniques to improve the accuracy and resolution of real-time oscilloscopes, and know when to avoid using DSP filtering techniques.