# Important Parameters in Choosing an Digital Storage Oscilloscopes (DSO) - Part 2

Please note that the probe must be included because it has its own rise time. Some firms used to specify the scope rise time at the probe tip. Any bandwidth printed onto a probe is to be taken with a grain of salt, because it will depend on the scope used and the correct calibration of the probe.

Many modern wideband DSO’s use software tricks to "lift" the frequency response by manipulating the signal after the a/d conversion, dispensing with the undistorted pulse response.

• Although DSO’s have to show a Gaussian response they remain sampling scopes. If the actual bandwidth is determined by the sampling rate, the pulse response will not anymore be Gaussian.

Any signal content above half the actual sampling frequency will be aliased. An analog scope is able to show signals far above its bandwidth, they are reduced in amplitude and rounded, but visible. This applies e.g. to wild oscillations of a circuit which will be obvious. DSO’s can not show anything above half the sampling frequency, i.e. only artefacts.

### 2.2 Artefacts, invalid digital data.

Whenever the sampling frequency is too low distortions and artefacts will crop up which mostly bear no resemblance to the signal. Only a low pass filter with sufficient damping at half the sampling frequency could avoid this. In principle, this would be possible if the sampling frequency were always constant, but DSO’s with small memories would require a low pass with variable cut-off frequency.

• While analog scopes do not distort signals at all, sampling scopes will show limitations and distortions due to the sampling process (see later).

DSO’s, however, will show the combined signal distortions of three processing steps: sampling, a/d conversion and d/a conversion.

• If a DSO displays artefacts it follows that all digital data derived are false, and so are all calculations based on such data! But how does one know that the display and the data are false? See later.

Note that the true rise time of 0.7 ns was shown eventually false by a whopping 6 orders of magnitude! Magnificent progress!

### 2.3 Resolution and accuracy.

• Most manufacturers claim that their DSO’s could improve their resolution from the usual 8 to 11 bits. Yes, this is true. What the manufacturers do not say, however, is that the higher resolution is achieved by averaging which is identical to low pass filtering, and that the bandwidth will be knocked down to a fraction of the nominal, e.g. from 100 to 1 MHZ! The innocent user will not be aware that he is now measuring with only 1 MHz bandwidth! They also do not say that averaging is limited to signals which repeat with the same shape. There is never something for nothing.

Figure 2.3: a to e show 5 screen photos of a 500 MHz DSO, the table below shows the digital display of the bandwidths, rise times and signal frequencies of a 100 kHz 0.7 ns rise time square wave at sweep speeds from 0.1 us/cm to 500 us/cm.

Except for a few very recent instruments with 12 bits with prices up to 40 K\$ all DSO’s have a maximum of 8 bits of vertical resolution; the stress is on the word “max.”, because the a/d converter range is seldom fully used. It is “effective bits” which count.

It is customary to advertise impressive theoretical high dynamic ranges of digitized signals like the 16 bits of the cd which should yield 96 dB. It is not mentioned that the distortions rise to 100 % when the signal approaches zero, so the lower bits can not be used. In case of the cd the lower 10 bits are for the birds, so only 6 bits or 36 dB dynamic range are usable. This is the reason why the music is compressed into those upper bits. The “low noise” is due to the fact that cd players short the output for low values of signal. Any good record has a dynamic range of 60 dB, top tape recorders achieve up to 80 dB. With analog signal processing, the resolution remains infinite until the signal disappears in the noise.

• With 8 bit DSO’s the trace remains jittery which causes the user to believe that his measuring object was noisy. With CCD’s this is still worse.

Figure 2.4: Digitized signals become the more distorted the smaller they become. Eventually, only the LSB will be switched on and off, i.e. there will be a rectangle, irrespective how the original signal looked like.

### 2.4 Reconstruction problems

There are many applications of the sampling technique where it is not necessary to reconstruct the waveform, e.g. if only the rms value of a signal is to be measured. However, with scopes reconstruction is the purpose. Just two sampling points per signal period (= Nyquist) give no hint to the waveform, any waveform may be drawn through these two points! From 10 points on a rough guess is possible, for a useful reconstruction many more points are required, a decent one needs about 100! This is just another way of expressing the need for a sampling frequency 5 to 10 higher than Nyquist or 10 to 20 times higher than the bandwidth. A single glimpse onto the screen of a sampling scope is enough to prove this.

Quotation Tektronix:

"For accurate reconstruction, using linear interpolation, the sample rate should be at least 10 times the highest frequency signal component!"

Quotation LeCroy:

"...with linear interpolation at least 10 points per cycle are necessary...” This statement pertains to the requirements of a barely acceptable reconstruction. If linear interpolation were forbidden, DSO’s would never have captured the market.

In the operating modes ETS and RS (see below) always sufficient points are gathered, however, in the operating mode RTS only the points which are obtained once are available, and this determines the performance of a DSO. The reason for the discrepancies in the manufacturers’ wordings is their effort for years to sidestep Nyquist and their statements at that time they had a special method to get away with 2.5 times the sampling rate instead of 10 times. This applies to the sin x/ x reconstruction which, however, is only appropriate for sine waves. The amount of data captured is mostly too much for the display, e.g. a 2,000 x 2,000 pixel vector display, so the samples acquired are binned down to 2,000 pixels. Decimation binning means that e.g. only every 50th sample is taken, this means that also the bandwidth is divided by 50. In order to see past events, the user must scroll through the memory and search.

Figure 2.5: Reconstruction errors resp. distortions caused by linear interpolation if the number of points is insufficient. The top picture shows the sampling points of a sine as they are obtained by a single acquisition in the operating mode “Real-Time Sampling”. This display is entirely useless, in spite of this all DSO’s draw interpolation lines through these points, the result is shown below and demonstrates how reliable DSO displays are if they are obtained from too few points! This display is just worthless and grossly misleading for the user.

### 2.5 Z-axis

• DSO displays are devoid of any information contained within the trace, the trace is equally bright everywhere.

In analog scopes the trace is intensity-modulated by its speed, in other words, those parts of the display where the trace is comparatively slow are brighter than those where its speed is higher. The display of a fast 1 kHz square wave, e.g., shows bright upper and lower parts while the transitions remain invisible. This is a very valuable information. A DSO will display the whole waveform equally bright which is false. Some newer extremely expensive DSO’s try to simulate this behaviour of analog scopes.

2.6 Repetition rate, capture rate

Even the oldest analog scopes featured repetition rates of 100 kHz, the last ones of 400 kHz. In analog scopes, the signal is always on the screen, but during the retrace time the trace is blanked, so it is not visible. DSO’s function entirely differently: after the signal was captured once, the DSO has to digitize and store it, the stored signal is mostly sampled again with a slow clock because the display can not accept the amount of information and d/a - converted. These processing steps take so much time, that the first and second DSO generations were only able to capture the signal at a rate below 100 Hz. In the time between the DSO remains blind to the signal, hence a DSO is least fit to detect rare events. These DSOs reacted so slowly that they could not even be used to adjust something, some needed minutes to fill the screen once. This so-called capture rate was increased over the years slowly so some hundred Hz. Tektronix introduced the only true advancement when they incorporated the whole signal processing of an analog scope so they advanced the capture rate in their most expensive products to over 400 kHz, they call it “Digital Phosphor” DSO’s are asleep for 99.9935 % of the time”.

## 3. How to identify false displays

### 3.1 Use a Combiscope

The Combiscope was invented around 1993 by Philips NL, it consisted of an analog scope which also contained the electronics of a DSO. By pushing a front panel button the function could be switched between analog and digital. The instrument was a 200 MHz 4-channel scope which also featured a unique autocalibration function: by pushing a button the instrument performed an almost complete self-calibration, even including the adjustment of the input attenuators. After Philips sold this business to Fluke, the oscilloscopes were discontinued. Hameg Instruments D introduced a whole line of Combiscopes and also used the term, the top of the line model was the 200 MHz, 2-channel type 2008 which used the same 14 KV, extremely bright and sharp crt from Philips. Production was terminated around 2010. If such an instrument is still available second-hand it would be a good buy, the original price was only around 2,000 E.

Figure 3.1: False stopped display of a 1 MHz square wave shown here as 151.82 Hz with a rise time of 358 us (correct: 0.7 ns). When running the frequency is shown as 847 kHz and the rise time as 8 ns. Actual sampling rate 100 MS/s

A 200 MHz Combiscope would not only be sufficient for most work on switching circuits, but it unites the best of both worlds. One would use it normally in the analog mode; the digital mode would only be needed if digitized signal data or signal storage were desired. The validity of the digital data can be verified any time by switching to the analog mode; in fact, this is the only sound method.

A Combiscope must be more expensive than an analog scope, but this was not the main reason why no other firms ever offered some. They were afraid of the easiness with which the shortcomings of their DSO’s and the superior quality of the analog display could be proven by just pushing a button...

### 3.2 Change the time base setting

According to the explanation above a DSO with an insufficient memory will have a different sampling rate and bandwidth in every lower time-base position. Hence the false displays will be different in each position.

• Whenever one suspects a false display change the time base setting and watch the display.

Figure 3.2: 500 MHz, 2 GS/s - DSO. 10 MHz sine wave, 80 % amplitude-modulated with 1 kHz. left 100 us/cm. Right 50 us/cm, 200 KS/s, best picture achievable, comes closest to the truth, but the waveforms inside the lobes are aliases

### 3.3 Watch the digital numerical displays

If e.g. a fast square wave is applied to the DSO, the rise time will be indicated on the screen. Switch the time base to slower positions and watch these numerical displays. Of course, the rise time of the signal does not change, hence the rise time display should not change either. But it will, and the rise time can be shown wrong by e.g. 6 orders of magnitude, see below. This is called progress.

### 3.4 Change the signal if possible

False displays will mostly change if e.g. the signal frequency is changed a bit, especially if the signal frequency and the sampling frequency happen to generate beat frequencies. Such false displays can look quite convincing, but their “frequency” may be off by orders of magnitude. 100 MHz can be shown as “1 kHz”!

### 3.5 Beware of modulated and similar signals

As will be explained in later chapters, the standard operating mode is ETS = Equivalent Time Sampling. In this mode, a screen display is composed of a multitude of signal reocurrencies in the very same shape. In the case of modulated and similar signals, each following signal period has a different shape.

If a continuous sine wave is sampled at the Nyquist limit, it will appear as an amplitude-modulated sine. 3.6 Watch for steps Watch whether the display shows steps: this is always a clear sign of insufficient resolution, there are too few points for a proper reconstruction, such displays are faulty, the data useless.

## 4. Analog oscilloscopes.

Briefly, an analog scope consists of a cathode ray tube with its high voltage supply (up to 24 KV), a wideband vertical amplifier for the signal and a time base which generates a sawtooth of variable speed which moves the trace from left to right. A delay line in the vertical delays the signal for some ten ns until the time base starts so the front of a pulse is visible which triggered the time base. It is the signal itself which deflects the trace in vertical direction. Analog scopes have a constant bandwidth and infinite resolution. Analog scopes were available in compact format and as plug-in instruments. Plug-ins can be exchanged like lenses in a camera; in addition to vertical amplifier and time base plug-ins there were sampling, spectrum analyzer, digital multimeter, counter, curve-tracer etc. plug-ins. This allowed to change the scope within minutes, the user got three instruments in one. Today, he has to buy separate instruments. In 1992 the production of the Tektronix 7000 series was terminated. The 7104 1 GHz type was the fastest analog scope ever. Since then some measuring tasks can not any more be fulfilled with scopes out of current production, e.g. there is no 10 uV/cm scope.

In second-hand shops, they are still available at a fraction of the original prices. A first-class scope set for a SMPS or similar lab would consist of: 7904A 600 MHz mainframe + 2 x 7A26 200 MHz 2-channel verticals + 7B92A dual time base with 0.5 ns/cm. Passive 10:1 and 100:1 probes, a 50 MHz DC/AC current probe and FET probes would complete the set. With a 7L13 spectrum analyzer, plug-in and a self-built LISN EMI measurements can be made.

## 5. Sampling oscilloscopes (SOs)

• All DSOs are by nature also sampling oscilloscopes because any a/d conversion must be preceded by sampling the signal. DSO’s can not be understood without understanding SO’s. In pure SO’s sampling happens directly at the 50 ohm input, the amplitude samples are amplified and displayed on the screen with infinite resolution as in analog scopes. DSOs’ inputs are identical to those of analog scopes, the signal is amplified and then sampled, a/d converted, stored. The stored data are binned or sampled again at a low rate, d/a converted and displayed.

Sampling is based on the stroboscopic principle, the first such apparatus was built in 1880 by Joubert. If the sampling frequency is the same or a multiple of the signal frequency, the difference frequency will be displayed which can be zero, in this case only horizontal lines will appear. This has an interesting consequence: the time scale can be extended to infinity.

The servo system was invented in 1898 by Callender. In 1952 McQueen in GB presented the first random sampling scope which already contained nearly all features of today’s instruments. In 1957 Sugarman built the first sampling scope capable of showing non-periodic signals. HP manufactured 1960 the first commercially available sampling scope (500 MHz, 10 mV/cm), 1962 the first 1 GHz scope followed, a two-channel unit with sampling probes, 1964 4 GHz and 1966 12.4 GHz scopes came in due course. Tektronix stayed behind until 1967 when the random sampling time base 3T2 appeared and 1, 4 and 14 GHz samplers.

Neither is the automatic measurement of signal parameters an invention of DSO manufacturers: also already in 1967 the Tektronix 567 and 568 measured automatically amplitude, time, time differences, frequency, rise, fall and delay times, averages etc., compared these with preset limits, displayed the values on digital displays with high accuracy and sent the data to external computers.

### 5.1 Principle

Sampling uses intermediate analog storage which is equivalent to pulse stretching, usually called sample and hold. This implies that energy is taken from the signal which will distort it more or less.

In practice, there are two methods:

• A switch between signal source and hold capacitor is closed for a very short time.
• A switch between signal source and hold capacitor is normally closed so the voltage on the hold capacitor follows the signal; in the moment of sampling the switch will be opened, the capacitor holds the last signal value.

Figure 5.1: Simple sampling circuit with reset as they were used in the first sampling scopes

Figure 5.1 shows the basic circuit. A second switch S 2 resets the hold capacitor as soon as the sampled value has been processed.

Two basic properties of sampling are obvious:

• During sampling, the signal is being averaged by the RC.
• The signal portions between the sample points are lost forever.

Figure 5.2: Influence of the width of the sampling pulses on the rise time and the waveform

If sampling is performed with a frequency very high with respect to the signal frequency a series of needle-shaped pulses is obtained as shown on the right, the envelope is the original signal waveform. Low pass filtering will delete the sampling frequency and its harmonics, so the signal’s waveform is approximately regained. Without filtering the signal frequency band will be grouped around the sampling frequency and its harmonics. Sampling is hence a process very similar to mixing or modulation, these expressions mean about the same just like multiplication; this is easy to understand if one realizes that a mixer or modulator also is nothing else but a switch which passes the signal at a rate given by the oscillator (= sampling) frequency. This also explains the generation of beat frequencies (artefacts, ghosts, aliases). The purpose of mixing, e.g. in a radio, is a frequency transformation of the hf signal into a frequency range where amplification and filtering are easier (intermediate frequency).

In a SO the signal’s waveform shall be preserved and later reconstructed.

Assumed the input circuit of a SO does not impair the bandwidth what then limits it? Figure 5.2 shows the influence of the width of the sampling pulses.

The sampling pulse of width 1 samples the input signal (front of an ideal square wave) at 3 times t1 to t3. During the closure of switch S1, the voltage across Cs assumes the average of the input voltage uE (t). The result uA (t) is a linear rise of duration τ for 0 to 100 %. As the rise time is defined from 10 to 90 % it follows:

$\large&space;tr=0.8\tau$

In case of a sine wave the well-known function:

$\large&space;\frac{sin\,pft}{pft}$      with    $\large&space;f_g=\frac{0.4}{t}$

will be obtained.

Evidently, the result will be zero if a whole signal period (or multiples) fits into the window, it is also evident that the result may become negative if the average is negative at the end of the sampling period. First-class samplers indeed do show this linear rise as in Figure 5.2 which does not at all convene with a Gaussian response. In spite of this the already known equation:

$\large&space;tr&space;\cdot&space;f_g&space;=&space;0.35$   because 0.35  = 0.8 x 0.44

also holds here.

• Even if there are no other influences, the limited width of the sampling pulses will limit the rise time to tr = 0.8 τ.

Figure 5.2 delivers another important parameter, this is the sampling efficiency. The charging of Cs is via Ri and follows an exponential. It depends hence of τ, Ri, Cs how many percent of the input voltage will remain on Cs. The efficiency is hence:

$\large&space;\eta&space;=&space;\frac{u_A(t_e+\tau)}{u_E(t_0)}&space;=&space;1&space;-&space;exp(\frac{-\tau}{R_iC_s})&space;=&space;\tau&space;/&space;R_iC_s$

The efficiency of typical samplers is 3 to 25 %; if the input impedance is 50 ohms it is constant. The efficiency decreases with increasing bandwidth. Any fluctuations of the sampling pulse width will enter the output.

### 5.2 Equivalent-Time-Sampling (ETS), principle of the SO

As mentioned the SO is based on the stroboscopic principle. The frequency-to-time transformation is best understood by looking at the so-called information volume shown in Fig. 5.3. By the use of the stroboscopic method, the shape of the volume is modified: it is compressed in the bandwidth axis and simultaneously extended in the time axis, the amplitude and volume remaining constant.

ETS and RS can only function if two preconditions are fulfilled:

• The signal must repeat itself until it has been reconstructed once.
• The signal shape must not change during the whole reconstruction time.

The sampling pulses are controlled in such a manner that only one sample is taken from consecutive signal periods, each sample is delayed by a fixed amount, such that, after completion of one sampling cycle, the same effect is obtained as if all samples had been taken from only one signal occurrence. Depending on the desired resolution, thousands or millions of signal repetitions will be necessary in order to reconstruct the signal once. Typically, SO’s require 10 us for one sampling cycle, the maximum “sampling frequency” is thus 100 KHz. If the signal frequency is higher cycles will be skipped.

Figure 5.3: Volume representing the information content and how it is deformed by the stroboscopic sampling process (Bandbreite = bandwidth, Zeit = time)

• In ETS mode the Nyquist theorem does NOT apply! There is no correlation between the signal and sampling frequencies. Sampling along the signal waveform can also be done manually or from a plotter. Bandwidth is limited only by the analog input resp. the minimum sampling pulse width.

The delay of the sampling pulses from cycle to cycle requires a trigger signal which is derived from the same point of the signal; triggering allows a variable repetition rate, i.e. the signal need not be periodic.

The trigger pulses cause a so-called fast ramp to start. A second signal generated in the time base is a staircase which is increased one step by each trigger pulse and moves the display one step in X direction.

Figure 5.4: Basic block diagram of a SO in ETS mode

A comparator compares the instantaneous values of fast ramp and staircase, if it switches a sampling pulse is triggered in the vertical channel of the SO. This is the same procedure as in a stroboscope.

The important parameter is the time increment T by which the sampling pulses are delayed, it is called the equivalent time scale. Standard SO’s cannot sample faster than each 10 us, i.e. that the individual samples on the screen are 10 us or more apart. If a screen display consists e.g. of 100 samples it takes 1 ms so fill it once. The real-time time scale would thus be 0.1 ms/cm, but this value is of no interest to the user. Only the “Equivalent Time Scale” is important and indicated on the front panel or the screen, this is the time scale related to the signal. As one does not read the time between two samples but from the raster the customary time scale is derived by taking the number of points per cm into account which are equivalent to the number of steps of the staircase per cm. If T = 1 ns, and if the SO is set to 10 points per cm the time scale will be 10 ns/cm.

The time scale is decreased by decreasing the height of the steps, this yields an arbitrary time stretching up to infinity, in this case the same point on the signal is sampled, so a horizontal line will appear on the screen according to the height of this point. This is a property of the stroboscopic method and does not constitute a particular performance. A SO or DSO is not better than another one because its time scale is faster. As with any scope, a fast time scale makes only sense up to the point where its own rise time is well resolved.

Which are the requirements other parts of a SO have to fulfill in ETS mode? As the samples are acquired not faster than every 10 us, a rise time of the vertical amplifier of 2 us will be sufficient which is equivalent to a bandwidth of 170 kHz. The fastest real time scale was 0.1 ms/cm or 1 ms for a full screen in X direction. Both are very low requirements. How many signal periods are necessary to reconstruct the signal once depends on the desired quality. At 100 points per screen and 10 us it takes 1 ms; during this time the 1 GHz signal will produce 10-3 x 10+9 = 10+6 periods of which only 100 are used. If the quality is increased to 100 points/cm, i.e. 1,000 points per screen, 10+7 periods will be needed.

This is a vital viewpoint which should be approached also from another side. A SO has the same vertical resolution as an analog scope, i.e. infinite resolution; in practice, the resolution is limited by the trace width and sharpness, far superior to any DSO. As the samples gained in ETS mode are identical to those which would be derived from a single signal cycle, 1,000 points per signal period (assumed one cycle would exactly fill the screen) would be equivalent to an apparent sampling frequency of 1,000 GHz if the signal frequency is 1 GHz!

• Because the normal operating mode of SO’s and DSO’s is ETS, manufacturers like to quote such fictitious figures in order to impress potential customers who tend to interpret these meaningless numbers as performance specifications.

100 points/cm are equal to 1 point/0.1 mm, even very good crt’s have wider traces such that the display will already look like a continuous trace. A SO delivers vertically and horizontally a signal quality equivalent to that of an AO, and this is a marked difference to DSO’s. There are no interpolations with resulting often gross distortions, and this is standard since 1960! As mentioned the signal can be sampled as slowly as desired, i.e. manually or controlled by an XY plotter. Screenshots are hence an old hat and no invention of DSO’s, not to speak of the enormous difference in quality between an analog plot and a printout.

It should be stressed again that the ETS mode strictly requires that the signal must repeat with the same shape until the screen has been filled. In many practical applications, this is not possible: examples are oscillating regulation loops, modulated signals, stochastic breakdowns of semiconductors etc. Such phenomena can only be captured in RTS mode.

### 5.3 Random Sampling

ETS suffers from the necessity of an analog delay line or a pretrigger if the pulse front shall be displayed, the same as in an analog scope. The horizontal system needs some time to start upon receipt of a trigger pulse. Delay lines with a bandwidth of 1 GHz were already incorporated in SO’s in the 1960’s, even two. Delay lines with shorter rise times are extremely bulky and heavy. In order to see the pulse front with faster samplers without a pre-trigger, Tektronix brought 1967 the first commercial “Random Sampling” time base 3T2 onto the market.

In order to see the pulse front without a delay line, the time base must be started before receipt of the trigger. RS functions entirely differently from ETS: the sampling time base runs freely and triggers continuously samples in the vertical channel; there is hence no correlation between the signal and sampling frequencies, this is like in RTS mode. Which parts of the signals are thus “hit” by the samples is random. The purpose of a scope is to reconstruct the signal waveform vs. time. In order to realize this, it is necessary to know the exact time position of each sample relative to a fixed point in time (trigger) and to position that sample correctly on the screen.

The vertical channel is identical to that in ETS mode - with one marked difference: the loop gain must be exactly equal to one, because any sample is not correlated to the preceding one. If the loop gain differs from one, the signal cannot approach its correct waveform, but the result will be a broad, scattered “sky of stars”.

The horizontal channel is a very complicated structure. Its function is basically to predict the occurrence of the next triggering slope and to start the time base early enough so that the pulse front is visible.

The signal representation in RS mode is not so good as in the ETS mode because the samples do not follow in due order nor are they equally well distributed. Depending on the precision of the reoccurrence of the signal the samples are more or less statistically distributed and move back and forth within the waveform.

In RS mode the likelihood of steady pictures of artefacts is low.

### 5.4 Real-Time Sampling (RTS)

This operating mode is not common with SO’s because they lack a memory.

## 6. DSOs

### 6.1 Summary of advantages of DSO’s.

1. Storage of single events.
2. Unlimited storage time.
3. it is possible to store long signal sequences for a later inspection in detail. This is especially valuable when searching for problems and rarely reoccurring phenomena.
4. Capture of events before a trigger by continuously writing into the memory and stopping this upon receipt of a trigger. This not only important in order to see the pulse front which triggered; depending on the size of the memory a wide time span before the trigger can be inspected. However, due to the rather low capture rate of most DSO’s, this is of doubtful value; only fast capturing DSO’s like the so-called DPO’s can use this fully.
5. Flicker-free display of very slow events. Analog storage scopes can also do this, but the display will disintegrate after some time because the stored charges will leak off.
6. All measurement results are available in digitized format and can be used for further digital processing within the scope or externally.

Despite their successful capturing of the whole market DSO’s are neither the successors of analog scopes nor are they “universal scopes”, also the claim that they “unite all advantages of analog and digital scopes” is objectively false. Reliable measurements of unknown signals can only be done with the most expensive DSO’s.

### 6.2 Operating modes ETS and RS

The operating modes ETS and RS are identical to those in SO’s, the only difference is the fact that each sample is a/d converted, stored, processed, d/a converted and displayed. Note that the highest bandwidth is obtained in these modes. With most DSO’s, especially low-cost models, the specified bandwidth is only achieved in these modes; what the manufacturers do not say is that these modes can only be used with repetitive signals!

### 6.3 Operating mode Real-Time Sampling (RTS).

Even the oldest SO’s and DSO’s featured RTS. Of course, DSO’s can never show the signal in real time, they remain sampling instruments, i.e. they reconstruct the signal from samples, long after it disappeared.

Quite obviously it is again the purpose to lead potential customers astray by insinuating these “Real Time DSO’s” could show the signal in real-time like analog scopes. This designation is false and misleading, the correct one is: Real-Time Sampling Oscilloscope because it only describes an operating mode.

RTS means that all samples displayed are derived from a single capturing event. A DSO is characterized by its performance in RTS mode.

Which sampling rate is required for RTS? Shannon - Nyquist only ask for a rate twice as high as the highest frequency in the signal. It is not as simple as that, with DSO’s there are two criteria for the sampling rate:

1. Fulfillment of the Shannon-Nyquist requirement. Again it should be stressed that the “highest frequency in the signal” is not mixed up with the bandwidth. Compliance with this requirement only protects from aliases, this is not sufficient for a usable display. In mathematician’s language, this is necessary, but not sufficient.
2. The sampling rate must be high enough to allow a usable reconstruction of an arbitrary non-sinusoidal signal. It is up to the user to determine what he considers acceptable resp. usable...

In ETS mode the resolution can be so high that the trace looks continuous, in RTS mode only the points are available which are obtained in one stroke. Assumed a 2 GS/s 200 MHz DSO would sample a 100 MHz square wave, this would yield 20 points or 2 per cm; one might suspect that these 20 points belong to a square wave, but the transitions would not be visible and the rise times could not be measured. A 200 MHz analog scope would show a rounded square wave continuous curve with infinite resolution, i.e., it is obvious that it is a square wave, no guesses necessary. If the DSO uses linear interpolation as usual, also a square wave would be shown, but the transitions would not be those of the square wave but those of the interpolation, i.e. unusable.

In order to display e.g. a 20 MHz signal with 100 points resolution, i.e. 1 point per mm, a sampling rate of 100 x 20 MHz = 2 GS/s is necessary. This example shows how fast seemingly impressive figures are deflated.

The erroneous perception Nyquist was good enough is one of the most severe misconceptions about DSO’s! The Nyquist theorem is not wrong, but pure theory, it is the most misunderstood theorem in electronics. It has caused many wrong decisions. Its application would require a so-called brick-wall low pass filter. Apart from the fact that this is not realizable, it would create strong pulse distortions. As mentioned earlier, the factor of 2 would mean that, after sampling, there were only 2 points on the screen through which any arbitrary waveform could be drawn. In Nyquist, there is the hidden assumption taken as an absolute limit and not applicable in practice. The inverse of Nyquist is also seldom understood: if a signal has been sampled with a given frequency, the reconstruction will definitely not contain any higher frequency than half the sampling frequency.

From the fact that a brick-wall filter is neither realizable nor tolerable in an oscilloscope, it follows that “oversampling” is mandatory. This is just another way of looking at it. Oversampling allows to insert a filter with the gradual slope of the Gauss curve. As mentioned earlier, a minimum attenuation of - 50 dB at half the sampling frequency is necessary. The Gauss response is that far down at about 4 x the bandwidth. Hence the sampling frequency must be > 8 x bandwidth if there is no filter. The factor of 10:1 ensures some reserve. While an analog scope still shows signals far above its bandwidth, although attenuated and rounded, this is not the case with sampling scopes; a sampling system has a hard limit at half the sampling frequency! If there are signals above that limit, they will show up as artefacts.

The dynamic range resp. the S/N ratio of an 8 bit a/d converter theoretically amounts to 256 : 1 or 50 (6n + 1.78) dB. In reality, assumed that another ± ½ LSB from other disturbances comes on top of the ± ½ LSB, the SNR = 40 (6n - 7.8) dB, i.e. a whole bit is lost in the a/d conversion. Still another way of looking at it assumes that an antialiasing filter must attenuate at least - 59 dB, if all signal content should be lower than the rms value of the quantizing noise which is q/ (2 x sqrt 3).