Help me understand FFT Span on Digital Storage Oscilloscopes - Page 1
Forum section: electronics / testgear

author: TK
author_rank: Super Contributor
timestamp: August 21, 2018, 01:36:59 am
content: I have both the Agilent 54622D (100MHz, 200MSa/s) and the Keysight DSOX1102G (70MHz, 2GSa/s) and there is something I need to understand. On both scopes, the FFT function has a SPAN of 100GHz, and the center Frequency setting goes up to 25GHz. How are these values possible and is there any way these devices can show signals on these ranges?

author: rhb
author_rank: Super Contributor
timestamp: August 23, 2018, 02:48:21 am
content: The short answer is they can't.The maximum span is controlled by the sample rate. The resolution bandwidth is controlled by the length of the FFT. It is *highly* unlikely that either scope has the analog BW to go that high.Shannon showed that the sample rate controlled the *bandwidth* which does not have to start at DC. But woe unto the user if they feed a signal with wider bandwidth than Nyquist with the anti-alias filters turned off using what is called "equivalent time" sampling.In short, they can't because it's not possible. I find it very distressing that what's left of HP is doing this sort of thing. The MSOX3104T is supposed to have a 1 GHz BW, which it does. But at the price of a 7% overshoot on a step response and only a 450 pS rise time instead of the 350 pS rise time and 2-3% overshoot one would expect from a high quality analog scope.How do you tell when a computer salesman is lying?His lips move.Unfortunately, this malady now afflicts the A list T&M companies.

author: TK
author_rank: Super Contributor
timestamp: August 23, 2018, 10:29:27 am
content: Quote from: rhb on August 23, 2018, 02:48:21 amThat is what I understood, but the 54622D has a sample rate of 200MSa/s and the DSOX1102G 2GSa/s and in both scopes, the SPAN goes to 100GHz. There must be something else. If they want to lie, why not use a figure like 10GHz which is also a high figure?

author: Mechatrommer
author_rank: Super Contributor
timestamp: August 23, 2018, 11:28:35 am
content: the upper limit FFT bandwidth is using formula:Quoteso for 200MSps, the upper limit is/should be up to 100MHz. for 1GSps sampler, the sensical upper limit is 500MHz.otoh FFT bin count is:QuoteFFT resolution bandwidth however will be according to formula:Quotefor example 200MSps @ 4MSamples... upper limit = 100MHz, RBW = 50Hz ... or if 1GSps @ 1MSamples... full span (upper limit) = DC to 500MHz, RBW = 1KHz.

author: TurboTom
author_rank: Super Contributor
timestamp: August 23, 2018, 11:46:37 am
content: Quote from: TK on August 23, 2018, 10:29:27 amI guess the answer is rather simple: The manufacturer uses a single software module to implement the FFT functionality into all his scopes and simply forgot to include configurable limits to match the different scopes's physical capabilities...

author: Berni
author_rank: Super Contributor
timestamp: August 23, 2018, 11:52:05 am
content: Technically its possible to get a higher sample rate in equivalent time sampling mode, but you can't really get anything useful out of that.

author: rhb
author_rank: Super Contributor
timestamp: August 23, 2018, 12:22:39 pm
content: Quote from: Berni on August 23, 2018, 11:52:05 amNo! It is *not* a higher sample rate. Equivalent time sampling works by bandpass filtering the signal so it meets the Shannon-Nyquist criterion. If you are band limited to 950-1050 MHz you can examine a 1 GHz signal by sampling at 200 MSa/S.There are radar band sampling scopes which are *very* useful. Those are what was used designing radar systems for many years. But the analog section has to be able to handle the BW of the signal. DSOs generally don't have the BW in the AFE to make equivalent time sampling usable even if they have a menu option for it.In summary, equivalent time sampling uses an alias of the signal to examine the signal.

author: nctnico
author_rank: Super Contributor
timestamp: August 23, 2018, 12:29:28 pm
content: Quote from: rhb on August 23, 2018, 12:22:39 pmNo. Equivalent time sampling works by sampling a repetitive signal at random (known) intervals which are then used to reconstruct the original signal. The bandwidth is DC to whatever the analog front-end and ADC supports.

author: Unknown
author_rank: Unknown
timestamp: August 23, 2018, 12:52:11 pm
content: FFTs on scopes seem to be universally shitty and badly implemented to the point of being a useless gimmick on some. You can't get any accurate measurements without knowing the response of the front end so you can apply corrections for the roll off. It is possible to collect data up to approx sample rate / 2 but without corrections it's just a novelty rather that a useful tool. Obviously 100GHz is a joke!

author: Berni
author_rank: Super Contributor
timestamp: August 23, 2018, 01:08:07 pm
content: Yes, exactly why i said you can't get anything useful out of that. You can get enough points to do a FFT to 100GHz but the spectrum there will be nothing but noise.

author: rhb
author_rank: Super Contributor
timestamp: August 23, 2018, 01:37:32 pm
content: Quote from: nctnico on August 23, 2018, 12:29:28 pmThey did make random sampling scopes, but equivalent time sampling is most often regular sampling at less than 2 samples per cycle, perhaps as little as a sample every N cycles. Any signal which is BW limited is by definition repetitive. In fact, the FFT of *any and all* sampled signals are repetitive over a period equal to the length of the series. Except in the case of mathematical abstractions like a Dirac functional, *all* signals are BW limited, and hence repetitive.You really do have to have the BP filter to limit the BW, otherwise you will have a mixture of signals which does not accurately describe the signal of interest. Shannon proved this long ago. I've not played with a sampling scope, but I suspect that many presume the user has sense enough to BP filter the input externally. But woe unto any who attempt to defy Shannon using regular sampling. Truly random sampling is different in that there is no aliasing in that case.This thread is a very clever application of equivalent time sampling:https://www.eevblog.com/forum/testgear/how-to-quickly-determine-your-dso-timebase-accuracy/msg1748363/#msg1748363As for the FFTs on DSOs, it all depends. Instek has a very nice FFT based spectrum analyzer in their MDO-2000E line, but they won't allow using it on the GDS or MSO versions. The FFT function supplied on the latter two works, but is *really* clunky to use because the interface is so poor. But they will give a span of 500 MHz and an RBW of 50 Hz. But you're limited to an RBW of 500 Hz in zoom mode. Up to FW 1.32 you can break out of the demo app for the MDO and use the SA function app on the MSO if it's got the AWG installed. I mentioned it to them, so in 1.34 they made sure you couldn't do that anymore :-(

author: Mechatrommer
author_rank: Super Contributor
timestamp: August 23, 2018, 03:17:52 pm
content: Quote from: rhb on August 23, 2018, 01:37:32 pmi think you are confusing betweenequivalent time samplinganddown sampling. the purpose of ETS is to perceptually increase sampling rate from limited ADC. down sampling is the other way (it can be done easily by increasing time base of DSO hence reducing sampling rate). or in other word, you are confusing between trying to observe or predict or reconstruct original signal (higher BW) and trying to capture lower BW from unnecessary high BW components. or as in the thread in your link, is to observe jitter correlation between 2 clocks or alignment (from my quick reading) its some nice trick, sort of signal mixing/superposition/addition technique, but its not to reconstruct the original signal. original signal is 10MHz, and as the OP stated, the signal sampled from 2.5MSps is aliased. in other word, its garbage, not the 10MHz signal. we are not doing that here...

author: rhb
author_rank: Super Contributor
timestamp: August 23, 2018, 05:12:02 pm
content: Well, from a mathematical perspective the equations are *exactly* the same. If you want to make up new names that's your business. I don't see a lot of use in it. 10 MHz sampled at 1/8th of Nyquist is no different mathematically from 10 GHz sampled at 1/8th Nyquist except for 3 zeros tacked on the ends of the values. If that makes a difference to you it's your problem. I can't help you.

author: Mechatrommer
author_rank: Super Contributor
timestamp: August 23, 2018, 06:07:22 pm
content: making new name is something, but misguiding people by telling this name is that name, which is not what it is is something else. you were talking about BW limiting (antialias filter), in ETS, you dont need the filter otherwise you will lose any information of the higher order. here we dont want to see a pure spectral FFT garbage. which is what you will get from the link you provided.

author: rhb
author_rank: Super Contributor
timestamp: August 23, 2018, 07:14:43 pm
content: I'll let you work out what happens if you regularly sample a 10 MHz tone and a 120 MHz tone using ETS at 200 MSa/S. You've only got 100 MHz of BW. Read Shannon's paper if you want further explanation. To get a valid result you must filter out one of the two tones. If you don't you'll get two tones 70 MHz apart rather than the correct value of 110 MHz apart.

author: nctnico
author_rank: Super Contributor
timestamp: August 23, 2018, 07:26:53 pm
content: Quote from: rhb on August 23, 2018, 07:14:43 pmThe thing is that ETS uses (known) random sample intervals to avoid aliasing. The samplerate in that case is just a given to show how fast the sampling process is. ETS and downsampling are two completely different 'processes'. The whole point of ETS is to under sample a repetitive signal from DC to <bandwidth> without aliasing problems.

author: KE5FX
author_rank: Super Contributor
timestamp: August 23, 2018, 08:15:04 pm
content: There is no requirement for ETS to involve random sampling. Random sampling is one way toimplementETS, but you can also use a vernier technique where the timebase is ramped slowly to fill in the sample record. The old-school Tek 7000 plugins supported both.'Downsampling' usually refers to decimation of a conventionally-sampled signal; 'subsampling' is probably a better synonym for ETS.As for 100 GHz FFT scales, your guess is as good as mine. I have no clue why Agilent does that.Maybe Daniel B. will see the thread and make some inquiries, assuming the answer isn't already buried in the docs somewhere.

author: rhb
author_rank: Super Contributor
timestamp: August 23, 2018, 09:14:16 pm
content: Quote from: nctnico on August 23, 2018, 07:26:53 pmThis is true *if and only if* the sampling is uncorrelated with the signal. Not all ETS scopes use random sampling. Sampling at a multiple of the signal period plus a fraction that varies in a regular fashion is quite common. For those you cannot escape Shannon.

author: egonotto
author_rank: Super Contributor
timestamp: August 23, 2018, 09:50:16 pm
content: Quote from: rhb on August 23, 2018, 01:37:32 pmHello,I dont believe that.If f(t) := sin(at) + sin(bt) for all t element IR and if a/b is not rational ?Best regardsegonotto

author: KE5FX
author_rank: Super Contributor
timestamp: August 23, 2018, 10:37:00 pm
content: Quote from: egonotto on August 23, 2018, 09:50:16 pmIf a number can be expressed as a/b, it's rational by definition, isn't it?

author: egonotto
author_rank: Super Contributor
timestamp: August 23, 2018, 10:55:35 pm
content: Quote from: KE5FX on August 23, 2018, 10:37:00 pmHello,rational numbers are those number which have the form a/b where a and b are whole numbers. Whole numbers are (.. -3,-2,-1,0,1,2,3,..)Best regardsegonotto

author: rhb
author_rank: Super Contributor
timestamp: August 23, 2018, 11:20:05 pm
content: If the signal is BW limited, then at some point it repeats. It might be a very long time, but it repeats. It's like the beat note between two tones. The repetition period depends upon how many discrete frequencies are involved and what their harmonic relationships are. But the sum of two sine waves of different frequencies is another sine wave. This is basic trigonometry. So it has to repeat. The longest repetition period is for the sum of two sine waves which are sin( wt) & sin(w+epsilon)t) for epsilon very small.In the case of the discrete Fourier transform, it repeats with a period of the length of the sample.If you're a young engineer or physical scientist, one of the most useful things you can buy is Ronald Bracewell's "The Fourier Transform and its Applications". It has a graphical dictionary of transform pairs. If you know those few pages well you can do Fourier transforms of complex signals on a bar napkin. It's engineering oriented, so it's not a substitute for a proper mathematical treatment of the transform, e.g. "Operational Mathematics" by Churchill. The problem with Churchill is it's integral transforms in general, not just the Fourier transform. Despite having probably 3 ft of books on the Fourier transform, when I went and looked I did not see any volume that really does a comprehensive modern treatment at the graduate level in mathematics. Nothing to equal Watson on the Bessel function. The best I know of to point to is Mallat's "A Wavelet Tour of Signal Processing" , 3rd ed.My undergraduate degree was English lit for which the math requirement was algebra and trig which I had taught myself in grades 6-8. Thus when I started taking serious math as a grad student I had the great advantage of only being expected to take a 12 semester hour course load instead of the 15 the engineering undergrads suffered with. That proved to be a huge advantage. I had far more time to work problems than the other students in my classes. Of course, by the time you hit Integral Transforms and Tensor Analysis, everyone is a grad student.

author: egonotto
author_rank: Super Contributor
timestamp: August 24, 2018, 12:03:14 am
content: Quote from: rhb on August 23, 2018, 11:20:05 pmHello,noBest regardsegonotto

author: rhb
author_rank: Super Contributor
timestamp: August 24, 2018, 12:11:10 am
content: ROFL!Obviously you've never learned to play a musical instrument, nor learned trigonometry or physics.Good luck!

author: JS
author_rank: Frequent Contributor
timestamp: August 24, 2018, 01:18:06 am
content: Quote from: rhb on August 24, 2018, 12:11:10 amNo, you are wrong, check it and come back... Summing two sine waves with different frequency gives an amplitude modulated signal, with the modulation at the difference of the frequencies and the carrier at the average of the frequencies if they are both the same amplitude, if not it shifts towards the higher amplitude one. The modulation index will depend on the ratio of the amplitudes as well.sin(w1*t)+sin(w2*t)=sin(w1-w2*t)*sin((w1+w2)/2*t) and that's not a pure sinewave.JS

author: egonotto
author_rank: Super Contributor
timestamp: August 24, 2018, 01:44:00 am
content: Quote from: rhb on August 24, 2018, 12:11:10 amHello,that very well may be.But we have the addition formulasin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)andsin(a-b) = sin(a)*cos(b) - cos(a)*sin(b)combinedsin(a+b) + sin(a-b) = 2*sin(a)*cos(b) (1)Now definea := 0.5*(f1*t + f2*t)andb := 0.5*(f1*t - f2*t)(1) now give us:sin(f1*t) + sin(f2*t) = 2*sin(0.5*(f1*t + f2*t))*cos(0.5*(f1*t - f2*t))You see that if f1 is not equal f2 the cos -term does not vanish and the sum is not a sinus.Best regardsegonotto

author: rhb
author_rank: Super Contributor
timestamp: August 24, 2018, 01:47:04 am
content: Quote from: JS on August 24, 2018, 01:18:06 amThat's exactly correct. And that amplitude modulated signal is a periodic sine wave. It's called a beat note both in radio and in music. I'm sure the latter is the older usage. The motivation for the 12 tone scale was to minimize dissonance. Hence "The Well Tempered Clavier".My statement was that a BW limited signal is periodic. I thought it sufficiently obvious it did not require a rigorous proof. After a long series of "ex cathedra" pronunciations, I don't think a rigorous proof would have made any difference.

author: JS
author_rank: Frequent Contributor
timestamp: August 24, 2018, 02:17:09 am
content: Quote from: rhb on August 24, 2018, 01:47:04 amIf the ratio of the two frequencies isn't rational, meaning w1/w2=n/m so n and m are integers, the signal is NOT periodic. The carrier is, the modulation is, but the modulated signal is not as the modulation will always land in a different place of the carrier, phase relation between the two signals will always be changing.Let's say f1 is 1 rad/s and w2 is π rad/s (pi rad/s)a(t)=sin(w1*t)b(t)=sin(w2*t)So our modulated frequency would be x(t)=a(t)*b(t)Find find the period of x(t) t1, such as t1≠t0=0 so w1*t=2*p*π and w2*t=2*q*π with q and p integers different than zero.(1) 1*t1=2*p*π(2) π*t1=2*q*π(1)/(2) 1/π=p/qq/p=π (pi)q is integer, p is integer. So if your statement is true, that all limited bandwidth signals are periodic then π(pi) is a rational number.JS

author: nfmax
author_rank: Super Contributor
timestamp: August 24, 2018, 07:31:07 am
content: But a sine wave is uniquely definedfor all timeby just three numbers - frequency, amplitude, and phase. Therefore the number of samples per second needed to reconstruct it over a time interval tends to zero as the time interval increases. Therefore the bandwidth tends to zero (using Shannon 'backwards'). This remains true for any finite number of sine waves summed together, whether the frequencies are rational numbers or not, even though the sum is not periodic when an irrational frequency is present.

author: rhb
author_rank: Super Contributor
timestamp: August 24, 2018, 12:57:10 pm
content: JSThank you. Very neat proof by counter example. However, there are an infinite number of rational approximations to an irrational number. So I shall amend my assertion to "all BW limited signals are quasi-periodic". I'd gotten rather weary of this thread, but your proof made it worthwhile.I pay close attention to the fine print in mathematics. While it generally doesn't matter, I have been bitten. Most notably by the general failure of writers to note that the discrete Fourier transform is defined on the semi-closed interval [-pi,pi). That nicety cost me considerable distress once while implementing code to do arbitrary resampling by Fourier transform. I'd done it routinely for years, but encountered an edge case. I had long series with missing data and was resampling the segments and reassembling them. I ran into a phase error at the end of a test case which took me over a week to resolve.In any case, for regular sampling, Shannon-Nyquist applies and ETS for a signal consisting of 10 & 120 MHz tones sampled at 200 MSa/S, one must BP filter to preclude aliasing. The discussion wandered off into the weeds from there as a consequence of ex cathedra pronouncements without proof.

author: Mechatrommer
author_rank: Super Contributor
timestamp: August 24, 2018, 01:44:33 pm
content: Quote from: rhb on August 24, 2018, 12:11:10 amno. square wave is summation of several sine waves with different freq (and phases). square wave is not sine wave... period. if you BW limit the 10MHz square, fair enough you'll get 10MHz sine (loosing all the higher order sine harmonics), but when you have non repetitive 10MHz square, BW limit it, you'll get 10MHz sine, but still non repetitive. so i dont know what you are talking about when you sayQuote from: rhb on August 23, 2018, 01:37:32 pmif a 10MHz square (100ns period, 50ns HI pulse) is happening non-repetitively on non integral of period (say happening in 100.3 period later), no practical BP filter will be able to cope with that (the signal will still pass) unless you capture 100+ period of signal and do SW processing on it all at once.Quote from: rhb on August 23, 2018, 07:14:43 pmi guess you read it wrongly. i have a suspicion from the way you are describing things..1) you are confusing between theoretical (infinitesimal) signal processing that Shannon et al usually explaining and practical signal processing that we usually deal with. be aware that some filters Shannon was explaining are never exist in this world such as Sinc filter or probably BP filter that you are describing.2) your description is only applicable in some other different field such as spectrum or purity analysis. where you have only one tone of interest. here on real time processing is different story, we want to capture as much tone as possible but we have a practical limit which is sampling speed. now we want to deal with that (increasing sampling speed) not degrading the captured signal even further.Quote from: rhb on August 23, 2018, 11:20:05 pmany FT derivatives are originating from the Original Fourier Transform (DFT or FFT is a "crippled" version of the OFT). if you are deluded from the original FT purpose, then i think something went wrong.Quote from: rhb on August 23, 2018, 11:20:05 pmreading theories is good, but when it separate us until we dont know what is real and what is abstract, then its no good. theories developed to fulfill reality requirements, but if its to delude reality, its off the chart.Quote from: rhb on August 23, 2018, 09:14:16 pmETS does not sample at MULTIPLE PERIOD + some fractions. ETS samples from TRIGGER POINT + some fractions. for non-integral period pulses, ETS still works, but your Shannon theory collapsed. not that Shannon is incorrect in his theory, its just the theory is not applicable here. do not confuse infinitesimal time (mathematical or non-causal) with real time (reality or causal) sampling.

author: nctnico
author_rank: Super Contributor
timestamp: August 24, 2018, 02:16:46 pm
content: Quote from: rhb on August 24, 2018, 12:57:10 pmAgain: ETS isn't regular sampling because the sampling interval is varied. Your really are confusing naming conventions here.

author: rhb
author_rank: Super Contributor
timestamp: August 24, 2018, 05:05:16 pm
content: Assume a stable, periodic trigger point and samples at evenly spaced delays from the trigger point and compute the Fourier transform of the sampling function. Then compute the transform of a signal which exceeds the Shannon-Nyquist BW limit. Then do the convolution of the two.In the case of random delays from the trigger point, there is no aliasing. But if the delays are at regular intervals there is. There is a nice demonstration of this with a Keysight X3104 in the thread I linked about measuring timebase accuracy.

author: Mechatrommer
author_rank: Super Contributor
timestamp: August 24, 2018, 06:58:35 pm
content: Quote from: rhb on August 24, 2018, 05:05:16 pmand then we get what?Quote from: rhb on August 24, 2018, 05:05:16 pmso? the random decimation need to be turned off anyway if FFT is activated.

author: tomato
author_rank: Regular Contributor
timestamp: August 24, 2018, 08:51:55 pm
content: Quote from: rhb on August 23, 2018, 11:20:05 pmAlgebra and Trig got you out of a math class in graduate school? Tough curriculum...

author: JS
author_rank: Frequent Contributor
timestamp: August 25, 2018, 05:28:51 am
content: Quote from: rhb on August 24, 2018, 12:57:10 pmNo, all BW limited signal aren't "quasi-periodic" either, whatever that means, I won't go into all the math here, but take a SINC(t) signal, the bandwidth has a very well defined span, a box in fact, but the signal has very little to do with a periodic signal, as it dies over time but slow enough to have a limited BW. What IS actually true, and you might be messing with that, is that any BW limited signal is infinite in time and any finite time signal has an infinite BW. Other thing that is true, is that periodic signals have discrete spectrum (BW?), so they can be represented by a Fourier series, so all their components fall in a product of the fundamental frequency. But there are, as proven in my other post, discrete spectrum signals which aren't periodic.While we are at this, one thing I didn't mentioned, for a sampled signal to be periodic, the sampled signal doesn't need to be periodic, but in the case it is, the period of the signal and the sampling period needs to be rational as well. Also, for doing over sampling not every frequency can be oversampled by any lower sampling rate than needed, so you end with narrower bands as you go up in ratio (f/fs) where the signal can be for a given sampling rate.JS

author: rhb
author_rank: Super Contributor
timestamp: August 25, 2018, 12:44:01 pm
content: JSQuite so. I made an off the cuff comment based on the BW - duration relationship without thinking closely enough about the details. I certainly should have recognized the sinc(w) case as an obvious counter example. Particularly as I was generating minimum phase BW limited impulse responses for the thread on developing FOSS DSO FW.

author: bicycleguy
author_rank: Frequent Contributor
timestamp: December 11, 2018, 08:33:02 pm
content: Quote from: TK on August 21, 2018, 01:36:59 amThis is old but may help someone. Can' help with the math... however if your DSOX1102G works similarly to my DSOX2014A:In the FFT mode directly adjusting the SPAN and CENTER indicates bogus 100Ghz and 25GHz as you noted.However, if you press MORE FFT/AUTO SETUP then go back to the FFT display you will see the auto selected Span and Center for the selected Horizontal sweep rate. As you spin the Horizontal sweep rate the SPAN will auto adjust and the CENTER will auto 1/2 the SPAN and the Acquisition display will show the samples/sec and the FFT Resolution will display. Once you adjust either the SPAN or CENTER this auto thing will stop happening.With the DSOX2014A the max auto SPAN is 16GHz with corresponding CENTER 8GHz.Code:Horizontal SPAN CENTER FFT Resolution AcquisitionComment(sec/div) (Hz) (Hz) (Hz) (Sample/sec) Samples2.00E-09 16.00E+09 8.00E+09 15.60E+06 2.00E+09Looks messy, can’t test 405.00E-09 6.40E+09 3.20E+09 6.25E+06 2.00E+09Looks messy, can’t test 10010.00E-09 3.20E+09 1.60E+09 3.13E+06 2.00E+09Looks messy, can’t test 20020.00E-09 1.60E+09 800.00E+06 1.56E+06 2.00E+09Looks messy, can’t test 40050.00E-09 1.00E+09 500.00E+06 977.00E+03 2.00E+09 1,000100.00E-09 1.00E+09 500.00E+06 488.00E+03 2.00E+09 2,000200.00E-09 1.00E+09 500.00E+06 244.00E+03 2.00E+09 4,000500.00E-09 1.00E+09 500.00E+06 61.00E+03 2.00E+09 10,0001.00E-06 1.00E+09 500.00E+06 30.50E+03 2.00E+09 20,0002.00E-06 1.00E+09 500.00E+06 15.30E+03 2.00E+09 40,0005.00E-06 500.00E+06 250.00E+06 7.63E+03 2.00E+09 100,00010.00E-06 250.00E+06 125.00E+06 3.81E+03 2.00E+09 200,00020.00E-06 125.00E+06 62.50E+06 1.19E+03 2.00E+09 400,00050.00E-06 62.50E+06 31.30E+06 954.00E+00 1.00E+09 500,000100.00E-06 31.30E+06 15.70E+06 477.00E+00 500.00E+06 500,000200.00E-06 15.60E+06 7.80E+06 238.00E+00 250.00E+06 500,000500.00E-06 6.25E+06 3.13E+06 95.40E+00 100.00E+06 500,0001.00E-03 3.13E+06 1.57E+06 47.70E+00 50.00E+06 500,0002.00E-03 1.57E+06 780.00E+03 23.80E+00 25.00E+06 500,0005.00E-03 625.00E+03 313.00E+03 9.54E+00 10.00E+06 500,000