NATONAL RADO ASTRONOMY OBSERVATORY Green Bank, West Vrgna SPECTRAL PROCESSOR MEMO NO. 25 MEMORANDUM February 13, 1985 To: Spectral Processor Group From: R. Fsher Subj: Some Experments wth an nteger FFT A number of experments wth the computer smulaton of the spectral processor multpler and accumulator are descrbed n ths memo. These experments were performed to develop a feel for the characterstcs of the spectrometer and to try to uncover any peculartes that mght be corrected n the early desgn stages. f you have any comments on my nterpretatons of the results or f you have any suggestons for other experments please let me know. n all of the followng experments the nput levels are referred to the A/D resoluton of 1 level = 1/63rd of ts total range (6-bt word) from +31 to -32. Unless otherwse noted, the A/D s assumed to produce an output of zero wth an nput between ± 0.5 levels. Output power word szes. The A/D and FFT multplers are assumed to use 16-bt, fractonal two's complement arthmetc wth the ten least sgnfcant bts from the A/D set to zero. The fractonal arthmetc and one-bt shfts assocated wth every add operaton prevents word overflow anywhere n the FFT from the strongest CW sgnal that can be accommodated by the A/D converter. A consequence of ths s that the average level of whte nose n the FFT s half of a bt (fr) lower n each successve FFT stage, and because of squarng to obtan the power spectrum each addtonal FFT stage reduces the average nose level n the accumulators by one bt. Wth an nput nose rms of 1 level the 1024-channel FFT produced an output power level of about 576 (assumng the bnary decmal s at the least sgnfcant end) or an average word wdth of about 9 bts. (The maxmum possble output word wdth
2 s 30 bts.) Droppng 3 FFT stages to produce 128 channels per nput wll produce an average power word wdth of 12 bts. The dstrbuton of power products for one FFT output n the 1024 channels s shown n Table 1. TABLE 1 Sngle FFT output power statstcs, 1024 channels, nput nose rms = 1 level. Power Level Number of Cumulatve Number Range Channels of Channels 0-31 60 1024 32-63 50 964 64-127 113 914 128-255 169 801 256-511 234 632 512-1023 237 398 1024-2047 126 161 2048-4095 34 35 4096-8191 1 1 Effects of roundng power words. To mnmze accumulator memory space the output power word sze must be reduced by droppng some of the least sgnfcant bts. Even f these words are rounded nstead of truncated some bas s ntroduced by word sze reducton because of the asymmetrc dstrbuton of values around the mean as shown n Table 1. Ths dstrbuton should be ndependent of nput ampltude and number of channels. Table 2 shows the results of the average of 256, 64 channel FFT operatons on the same gaussan random nose wth dfferent numbers of bts rounded n the power words. The average word wdths under these condtons was about 13 bts. Two effects of mportance to the spectrometer are evdent n Table 2.
TABLE 2 Effects of roundng output power word. Measured Rms/Avg Number of Average Level rms of Degra- Bts Dropped Level Reducton Spectrum daton 0 8824.4 ---- 470.4---- 10 (- 1024) 8821.6 0.03% 469.3 1.00 11 (- 2048) 8811.5 0.15% 470.4 1.00 12 (+ 4096) 8762.8 0.70% 486.8 1.04 13 (- 8192) 8546.0 3.15% 507.3 1.11 14 (+16384) 7670.0 13.08% 569.2 1.39 15 (+32768) 5274.0 40.23% 705.9 2.51 The frst s the obvous one of ncreased nose due to loss of low order bts. The second s the reducton n average level due to roundng an asymmetrcally dstrbuted set of numbers. Ths wll cause a dstorton of a curved nose spectrum because each channel wll have a dfferent average level, and roundng wll reduce the level of channels wth low output more than those wth hgh output. Ths dstorton s predctable and could be corrected usng only the nformaton n the spectrum tself assumng that the nose power s constant durng the ntegraton. Roundng of the power word should be done at least two bts below the average. f we assume that the lowest nput nose rms = 1 A/D level under normal operaton, a 128 channel spectrometer could drop 10 bts and a 1024 channel confguraton could drop 7 bts. Ths would allow some operaton of about half of ths nput level (-6 db) f reduced senstvty and spectrum dstorton are less mportant than dynamc range, and operaton at hgher nput levels would provde a reasonable margn above the roundng lmt to produce a stable spectral shape. For example, f the nput rms = 5 A/D levels the roundng would be about 6 1/2 bts below the average, and the ampltude dstorton would be about 0.001% or about 0.6 mk for a 50 K system temperature whch would be reached n 50 hours of ntegraton on a 1024 channel, 40 MHz spectrum. n memo number 24 suggested a 24-bt accumulator word wdth, but the assumpton of the lowest order bt just coverng the average power level wth 1 level nput rms has now been changed by 2 bts. The next ncrement n word wdth s 28 bts so the extra two bts could be used to cover most of the added range needed by the cases of fewer FFT stages. now favor a 28-bt accumulator word wdth.
4 Senstvty loss due to quantzaton of nput waveform. The loss n senstvty due to quantzaton of the nput waveform has been well documented by others, but out of curosty ran a few tests of ths wth the spectrometer smulaton and the results are shown n Table 3. The rato of output nose power to the standard devaton of ths power remans constant wth varaton of the nput quantzaton, but the relatve ampltude of a CW sgnal decreases wth ncreasng quantzaton so the sgnal to nose rato of the CW sgnal was measured. Two cases are shown n Table 3. One uses the normal ± 0.5 level A/D quantzaton and the other changes the A/D output from zero to one at zero volts nput (zero-slcng). The sne wave peak ampltude was approxmately equal to the nose rms. At low nput levels the zero-slcng A/D has an advantage over the ± 0.5 level case because the state changes stll occur at very low sgnal levels n the former. At normal operatng levels the dfference between the two cases s very small. The ± 0.5 slcng case has the advantage of a much smaller DC channel ampltude that would be easer to handle n the accumulator. Note that the relatve sgnal to nose ratos n Table 3 are themselves subject to some random error even though the nput sgnal s dentcal n all cases. TABLE 3 CW sgnal-to-nose ratos vs. quantzaton level. nput nose Relatve CW sgnal to nose rms n + 0.5 level Zero A/D level slcng slcng 4... 1.0 (ref) 1.0 (ref) 1... 0.95 0.93 0.5... 0.68 0.76 0.4... 0.62 0.68 0.3... 0.46 0.58 0.2... 0.24 0.57 CW harmoncs due to nput quantzaton. A symmetrc A/D converter wll produce odd harmoncs when samplng a pure CW sgnal due to quantzaton errors, and these harmoncs or ther alases wll appear n the spectrum of the A/D output. The addton of nose to the CW sgnal wll tend to destroy the coherence of these harmoncs, and to determne the harmonc ampltude reducton that can be expected from nose
5 a number of accumulated spectra were computed wth dfferent CW sgnal and nose levels. The results are shown n Table 4. TABLE 4 CW harmoncs as a functon of quantzaton and added nose. Relatve harmonc strength Harmonc CW rms n A/D levels Nose rms n A/D levels Number (no nose) (CW rms = 0.7 0.7 1.1 3.0 0.2 0.5 db db db db db Fundamental 0 0 0 0 0 3-16 -21-24 -27 <-40 5-25 -17-23 <-40 <-40 7-18 -20-28 -24 <-40 9-18 -27 <-40 11-24 -40 <-40 13-47 <-40 <-40 15-26 <-40 <-40 17-24 <-40 <-40 The measurement of harmonc strengths weaker than -40 db would have taken too much computer tme, but t seems safe to assume that wth nose rms values greater than 1 A/D level the harmoncs wll be much weaker than -50 db. Nose ampltude dstorton n the presence of a strong CW sgnal. Wth a one bt A/D converter a CW sgnal wth a strength comparable to or stronger than the nose power wll severely dstort the sample statstcs and reduce the measured spectral nose power n proporton to the CW strength. A smlar but smaller effect must happen wth a multbt A/D so a few relatvely extreme tests were performed wth nose n the presence of a CW sgnal wth the spectrometer smulaton. The results are shown n Table 5. Wth nose rms values greater than 1 A/D level there appears to be very lttle effect on the measured nose power from strong CW sgnals.
TABLE 5 Nose spectrum dstorton from CW sgnals. Nose rms CW rms Nose power n A/D levels n A/D levels suppresson Zero slcng 0.3 4.0 15% 1.0 4.0 <0.02% +± 0.5 slcng 0.3 4.O 37% 1.0 4.0 <0.01% Overflow n the FFT arthmetc. sad n the frst secton that the FFT arthmetc s such that overflows are avoded. Ths s true for strong coherent sgnals but not strctly true for strong nose. n the thrd and followng FFT stages there are nstances where the sne and cosne components of the rotaton coeffcents are both near 0.7, and f both real and magnary components of the B nputs of such a butterfly are near one the sum of the sne and cosne products wll be greater than one; hence, an overflow wll occur. The probablty of overflow depends on the number of large random nose data values. Table 6 shows the fracton of overflows per butterfly operaton n the thrd FFT stage as a functon of nput nose level. Because of the decrease n average nose levels n later stages no overflows were seen n any stage but the thrd. The nose levels shown n Table 6 are much hgher than would be used n normal operaton, but ths smulaton could not process enough data to measure very small overflow probabltes so we have to rely on an extrapolaton of the hgh level results. The thrd column n Table 6 gves the fracton of data ponts whch are equal to the maxmum possble value from the A/D converter. Ths fracton should be a good predctor of the overflow probablty, and, n fact, the number of overflows measured s proportonal to the fourth power of the fracton of maxmum A/D values as would be expected from the fact that four large nput data ponts n the rght places are requred to cause an overflow.
TABLE 6 Overflow probablty vs. nput nose level. Nose rms No. of overflows per Fracton of n A/D levels butterfly operaton maxmum A/D levels 3000 4.3 (±.7) x 10_ 0.99 600 3.1 (±.6) x 10-3 0.96 300 2.1 (±.5) x 10- _3 0.91 200 1.7 (±.5) x 10- _3 0.87 100 1.0 (±.3) X 103 _3 0.75 50 0.20 (±.05) x 10 0.52 25 <0.02 x 10-3 0.20 Usng the fourth-power law derved from the data n Table 6 the probablty of havng an overflow n one second wth the spectrometer acceptng a 40 MHz bandwdth and a nose rms of 10 A/D levels s 10-6. Wth a nose rms of 15 levels the probablty s 0.13 under the same condtons. Even the 10 level rms s hgher than we would normally put nto the spectrometer so the overflow probablty s -acceptably small wthout reducng the nose level n the thrd stage. To see what effect addng a CW sgnal to the nput nose would have on the overflow statstcs a 40-level peak-to-peak sne wave was added to the 50-level rms nose case. The result was a 50% decrease n the overflow probablty because the nose statstcs were based aganst havng all of the requred hgh level nput condtons at the susceptble butterfles to cause an overflow. Spectral dstorton due to A/D quantzaton. The effects of severe quantzaton n one and two bt autocorrelaton spectrometers have been studed n consderable detal by others. Snce the autocorrelaton and Fourer transform spectrometers are equvalent n ths respect we can use the work connected to autocorrelators as a gude n predctng the errors that can be expected n an FFT devce. My thanks go to John Granlund for some very helpful dscussons n ths area. Quantzaton error can be looked at n two ways. t reduces the relatve ntensty of low level correlatons wth respect to perfect correlaton at zero delay n the autocorrelaton functon of an RF sgnal. Ths can be seen n Fgure 1 whch s a plot of correlaton values for quantzed vs. unquantzed sgnals. Each curve represents a dfferent degree of quantzaton. Vewed n the spectral doman the quantzaton nonlnearty generates harmoncs and mxng products of all frequency components n the nput spectrum. These products or ther alases are
8 lad over the true spectrum. f the unquantzed sgnal s nose wth a reasonably flat spectrum wthn the sampled passband the error spectrum wll be very nearly whte and wll appear mostly as a DC offset n the true spectrum. n the autocorrelaton functon ths means that the slope at low correlatons n Fgure 1 s not unty and, hence, does not pass through the zero delay correlaton pont. The degree to whch the error spectrum s not perfectly whte depends on the complexty of the true nput spectrum and the sze of the quantzaton errors. n autocorrelaton terms a complex spectrum wll have large correlatons at delays beyond zero, and large quantzaton errors wll produce more curvature n the true vs. quantzed correlaton curve n Fgure 1. Lookng for quantzaton dstortons n a smulated nose spectrum takes an enormous amount o computer tme to reduce random errors to levels to be expected n the operaton of the spectral processor. Autocorrelaton functons can be computed much faster than a spectrum can be transformed so the approach taken has been to compute true vs. quantzed correlaton curves for several degrees of quantzaton usng computer generated nose. These curves gve the slope and amount of nonlnearty at low correlaton levels. The autocorrelaton functons have been calculated for two nput spectra, and these functons have then been dstorted by the correlaton error curves and transformed back nto the frequency doman to look for spectral dstortons. Two rather severe passband shapes shown n Fgure 2 have been used n these calculatons. The top curve has a maxmum autocorrelaton value of 13% relatve to the zero delay value, and the largest value for the bottom curve s 21%. Except for the case of a very strong CW sgnal n the passband t s unlkely that the autocorrelaton values n practce wll be greater than the ones used n these tests. The slopes of the low end of the curves n Fgure 1 are shown n Fgure 3 where the slope s plotted as a functon of nose ampltude relatve to the quantzaton nterval. Even when the nose rms s fve quantzaton levels the slope s suffcently dfferent from unty to requre a correcton n the zero level and gan of the computed spectrum. Upper lmts on the curvatures of the lnes n Fgure 1 were estmated by comparng the fts of lnear and quadratc curves on the autocorrelaton values from 0.0 to 0.5 and hgher. For nose rms levels greater than 1.0 no sgnfcant dfference between the straght lne and quadratc fts could be seen up to an autocorrelaton level of about 0.7. For these nose levels a conservatve upper lmt on the quadratc coeffcent s 0.001. For nose rms levels of 0.8 and 0.5 the upper lmts on the quadratc coeffcent are 0.005 and 0.01, respectvely. These lmts are set by the nose n the calculatons of the autocorrelaton coeffcents, and the lnearty of the curves may be consderably better than ths.
The autocorrelaton functons derved from the two bandpasses n Fgure 2 were modfed by a transfer functon that contaned only a quadratc term (a 3 x 2 ), and wth a 3 = 0.001 the peak fractonal error 4 n the lower bandpass curve was about 10 except near the left edge of the bandpass where the error rose to 2 x 10-4. The error n the top bandpass was 5 x 10 - s rsng to 10-4 near the left edge usng the same quadratc coeffcent. An error of 10-4 s equvalent to the rms nose n a 1024-channel, 40 MHz spectrum after 40 mnutes of ntegraton. Ths sze of error would be detectable, but f the nput power to the A/D converter does not change very much between sgnal and reference the error wll cancel n the dfference. Also, there s a good chance that the autocorrelaton transfer functon s even more lnear than supposed here so the spectrum dstorton may be qute a bt less than 10-4. Zero offset and gan correctons wll certanly need to be appled to the FFT spectrum. f a hgher order correcton for quantzaton effects s necessary the averaged spectra could be Fourer transformed to an autocorrelaton functon n the spectral processor computer, corrected, and transformed back nto the frequency doman. Both the zero and hgher order correctons depend on the nput level, and, n the absence of an exact set of equatons for the correctons, the autocorrelaton transfer functon would have to be calbrated wth long ntegratons wth the spectral processor hardware under dfferent quantzaton condtons. At least a few of these calbratons should be done whether the second order correctons are to be performed or not. JRF/cjd Attachments Fgures 1, 2, 3
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