Understanding. FFT Overlap Processing. A Tektronix Real-Time Spectrum Analyzer Primer

Understanding FFT Overlap Processing A Tektronix Real-Time Spectrum Analyzer

Contents Introduction....................................................................................3 The Need for Seeing Faster Time-Varying Signals.....................................................3 Expand Your View...............................................................................3 Figure 1. Similar to Zoom...................................................................3 How it Works...................................................................................4 Overlapping Many FFTs........................................................................4 Figure 2. Overlap FFT Processing..............................................................4 Some Comparisons.............................................................................5 Figure 3. Non-overlapped Spectrogram.........................................................5 The Overlapped FFT.............................................................................5 Figure 4. The Overlapped FFT Spectrogram.....................................................5 A Pseudo-Random Modulated Pulse..............................................................6 Figure 5. Overlapped FFT Spectrogram.........................................................6 Figure 6. Frequency vs. Time - Pseudo-Random Hopper...........................................6 Stretching Time.................................................................................7 Figure 7. Time Overlap.....................................................................7 Figure 8. Seeing Time-Varying RF.............................................................7 Spectrogram Risetime...........................................................................8 The Tektronix RTSA FFT Overlap.................................................................8 FFT Window Effects on Time Resolution...........................................................8 The Effect of Span and Sample Rate..............................................................8 Measuring the Spectrogram Risetime..............................................................9 Table 1. Spectrogram Effective Risetime........................................................9 Effects of Using No FFT Window.................................................................9 Figure 9. Rectangular FFT Window............................................................9 Figure 10. Blackman-Harris 4B Window - the Default..............................................9 Figure 11. Moiré Patterns in Spectogram......................................................10 The New Artifacts...........................................................................10 Measuring Time Events.........................................................................10 Figure 12. Measuring Time Events............................................................10 Time Correlated Multi-domain Analysis............................................................10 Amplitude Effects..............................................................................11 Short Pulses................................................................................11 Figure 13. The Lower Frequencies are Shown Lower in Amplitude Here...............................11 Table 2. Amplitude Reduction of Short Pulses...................................................11 Figure 14. Overlapped FFT Frames and a Single Pulse............................................12 Figure 15. FFT Frames with Window Function Applied............................................12 The Blackman-Harris Window..................................................................13 Figure 16. BH-4B Window Plot..............................................................13 Figure 17. One Marker in Time Displaying Multiple Frequency Hops..................................13 Calculating the Amplitude Reduction.............................................................13 Other Windows.............................................................................14 Super-short Pulses...........................................................................14 Figure 18. 250 ns Pulse Easily Measurable in the Time View........................................14 Figure 19. Tiny Pulse is Not Visible in Spectrum.................................................14 Figure 20. The 250 ns Pulse is Now Visible....................................................15 What Overlap Can Not Do.......................................................................16 Conclusion:...................................................................................17 Appendix A. Pulse Generation Files..............................................................18 Table 3. Demonstration File names...........................................................18 2 www.tektronix.com/rsa

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. 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 Figure 1. Similar to zoom. The Need for Seeing Faster Time-Varying Signals As faster time-varying frequency signals are becoming more widespread, Tektronix has responded to 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 than ever before. Overlapped FFT works somewhat like a Zoom for the Spectrogram. It does this by effectively stretching the time scale. While it does overlap small time events, the greater visibility provided in the Spectrogram enables much greater visibility of frequency changes with time. The lower Spectrogram in Figure 1 gives visibility to the transient time behavior. In this case, two separate frequency steps that appear as only one in the upper Spectrogram. The entire lower Spectrogram is contained within only 5 frames in the upper Spectrogram. www.tektronix.com/rsa 3

Signal Captured in the Time Domain Acquired Signal Data Transformed into FFT Frames, No Overlap Processing Acquired Signal, Post-Processed with Overlap FFTs Overlap Interval Samples FFTs Overlap Samples Figure 2. Overlap FFT Processing. How it Works Overlapping many FFTs Previous generation RTSAs processed the data from the A/D converter mostly in a straight sequential manner. The first 1024 bytes went into the first FFT (first frame), the second 1024 bytes into the second FFT (second frame), etc. This new capability allows the user to select the number of data points of overlap from 0 to 1023. This means that at 0 overlap we use the sequential way with each FFT adjacent to the previous one. The overlap number specifies how many of the last data points in any FFT frame also become the beginning points included in the next frame. When 1023 overlap is selected, the second FFT frame will have only one new data point at the end, while only one point from the previous FFT will be dropped off the beginning. 1023 data points are shared between these adjacent frames. Figure 2 illustrates the difference between FFT processing with 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, middle image, is using a 256 sample FFT Interval with 768 samples overlapped on each frame. Each frame starts 5 microseconds after the previous one and shows a 15 MHz span. Narrower spans would result in each spectrum taking up more time, while simultaneously seeing frequency components that are closer to each other. With this technique, a very short spectral event (particularly one that does not even last as long as one FFT frame) can be seen, even if at reduced amplitude, in many FFT spectra that are displayed adjacent to each other. The advantage this provides is visibility of the very-short time variations within a signal. The disadvantage is that since the FFTs are overlapped in time, the various frequency events will also appear in the Spectrogram display to be overlapped by the same amount. This effect will prevent relative timing measurements between spectral events from enjoying the same increase in resolution as does the spectral visibility. 4 www.tektronix.com/rsa

Figure 3. Non-overlapped Spectrogram. Figure 4. The overlapped FFT Spectrogram. Some Comparisons In Figure 3, we see a Spectrogram that includes a radar pulse. This pulse is frequency-hopped with five steps across 32 MHz. Including all five hops, it is 8 microseconds long. Using a 36 MHz span, this is about 1/3 of one FFT spectrum frame of 20 microseconds. And, each hop lasts only about 1/12 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. For triggering on very short pulses, the power trigger is preferred. The Overlapped FFT In Figure 4, the Tektronix RTSA with 36 MHz bandwidth shows this same hopped pulse using overlap FFT processing where the FFT frame-to-frame time interval is reduced from 1024 (no overlap) to 16 samples with 1008 sample overlap. The result is dramatic. Even though the pulse is still shorter than one frame, the individual frames are sequentially positioned 320 nanoseconds apart. This clearly shows the steps that are increasing in frequency throughout the pulse, and the approximate time of each step. The previously mentioned pulse was generated by a Tektronix AWG710B Arbitrary Waveform Generator. The files for this pulse and all the others used in this document are listed in Appendix A. www.tektronix.com/rsa 5

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 hop times, and the same frequency spacing as the previous pulse. But instead of a linearly increasing frequency with each hop, the hops are made randomly throughout the bandwidth. In Figure 6, we see that the time-overlap effect of the Overlapped FFTs makes it important to use the Frequency vs. Time display to measure timing on the hops. The figure shows the benefits of this display as there is an apparent reversal of time in Figure 5 and time is shown sequentially. 6 www.tektronix.com/rsa

Figure 7. Pulses app to be overlapped in time. Figure 8. Seeing time-varying RF. Stretching Time The effect that overlapping really brings is not really a zoom, but is much more a stretching of time. This gives much improved visibility of time-variant phenomena, but the events 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 overlap of the FFT frames, the individual steps appear to be overlapped in time. This overlap is not real, but is due to the overlap of the FFT frames. Figure 8 shows the marker placed at the end of the same middle hop. These pictures clearly show the frequency and time-varying nature of this pulse, which is entirely within the frame time of one FFT. This visibility would not be possible without overlapping FFT frames. The effect of stretching time has made all of the hops of this pulse appear to be mostly concurrent in time, while we know that these hops are all separate in time. One of the biggest contributors to this effect is the fact that these particular events are all shorter than one FFT frame. The FFT process is not an infinitely short slice of time. It requires processing multiple cycles of the input frequencies in order to separate them from each other. Therefore, if there are several different frequencies that are all within this frame, 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 within the one FFT frame. www.tektronix.com/rsa 7

Spectrogram Risetime Another effect that limits the ability to actually increase the time-resolution in Spectrogram is the apparent risetime of the Spectrogram when using overlapped FFTs. This also increases the time-overlapping of the separate hop segments of this pulse display. The Tektronix RTSA FFT Overlap When using non-overlapping FFTs, each spectrum is produced from 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 between two sequential frames, it would contribute no energy to the first frame, and fully contribute to the next. This implies that when using the Spectrogram, time events can be measured with a resolution of 20 microseconds. One might be tempted to extrapolate that if the fully overlapped FFTs were used (such that the time between the start of one spectrum and the start of the next one is 20 nanoseconds), that the time resolution would be 20 nanoseconds. But it is not. Some understanding of the process may help one be aware of the potential to misinterpret the results. FFT Window Effects on Time Resolution The A/D converter is continuously digitizing at a 102.4 MHz rate which provides both I and Q samples at a 51.2 MHz effective rate, filling the memory record. Each FFT will be provided with 1024 contiguous samples from somewhere within 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 overlapped FFTs. The first FFT that we will look at is the last one before a short burst of RF. This FFT has no power contribution from the RF burst, since the burst has not yet started. The second FFT is overlapped by 1023, and therefore starts one sample point later. It contains only one sample point of power from the burst. Each subsequent frame will contain one more additional sample of the burst until finally we will have one frame that contains the entire burst (if the burst is shorter than a frame), or until we have a frame that is entirely filled with the burst (if the burst is equal to or longer than the 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, at most, contain 1/1024 of the possible full power of the burst. This is in fact explained by Parseval s Theorem, which states that an FFT will produce a spectrum that displays the power level of a coherent signal that increases in direct proportion as the square of the number of samples of that signal which are 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. This reduces the amount of contribution that the samples at the ends of the Frame are allowed to make to the spectrum (reduces to essentially zero at the very end samples). The purpose 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 spectrum that contains the one sample of the burst will essentially have no spectrum of the signal of interest. The next spectrum will have very little more. It is not until we get a spectrum that is nearly one-quarter full of the burst that we will have significant amplitude of the burst displayed. This slows the effective risetime of the Spectrogram, and its resultant time resolution. The Effect of Span and Sample Rate Spectrogram risetime is also dependent on the selected span (which determines the actual effective sample rate used and therefore the noise bandwidth). This risetime is not dependent on the amount of FFT overlap. It is dependent solely on the selected span and its resultant effective sample rate. Note that there is an equivalent falltime and time smearing that occurs at the end of a pulse or other frequency event. Both ends of a pulse have the same effect applied to them. 8 www.tektronix.com/rsa

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 Spectrogram risetime measurements. 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 Table 1. Sprectrogram effective risetime. Note: These measurements were done using the 10% and 90% voltage points on an RF pulse. This risetime limit is limiting the time resolution available when attempting to measure the timing of spectral events using the Overlapped FFT Spectrogram. Effects of using no FFT window. Since the FFT window is such a large part of the risetime limitation in the Spectrogram, we should examine what will happen 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 the Spectrogram. But we also know that we will now get the effects of the abrupt ends of the frame. Figure 9 shows the result. The start and end of the individual hops are still overlapped in 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 window again. Using the rectangular window the Effective Risetime has indeed gotten better. For the 36 MHz span with the Blackman-Harris 4B (BH-4B) (Figure 10) the risetime was 7.2 microseconds. The risetime we have with the filter removed is 1.27 microseconds! Quite an improvement. www.tektronix.com/rsa 9

Figure 12. Measuring time events. Measuring Time Events Figure 11. Moiré patterns in Spectogram. The new artifacts When using the Rectangular window, the time-varying hops are more visible, but the artifacts from the edges of the hops are now quite objectionable. They will cover up any small signals that we might need to see. In fact, the end effects of just one of the hop segments show up as significant Moiré patterns in the Spectrogram as shown in Figure 11. The end effects are formed due to the pulse effectively starting only when the FFT frame includes some samples of it, and ending when the pulse itself ends. Without the Window filter, this means that as you move through the Overlapped 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 was seen by the prior FFT, the sinex/x pulse spectrum changes its periodicity and the pattern is formed in the Spectrogram. So what do we do if we want to measure time? The Tektronix 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. For this frequency-hopped pulse, we choose the Frequency Demodulation (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, the markers will also accurately measure the 8 MHz frequency step size. On the other hand, we see that without the filter, all of the frequency components of this hopped pulse are timestretched into a portion of time at least as long as one FFT frame without any one frequency step being reduced any more than any other. They are all equally reduced by the shortness of the pulses. And, the Overlapped FFT allows us to select an FFT window exactly in this time position. The upper spectrum view in Figure 9 shows this, with all of the hop components shown at equal amplitudes instead of being tilted. 10 www.tektronix.com/rsa

Amplitude reduction seen for short pulses in 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 Figure 13. The lower frequencies are shown lower in amplitude here. Table 2. Amplitude reduction of short pulses. Amplitude Effects Short Pulses As previously mentioned, pulses or other RF events that are shorter than one FFT frame will be converted to spectrum information at a reduced amplitude that is proportional to the amount of the FFT frame that such signals occupy (Parseval s Theorem). Look at Figure 13. The spectrum that is produced by the one FFT that contains the pulse is clearly varying in amplitude. The upper-most frequency shows as about -15 dbm, the next lower one appears to be -21 dbm, the next two are -31 and -45 dbm, and the last one is not even visible below the noise. Yet we know that all of the hop segments are the same amplitude. The overall amplitude reduction is due to the individual segments being shorter than one FFT frame. The amplitude difference between the separate segments is due to their position within the FFT window filter (position in the frame). Since the pulse was completely asynchronous to the FFT frame, the one we measured happened by chance to be located near the beginning of the frame. Therefore the first hop (the lowest frequency one) was severely reduced due to the window function applied to the FFT. This reduction becomes smaller as the hops occur later in the frame, and therefore closer to the middle of the window function. This table has correction values for several short pulse lengths. For this table, we are using a Real-Time bandwidth of 36 MHz. This bandwidth uses an FFT frame length of 20 microseconds. Pulse lengths that are equal to or longer than the frame time will be measured correctly for amplitude. This table lists the errors that will cause reduced amplitude to be reported for short pulses. The calculations used for this table assume that the pulse is centered in the frame and that the window used is either none, or BH-4B, as noted in the column heading. Another assumption is that there is no scalloping error due to the incoming signal being a different center frequency than the analyzer is tuned to. Scalloping error occurs when a signal is not exactly in the center of an FFT and can cause a few tenths of a db additional error. These errors can 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. www.tektronix.com/rsa 11

RF Pulse RF Pulse First FFT Frame Second FFT Frame No Window Function applied FFT Frames have the Window Function applied Third FFT Frame Fourth FFT Frame Later FFT Frame Figure 14. Overlapped FFT frames and a single pulse. Figure 15. FFT frames with window function applied. In Figure 14, we see a pulse of RF at the top. Beneath that we have drawn four lines that represent four sequential overlapped FFT frame times. Each of these frames is further along, and contains more of the RF pulse. It can be seen that the power contained in each frame is more than the frame before. Then at the bottom is a frame (taken considerably later in time) which contains the entire pulse in its middle section. This frame will show the amplitude of the pulse, 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 from the pulse being at one edge or other of the filter, while the amplitude of a pulse in the middle of a frame will be only slightly reduced. The correct amplitude will be measured for an RF signal which is equal to or longer than one frame, and is continuous throughout the frame. The amplitude reported by the FFT process is normalized to the response of the filter used for the window function. Next we will look at the specific filter used here. 12 www.tektronix.com/rsa

Figure 16. BH-4B Window plot. 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 samples without this filter, there would be a number of spectrum artifacts generated due to the sudden start of the spectrum as well as the sudden stop. The solution is to use this filter (or a similar filter) to reduce the response from the ends of the frame to zero. Then, to gradually increase the contribution of 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 between the filtered frames and the unfiltered frames for the table of pulse width versus displayed amplitude (Table 2). Calculating the Amplitude Reduction Figure 17 shows that if we use overlapped FFTs, and we position the marker on or about the middle of the stretched display, that we will see a spectrum that is the result of an FFT where the pulse is centered in the frame. Here we see that all of the frequency hops are contained within the single frame, and are centered in the window-function applied to the frame. They all read between -13 dbm and -16 dbm. Since each hop is 1.6 microseconds long, and the full frame is 20 microseconds long, we expect to see the power recorded in the Spectrogram to be reduced by Parseval s Theorem. The calculation is 20Log(1.6/20) = -21.9 db. We must also account for the correction that the power readout for the 20 microsecond frame already has due to the FFT window 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 narrow pulse amplitude. The center pulse reads -10.071 dbm and the correction is to make up for a reduction of 13.66 db. The correct amplitude for this pulse is +3.589 dbm. When this pulse was stretched to be longer than an FFT frame, it measured +3.51 dbm. This is a very close agreement between the practical measurement and the theoretical calculation. Calculating amplitude correction for pulses that are not centered in the window is much more difficult, and requires using the BH-4B curve separately for each side of the filter. www.tektronix.com/rsa 13

Measurement of Pulse Width Trigger Point Figure 18. 250 ns pulse easily measurable in the Time view. Figure 19. 250 ns pulse is not visible in Spectrum. Other Windows There are several different windows available in the Tektronix RTSA products. This document has explained the issues with the BH-4B and the Rectangular. Other windows will have greater or lesser time and amplitude effects than the default one. If you are using one of these others, you will need to re-calculate the amplitude and apparent Spectrogram risetime effects. The selection of the best window function for your particular signal is a subject beyond this paper. Super-Short Pulses Pulses that are significantly shorter than one FFT frame will not be visible on either the Spectrogram, or in a Spectrum View. Such pulses can only be triggered by the power trigger, and the FFT frame will be triggered and started exactly coincident with the pulse. This positions the pulse at the start of the FFT frame, and consequently at one end of the window filter. The Power vs. Time display is used to measure short pulses. The pulse measurement suite is specified to characterize pulses as short as 400 nanoseconds. For pulses shorter than 400 ns, time correlated multi-domain measurements work well. The Power trigger works on extremely short pulses. It simply looks for any digitized samples that are above the selected threshold. In Figure 18 we see that markers on the 250 nanosecond pulse report a measurement of 254 nanosecond width. The pulse amplitude is already very low due to its narrow width. But since the pulse is at one edge of the frame, the window function will additionally reduce the amplitude below the noise. See Figure 19. This pulse is 250 nanoseconds long, only a little more than one percent of one frame. If we had not already seen it in the time domain, then the only way we would suspect that it is here is the fact that the analyzer triggered, and the Spectrogram shows the trigger point half-way up its view, without any signal apparent there. This is a case where Overlapping FFTs make the difference between no spectral visibility at all of this pulse, and great visibility. Compare Figure 19 with Figure 20, where we now have set the overlap to start each frame 16 samples after its neighbor. 14 www.tektronix.com/rsa

There are now 64 FFT frames that include this pulse. Of these, there are five that have the pulse so close to their center that they provide measurements of the amplitude that are within 0.2 db of each other. The key to setting the overlap is to have enough to find the smallest pulse, or to show time-variant phenomena, but not to have more than necessary. Excess overlap causes excess time-smear of the spectral events. Trigger Point Figure 20. The 250 ns Pulse is now visible. In Figure 20, we can now clearly see both the Spectrogram and Spectrum when we can select a spectrum that has the pulse in the middle of the window. Only FFT Overlap can do this. Note the time delay visible between the Trigger point (top of the white bar on the left) and the beginning of the visible pulse. This is because the trigger happens at the first (nonoverlapped) frame that contains the signal. This frame has no appearance of the signal, since the Window filter reduces it to zero. Then, as the overlap progresses, the frames have more and more of the signal until there is a visible amount. It is the previously mentioned risetime that causes this apparent delay. www.tektronix.com/rsa 15

What Overlap Can Not Do Overlap of FFT processing only works in the Real-Time Spectrum Analysis mode where it can digitize a continuous record 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 different RF converter settings and therefore overlapping FFT capability cannot work. In our example, the Real-Time bandwidth is 36 MHz. Overlap can not provide measurement resolution as much as it provides visual improvement. There is still an Effective Risetime limitation even if we remove the Windowing Filter, and there is still the time-stretching effect. Both of these limitations can be overcome by using the time domain displays in the RTSA. 16 www.tektronix.com/rsa

Note Conclusion Overlapped FFT provides a huge (~2000x) increase in visibility of short time varying RF phenomena. It can show you multiple time-varying events that are shorter in time than one standard (non overlapped) FFT frame, enabling you to see extremely short pulses that would not be visible on previous generation RTSAs. Overlapped FFT is just part of the powerful analysis capabilities that the Tektronix RTSA products offer and to get the most complete and accurate results, overlapped FFT must be used in conjunction with the other in-depth analysis modes. For extremely short time RF phenomena, the Time or Demod modes make accurate time and power 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 for the Spectrogram enabling engineers to identify extremely short RF events that they could not see before. www.tektronix.com/rtsa 17

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.sta 9, 10, 12, 13, & 17 Figures 18, 19 & 20 IF_Pulse250ns.EQU QuartUsPulse.iqt QuartUsPulseSpec.sta QuartUsPulseFM.sta (No Figure for this Pulse Train) IF_Pulse250nsTrain.EQU Figures 5 & 6 IF_HOP_RND_LAP.EQU RND-HOP.iqt RND-HOP-SA.sta RND-HOP-SA-LAP.sta RNS-HOP-FM.sta Table 3. Demonstration file names. Appendix A: Pulse Generation Files The Tektronix AWG710B Arbitrary Waveform Generator was used to provide all of the signals in this document. There are 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 and sequence files. There are also data files which contain the signals digitized by the RTSA. And, there are setup files that can preset the RTSA settings to show the measurements as they were done for the screenshots presented here. Table 3 lists these filenames a well as a few more that can help demonstrate the concepts presented here. The files can be found on tektronix.com in the software and drivers section by searching for Overlap FFT files. 18 www.tektronix.com/rsa

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