As the popularity of Digital Storage Oscilloscopes has grown, a need has arisen for understanding their operating modes and performance characteristics. This technical brief describes two of the fundamental modes of waveform acquisition utilized in Tektronix products. Knowledge of the benefits and trade-offs of Real-Time and Equivalent-Time sampling will make it easier to choose and use a Tektronix digital storage oscilloscope.

Appended to this tech brief is an explanation of the Sin(x)/x interpolation method that Tektronix DSOs use to produce high resolution timing and amplitude measurements and extremely accurate displays.

The performance of the DSO continues to evolve toward an analog-type performance level. Higher sample rates allow mid-range digital scopes to acquire single shot waveforms with a level of timing accuracy rivaling the capabilities of premium DSOs. The standard sampling rate for these reasonably priced scopes has grown exponentially from a 50 or 100 MS/s rate to 2 GS/s.

### Analog and Real-Time Bandwidths

To create a waveform accurately, the DSO must gather a sufficient number of samples after the initial trigger. In theory a digital scope needs at least 2 samples per period (one full cycle of a regular waveform) to faithfully reproduce a sine wave; otherwise the acquired waveform will be a distorted representation of the input signal. In practice, using Tek's Sin(x)/x interpolation in the TDS Series scopes, a DSO needs at least 2.5 samples per period.

This requirement usually limits the signal frequency a digital scope can acquire in real-time. Because of this limitation in real-time acquisition, most DSOs specify two bandwidths - Analog and Real-Time. The *Analog Bandwidth*, defined by the circuits composing the input path of the scope, represents the highest frequency signal a DSO can accept without adding distortion. The second bandwidth, called the *Real-Time Bandwidth*, defines the maximum frequency the DSO can acquire by sampling the entire input waveform in one pass, using a single trigger, and still gather enough samples to reconstruct the waveform accurately. The following equation describes the real-time bandwidth:

For some DSOs, the real-time bandwidth theoretically exceeds the analog bandwidth. But since the input path distorts any signal above its frequency limit, the real-time bandwidth can only be equal to or less than the analog bandwidth. Even though a DSO may sample at a higher bandwidth than its analog bandwidth, the analog bandwidth establishes the highest frequency the scope can accurately capture.

### Equivalent-Time Sampling

When a DSO uses *equivalent-time sampling*, it can acquire any signal up to the analog bandwidth of the scope regardless of the sample rate. In this mode, the scope gathers the necessary number of samples across several triggers. The input signal must be repetitive to generate the multiple triggers needed for equivalent-time sampling. In equivalent-time, a slower, lower-cost digitizer provides the same accuracy on repetitive waveforms as a higher cost DSO with a faster sampler. For example, the TDS 460 offers a 350 MHz bandwidth with only a 100 MS/s sampling rate on each of its four channels.

The TDS 400 and TDS 500 Series scopes use a common method called random equivalent-time sampling. Although these scopes acquire samples sequentially after each trigger, each acquisition starts at a different time with respect to the trigger. Figure 1 depicts how random equivalent-time sampling works.

**Figure 1.** Random equivalent-time sampling digitally reconstructs a waveform using several trigger events.

Because equivalent-time sampling requires a repetitive signal, it has certain restrictions. A DSO in equivalent-time cannot create a meaningful display from a single-shot event. Also, the signal must repeat identically each time or the displayed waveform will be distorted. Figure 2 illustrates what happens to a display when a repetitive signal changes over time. The scope creates sharp vertical lines, or hashing, indicating the differences in the signal across multiple acquisitions. A viewer could easily misinterpret these lines and conclude that they represent high-frequency noise riding on the signal.

**Figure 2.** When a signal that changes over time is acquired in equivalent-time, the display has sharp vertical lines indicating the modulation in the signal. This particular type of distortion can easily look like noise to the user.

Some scopes perform equivalent-time sampling exclusively and can accept only repetitive signals. Because these scopes are limited to equivalent-time, they either dramatically increase their accuracy or bandwidth or offer significantly lower cost compared to a real-time digitizing scope.

### Real-Time Sampling

When a DSO operates in real-time or single-shot mode, it attempts to gather all the samples for a waveform with one trigger event (Figure 3). Because this mode uses only one trigger from the input signal, real-time sampling treats both repetitive and single-shot waveforms as one time events.

**Figure 3.** Real-time sampling captures a complete waveform with a single trigger event.

By using DSOs with higher sample rates one can acquire higher-bandwidth signals in real-time. For example, an engineer wants to acquire and store a single-shot 50 MHz signal. Using a scope with a 400 MHz analog bandwidth and a 1 GS/s sample rate, creating a real-time bandwidth of 400 MHz, he can easily acquire the signal in real-time.

However, if the engineer chooses a digital scope with an analog bandwidth of 400 MHz and a 100 MS/s sampler, he cannot accurately acquire the 50 MHz signal in real time. Although this scope, like the first one, has an analog bandwidth of 400 MHz, its maximum sample rate of 100 MS/s limits the real-time bandwidth to only 40 MHz.

### TDS 600: Real-Time Scopes

The TDS 620 and TDS 640 digital scopes have a 500 MHz bandwidth and 2 GS/s sample rate. Their theoretical real-time bandwidth is 2 GS/s divided by 2.5 = 800 MHz. Since the TDS 600 scopes cannot pass signals higher than 500 MHz without distorting them, their real-time bandwidth equals their analog bandwidth. Because the two bandwidths are the same, these scopes can easily acquire signals in real-time up to the analog bandwidth of the scope. Digital scopes only require equivalent-time sampling when the real-time bandwidth is lower than the analog bandwidth. Since the TDS 600 scopes can acquire signals up to the bandwidth of the scope with one trigger event, they offer only real-time sampling.

To demonstrate the TDS 600 Series DSO's powerful acquisition capability, Figures 4a-c graphically depict the differences between real-time and equivalent- time sampling. A calibrated pulse generator created a 1 ns rise time pulse as a single-shot event and as a repetitive waveform. For reference, Figure 4a shows a display of this pulse captured by a Tektronix 2400 analog scope. In Figure 4b, the TDS 540 acquires a repetitive version of the same pulse with equivalent-time sampling. Multiple acquisitions were required to capture the signal. In Figure 4c, the TDS 640 displays the same pulse with real-time sampling. Thanks to its 2 GS/s sample rate, the TDS 640's waveform exhibits the same rise time, amplitude, and visual characteristics as the analog display in Figure 4a. Although the TDS 540 and TDS 640 both have 500 MHz bandwidths, the high-speed, real-time sampling of the TDS 640 clearly delivers a more analog-like representation of the input signal.

**Figure 4.** These three screen captures demonstrate the differences between real-time and equivalent-time sampling.

**Figure 4a.** For reference, the pulse captured by a Tektronix 2465BDV analog scope

**Figure 4b.** Using equivalent-time sampling, the TDS 540 digital scope acquires a repetitive version of the pulse.

**Figure 4c.** With real-time sampling, the TDS 640 displays the pulse after one trigger event. Note how close this waveform's appearance is to the analog display of the signal in Figure 4a.

### Reconstruction Techniques for Waveform Display

Whether a digital scope acquires a waveform in real-time or equivalent-time, interpolation displays the acquired signal more clearly. When a scope interpolates, it draws lines between the samples on the display, creating a continuous waveform instead of a string of individual points. Figures 5a and 5b show the difference interpolation makes in creating a more realistic display.

**Figure 5.** Interpolation helps create more meaningful waveform displays.

**Figure 5a.** When a DSO displays only sample points, the user can have trouble determining the actual waveform shape.

**Figure 5b.** Interpolation connects sample points and creates a more intelligible display.

Most DSOs offer two types of interpolation: linear and sine. Linear interpolation draws lines between the samples using a straight-line fit. This method works well with pulses and digital signals but may produce distortions on sine waves. Sine interpolation connects the samples using a curve fit. Ideal for sinusoidal signals, this approach can produce apparent overshoot or undershoot when displaying pulses.

Tektronix DSOs offer a modified sine interpolation method that eliminates the inaccuracies when displaying pulses. The Sin(x)/x method uses an adaptive prefilter to locate and compensate for fast signal transitions. Although this method requires more calculations than linear interpolation, the TDS Series scopes with their custom digital signal processor update their screens quickly in both real-time and equivalent-time modes. Figures 6a and 6b demonstrate linear and sine interpolation.

**Figure 6.** Some DSOs have two types of interpolation.

**Figure 6a.** Linear interpolation uses a straight-line fit to draw lines between samples.

**Figure 6b.** Sin(x)/x interpolation is a modified sine interpolation method that connects samples using a curve fit.

### Conclusion

In real-time, a scope's digitizer samples the entire input waveform in one pass, with a single trigger. The term "real-time" arises because acquisition and display occur in the same time frame. Real-time digital scopes are ideal for single-shot applications. Real-time sampling generally results in fewer complicating defects, such as aliasing or distortion, which can occur with equivalent-time sampling.

Random equivalent-time sampling takes advantage of the nature of a repetitive signal by using samples from several trigger events to digitally reconstruct the waveform. Since sampling occurs on both sides of the trigger point, pretrigger capability is very flexible. Because repetitive signals are being sampled, the bandwidth of an equivalent-time scope can far exceed its sample rate.

### Appendix

#### How Sin(x)/x Interpolation Works

The TDS Series scopes expand waveforms by using a digital signal processing technique that reduces the sample rate requirements for sine waves to about 2.5 per cycle. This method of interpolation produces higher resolution timing and amplitude measurements than linear interpolation, as well as more accurate displays. The following discussion explains the technique, which is essentially a linear filtering process.

When the TDS oscilloscope acquires a continuous-time input signal

at a sampling rate with period T, it saves the acquisition as a sequence of equally spaced samples:

where *N* is the selected record length.

(Note that

must be bandlimited to a frequency

to prevent aliasing.)

With approximations of *t* and *n* extending to

, the Fourier transforms of

and *x(n)* are

and

, respectively.

Figure 7 shows the relationship between

and

when

[[nid:]]

**Figure 7.** This illustration shows: a. The magnitude Fourier transform of the input signal

* *

*, b. The magnitude Fourier transform of the saved-waveform sequence x(n).*

Note that:

The goal of interpolation is to create a sequence that would result from sampling the input signal at a faster rate. If the sampling rate is increased by an integer factor *L* , the sampling period is decreased to

.

Therefore, the desired sequence is:

.

The Fourier transform of *y(n)* must be:

.

Figure 8 shows

for the case where *L =* 4. Comparing Figure 8 to Figure 7b suggests the use of a filter to create the higher sample-rate sequence by adjusting the gain of

and removing the unwanted images centered at:

.

**Figure 8.** This illustration shows the magnitude Fourier transform of the sequence y(n) with L=4. This sequence is the one that would be obtained by sampling the input signal

* *

*four times faster.*

This is just what the TDS Series scopes do in a two-step process. Refer to Figure 9.

**Figure 9.** The TDS 600 expands waveforms in two steps, as illustrated here for L=4. The intermediate sequence v(n) results from effectively sampling the acquired sequence x(n) four times faster. When properly filtered, v(n) produces the desired sequence y(n).

The first step toward obtaining the desired sequence *y(n)* is, in effect, to sample the saved-waveform sequence *x(n)* at the higher sampling rate, which is achieved by inserting *L -* 1 zero-valued points after each acquired sample in *x(n)*. The resulting sequence is:

The sequence *v<n)* has sampling period T'.

The Fourier transform of the new sequence *v(n)* turns out to be identical to the Fourier transform of the sequence shown in Figure 7b, that is:

However, the Nyquist frequency has been increased by the factor *L* , which makes it possible to obtain *y (n)* by applying a suitable digital low-pass filter to *v (n)* .

Ideally this filter must:

- be periodic with period

- have a gain of
*L* , and - reject frequencies between

and

Such a filter has Fourier transform of

.

The TDS Series waveform interpolators approximate this ideal filter with a Finite-duration Impulse Response (FIR) digital filter. This extremely efficient filter delivers high throughput for displaying and measuring expanded waveforms. Furthermore, the FIR filter accurately passes frequency components up to 80% of the Nyquist frequency. The resulting useful storage bandwidth (USB) is:

The TDS Series interpolator filter is also quite powerful for pulse-type waveforms. The filter provides nearly perfect expansion as long as the rise and fall time of the pulse is greater than 1.7 times the sampling period.

Pulses with faster edges, however, contain frequency components near the step transition that exceed the Nyquist criterion. These high-frequency components cause aliasing and can never be accurately recovered. They appear in the interpolated waveform as several cycles of ringing before and after the step. For optimum interpolation, additional processing is needed.

The TDS Series scopes provide an adaptive pre-filter that locates fast transitions and significantly reduces their high-frequency content. In addition, the TDS 600 Series 500 MHz bandwidth has a calculated 800 ps rise time, providing effective anti-aliasing for steps acquired at the scope's maximum 2 GS/s sampling rate.

*Tektronix is a registered trademark of Tektronix, Inc.*