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UnderstandingFFT Overlap Processing

A Tektronix Real-Time Spectrum Analyzer Primer

RTSA FFT Overlap Processing PrimerPrimer

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3The Need for Seeing Faster Time-Varying Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Expand Your View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Figure 1. Similar to Zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

How it Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4Overlapping Many FFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

Figure 2. Overlap FFT Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

Some Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5Figure 3. Non-overlapped Spectrogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

The Overlapped FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5Figure 4. The Overlapped FFT Spectrogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

A Pseudo-Random Modulated Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Figure 5. Overlapped FFT Spectrogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Figure 6. Frequency vs. Time - Pseudo-Random Hopper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

Stretching Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Figure 7. Time Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Figure 8. Seeing Time-Varying RF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

Spectrogram Risetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8The Tektronix RTSA FFT Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8FFT Window Effects on Time Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8The Effect of Span and Sample Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8Measuring the Spectrogram Risetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Table 1. Spectrogram Effective Risetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Effects of Using No FFT Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Figure 9. Rectangular FFT Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Figure 10. Blackman-Harris 4B Window - the Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Figure 11. Moiré Patterns in Spectogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

The New Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

Measuring Time Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10Figure 12. Measuring Time Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

Time Correlated Multi-domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

Amplitude Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11Short Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

Figure 13. The Lower Frequencies are Shown Lower in Amplitude Here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11Table 2. Amplitude Reduction of Short Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11Figure 14. Overlapped FFT Frames and a Single Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12Figure 15. FFT Frames with Window Function Applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

The Blackman-Harris Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13Figure 16. BH-4B Window Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13Figure 17. One Marker in Time Displaying Multiple Frequency Hops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Calculating the Amplitude Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13Other Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14Super-short Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Figure 18. 250 ns Pulse Easily Measurable in the Time View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14Figure 19. Tiny Pulse is Not Visible in Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14Figure 20. The 250 ns Pulse is Now Visible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

What Overlap Can Not Do . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

Conclusion: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17Appendix A. Pulse Generation Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

Table 3. Demonstration File names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

Contents

Figure 1. Similar to zoom.

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As faster time-varying frequency signals are becoming more widespread, Tektronix has respondedto the need to provide more visibility of very short-time events with Real-Time Spectrum Analyzers(RTSA) employing FFT frames that can be overlapped fully.

In this primer, we will show the analysis benefits of this technique. We will also explore how it works,and how you can use it most effectively to see Time-Varying RF signals with greater clarity thanever before.

This primer assumes the reader understands the basic fundamentals of how a RTSA operates. Educational information describing the basics of Real-Time Spectrum Analysis can be found at www.tektronix.com/rsa.

The Need for Seeing Faster Time-Varying Signals


Overlapped FFT works somewhat like a “Zoom” for theSpectrogram. It does this by effectively stretching the timescale. While it does overlap small time events, the greatervisibility provided in the Spectrogram enables much greatervisibility of frequency changes with time.

The lower Spectrogram in Figure 1 gives visibility to the transient time behavior. In this case, two separate frequencysteps that appear as only one in the upper Spectrogram.The entire lower Spectrogram is contained within only 5frames in the upper Spectrogram.

Expand Your View

Top Spectrogram shows no overlap Frame duration = 20 µs

768 FFT points overlap (FFT interval - 256 points)Frame duration = 5 µs

960 FFT points overlap(FFT interval - 64 points)Frame duration = 1.25 µs

Signal Captured in the Time Domain

Acquired Signal Data Transformed into FFT Frames, No Overlap Processing

Acquired Signal, Post-Processed with Overlap FFTs

FFTs Overlap Samples

Overlap Interval Samples

Figure 2. Overlap FFT Processing.



Overlapping many FFTs

Previous generation RTSAs processed the data from the A/Dconverter mostly in a straight sequential manner. The first1024 bytes went into the first FFT (first frame), the second1024 bytes into the second FFT (second frame), etc.

This new capability allows the user to select the number ofdata points of overlap from 0 to 1023. This means that at 0overlap we use the sequential way with each FFT adjacentto the previous one. The overlap number specifies howmany of the last data points in any FFT frame also becomethe beginning points included in the next frame. When 1023overlap is selected, the second FFT frame will have onlyone new data point at the end, while only one point fromthe previous FFT will be dropped off the beginning. 1023data points are shared between these adjacent frames.Figure 2 illustrates the difference between FFT processingwith and without overlap.

This also means that as each FFT spectrum is displayed, it contains some information from the previous spectrum.

The example of overlap FFT processing in Figure 1, middleimage, is using a 256 sample FFT Interval with 768 samplesoverlapped on each frame. Each frame starts 5 microsecondsafter the previous one and shows a 15 MHz span. Narrowerspans would result in each spectrum taking up more time,while simultaneously seeing frequency components that arecloser to each other.

With this technique, a very short spectral event (particularlyone that does not even last as long as one FFT frame) canbe seen, even if at reduced amplitude, in many FFT spectrathat are displayed adjacent to each other.

The advantage this provides is visibility of the very-shorttime variations within a signal. The disadvantage is thatsince the FFTs are overlapped in time, the various frequencyevents will also appear in the Spectrogram display to beoverlapped by the same amount. This effect will prevent relative timing measurements between spectral events from enjoying the same increase in resolution as does thespectral visibility.

How it Works



Figure 3. Non-overlapped Spectrogram. Figure 4. The overlapped FFT Spectrogram.

In Figure 3, we see a Spectrogram that includes a radarpulse. This pulse is frequency-hopped with five steps across32 MHz. Including all five hops, it is 8 microseconds long.Using a 36 MHz span, this is about 1/3 of one FFT spectrumframe of 20 microseconds. And, each hop lasts only about1/12 of one FFT frame. Therefore, it exists in one frame only,and we can not see much detail in the hopping pulse.

The RTSA can trigger on and capture such a pulse. But,displaying it in a Spectrogram has been a bit difficult. Fortriggering on very short pulses, the power trigger is preferred.

The previously mentioned pulse was generated by a TektronixAWG710B Arbitrary Waveform Generator. The files for thispulse and all the others used in this document are listed inAppendix A.

In Figure 4, the Tektronix RTSA with 36 MHz bandwidthshows this same hopped pulse using overlap FFT processingwhere the FFT frame-to-frame time interval is reduced from1024 (no overlap) to 16 samples with 1008 sample overlap.The result is dramatic. Even though the pulse is still shorterthan one frame, the individual frames are sequentially positioned 320 nanoseconds apart. This clearly shows the steps that are increasing in frequency throughout thepulse, and the approximate time of each step.

Some Comparisons The Overlapped FFT



Figure 5. Overlapped FFT Spectrogram Pseudo Random Hopping. Figure 6. Frequency vs. Time – Pseudo-Random Hopper.

A Pseudo-Random Modulated Pulse

A Pseudo-Random hopped pulse waveform is also available.This is seen in Figures 5 and 6. This uses the same hoptimes, and the same frequency spacing as the previouspulse. But instead of a linearly increasing frequency with eachhop, the hops are made randomly throughout the bandwidth.

In Figure 6, we see that the time-overlap effect of theOverlapped FFTs makes it important to use the Frequencyvs. Time display to measure timing on the hops. The figureshows the benefits of this display as there is an apparentreversal of time in Figure 5 and time is shown sequentially.



Figure 7. Pulses app to be overlapped in time. Figure 8. Seeing time-varying RF.

The effect that overlapping really brings is not really a “zoom,”but is much more a “stretching” of time. This gives muchimproved visibility of time-variant phenomena, but theevents are all stretched out together.

In Figure 7, we have placed the marker at the beginning of the middle hop of the pulse. Note that due to the overlapof the FFT frames, the individual steps appear to be over-lapped in time. This overlap is not real, but is due to theoverlap of the FFT frames.

Figure 8 shows the marker placed at the end of the samemiddle hop. These pictures clearly show the frequency andtime-varying nature of this pulse, which is entirely within theframe time of one FFT. This visibility would not be possiblewithout overlapping FFT frames.

The effect of stretching time has made all of the hops ofthis pulse appear to be mostly concurrent in time, while we know that these hops are all separate in time. One ofthe biggest contributors to this effect is the fact that theseparticular events are all shorter than one FFT frame.

The FFT process is not an infinitely short slice of time. Itrequires processing multiple cycles of the input frequenciesin order to separate them from each other. Therefore, ifthere are several different frequencies that are all within thisframe, they will all be reported by the FFT. And, this FFT will create only one spectrum. Therefore, even if these frequency hops are all separate in time, they will all be displayed together in the one spectrum if they exist withinthe one FFT frame.

Stretching Time



Another effect that limits the ability to actually increase thetime-resolution in Spectrogram is the apparent “risetime” ofthe Spectrogram when using overlapped FFTs. This alsoincreases the time-overlapping of the separate hop seg-ments of this pulse display.

The Tektronix RTSA FFT Overlap

When using non-overlapping FFTs, each spectrum is producedfrom the next frame that consists of 20 microseconds of data.(1024 samples multiplied by 20 ns per sample). Therefore, if a signal were to be turned ON just at the point betweentwo sequential frames, it would contribute no energy to thefirst frame, and fully contribute to the next. This implies thatwhen using the Spectrogram, time events can be measuredwith a “resolution” of 20 microseconds.

One might be tempted to extrapolate that if the fully over-lapped FFTs were used (such that the time between thestart of one spectrum and the start of the next one is 20nanoseconds), that the time resolution would be 20nanoseconds. But it is not.

Some understanding of the process may help one be awareof the potential to misinterpret the results.

FFT Window Effects on Time Resolution

The A/D converter is continuously digitizing at a 102.4 MHzrate which provides both I and Q samples at a 51.2 MHzeffective rate, filling the memory record. Each FFT will beprovided with 1024 contiguous samples from somewherewithin the memory record (a frame). In the example we have been using, the 36 MHz span has a frame length of 20 microseconds.

Let’s more closely examine what happens with overlappedFFTs. The first FFT that we will look at is the last one beforea short burst of RF. This FFT has no power contributionfrom the RF burst, since the burst has not yet started. Thesecond FFT is overlapped by 1023, and therefore startsone sample point later. It contains only one sample point ofpower from the burst. Each subsequent frame will containone more additional sample of the burst until finally we willhave one frame that contains the entire burst (if the burst isshorter than a frame), or until we have a frame that is entirelyfilled with the burst (if the burst is equal to or longer thanthe 20 microseconds of a frame).

As each frame contains more power than the previous frame,we see that the second frame we are examining can, atmost, contain 1/1024 of the possible full power of the burst.This is in fact explained by Parseval’s Theorem, which statesthat an FFT will produce a spectrum that displays the powerlevel of a coherent signal that increases in direct proportionas the square of the number of samples of that signal whichare included in the FFT. The formula is:

Power(dB)=20 Log (SignalSamples / TotalSamples)

For additional information on this phenomenon, see the‘Amplitude Effects’ section of this paper.

There is a second effect due to the FFT window. Thisreduces the amount of contribution that the samples at theends of the Frame are allowed to make to the spectrum(reduces to essentially zero at the very end samples). Thepurpose of this time-filter is to eliminate the “end effect” of having an abrupt start and/or an abrupt end to the signal that is in the frame. This window further reduces the contribution from the signal that is seen when any portion of that signal is present near one end or the other of the frame (See Figure 7 and Figure 8).

These two effects together mean that the first spectrumthat contains the one sample of the burst will essentially haveno spectrum of the signal of interest. The next spectrum willhave very little more. It is not until we get a spectrum that isnearly one-quarter full of the burst that we will have significantamplitude of the burst displayed.

This slows the effective “risetime” of the Spectrogram, andits resultant “time resolution.”

The Effect of Span and Sample Rate

“Spectrogram risetime” is also dependent on the selectedspan (which determines the actual effective sample rateused and therefore the noise bandwidth). This risetime isnot dependent on the amount of FFT overlap. It is dependentsolely on the selected span and its resultant effective sample rate.

Note that there is an equivalent “falltime” and time smearingthat occurs at the end of a pulse or other frequency event.Both ends of a pulse have the same effect applied to them.

Spectrogram Risetime



Figure 9. Rectangular FFT Window. Figure 10. Blackman-Harris 4B Window - the default.

Measuring the Spectrogram Risetime

The following table (Table 1) gives the results of Spectrogramrisetime measurements.

Note: These measurements were done using the 10% and90% voltage points on an RF pulse.

This “risetime” limit is limiting the “time resolution” availablewhen attempting to measure the timing of spectral eventsusing the Overlapped FFT Spectrogram.

Effects of using no FFT window.

Since the FFT window is such a large part of the risetimelimitation in the Spectrogram, we should examine what willhappen if we remove this filter. One of the filter selections is“Rect” (Rectangular filter, or like having no filter at all).

We expect this will improve the apparent “risetime” of theSpectrogram. But we also know that we will now get theeffects of the abrupt ends of the frame. Figure 9 shows the result.

The start and end of the individual hops are still overlappedin time. This is unavoidable with Overlapped FFTs. But now we see a much more crisp beginning and end of each segment. Compare this to Figure 10 - the default windowagain. Using the rectangular window the “Effective Risetime”has indeed gotten better. For the 36 MHz span with theBlackman-Harris 4B (BH-4B) (Figure 10) the risetime was 7.2microseconds. The risetime we have with the filter removed is 1.27 microseconds! Quite an improvement.

Table 1. Sprectrogram effective risetime.

Span / NBW Overlap

1023 1022 992

10 MHz / 25 kHz 27.0 µs 28.0 µs 31.0 µs

20 MHz / 50 kHz 14.6 µs 14.4 µs 14.1 µs

36 MHz / 100 kHz 7.2 µs 7.2 µs 7.1 µs

The new artifacts

When using the Rectangular window, the time-varying hopsare more visible, but the artifacts from the edges of the hopsare now quite objectionable. They will cover up any smallsignals that we might need to see. In fact, the end effects ofjust one of the hop segments show up as significant Moirépatterns in the Spectrogram as shown in Figure 11.

The end effects are formed due to the pulse effectivelystarting only when the FFT frame includes some samples of it, and ending when the pulse itself ends. Without theWindow filter, this means that as you move through theOverlapped FFTs, you will see a sharp narrow pulse in the first FFT Frame, and a slightly wider pulse in each subsequent frame until the entire pulse is within one frame.As each frame analyzes the slightly wider pulse than wasseen by the prior FFT, the sinex/x pulse spectrum changesits periodicity and the pattern is formed in the Spectrogram.

On the other hand, we see that without the filter, all of the frequency components of this hopped pulse are time-stretched into a portion of time at least as long as one FFTframe without any one frequency step being reduced anymore than any other. They are all equally reduced by theshortness of the pulses. And, the Overlapped FFT allows usto select an FFT window exactly in this time position. Theupper spectrum view in Figure 9 shows this, with all of thehop components shown at equal amplitudes instead ofbeing ‘tilted.’

Measuring Time EventsSo what do we do if we want to measure time? TheTektronix RTSAs are optimized to correctly measure time-varying phenomena. Simply use the Time or Demodulation modes.

The RTSA also has time correlated multi-domain analysis.This is how we accurately measure the time-variations. Forthis frequency-hopped pulse, we choose the FrequencyDemodulation (FM) mode (Frequency vs. Time).

Time Correlated Multi-Domain Analysis

Figure 12 shows Frequency vs. Time in the bottom window(FM Demodulation is selected). Delta markers are placed at the transition points of a single hop portion of the pulse.The readout shows the hop time of 1.602 microseconds.

Since they are placed on the corners of the steps, themarkers will also accurately measure the 8 MHz frequencystep size.



Figure 11. Moiré patterns in Spectogram.

Figure 12. Measuring time events.

Amplitude EffectsShort Pulses

As previously mentioned, pulses or other RF events that areshorter than one FFT frame will be converted to spectruminformation at a reduced amplitude that is proportional tothe amount of the FFT frame that such signals occupy(Parseval’s Theorem).

Look at Figure 13. The spectrum that is produced by theone FFT that contains the pulse is clearly varying in ampli-tude. The upper-most frequency shows as about -15 dBm,the next lower one appears to be -21 dBm, the next twoare -31 and -45 dBm, and the last one is not even visiblebelow the noise.

Yet we know that all of the hop segments are the sameamplitude. The overall amplitude reduction is due to theindividual segments being shorter than one FFT frame.

The amplitude difference between the separate segments is due to their position within the FFT window filter (positionin the frame). Since the pulse was completely asynchronousto the FFT frame, the one we measured happened bychance to be located near the beginning of the frame.Therefore the first hop (the lowest frequency one) wasseverely reduced due to the “window” function applied tothe FFT. This reduction becomes smaller as the hops occurlater in the frame, and therefore closer to the middle of thewindow function.

This table has correction values for several short pulselengths. For this table, we are using a Real-Time bandwidthof 36 MHz. This bandwidth uses an FFT frame length of 20 microseconds. Pulse lengths that are equal to or longerthan the frame time will be measured correctly for amplitude.This table lists the errors that will cause reduced amplitudeto be reported for short pulses.

The calculations used for this table assume that the pulse iscentered in the frame and that the window used is eithernone, or BH-4B, as noted in the column heading.

Another assumption is that there is no scalloping error dueto the incoming signal being a different center frequencythan the analyzer is tuned to. Scalloping error occurs whena signal is not exactly in the center of an FFT and cancause a few tenths of a dB additional error. These errorscan be manipulated by changing filters.

There are two columns for the BH-4B window - Theory and Measured. The measured results are extremely close to the theoretical.



Figure 13. The lower frequencies are shown lower in amplitude here. Table 2. Amplitude reduction of short pulses.

Amplitude reduction seen for short pulsesin the middle of a 20 microsecond FFT frame

(normalized to full-frame amplitude)

Pulse Length Rectangular Filter Blackman-Harris BH-4B(microseconds) (none) BH-4B (measured) (theory)

20 or more 0 dB 0 dB 0 dB

8 -7.96 dB -1.61 dB 1.45 dB

4 -13.98 dB -5.72 dB 5.74 dB

2 -20.00 dB -11.29 dB 11.37 dB

1 -26.02 dB -17.23 dB 17.18 dB

0.5 -32.04 dB -23.19 dB 23.34 dB

0.25 -38.06 dB -29.11 dB 29.62 dB



RF Pulse

First FFT Frame

Second FFT Frame

Third FFT Frame

Fourth FFT Frame

Later FFT Frame

No WindowFunction applied

Figure 14. Overlapped FFT frames and a single pulse.

FFT Frames have the WindowFunction applied

RF Pulse

Figure 15. FFT frames with window function applied.

In Figure 14, we see a pulse of RF at the top. Beneath thatwe have drawn four lines that represent four sequentialoverlapped FFT frame times. Each of these frames is furtheralong, and contains more of the RF pulse.

It can be seen that the power contained in each frame is morethan the frame before. Then at the bottom is a frame (takenconsiderably later in time) which contains the entire pulse inits middle section. This frame will show the amplitude of thepulse, the highest seen in any of these frames.

These FFT frames do not have any window filter in use.

In Figure 15, we see the same pulse and FFT frames with the addition of a time windowing filter. This shows a greater reduction in pulse amplitude which results fromthe pulse being at one edge or other of the filter, while theamplitude of a pulse in the middle of a frame will be onlyslightly reduced.

The correct amplitude will be measured for an RF signalwhich is equal to or longer than one frame, and is continuousthroughout the frame. The amplitude reported by the FFTprocess is normalized to the response of the filter used forthe window function. Next we will look at the specific filterused here.

Figure 17. One marker in time displaying multiple frequency hops.

The Blackman-Harris Window

The default FFT time window for the Tektronix RTSA products is the Blackman-Harris BH-4B window. Figure 16 is a plot of the normalized response of such a filter.

If the FFT were to be performed on a frame of input sampleswithout this filter, there would be a number of spectrum artifacts generated due to the sudden start of the spectrumas well as the sudden stop. The solution is to use this filter(or a similar filter) to reduce the response from the ends ofthe frame to zero. Then, to gradually increase the contributionof the samples that are nearer to the middle of the frame.The horizontal axis of this plot is the frame of 1024 samples.

It is from this plot that we calculated the difference betweenthe filtered frames and the unfiltered frames for the table ofpulse width versus displayed amplitude (Table 2).

Calculating the Amplitude Reduction

Figure 17 shows that if we use overlapped FFTs, and weposition the marker on or about the middle of the “stretched”display, that we will see a spectrum that is the result of anFFT where the pulse is centered in the frame. Here we seethat all of the frequency hops are contained within the singleframe, and are centered in the window-function applied tothe frame. They all read between -13 dBm and -16 dBm.

Since each hop is 1.6 microseconds long, and the full frameis 20 microseconds long, we expect to see the powerrecorded in the Spectrogram to be reduced by Parseval’sTheorem. The calculation is 20Log(1.6/20) = -21.9 dB. Wemust also account for the correction that the power readoutfor the 20 microsecond frame already has due to the FFTwindow filter (Table 2 BH-4B filter response) = +8.24 dB, for a total of -13.66 dB.

Using this, we can manually correct our readings for narrowpulse amplitude. The center pulse reads -10.071 dBm andthe correction is to make up for a reduction of 13.66 dB. The correct amplitude for this pulse is +3.589 dBm. Whenthis pulse was stretched to be longer than an FFT frame, itmeasured +3.51 dBm. This is a very close agreement betweenthe practical measurement and the theoretical calculation.

Calculating amplitude correction for pulses that are not centered in the window is much more difficult, and requiresusing the BH-4B curve separately for each side of the filter.



Figure 16. BH-4B Window plot.



Figure 19. 250 ns pulse is not visible in Spectrum.Figure 18. 250 ns pulse easily measurable in the Time view.

Other Windows

There are several different windows available in theTektronix RTSA products. This document has explained theissues with the BH-4B and the Rectangular. Other windowswill have greater or lesser time and amplitude effects thanthe default one. If you are using one of these others, youwill need to re-calculate the amplitude and apparentSpectrogram “risetime” effects.

The selection of the best window function for your particularsignal is a subject beyond this paper.

Super-Short Pulses

Pulses that are significantly shorter than one FFT frame willnot be visible on either the Spectrogram, or in a SpectrumView. Such pulses can only be triggered by the power trigger,and the FFT frame will be triggered and started exactlycoincident with the pulse. This positions the pulse at thestart of the FFT frame, and consequently at one end of thewindow filter.

The Power vs. Time display is used to measure short pulses.The pulse measurement suite is specified to characterizepulses as short as 400 nanoseconds. For pulses shorterthan 400 ns, time correlated multi-domain measurementswork well.

The Power trigger works on extremely short pulses. It simply looks for any digitized samples that are above theselected threshold.

In Figure 18 we see that markers on the 250 nanosecondpulse report a measurement of 254 nanosecond width.The pulse amplitude is already very low due to its narrowwidth. But since the pulse is at one edge of the frame, thewindow function will additionally reduce the amplitudebelow the noise. See Figure 19.

This pulse is 250 nanoseconds long, only a little more thanone percent of one frame. If we had not already seen it inthe time domain, then the only way we would suspect thatit is here is the fact that the analyzer triggered, and theSpectrogram shows the trigger point half-way up its view,without any signal apparent there.

This is a case where Overlapping FFTs make the differencebetween no spectral visibility at all of this pulse, and greatvisibility. Compare Figure 19 with Figure 20, where we nowhave set the overlap to start each frame 16 samples afterits neighbor.

Measurement of Pulse Width

Trigger Point

There are now 64 FFT frames that include this pulse. Ofthese, there are five that have the pulse so close to theircenter that they provide measurements of the amplitudethat are within 0.2 dB of each other.

The key to setting the overlap is to have enough to find thesmallest pulse, or to show time-variant phenomena, but notto have more than necessary. Excess overlap causesexcess time-smear of the spectral events.

In Figure 20, we can now clearly see both the Spectrogram andSpectrum when we can select a spectrum that has the pulsein the middle of the window. Only FFT Overlap can do this.

Note the time delay visible between the “Trigger point” (topof the white bar on the left) and the beginning of the visiblepulse. This is because the trigger happens at the first (non-overlapped) frame that contains the signal. This frame hasno appearance of the signal, since the Window filterreduces it to zero. Then, as the overlap progresses, theframes have more and more of the signal until there is a visible amount. It is the previously mentioned “risetime” that causes this apparent delay.



Figure 20. The 250 ns Pulse is now visible.

Trigger Point



What Overlap Can Not DoOverlap of FFT processing only works in the Real-TimeSpectrum Analysis mode where it can digitize a continuousrecord without changing the hardware setting for the RTSA.For spans wider than the Real-Time bandwidth of an RTSA,one must piece together multiple acquisitions using differentRF converter settings and therefore overlapping FFT capabilitycannot work. In our example, the Real-Time bandwidth is36 MHz.

Overlap can not provide measurement resolution as muchas it provides visual improvement. There is still an “EffectiveRisetime” limitation even if we remove the Windowing Filter,and there is still the time-stretching effect. Both of theselimitations can be overcome by using the time domain displays in the RTSA.

RTSA FFT Overlap Processing PrimerPrimer Note

17www.tektronix.com/rtsa

Conclusion

Overlapped FFT provides a huge (~2000x) increase in visibility of short time varying RF phenomena. It can showyou multiple time-varying events that are shorter in timethan one standard (non overlapped) FFT frame, enablingyou to see extremely short pulses that would not be visibleon previous generation RTSAs.

Overlapped FFT is just part of the powerful analysis capabilities that the Tektronix RTSA products offer and toget the most complete and accurate results, overlappedFFT must be used in conjunction with the other in-depthanalysis modes. For extremely short time RF phenomena,the “Time” or “Demod” modes make accurate time andpower measurements without being influenced by some of the trade-offs that occur during the overlap FFT process.

Overlapped FFT is an extremely effective time-enhancer forthe Spectrogram enabling engineers to identify extremelyshort RF events that they could not see before.

Product Figure Number AWG File Name RTSA Data File RTSA State File

RSA3408A Figures 3, 4, 7, 8, IF_HOP_LIN_LAP.EQU Hop-1.iqt Hop-1.sta9, 10, 12, 13, & 17

Figures 18, 19 & 20 IF_Pulse250ns.EQU QuartUsPulse.iqt QuartUsPulseSpec.staQuartUsPulseFM.sta

(No Figure for this IF_Pulse250nsTrain.EQUPulse Train)

Figures 5 & 6 IF_HOP_RND_LAP.EQU RND-HOP.iqt RND-HOP-SA.staRND-HOP-SA-LAP.staRNS-HOP-FM.sta



Appendix A: Pulse Generation FilesThe Tektronix AWG710B Arbitrary Waveform Generator wasused to provide all of the signals in this document. Thereare Equation Files available for all of these waveforms.These Equation Files will need to be loaded onto the AWG,and then compiled into the necessary waveform andsequence files.

There are also data files which contain the signals digitizedby the RTSA. And, there are setup files that can preset theRTSA settings to show the measurements as they weredone for the screenshots presented here.

Table 3 lists these filenames a well as a few more that canhelp demonstrate the concepts presented here. The filescan be found on tektronix.com in the software and driverssection by searching for Overlap FFT Primer files.

Table 3. Demonstration file names.

Contact Tektronix:ASEAN / Australasia / Pakistan (65) 6356 3900

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United Kingdom & Eire +44 (0) 1344 392400

USA 1 (800) 426-2200

For other areas contact Tektronix, Inc. at: 1 (503) 627-7111

Updated 15 June 2005

For Further InformationTektronix maintains a comprehensive, constantly expanding collection ofapplication notes, technical briefs and other resources to help engineersworking on the cutting edge of technology. Please visit www.tektronix.com

Copyright © 2005, Tektronix, Inc. All rights reserved. Tektronix products are covered by U.S. and foreignpatents, issued and pending. Information in this publication supersedes that in all previously published material. Specification and price change privileges reserved. TEKTRONIX and TEK areregistered trademarks of Tektronix, Inc. All other trade names referenced are the service marks,trademarks or registered trademarks of their respective companies. 06/05 EA/WOW 37W-18839-0

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