How to extend SNR in wideband oscilloscope-based pulsed RF measurements

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In urban slang, signal-to-noise ratio (SNR) is a simple enough concept: the ratio of useful to useless information. We all know people whose SNR is not as high as we might hope. Unfortunately there’s no technology yet available to boost their SNR.

So engineers can be happy that’s not true for RF signals. We can now extend SNR in wideband oscilloscope-based RF measurements through what’s known as “processing gain.” Digital down-conversion lets you see small pulsed RF signals next to large signals by reducing the noise level in a particular measurement—whether it’s RF pulse envelope characteristics or frequency or phase shift across a pulse.

Increase in pulsed RF capture dynamic range

So how does it work? The trick is adding vector signal analysis (VSA) software. VSA in conjunction with an oscilloscope can extend the SNR. First VSA shifts a captured signal down to baseband I/Q. Then it bandpass filters the acquired oscilloscope data and finally resamples the data at a lower sample rate. The result is lower noise, higher dynamic range, and a wider SNR.

Let’s look at an example: An 8 GHz-wideband oscilloscope captures a pulse train in which a large pulse is immediately followed by a small pulse that is 50 dB down from the first pulse. This corresponds to being 100,000 times lower in power and ~316 times smaller in voltage (sqrt[100,000]) than the first pulse. The two-pulse sequence then repeats.

The large pulse has a +6 dBm power level (~1.4 mW), which results in a peak voltage of around 633 mV into 50 ohms. This can be represented as a -4 dBVpk level (20log 0.633). It also corresponds to a 1266 mV peak-to-peak signal into 50 ohms.

In contrast, the small pulse, being 316 times smaller in voltage, is only 4 mV peak to peak (-44 dBm, -54 dBVpk).

The VSA software, which also controls the oscilloscope front-end sensitivity, is set to +6 dBm (633 mV peak). This corresponds to an oscilloscope vertical range of 1266 mV.  There are eight vertical divisions, so this also corresponds to a ~160 mV/div setting.

At the full 8-Hz bandwidth for this ~160 mV/div setting, the broadband RMS noise for the 8 GHz bandwidth oscilloscope is around 5 mV, interpolating from a noise chart in the data sheet, as shown in Table 1.  The 5 mV of noise translates roughly into a peak-to-peak noise that is three times the RMS noise (assuming Gaussian noise). In other words, we’re looking at 15 mV of peak-to-peak noise.

Table 1. 8-GHz bandwidth oscilloscope RMS noise levels at various V/div settings

The small pulse (4 mV p-p) is masked by the noise in the measurement (15 mV p-p). (Think how easily a big-mouth can drown out softer-spoken colleagues.) The small pulse can’t be well-discerned in the full 8-GHz measurement of the oscilloscope, with a linear scale and no averaging, as shown in Figure 1.

Figure 1. 8-GHz bandwidth oscilloscope capture of +6 dBm pulse next to a 50 dB down pulse (2nd pulse cannot be seen)

 

Import of real-time captured pulsed RF signals into analysis software and digital down-conversion

Basic pulsed RF measurements can be made natively on a high-bandwidth oscilloscope. And there are certainly times that measurements on directly sampled signals are desired. But this isn’t one of those times. Instead we’re looking for advantages available through external signal processing and analysis on captured signals.  For example, through a process called digital down-conversion, it’s possible to make a range of RF pulse measurements with higher accuracy. That’s due to the lower noise present by using processing gain. Let’s take a closer look.

Figure 2 shows the basic process of digital down-conversion.  Through digital signal processing, the oscilloscope samples are multiplied by the sine and cosine of an imaginary oscillator of frequency f_c, where f_c is generally chosen to be the center frequency of the signal of interest. In effect, we’re “tuning” to the frequency of the input signal. This process converts the time samples into real and imaginary number pairs that completely describe the behavior of the input signal. To reduce noise, these samples can be low-pass filtered and then re-sampled at a lower rate to reduce the size of the data set and allow FFT processing of the data at a later stage. The resulting digitally down-converted samples can then be placed into memory for further processing.

Figure 2. Oscilloscope-captured samples input to VSA software for digital down-conversion

 

Some important demodulation information comes from this digital down-conversion process. First, consider what happens when the digital local oscillator frequency F_c is equal to the carrier frequency of a modulated signal. The output of the digital filters, which includes the real part I(t) and imaginary part Q(t), consists of time-domain waveforms that represent the modulation on the carrier signal.

Do you want that in math? Here’s a representation of the captured input signal:

=  A(t) * Cos[2pf_ct +q(t)]

where the following equation describes the amplitude modulation:

And the equation here describes the phase modulation:

Displaying the I-Q results in terms of magnitude coordinates gives us a view of the amplitude modulation.  Displaying the I-Q results in terms of phase coordinates offers a view of the phase modulation. Taking the derivative of phase modulation yields the frequency modulation.

By adjusting the width of the low-pass filters, you can set a defined span around the center frequency where the filter width is just wide enough to pass the signal of interest, but narrow enough to filter out a lot of the noise.

 

Results of digital down-conversion and processing gain on the 50-dB down RF pulse

So in short, processing gain “tunes” to the center frequency of the signal and “zooms” into the signal to analyze the modulation.

In this example, the original 8-GHz-wide measurement with the associated noise is reduced to a 500-MHz wide measurement, centered on the 3.7-GHz carrier with an instantaneous measurement bandwidth slightly wider than the width of the signal modulation.  This corresponds to an improvement in SNR as follows:

10log*(ScopeBW/Span) = 10log*(8E+09/500E+6) = 12 dB.

Taking advantage of this processing gain, combined with VSA software’s ability to have a log magnitude scale, and using averaging, the 50-dB down pulse is now visible, as shown in Figure 3.

Figure 3. 50-dB down pulse seen with VSA software “Center Frequency” and “Span” set to 3.7 GHz and 500 MHz

 

The improvement in SNR realized through narrowing down the span is depicted graphically in Figure 4.

Figure 4. Plot of SNR achievable in time view verses span adjustment in VSA software

 

 

You can draw a similar plot to see improvement in dynamic range possible when measuring narrow band signals, as shown in Figure 5.

Figure 5. Plot of dynamic range in FFT vs. resolution BW setting in VSA software

 

Here the dynamic range improvement when measuring narrowband signals in an FFT view is described as:

10log*(ScopeBW/RBW)

This does not describe the spur-free dynamic range (SFDR) or harmonic distortion characteristics of the oscilloscope response, but it does give an idea of where the noise floor will be in an FFT measurement.  As the resolution bandwidth is decreased, and the noise is divided among smaller time buckets, the noise floor drops.

This graph does not account for limitations due to various spurs, so the spur-free dynamic range (SFDR) remains limited to around 50 dB.

 

Conclusion

Through the process of digital down-conversion, the SNR of an oscilloscope measurement can be significantly improved as a function of how much that measurement can be “spanned down” from the initial DC to 3dB bandwidth of the oscilloscope.  As our example showed, a 50-dB down pulse, not even visible on a normal scope screen, can be clearly seen once processed by VSA software and then displayed in a log-magnitude scale. This approach can be very helpful to speed system validation measurements on Aerospace/Defense pulsed-RF signals. With this process, you can significantly improve measurement accuracy when evaluating the spectral, pulse envelope, frequency chirp, and phase shift characteristics of an RF pulse train.

Solving Signal Integrity Issues with I²S Protocol Triggering and Decode Software

By Rachel Beddor

The surge in low-speed serial buses in the consumer market has brought about the need for accurate protocol decoders. Traditionally, decoding a serial bus would mean a trip back to your introductory engineering class— lots of counting 1s and 0s. This method is tedious and prone to errors, so a better option is to use technology to decode serial buses. As oscilloscopes become the all-in-one lab instrument (some InfiniiVision scopes even include a function generator), separate protocol analyzers are no longer needed. I was able to explore this as an intern with Keysight this summer when I worked on the release for the I²S Protocol Triggering and Decode Software for Infiniium oscilloscopes.

The three crucial signals in an I2S bus: data (yellow), word select (blue), and clock (red).

The I²S bus is used to transfer data within audio systems. It’s a straightforward protocol consisting of a data line, word select signal and clock. From cars to laptops, the I²S bus is becoming pervasive in a range of industries, and this makes it an exciting protocol to work on as an intern. In August, Keysight released the I²S Protocol Triggering and Decode Software for Infiniium (Option N8811A). This Infiniium add-on features I²S serial bus hardware triggering*, I²S protocol decoding, and user-selectable signal alignment selections including support for time division multiplexed signals (TDM).

The increasingly popular TDM I²S signals allow for multiple lines of data to be sent over the same bus. For example, this technology could be used in an automobile, where digital audio data can be sent to front speakers and a rear woofer through the same bus. Since TDM signals might include several channels in the same packet, these signals are particularly difficult to debug without advanced software.

Using TDM technology, four unique channels of data are sent through a single I²S data line.

In addition to TDM alignment, the I²S Protocol Triggering and Decode Software supports standard I²S, left-justified and right-justified signals. Any decode only requires three inputs: the data, word select, and clock line of an I²S bus. On a Keysight MSO, these can be either analog or digital inputs.

The protocol search tool makes finding errors effortless. Trigger on specific packets, conditions and errors.

Throughout the industry, protocol decoding has never been simpler, more flexible or more reliable. With this new technology, you can be confident in your ability to decode any I²S signal – including TDM – and customize your decode to suit your specific application. Want to read your results in Hex? It’s just a tap on the screen. Want to use a digital clock signal but an analog data signal? It’s a touch of a button. Want to turn your instrument into an alarm clock? Well…you might want to check out this video first https://youtu.be/4sqmvzxFISE.

In all seriousness, ultimately what I found most impressive about Keysight’s software throughout my internship experience was the ease of use. I came to the office with only two years of engineering school and I knew absolutely nothing about serial buses—I complained that a spec that used the term “slave mode” wasn’t PC.

To be honest, the Infiniium oscilloscope software made my job really easy — it’s not just accurate, it’s intuitive. Every scope has a built-in help menu and demos for protocol decode applications so I could figure everything out on my own.

You really can’t mess this up.

Keysight’s I²S Protocol Triggering and Decode Software for Infiniium oscilloscopes may be the best I²S protocol application. But it wasn’t programmed for a serial bus, it was programmed for people. That’s what sets Keysight apart, and that’s how you’re going to solve your signal integrity problems.

 

*S-Series only

Finding middle ground in the conflict between wide analysis bandwidth and long capture times

We are moving to a new blogging platform this month. We’ll be posting to both platforms during the transition, but come check out our new location  https://community.keysight.com/community/keysight-blogs/oscilloscopes to see our future posts.

 

Battlefield scenarios can set up competing forces for radar/EW designers as well. Many seconds or minutes may go by as a scenario plays out with a clear winner and loser. But emulating this scenario with multi-emitter signal sources and multi-channel receivers is nontrivial. Designers need wide analysis bandwidth in measurements on hardware; they also need to evaluate a significant time period of system activity. Given this conflict, this class of pulsed RF, microwave, and mmWave applications presents a challenge.

On the signal source side, the technique of using pulse descriptor words (PDWs) is changing the game with regard to throughput and real-time signal creation. On the receiver side, if direct digitization techniques are used for amplitude and phase flatness advantages, as is the case when using some high-bandwidth oscilloscopes, the related high-speed sampling approach will burn through acquisition memory very quickly. But “segmented memory” can save the day: Signals of interest are placed into memory segments, and the receiver ignores the time when signals of interest are not present, as shown in Figure 1.

Figure 1. The segmented memory approach, where signals of interest are stored into memory segments

This blog post explores how segmented memory in wideband oscilloscopes can be used through pulse analysis software. We’ll address the application area of Radar/EW in terms of pulse amplitude, frequency, and phase measurements, and how you can optimize accuracy.

Oscilloscope segmented memory helps you achieve long target-time capture in pulsed RF applications

In most basic pulsed RF measurements with an oscilloscope, you take measurements on a single RF pulse from a pulse train or on a limited number of pulses. And that makes sense—a fast sample rate (adequate to capture carrier plus modulation without aliasing) uses up the scope memory depth quickly. Consider an example where a pulsed RF signal has a 15-GHz carrier frequency and 2-GHz-wide modulation.

The oscilloscope must sample fast enough to handle the modulated 15-GHz RF pulse signal.  That requires a sample rate of at least ~ 2.5 x 16 GHz, or 40 GSa/sec. To have some margin beyond the 2-GHz modulation on the carrier, and to avoid the roll-off of the scope bandwidth, the next highest sample rate selectable is the full 80 GSa/sec of the oscilloscope for 33-GHz bandwidth capture.

Using a standard capture approach, where all samples simply go into the available memory regardless of what signals are present, and using the full 2-Gpts memory depth available, that corresponds to 25 msec of capture time:

(2 GSa) / (80 GSa/sec) = 25 msec

But let’s consider a different example, where a pulse train has a pulse repetition interval of 100 μusec (a pulse repetition rate [PRI] of 10 kHz) and 1-usec wide pulses. The related scope capture includes close to 250 pulses based on the following calculation:

(25 msec) / (100 μsec / pulse) = 250 pulses

By using oscilloscope segmented memory, you can dramatically increase the number of pulses captured.  With segmented memory mode, the 2 Gpts of memory depth can be broken into smaller segments. Each segment gets filled with captured trace after a trigger condition is met. In this case, the trigger event is still the beginning of the RF pulse, and segments can be defined to be a little longer than the longest pulse captured.  For example, you can use a 1.2-μsec-wide segment size can to capture the 1-usec-wide pulses, for example.

The segmented memory capture can be set up to achieve 1.2-μsec wide segments where the memory depth is chosen to be 96 kpoints and 32,768 segments, as shown in Figure 2.

Figure 2. Segmented memory setup using 1.2-usec wide segments for 1-usec-wide pulse capture

The calculation for the required segment memory depth is simple. If you know that the sample rate is 80 GSa/sec and you want a 1.2-μsec segment length, then:

 

(80 GSa/sec) x (1.2 μsec) = 96,000 samples

With this choice, up to 32k segments can be selected.  Press the “Single” capture button, and 32k pulses are captured and brought into 32k segments. That corresponds to 3.3 seconds of target activity time. Is this gapless capture? No, but it is capture that focused on capturing RF pulses and ignored the time when no signal is present. Contrast this with Real Time Sampling Mode, which had 25 msec of gapless capture of 250 RF pulses.

The segmented capture can be seen in Figure 3, taken on a pulsed RF signal with a 15-GHz carrier and 2-GHz-wide linear FM chirp modulation. You can even use the “Play” button to play back the 32k segments. Statistics are calculated on the 32k pulses that were captured.

Figure 3. 33-GHz oscilloscope segmented memory capture of 32k pulses into 32k segments, 1.2 usec per segment

You can make similar measurements on lower frequency signals using a mid-range 8-GHz bandwidth oscilloscope. With 20-GSa/sec sampling rate on two inputs channels and 800 Msamples of memory depth, a “Single” capture can be spread across multiple memory segments.  These oscilloscopes offer 10 GSa/sec sampling across four channels as well.

There are also oscilloscopes with 63 GHz of bandwidth on two channels, with 160 GSa/sec sampling rate and a 2 Gpts memory depth.  They offer 80 GSa/sec sampling rate capture on four channels with a 2-Gpts memory depth.

Enhance measurements with oscilloscope segmented memory combined with pulse analysis software

You can control segmented memory with vector signal analysis (VSA) software. VSA lets you conduct statistical pulse analysis on many RF pulses captured into segmented memory. For example, you can perform analysis on digitally down-converted oscilloscope samples, where the format is now baseband I/Q, the measurement has been tuned to the center frequency, and a frequency analysis span is chosen to be just a little wider than the signal spectral width. This allows processing gain to reduce noise in the measurement.

After noise reduction, many measurements can be taken on the I/Q data, including how the amplitude, frequency, and phase change across an RF pulse. Figure 4 shows an example of these measurements, where memory segments 3, 4, and 5 and the pulses contained in those segments are being analyzed.

In this example, the linear FM chirp-frequency shift across the RF pulse is measured and compared to a best-fit linear ramp. (Check the right pane center).  The difference between the measured pulse and the best-fit straight line ramp is calculated and displayed (horizontal trace with noise).  You can see that the measured ramp and reference ramp have very little difference between them. The error trace is displayed with a 1 MHz/div scale and around 500-kHz peak deviation; the Freq Error RMS in the right bottom right shows around 300 kHz of frequency error.

In a similar way, the phase shift across a pulse is compared to a best-fit parabolic phase shift (see right top pane), characteristic of linear FM chirp modulation on radar pulses. You can zoom in on the difference between the measured and reference to see how much a target system is deviating from the ideal. Here we see around +8 and -5 degrees peak deviation and a Phase Error RMS of 2 degrees, as shown in the bottom right table of Figure 4.

Figure 4. Pulse analysis software calculations based on measurements taken on oscilloscope segmented memory

The left center pane shows the spectral content of the RF pulse, the left upper pane displays a view of RF pulse envelope amplitude, and the left lower pane shows the difference between the measured amplitude envelope and a best-fit straight-line reference signal.

Finally, you can perform statistical analysis on the measured parameters on the number of pulses captured into segments. In Figure 5, the statistical analysis can be seen in the pulse table based on capture of 1000 memory segments.

Figure 5. Statistical analysis on 1000 memory segments

 

Summary

When directly capturing wideband pulsed RF signals, the fast sampling rate required can make the capture of many pulses a challenge. The available acquisition memory gets eaten up quickly. Segmented memory is one way to address this problem by acquiring RF pulses into memory segments, and then turning off the acquisition during “quiet” time until the next RF pulse occurs.

Pulse-analysis software can both control a segmented memory capture and digitally down-convert captured signals into baseband I/Q data. This effectively tunes the measurement to a specific carrier frequency with a frequency measurement span slightly wider than the signal under test—reducing noise and increasing measurement accuracy. The time required for system validation decreases thanks to the capability to compare actual, measured pulse characteristics against ideal, relative, best-fit reference signals for amplitude, frequency, and phase. With that, you can identify issues in signal creation or system performance, and overcome the challenges that battlefield scenarios present.

Keysight Oscilloscopes Blog is moving!

Thank you for supporting the Keysight Oscilloscopes blog! We are moving to a new blogging platform this month. Come check out the same great content in our new location: https://community.keysight.com/community/keysight-blogs/oscilloscopes.

 

We’ll see you tomorrow with our first post to the new blog!

 

High Bandwidth Oscilloscope + Analysis Software = Powerful, Wideband RF Measurement Suite

A change is coming in the tools for measurements in both pulsed RF aerospace/defense and I/Q vector-modulated communications application. Whether for multi-channel analysis or for wider analysis bandwidth, high-bandwidth oscilloscopes are taking the place of traditional spectrum and signal analyzers. That’s because they can handle signals with spectral content beyond 1 or 2 GHz. These signals are being created to support the higher resolution requirements in radar systems and move the vast amounts of information in new communications systems.

So how do you create a powerful, wideband RF measurement suite? By coupling a high-bandwidth real-time oscilloscope with RF analysis software. Once you’ve married the two, you achieve a number of enhancements:

• Noise reduction through digital down-conversion
• A wide range of vertical scaling options, including linear and log magnitude
• Key RF measurements including occupied bandwidth (OBW) and power spectral density (PSD)
• Vector demodulation options for communications formats like QAM16
• Analog demodulation options including AM, FM and PM
• Set-up of segmented memory capture
• Statistical pulse analysis

 
Pulse amplitude, frequency, phase, and FFT measurements

For radar and electronic warfare applications, it’s helpful to perform a variety of measurements on many pulses. This includes things like amplitude variation, frequency, and phase shift across pulses, and a view of the spectrum of signals. For applications such as aircraft warning receivers, you also want the capability to measure time difference and phase difference between pulses associated with the capture of a wave front by multiple antennas on an aircraft. Let’s consider some of these measurements.

In the simplest case, you can measure the basic pulse amplitude, frequency shift, and phase shift across the measured RF pulse. The RF pulse train is sampled by the oscilloscope and then digitally down-converted to reduce noise and allow further signal processing.

For example, in Figure 1, a 15-GHz carrier, 2-GHz-wide linear FM chirped RF pulse signal is shown after vector signal analysis (VSA) processing. Here’s what the image shows:

• 2 GHz-wide spectral content of the signal (upper left);
• Real part of the down-converted I/Q data (lower left);
• 2-GHz-wide linear FM frequency chirp seen across the RF pulse (upper right);
• parabolic phase shift seen across the RF pulse (lower right).

These measurements are taken in the “Vector” measurement mode.

Basic vector mode analysis of FFT, real part of I/Q, FM chirp, and phase shift across pulse seen

 

Single channel, segmented memory capture, statistical RF pulse analysis

The next level of analysis requires a shift into “Pulse Analysis” mode. Here we use multiple oscilloscope channels to capture RF pulse signals into segments of oscilloscope memory. These are digitally down-converted into baseband I/Q signals, and then evaluated for single and multiple channel pulse analysis. For single-channel measurement, you can make three comparisons:

• the linear FM frequency chirp to an ideal, best-fit linear FM chirp signal;
• the phase shift across a pulse to a best-fit parabolic phase shift profile;
• the amplitude of the pulse envelope to a best fit ideal straight-line best fit reference.

In Figure 2, you’ll see these comparisons being made between measured to reference, and then the “error” between the measured and reference is expanded in vertical scale for a close view.

A pulse table also displays RF pulse parameters, including an RMS error calculation between the measured frequency or phase across the pulse, compared to a best-fit reference signal. It’s also possible to show statistics for the measurements over all the pulses.

Single-channel spectrum, amplitude, phase, and frequency measurements vs. best-fit reference signals

 

Dual-channel delta pulse amplitude, frequency, and time-delay measurements

You can also make “two-channel delta” measurements, as shown in Figure 3. These measurements are becoming increasingly important in applications such as aircraft warning receiver testing, where multiple signals are being captured from multiple antennas. The time delay and frequency difference of arrival between wave fronts must be measured for angle-of-arrival calculations.

Notice in this example a 1 nsec time delay being measured between two RF pulses. You’ll also see a 0.2-dB difference in amplitude and a 16-kHz difference in frequency, on average.

Pulse analysis is also performed on three of ten captured pulses that are being placed into oscilloscope memory segments. The parabolic phase shift across pulses (lower left), the linear FM chirp frequency shift across pulses (middle right), and the pulse envelope of pulses (upper left) are superimposed for signals coming into two oscilloscope channels. As in the previous example, each scope-channel measured signal can have the measured, reference, and error signal calculations made. Finally, the FFT spectral content for both scope-channel captures of the two pulse trains (center left) is also shown.

Two-channel measurements of RF pulse characteristics including time, amplitude, and frequency difference between two channels

 

Cross-correlation between pulses for precise time-delay measurements between RF pulses

In the aircraft warning receiver example mentioned previously, you can determine very precise measurements of time delay between RF pulses captured on different antennas on an aircraft by using a cross-correlation measurement between pulses. In Figure 4, a 50-psec time difference of arrival (TDOA) is being measured between two RF pulses captured on two scope input channels. Here pulses have a 10-GHz carrier, 100-MHz-wide linear FM chirp modulation, and a 1-usec width. In this measurement, you can first remove the channel-to-channel skew between oscilloscope channels, including cable delays at the temperature measurements will be taken, through de-embedding. Then a measurement can be made to see the actual time shift between the captured signals. Measurements show a mean delay of 50 psec, with a peak-to-peak variation in delay between 47 psec and 53 psec.

Two-channel cross-correlation measurement for precise time delay between pulses

 

Math function used to measure phase shift between two RF pulses

The difference in phase between two RF pulses is also critical in a variety of radar/EW/warning receiver-oriented applications. Through the use of math functions, the measured phase across one pulse can be subtracted from the measured phase across a second pulse, measured on two oscilloscope input channels. We can measure the same two linear FM chirp signals from the last example to view the phase shift between the two pulse trains by comparing related pulses. Again this might be seen from two antennas on an aircraft. The time shift has now been set to zero on an arbitrary waveform generator, but a 25-degree phase shift is being introduced between the two signals. A capture shown in Figure 5, top center trace C, and related blue marker 1, show this 25-degree phase shift in a mean measurement in lower right Trace D, as well as only a 0.8-degree standard deviation and a 0.7 variance.  These are average values over the width of the pulses.

Two-channel phase difference measurement between two RF pulses

 

Summary

More radar/EW/warning receiver applications are driving toward wider modulation bandwidths to increase range and angle-of-arrival precision capability in related systems. At times, this extends beyond 1-GHz modulation bandwidths. Designers increasingly use wideband oscilloscopes as RF receivers to evaluate related wideband signals when validating their hardware prototypes. Although scope measurements directly are of interest, it’s often advantageous to use analysis software to digitally down-convert captured wideband signals to reduce noise and allow more in-depth analysis of baseband I/Q signals. By combining a wideband scope and VSA software with appropriate techniques, you can readily make angle-of-arrival calculations for a variety of systems.

Fourier Series of a Square Wave (and Why Bandwidth Matters)

By: Drew Hanken

Have you ever wondered, “What is this ringing I am seeing in my signal? Why is there preshoot and overshoot on a simple square wave? How is a preshoot possible when it appears to be precausal to downstream information?”

I have been an R&D engineer here at Keysight/Agilent Technologies for the last five years. Fourier series is a topic that was covered in a recent graduate class as a method for solving partial differential equations. I’ll explain the occurrence of this ringing from the perspective of the underlying theory, and then relate it back to using an oscilloscope.

In short, this ringing is a phenomenon that presents itself because of the method an oscilloscope uses to construct a signal – by summing the frequency components of the signal. This method of building a signal is known as Fourier series. It is the most useful way for an oscilloscope to process and measure a signal because it deconstructs the signal into its frequency components for analysis. The inherent discontinuity of a square wave presents some problems with this reconstruction method that can be understood by exploring the mathematical theory. Understanding this can help you to select the right oscilloscope for your measurement needs. It’s also worth noting that this method is used by all oscilloscopes, it’s not just a Keysight method.

First off, you need to understand that any arbitrary function f(x) can be constructed by the sum of simple sine or cosine functions that vary in amplitude and frequency. More specifically, any function of x, f(x), can be built using an infinite series of sine waves with coefficients A_n and increasing frequencies (n*pi).

Sine and cosine functions can both be used for Fourier series because they both have the property of orthogonality. There are a few other functions that satisfy the orthogonality requirement including Bessel functions and Legendre polynomials. While these functions are useful for problems in cylindrical or polar coordinates, they do not apply to this discussion. The definition of orthogonality for a function φ is that it must satisfy the following condition:

Using the above definition, it is possible to solve for the coefficients (A_n) for any function, and build the function, using orthogonal functions.

You can see that any function can be constructed using an infinite series of terms, and approximated by a finite number of sine waves. Every function has a unique set of coefficients, A_n, that are substituted into the sum. The values of these coefficients determine the function that will be reconstructed.  It is easy to picture changing the amplitude of a sine wave by multiplying it by a coefficient. This would be a Fourier series with only one term, and would return the desired function with the magnitude changed. Let’s look at constructing a linear line using sine and cosine functions. This is tougher to picture because a line is not oscillatory, but the addition of multiple sine or cosine terms will begin to take the shape of a line.

You can see that the more terms that are summed, the closer the approximation becomes to the actual function. Theoretically, if an infinite number or terms are used, the Fourier series will cease to be an approximation and take the exact shape of the function.

Now, let’s take a look at a square wave and how it appears when constructed using Fourier series the same way an oscilloscope would. We will first write a step function of length (L) that, when repeated periodically, will be our representation of a square wave.

As stated earlier, this function can be rewritten as an infinite series of an orthogonal function φ:

With choosing a sine wave as the orthogonal function in the above expression, all that is left is to solve for the coefficients to construct a square wave and plot the results.

One important takeaway from this formula is that the series composition of a square wave only uses the odd harmonics. This stems from the fact that a square wave is an odd function, which has important implications on measuring signals of this sort. Given a 1 Gb/s square wave, the bandwidth of the measurement device must now exceed 3 Gb/s to capture more information than the primary frequency. Each incremental harmonic that is captured will start to look more like a square wave with faster rising and falling edges as seen below.

The rise time of the plotted signals gets smaller as the number of terms increases. The highest term in the Fourier series will correspond to the highest frequency that is used to construct the signal. Thus, the rise time is dictated by this last term, which in turn dictates highest frequency. An ideal square wave will have a zero rise time – but that would take infinite bandwidth to reproduce with this method. This square wave’s discontinuity is the heart of the problem, and is the reason for the preshoot and overshoot seen above. Taking a closer look at these areas of the wave, you can see that the ringing in the signal does not change in magnitude with the number of terms and remains at roughly 18% of the amplitude. It will, however get thinner as more terms are used and be a smaller source of error. This ringing caused by a discontinuity is referred to as Gibbs Phenomenon, and is unavoidable when the signal is properly constructed using Fourier series. That is not to say that other sources of ringing are not present, but it is important to be aware of this behavior at discontinuities.

Given this information, you can see why it is important to select the right oscilloscope based on your measurement needs. If you need to capture the slew rate of your transceiver, or to open the area of your eye diagram, you may need a higher bandwidth oscilloscope to capture the frequency content of the higher order harmonics and to reduce the effects of ringing on your capture.

Learn more about oscilloscopes available from Keysight.

New tools make short work of wideband RF measurements

Wideband RF measurements are changing and along with them, the tools you need to make sense of the signals. Today’s radar systems require higher target tracking resolution, communications systems require higher data throughput—and to meet the demands, you require wider modulation schemes on related signals to validate prototypes and production units.

Gone are the days when an instantaneous measurement bandwidth of 510 MHz, the longtime standard in signal and spectrum analyzers, could handle this modulation bandwidth. Some systems have crossed beyond 1- GHz and even 2-GHz-wide formats. You need a different approach to make high-quality, insight-providing wideband RF measurements.

How different an approach? One that uses high-bandwidth, real-time oscilloscopes. Digitizers and oscilloscopes offer enough bandwidth and sample rate to directly sample the carrier plus modulation either alone or with the use of down-converters in front of the scopes.

The trick is knowing what to use when. One way to consider your options for wideband measurements is to plot the possibilities on a chart. The vertical axis represents analysis bandwidth of the solution and the horizontal axis representing carrier frequency that you can measure. Don’t worry about doing the plot—we’ve taken care of it:

Applicable tools as a function of signal carrier frequency and spectral width

As you see, classic signal analyzers offer analysis bandwidths up to 1 GHz and handle carrier frequencies up to around 50 GHz. As an alternative, mid-range oscilloscopes offer bandwidths in the 8-GHz range, letting you measure signals with carrier frequencies approaching 8 GHz, and with very wide modulation bandwidths, approaching 8 GHz. As long as the carrier plus modulation spectrum fits within the oscilloscope bandwidth, you can make meaningful measurements.

But even that may not be enough. In wideband aerospace/defense applications, including electronic warfare, radar, and surveillance, signals of interest may have carrier frequencies higher than 8 GHz. Cue the high-performance oscilloscopes. These scope families have higher bandwidths up to 33 GHz and 63 GHz and, as you might guess, corresponding higher prices. But they offer impressive performance in areas like frequency response flatness and low noise. An alternative is to place a down converter in front of a mid-range oscilloscope. You pay less but can handle high carrier-frequency signals with wideband modulation—provided you’re willing to make some tradeoffs in amplitude and phase linearity.

As a first down-converter option, you can place a standard signal analyzer in front of a mid-range oscilloscope and use the IF down-conversion path in the signal analyzer. You’ll typically need calibration to flatten the overall system amplitude and phase response over frequency. But a solution like this can address a wide range of carrier frequencies, typically up to 50 GHz.

A second down-converter option is to place a lower cost harmonic mixer in front of a mid-range scope. This results in a “banded” solution: Very high carrier frequencies can be analyzed, but there is generally a “band” of carrier frequencies that a particular harmonic mixer can handle. That makes this option especially convenient for applications like 5G, Wigig, and automotive radar.

Typical RF performance for high-bandwidth real-time oscilloscopes

So what do you need to know before making FFT or wideband RF measurements with an oscilloscope or scope combined with vector signal analyzer (VSA) software? You need to know that the RF characteristics can have a major influence on the measurement results—so you’ll need to evaluate this first.

Today you can find oscilloscopes that incorporate amplitude and phase correction for excellent absolute amplitude accuracy and low deviation from linear phase across their frequency range. This in turn contributes to high-quality RF measurements. These oscilloscopes also offer excellent noise densities, in the vicinity of -160 dBm per hertz, and high dynamic range and signal-to-noise ratios, considering the wide bandwidth capability they offer.

What does that do for you? You can look at wideband signals with very small amplitude adjacent in time to large signals. You can also boost scope sensitivity to measure isolated, small-amplitude signals. The time-base circuitry in these oscilloscopes also means good, close-in phase noise, which corresponds to low jitter in very deep memory traces. If you want more details, see the RF characteristics of a high-performance 33-GHz oscilloscope in Table 1.

Table 1. Typical RF performance in a high-bandwidth oscilloscope

Wideband pulsed RF time-domain measurements of envelope, frequency, and phase chirp

Now that we know what our high-bandwidth scope is capable of, let’s see how it handles time-domain measurement and analysis of wideband pulsed-RF signals with no help. The choice of which oscilloscope to use depends on the maximum frequency content of the carrier plus modulation. Consider an example where a signal under test is supposed to have 1-usec-wide pulses, with a pulse repetition interval of 100 usec. It also has an RF carrier frequency of 15 GHz and linear FM chirping that is 2-GHz wide.

Figure 2 shows a variety of measurements on a single RF pulse, including envelope parameters and the frequency chirp across the pulse. Stable triggering on this pulse is accomplished with trigger “holdoff” set to a value slightly longer than the 1-usec RF pulse width.

Figure 2. Time-domain measurements on 1-usec wide, 15-GHz carrier, 2-GHz-wide linear FM chirped RF pulse with a 33-GHz bandwidth oscilloscope

 

To make amplitude measurements, we use the “Envelope” math function and then pulse measurements are dropped down onto the visible RF pulse envelope.  A “Frequency” measurement is dropped down onto the RF pulse (not onto the envelope), and a “Measurement Trend” math function is defined with the frequency measurement as a source. Next we perform a smoothing math function on the measurement trend with the resultant linear ramp display of the linear FM chirp modulation, also shown in Figure 2. The oscilloscope magnitude linearity over the frequency span of interest has a direct effect upon the quality of the envelope measurement. To see the effect, take a look at the magnitude plot over frequency of the 33-GHz bandwidth scope in Figure 3.

Figure 3. Typical magnitude linearity over frequency on four individual 33-GHz channels

Wideband pulsed RF-gated FFT measurement of spectrum

You can create a wideband FFT by defining an “FFT Magnitude” math function with “Rectangular” windowing. Then create a time-gated FFT using the (you guessed it) “Timing Gate” math function. Once the time-gating math function is defined, you can define an FFT math function that is calculated from the time record within the time gate, as shown in Figure 4.

Figure 4. View of normal and time-gated FFT and display with time gate at the beginning of the RF pulse

Wideband pulsed-RF time- and frequency-domain measurements with a scope plus VSA software

But that’s not all. You can further enhance RF and FFT measurements made with high-bandwidth oscilloscopes by importing scope-captured signals into VSA software. Some advantages of using VSA software include:

• many built-in RF measurements;
• ability to bandpass-filter oscilloscope input samples and decimate prior to the FFT calculation to reduce noise and speed the calculation;
• variety of digital and analog demodulation options like QAM16 and FM demodulation;
• time-domain baseband view of pulse with reduced noise through processing gain;
• frequency and phase shift across the pulse through a demodulator.

If the oscilloscope-captured data is imported to VSA software, it can be digitally down-converted into I and Q baseband data, bandpass-filtered, and then resampled. This can greatly decrease the amount of noise in the measurement. Essentially the process is “tuning” to the center frequency of the signal and “zooming” into the signal to analyze the modulation.  This is also referred to as “processing gain.”

In this example, the original 8-GHz-wide measurement with the associated noise is reduced to a 500-MHz-wide measurement, centered on the 3.7-GHz carrier with an instantaneous measurement bandwidth slightly wider than the width of the signal modulation. This corresponds to an improvement in signal-to-noise (SNR) ratio of:

10log*(ScopeBW/Span) = 10log*(8E+09/500E+6) = 12 dB.

SNR is improved by 10log*(ScopeBW/Span).

By taking advantage of this processing gain, combined with the VSA software’s capability to use a log-magnitude scale, and using averaging, you can now see the 50-dB down pulse, as shown in Figure 5. It wasn’t visible in the scope display with the 8-GHz wide measurement.

Figure 5. 50 dB down pulse seen with VSA software “Center Frequency” and “Span” set

The secret to long target-time capture and statistical pulse analysis

 When an oscilloscope samples a wideband RF signal, it must do so at a fast enough rate to accurately capture the carrier plus modulation. Often a very fast sample rate is required. In a normal real-time sampling mode, the oscilloscope memory will not allow for a long capture period.

But there is a work-around: oscilloscope segmented memory. This can greatly increase the target activity time when there is a low-duty-cycle signal, such as a common pulsed RF radar signal. The scope memory is divided into smaller segments of fixed time width, chosen to be a little wider than the widest RF pulse. The scope triggers on an event, such as the beginning of the RF pulse, and then places one RF pulse in a memory segment. The scope then stops capturing data, rearms the trigger, and waits for the next RF pulse. A second RF pulse is put into the second segment of memory. This process continues until all the scope memory segments are used.

Modern pulse-analysis software can let you take advantage of the scope segmented memory and then offers built-in measurements for pulsed RF signals. Figure 6 shows a capture of many RF pulses via segmented memory, combined with pulse-parameter measurements in the pulse-analysis software. Here a 1-GHz linear FM chirp and related phase shift across pulses is compared to a best-fit ideal linear ramp and ideal parabola, respectively. A close-up view is made of the delta between measured and reference for frequency in trace S and for phase in trace J.

Figure 6. Pulse analysis software calculations based on measurements taken on oscilloscope segmented memory

Conclusion

The bandwidth limitations of signal and spectrum analyzers are driving designers to use digitizers and oscilloscopes, with or without down-converters. Math functions like envelope, measurement trend, and FFT all prove helpful in understanding target system operation and issues. Combining an oscilloscope with VSA software creates a powerful RF-measurement suite to perform measurements, including demodulation, extended SNR time-domain views, and statistical RF pulse analysis. Yes, there’s a tradeoff between dynamic range/SNR and the instantaneous bandwidth available, but you can still access many useful wideband measurements to evaluate a prototype or production unit.

 

Insider Tips for Using an Oscilloscope with Touch

Author: Chris Felder

As one of the Keysight R&D engineers who developed Project Echo, the touch screen and interface for Keysight InfiniiVision oscilloscopes introduced on the 4000 X-Series in 2012, I know these oscilloscopes from the inside out; literally. Here are a few creative shortcuts we have built into the oscilloscope interface to help you get more out of the scope.

As Keysight was designing the first touch interface, which is used on the Keysight 3000T, 4000-X, and 6000-X Series scopes today, we conducted extensive usability testing to ensure the touch screen and interface design enhanced the existing interface and the scope could be entirely driven using the touch screen. While touch can provide many benefits, we also wanted to be sure that it did not impair the usability for those not using the touch feature. Even if you prefer to drive the scope using the front panel keys and knobs, using touch in minor ways may greatly accelerate your tasks.

Let’s start with the “main menu” button in the upper left corner.

All of the oscilloscope’s menus and dialogs are accessible through this menu.  There are some handy shortcuts along the left side, and you can manipulate several feature states directly through this menu (channels, cursors, measurements, etc.).  The Applications menu gives a list of your licensed and installed oscilloscope applications, but also lists unlicensed applications – handy if you’d like to explore and read about all the capabilities built into your scope.

From the main menu, we move on to the status area along the top of the graticule; this area hold lots of readouts that show the state of the oscilloscope, and all of them are touchable.  Touch the scale or delay values in the Horizontal grouping, for instance, and you’ll get this handy popup:

 

From here, you can step the values using buttons, or touch the values once more to get a numerical keypad for direct entry.  If you want to change other timebase settings, you can press the gray ‘H’ button in the status area for a quick shortcut to the Horizontal softkey menu.

 

In some areas, we’ve added more significant shortcuts for the most common tasks.  Touch the trigger status indicator, for example, and you immediately toggle from Auto mode to Normal mode, and vice-versa:

 

The sidebar along the right side of the screen is another area we’ve really optimized for touch.  Any dialog box with a series of dots in the upper left (what we call a “gripper”) can be repositioned by dragging it from the title bar area; the same is true of sidebar tabs.  Any tab can be grabbed using the grippers, undocked, and positioned anywhere you like.  You can even re-dock the tab in a half-height mode, allowing you to see two tabs at once:

Like the status area, sidebar tabs are filled with touch shortcuts.  You can touch the analog channel input information in the Summary tab to quickly perform a slew of front end and probe configuration settings:

Titles in the sidebar tab look a bit like buttons for a reason – they all have handy shortcut menus when you touch them.  Touching the title in the Cursors sidebar, for example, lets you directly change mode and source settings without needing to travel to the Cursors Menu:

 

 

 

 

In the Measurements tab, you can touch individual installed measurements to track, clear, or reset them:

 

 

The softkey menu area along the bottom of the screen frequently includes readouts for status items related to the current menu, and…you guessed it…they’re all touchable! If you have the WaveGen (waveform generator) option enabled on your scope, the Waveform Generator Menu contains a particularly handy shortcut; if you touch the “Gen Out” area, you get a comprehensive control stack for the selected WaveGen, from which you can change a variety of settings without bouncing between multiple softkey menus:

Like all dialog boxes, this dialog can be re-positioned by dragging it within its title bar area.  You can also use the blue Menu button to configure dialog boxes to use a transparent background.  Now you can position and configure dialog boxes and sidebar tabs as you wish!

We strive to follow the rule, “everything is touchable” and we’re constantly adding new shortcuts and convenience menus with every software release.  We always welcome your suggestions and feedback – comment here to let us know what we can do to make your oscilloscope experience more efficient.

Welcome to Scope Month!

Welcome to Keysight’s inaugural Scope Month – an entire month focused on oscilloscope measurement tips, new content and oscilloscope giveaways! Join us throughout March for great oscilloscope resources and activities, including:

• New Oscilloscope Learning Center
  • You already have the expertise and ability to perform your job at a high level. Keysight wants to help by offering you the tools and information you need. As part of this, we are launching the Oscilloscope Learning Center where you can access a wide variety of information – from basic to advanced. Find application notes, measurement tips, videos, articles, and webcasts all in one location.
  • Content will be updated often so check back for the latest information and answers to your questions.
  • www.oscilloscopelearningcenter.com
• Scope-a-Day Giveaway –Over $500,000 in daily oscilloscope giveaways!
  •  This is the largest scope giveaway we have ever done, with over $500,000 USD in oscilloscopes being given away during Scope Month!
  • We will be giving away one MSOX3104T oscilloscope per day during Scope Month.
  • We will also be compiling the entries each week during Scope Month and picking a weekly winner via a live video stream. These weekly winners will be awarded an MSOX4104A oscilloscope.
  • Submit your entry and view the legal terms and conditions at www.scopemonth.com. And don’t forget to enter every day.
• “Test to Impress” Contest – win a fully-loaded 6 GHz scope, valued at over $70,000!
  • Throughout Scope Month, you can submit a 1-2 minute video describing how you have used (or would use) InfiniiVision oscilloscopes to impress; by solving a measurement challenge, working on a cool application, etc.
  • During the first two weeks of April, Keysight will post these videos and you,with the rest of the engineering community, will be able to vote on your favorite submissions to decide the winner (announced April 15^th).
  • The winner will receive a 6 GHz 6000 X-Series oscilloscope, four N2752A InfiniiMode 6 GHz differential probes, and a 6000X application bundle – valued at over $70,000!
  • Submit your Test to Impress entry . You can also see the legal guidelines and eligible countries.
• 2-Minute Guru Video Series
  • In addition to the Oscilloscope Learning Center, we are creating a video series called the “2-Minute Guru” which will provide information regarding scope basics. These will be launched throughout Scope Month and will continue beyond the month as well. Check the blog frequently to be notified when the next episode is available.
• New Limited-Time offer – Get an MSO for the Price of a DSO!
  • Until September 30^th, 2016, you can purchase an MSO model of select Keysight InfiniiVision Series oscilloscopes for the price of a DSO.
  • Visit Keysight MSO offer for more information.
• Webcast Series
  • Throughout 2016, Keysight will be hosting a series of oscilloscope and measurement webcasts. You can view and sign up for the entire series, or watch previously recorded webcasts on your schedule by scrolling to the bottom of that page.
• “Will It” Videos
  • InfiniiVision oscilloscopes are known for their instrument integration – offering you more value with your purchase by embedding other instrument functionality into the oscilloscope. Our marketing team has taken this one step further and has asked our R&D team to add several new features to our scopes. The question now is “will the oscilloscope be able to perform the tasks?” Watch these humorous videos every Friday during Scope Month.

And much, much more…

 

We hope you enjoy this exciting event, are able to join in some of the fun and also learn some new things as well. Stay up-to-date on this blog and our social media pages to see what else we have coming!

Meet the Team!

Welcome to the new Keysight Oscilloscopes blog! We will be posting regularly on a wide variety of topics involving test and measurement: everything from industry updates to application news to oscilloscope tips and tricks.

To begin, we wanted to introduce the blogging team so you know their areas of expertise and where they will focus their posts and communications.

Daniel Bogdanoff – Daniel has a degree in Electrical Engineering from Texas A&M University (whoop!) and works closely with oscilloscopes.  He will focus on helpful tips and techniques you should consider when working with benchtop equipment. In addition to working at Keysight, Daniel is also a Contributing Technical Expert for Electronic Design.

 

Takuya Furuta – Taku has a wide range of experience and in-depth oscilloscope product knowledge, including oscilloscopes from other manufacturers. His posts will cover the history of oscilloscopes, internal scope architecture and front end filters, and perhaps even some high speed digital applications.

 

Johnnie Hancock – Johnnie began his career as an analog hardware designer and has been around as long as dirt. He has tremendous knowledge across a wide variety of applications with his most recent areas of focus being on oscilloscope-based power supply and automotive serial bus measurements. In his spare time Johnnie enjoys spending time with his four grandchildren and beautiful wife of 40 years.

Mike Hoffman – Mike has a great deal of experience around oscilloscope tips and tricks as well as deep technical knowledge. He will focus most of his topics around these areas and will also cover industry news.

 

Kenny Johnson – Kenny is an R&D engineer currently disguised as a marketing engineer. He spent a significant amount of his R&D career working on oscilloscope probes (design and project management) and has 17 patents related to this work. When not earning a paycheck he enjoys running, biking and canyoneering (just like 127 Hours but without the arm thing).

 

Don Schoenecker – Don is a very technical member of the team who will cover a broad range of topics – everything from modular trends to application-specific posts. With over 30 years of experience in the use of test tools to improve product development, Don brings stories and a message of encouragement to help you improve your designs. As a Texas A&M Aggie from Colorado (whoop!), he is happy to be back in the mountains.

 

Robert Lashlee – Robert will write on a wide variety of topics and cover industry trends as well as new oscilloscope measurements and applications.