Windoing High-Resolution DC Data art Josh Carnes pplications Engineer, ational Seiconductor Corp. bstract nalyzing data fro DCs requires the use of indoing functions for spectral estiation and analysis but different indos suit different purposes. ational Seiconductor s ne WaveVision 5 softare provides a faily of atheatically siple indoing functions that spans a fundaental tradeoff and allos the flexibility to eet a ide range of user applications. art of this article presents the Cosine-Su faily of indoing functions available in WaveVision 5, the application of these indos, and an eleentary discussion on the atheatics behind the calculation of perforance etrics such as the SR. Introduction art of this article included a discussion on finite-tie sinusoidal segents and described ho data captured fro analog-to-digital converters (DC) cannot be iediately transfored into the frequency doain due to the resulting spectral leaage of the signal. The previous installent also presented frequency coherency as a preferred solution for avoiding spectral leaage but explained ho it is ipractical in any test scenarios. pplying indoing functions as introduced as the ore coprehensive solution to the spectral leaage proble and a fe coon indoing functions ere presented along ith their iportant features. art also shoed that no single indoing function can be used for all applications and that the ost coon indos are not sufficient for analyzing high resolution DCs for arbitrary input frequencies due to insufficient indo dynaic range. art oves the context to indoing functions used in ational Seiconductor s ne DC evaluation softare platfor called WaveVision 5. Using WaveVision 5, a user can tae advantage of a flexible faily of Cosine-Su indoing functions to analyze the perforance of DCs that have a ide span of speeds and resolutions. These indos can be applied to data captured fro a variety of different applications. The atheatical foundation of perforance calculations in WaveVision 5 is also presented. The reader ill notice that this installent is ore atheatically intensive than the previous picture-friendly installent. The intention here is to provide intuitive understanding as ell as a ore concrete foundation of spectral analysis and provide the atheatical bacground that can be used in practice. Faily of Cosine-Su Windos WaveVision 5 softare allos the user to test a ide variety of DCs ith different resolutions fro 8-bits to above 6-bits. Due to the ide range of indoing
requireents, WaveVision 5 offers a faily of optiized Cosine-Su indos, see [], to exercise the tradeoff beteen ain lobe bandidth (frequency resolution) and sidelobe suppression (dynaic range). The indo faily is expressed by () here the indos differ by the nuber of cosine ters and the coefficients used in the suation. Each coefficient is calculated so that the axiu values of all the indo side lobes are nearly flat across the spectru. ctually coputing the coefficients is beyond the scope of this article, but a detailed explanation can be found in []. The coputational coplexity of creating a high order, -ter indo of this faily ay see arduous as a suation of cosinusoids in tie, but an -point DFT of an -point indo reveals that this faily of indos can be succinctly described in the sapled frequency doain. Given by (), the DFT spectru contains a liited nuber of non-zero ters hose agnitudes correspond to the coefficients, aing the application of the indo straight-forard as convolution in the frequency doain. W n 0-0 - cos n for 0 n - for 0 n - () () By appending 5 zeros to an -point indo and perforing a 6-point DFT of the indo, one can observe a ore continuous approxiation of the indo spectru to reveal its side lobe suppression. Figure copares the spectru of the = -ter, 4-ter, 6-ter, 8-ter, and -ter indos provided by the WaveVision 5 softare and Table copares the ain lobe 3-dB bandidth versus the side lobe suppression. Figure : Frequency spectru of cosine-su indos
Table : ain lobe bandidth and axiu Side Lobe Suppression Cosine-Su Windo Order ain Lobe 3-dB BW [bins] axiu Side Lobe Suppression [db] -ter.3 43. 4-ter.87 98. 6-ter.30 53.6 8-ter.65 07.5 -ter 3. 89.6 Optiized for axiu side lobe suppression, the -ter and 4-ter indos presented here are also non as the Haing and Blacan-uttall [3] indos respectively. The optiization given by lbrecht in [] is generalized for any nuber of cosine ters and contains the Haing and Blacan-uttall indos as ebers of the faily. Choosing a Windo for a articular pplication Supplying this faily of indos in WaveVision 5 allos the user to ae a ell-infored choice of hich indo to use depending on the application. ll the ystery surrounding the indos and their ost iportant effects is succinctly suarized in the above figure and coparison table. s an exaple, ational Seiconductor s DC455 has an expected SR of 7.3 db relative to full scale (dbfs) and saples at 55 -saples per second (SS). What is the best indo for analyzing data fro this DC? With a ~3 saple record the corresponding noise poer spectral density is 3.3 dbfs/bin, therefore the 6-ter indo is sufficient for this application because the side lobe suppression is 53 db. This choice liits the frequency resolution to 55Hz / 3-saples * * (indo order) = ~57 Hz. The indo order is iportant here because it is tied directly to the ain lobe idth such that the ain lobe of the -ter indo eets the side-lobe level bins aay fro the lobe center as seen in Figure. n 8-bit, GSS DC08000 does not have such a high dynaic range requireent. With an expected SR of 48.5 dbfs, the noise poer spectral density is 90.6 dbfs/bin for a ~3 saple data record. The 98. db side-lobe suppression of the 4-ter indo is sufficient for this application, liiting the frequency resolution to 44 Hz. The consequence of using a indo ith insufficient dynaic range for an application is exeplified in the coparison of Figure. Data captured fro a high speed, 6-bit converter is indoed ith the Flattop indo and the 6-ter Cosine-Su indo. The coparison clearly sho the influence of the side-lobes hen the Flattop indo is applied, deonstrating the indo s inadequate dynaic range. ote that the spectru does not have distinct side lobes as in Figure. This is because the leaage around the fundaental lobe is ade up of one sapled point per side lobe in the sapled spectru. The result of using the Flattop indo in this case for spectral analysis ill be a degraded signal-to-noise ratio (SR) as the side-lobe poer ill be interpreted as additional noise in the spectru.
Figure : The inadequate dynaic range of the Flattop indo for a 6-bit DC lotting and nalyzing the Windoed Data Spectru fter data fro a single tone DC test is appropriately indoed to give sufficient side lobe suppression and after the discrete Fourier transfor (DFT) is perfored, the spectru can be plotted and calculations of perforance etrics can be ade. Here e present to eaningful spectru noralization schees and explain the calculation of iportant values such as the fundaental tone poer and total noise poer. One schee is used for easy visualization of the spectru and the other is used for perforance calculations. When plotting the spectru, the agnitude is typically noralized in decibels so that 0 db corresponds to the axiu possible sinusoidal output fro the converter. This noralized spectru for a B-bit converter hose output values range fro - (B-) to (B-) - is achieved ith (4) for an -saple data record x[n] and indoing function [n]. In this case the noralization factor is the indo s coherent gain factor defined in (3). oralization by the indo ean forces the pea value of the fundaental lobe to have the sae value for all indos, aintaining visual consistency. This spectru noralization schee is used in WaveVision 5 for plotting because a consistent fundaental pea is visually ore pleasing and less confusing to ost users. This schee is not used to ae perforance calculations. n 0 n W 0 (3) This ignores scalloping loss here the pea of the lobe varies depending on the location of the signal frequency beteen DFT bins. Variation due to scalloping loss reduces for higher order Su-Cosine indos and is less than 0.9dB for the 4-ter indo.
X 0 log 0 B DFT n x n [ n] (4) Windoing a data record can change the total poer in the spectru due to attenuation at the endpoints, so a different noralization schee is used for etric calculations to conserve the total signal poer hen the indo is applied. This noralization factor,, is the incoherent gain that as briefly entioned in the previous installent of this article. The incoherent gain is given by (5) and specifies ho the total poer of a signal changes hen the indoing function is applied to the data record here x[n] is the data record, [n] is the indo function, W[] is the DFT of [n], and is the nuber of points in the data record and indo. This noralization ensures that the total poer is conserved hen the indo is applied and is equivalent to scaling the indo so that its incoherent gain is unity. The noralized spectru is given by (6) and is used in WaveVision 5 to ae all perforance calculations. n additional sqrt() factor is included in the denoinator to consider the full dual-sided spectru, not just the single-sided spectru. erforance calculations that are reported as relative to full scale are often of interest. These values, given in units of dbfs, are easily calculated ith the noralized spectru of (6) because only the suing of poer is required ithout needing to no the total poer of the fundaental tone. In this case, additional noralization and the calculation of ratios are not required for etrics lie the SR, THD, and SID. X n n 0 0 B DFT n W x n [ n] (5) (6) o that the spectru is properly noralized in (6) to ae perforance calculations, the spectru ust be divided into frequency regions that represent the fundaental, haronic distortion, or noise. The spectru contains a large lobe at the fundaental location and saller lobes at the haronics, so one ust as the question: Ho uch bandidth is sufficient to contain the poer of a ain lobe? Once this is non the SR, SID and THD are easy to calculate. helpful characteristic of the presented cosine-su faily is that the ain lobe of a indo s frequency spectru extends bins fro the lobe axiu before reaching the axiu side lobe level here is the order of the indo. Therefore, the approxiation can be ade that the poer of a ain lobe spans + bins after applying the indo. Using this approxiation, the WaveVision 5 softare identifies the bin ith the largest poer as the fundaental frequency and sus the poer in a K+ bin bandidth to find the total poer of the fundaental. Haronics of the fundaental are treated siilarly. Bins that are not part of a fundaental or haronic are treated as noise. To
eep the analysis flexible, the user is alloed set K to any value in the FFT Control Options of the WaveVision 5 softare, but it is recoended to set K =. The poers of the fundaental tone and total noise are calculated ith the dualsided spectru in (7) and (8) respectively here F bin is the bin containing the pea value of the fundaental lobe, + is the fundaental lobe idth, is the nuber of points in the spectru, and X [] is the signal spectru given by (6). The suation of (8) includes all noise bins in the dual-sided spectru or equivalently fro DC up to the sapling frequency. With the incoherent gain noralization schee described above, the SR in units of dbfs is the negative value of noise. fund 0 log 0 F bin F bin X F bin F bin X [dbfs] (7) noise 0 log 0 X {oisebins} [dbfs] (8) The ratio of and, given by (9), is an iportant feature of the indo itself called the processing gain, G []. Figure 3 shos the zooed spectra of a coherent frequency signal having no indo (Rectangular indo), the 4-ter, and the -ter Cosine-Su indo. The change in the pea aplitude after the indo is applied is identically the processing gain in [db]. Rectangular indo gain has a processing gain of unity hereas shaped indos have a processing gain less that one. One consequence of indoing that can lead to confusion is that the dynaic range in decibel units beteen the pea of the fundaental lobe and the average noise level ill be different depending on the chosen indo. In the case of noralization schee shon in Figure 3, the average noise level stays the sae hile the pea value of the fundaental lobe changes. lternatively, the noralization schee locs don the pea value of the fundaental lobe and causes the noise poer density to ove around for different indos depending on the indo s processing gain, G. The choice to display spectra in WaveVision 5 ith the schee causes the observed average noise level to be higher than the calculated level that is reported hen indoing functions are used. The higher order indos have saller processing gains and therefore have a larger difference beteen the observed and calculated average noise level. With a single data capture this difference is not observable due to the large variation of noise poer in adjacent bins but the effect is ore apparent hen applying FFT averaging, a poerful feature of the WaveVision 5 softare. Despite the visual inconsistency in the noise floor, the calculated perforance etrics are not affected. lthough scalloping loss affects the ain lobe pea value, it affects the total poer ithin the ain lobe by less than 0.0 db for the 4-ter and higher order indos aing this expression suitable for finding the fundaental poer.
G n 0 W 0 n n n 0 0 W (9) Figure 3: The processing gain of the 4- and -ter Cosine-Su indos Using the Cosine-Su Windos Table gives the standard coefficients for the 4-ter and 6-ter indos hich are ost appropriate for today s state of the art 0-bit to 6-bit DCs. Using the coefficients and (), the indos ay be calculated and applied to data records that have been iported into data analysis softare lie atlab. Table : Su-Cosine Windo Coefficients Coefficient Su-Cosine Windo 4-ter 6-ter a 0 3.6358967707608e-0.9355789500797e-0 a 4.89774374507e-0 4.5935773474506e-0 a.365995397869e-0.0464746396e-0 a 3.0640553003e-0 4.7960905837e-0 a 4 5.069646859393e-03 a 5.375555679558877e-04 Conclusion DC softare evaluation platfors supplied to custoers are designed to be as user friendly as possible, but pitfalls are alays present. roviding easy-to-use options
ill greatly reduce the potential for frustration, but education ust also be provided to avoid erroneous evaluation results. When cobined ith this article, ational Seiconductor s ne WaveVision 5 softare accoplishes both. The faily of Cosine- Su indos provided in the softare aes selection of the appropriate indo for an application as siple as looing at a chart due to each indo s basic for and intuitive results. fir, atheatical bacground has also been provided here to increase a user s understanding of the analyses built into WaveVision 5 as ell as to allo the septics to crunch the nubers theselves. References [] H.-H. lbrecht, Faily of Cosine-Su Windos For High-Resolution easureents, IEEE roc. coustics, Speech, and Signal rocessing, vol. 5, pp. 308-3084, ay 00. [] F. J. Harris, On the Use of Windos for Haronic nalysis ith the Discrete Fourier Transfor, roceedings of the IEEE, vol. 66, pp. 5-83, Jan. 978. [3]. H. uttall, Soe Windos ith Very Good Sidelobe Behavior, IEEE Trans. coustics, Speech, and Signal rocessing, vol. 9, pp. 84-9, Feb. 98. bout the uthor Josh Carnes is an applications engineer ith ational Seiconductor s Strategic Signal ath Group, based in Ft. Collins, Colorado. He received his BSEE and SEE degrees fro Oregon State University in 004 and 007, respectively, ith research focusing on lo-voltage pipelined DC design techniques. His interests include cellular base station subsystes, ireless counications, as ell as autoated testing and analysis of DCs.