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Fourier S/N Estimation

The Fourier S/N Estimation item in the DSP menu opens the Fourier Signal/Noise Analysis procedure: Although this procedure is designed primarily for S/N estimates, it is a full Fourier analysis procedure with data tapers, different Fourier transform algorithms, variable zero padding, different plots, and critical limits for white and red noise. The data stream should be uniformly sampled (constant sample increment).

Preprocess

The Remove Trend will force the endpoints of the data processed from the Fourier algorithm to begin and end at 0 by subtracting a linear trend.

Window

PeakLab offers a broad selection of data tapering windows to minimize spectral leakage. The adjust field is used to set the spectral width, and thus the dynamic range, of adjustable windows. This field will be disabled for fixed windows.

The spectrum is normalized so that its power equals the power of the input data, regardless of the data window used. In PeakLab, a windowed FFT spectrum will contain the same power as a non-windowed spectrum, and both will match the power in the input data.

Transform

AutoSignal offers the following FFT algorithms:

Prime Factor

Chirp-Z

Best Exact N

The Best Exact N composite algorithm is the default. If the data size is a power of 2, the FFT Radix 2 algorithm is used. If not and the size is included in the prime-factor set, then the Prime Factor procedure is used. Otherwise, the Mixed Radix algorithm is used if the largest prime <= 509 and the Chirp-Z is used if the largest prime >= 521. This produces the fastest possible exact n FFT.

Nmin

The initial Nmin value will be the data size. To zero pad, enter any value greater than the data size. You may also select from a power of 2 sequence in the drop down box. Note that the actual size of the FFT may be greater than this value if the FFT Radix 2 or Prime Factor algorithm is used.

Increasing the zero padding will increase the number of frequency channels, which for small size data sets can aid in more accurately determining the center frequencies of spectral peaks. This will not change the basic shape of the spectrum, however. If a given peak is defined by only three frequency bins when an exact 64 point FFT is made, a 1024 point FFT will basically fill in this same shape. It is a kind of interpolation since zero padding cannot sharpen the peaks. To achieve this sharpening with an FFT, a longer set at the same sampling rate would be required. For data that are rapidly changing, or when the time series is limited in size, a non-FFT procedure is usually required for good spectral resolution.

When a data tapering window is used, very little spectral leakage arises from the zero-padding.

Plot

The frequency domain information can be plotted in a variety of formats. In the following table, Re is the real component of the FFT at a given frequency, Im is the imaginary component, n is the data set size, dx is the sampling interval, and var is the variance of the data series.

Real, abs(Re)

Imaginary, abs(Im)

Magnitude, sqrt(Re*Re+Im*Im)

Phase, sine-based, Pi/2+atan(Im/Re)

Mag/Phase, dual plot, magnitude in Y, phase in Y2

Amplitude, 2.0*sqrt(Re*Re+Im*Im)/n

Ampl/Phase, dual plot, amplitude in Y, phase in Y2

dB, decibels, 10.0*log10(Re*Re+Im*Im)

dB Norm, decibels, normalized to 0 for frequency channel with maximum power

PSD SumSq, Power as Sum Squared Amplitude, 2.0*(Re*Re+Im*Im)/n

PSD MeanSq, Power as Mean Squared Amplitude, 2.0*(Re*Re+Im*Im)/n/n

PSD TimeInt, Power as Time-Integral Squared Amplitude, 2.0*dx*(Re*Re+Im*Im)/n

Variance, Power normalized by variance, (Re*Re+Im*Im)/n/var

In an amplitude plot, you see the actual amplitude of sine components. In a normalized decibel plot, the highest peak is at 0dB, a peak at -3dB would have half the power, and a peak at -6dB would have half the amplitude. The PSD TISA (time-integral squared amplitude power) is the actual integral under the curve defined by the square of the raw data.

Peaks

The spectral peaks are identified by a local maxima detection algorithm. Both the amplitude and the frequency locations of the detected peaks are based upon a cubic spline bin interpolation procedure.

The sig item sets the target number of peaks (signal components) to detect. Up to 50 peaks can be detected. Peaks are ranked by interpolated amplitude. Note that this target signal component count may not be realized as fewer peaks than this target may be detected. Note also that the frequency analysis and linear sinusoidal fits reported in the Numeric Summary use the component count and frequencies from this peak identification.

The wid item sets the bin width tolerance for defining a peak. A peak must exist across this number of FFT bins to be counted. The default is a single bin. The Display Maxima option is used to step through the options for displaying spectral peak labels: frequencies, spectral magnitudes, both frequencies and spectral magnitudes, or none.

AR(1) Background

PeakLab offers peak-type critical limits to determine the statistical significance of the largest peak present in the spectrum. These limits are computed for all windows, including those with adjustable parameters. The default background used for this null hypothesis is white (Gaussian or normally distributed) noise, AR(1)=0.0. A lag-1 autoregressive spectrum can also be specified. An AR(1) coefficient greater than 0.0 can often model red noise (where the noise power decreases with increasing frequency). The AI Expert option will set the AR(1) value to the lag-1 normalized autocorrelation. This should only be used as a preliminary estimate. The Show Significance Levels option is in the graph's toolbar. This button is used to toggle the significance levels on and off.

Numeric

The Numeric button opens the Signal Noise Analysis window.

Signal Noise Analysis

Algorithm: Best Exact N

Window: None

Size      Size      Time      Time      Time      Fourier      Fourier      Data

Data      FFT      SNRdb      Digits      ppm      SNRdb      Digits      ID

161      161      41.3216935      2.06608467      8588.46056      58.5532309      2.92766154      s219

161      161      51.6809324      2.58404662      2605.87379      61.3382564      3.06691282      s220

161      161      41.9321351      2.09660675      8005.58819      65.2179760      3.26089880      s222

161      161      43.7644836      2.18822418      6482.99698      62.7763699      3.13881849      s223

161      161      45.0613650      2.25306825      5583.82435      63.0190151      3.15095076      s224

161      161      45.4617571      2.27308786      5332.27015      64.6110313      3.23055156      s225

161      161      45.5542386      2.27771193      5275.79691      65.2302258      3.26151129      s226

161      161      47.2975383      2.36487691      4316.41394      63.2402817      3.16201409      s227

161      161      47.4876148      2.37438074      4222.98228      61.9017618      3.09508809      s228

161      161      41.6075109      2.08037555      8310.44834      63.7479261      3.18739630      s229

161      161      55.0072352      2.75036176      1776.79875      65.4612237      3.27306119      s230

161      161      52.1091906      2.60545953      2480.50707      64.0886330      3.20443165      s231

Avg      46.5238079      2.32619040      5248.49678      63.2654943      3.16327472      All Data

The PeakLab S/N Analysis consists of both a time domain method which estimates the amount of white noise in the data, and a Fourier analysis procedure which measures the noise floor in the data.

The time domain procedure estimates the level of white (Gaussian random) noise in the data. For each point, a time-domain interpolation is made using adjacent points on each side of that data point, and the difference between that interpolation and the data is treated as the noise predictor at that data value. The average of those absolute differences is input in a predictive model that was built using a large Monte Carlo simulation consisting of different sizes of data and different levels of known white noise.

In this procedure the Time ppm is this computed white noise value based on the average absolute errors in the procedure. In the first sample in the above table you can read the S/N estimate as there existing 8588 ppm or 0.86% white noise in the signal. If that level of noise is converted to significant digits, -log10(0.008588), we have the Time Digits value of 2.066 significant digits. If those digits are converted to dB, we have -20*log10(0.008588), the Time SNRdb of 41.32 dB.

The Fourier procedure measure the noise floor in the Fourier domain. At the highest frequencies, it is assumed the signal consists almost exclusively of noise, a fair assumption for chromatographic and spectroscopic data. In the first sample above, the noise floor occurs at the Fourier SNRdb of 58.55 db in the upper frequencies of this data. This corresponds with 2.927 digits precision, the Fourier Digits in the above table.

The time domain estimate doesn't specify that the noise is white, independent of signal and essentially constant at all frequencies in the Fourier domain. It only means, for this first sample, that 0.86% white noise corresponds with the overall measure of noise in the data. The noise may well be red, higher at lower frequencies where the most of a chromatographic or spectroscopic signal exists and lower at the upper frequencies where no signal exists. For example, if you look at the critical limit lines for an AR-1 (autoregression order 1) red noise of 0.6, you see a pattern in the critical limits that looks a lot like the pattern in the Fourier spectrum of data from about 0.1 upwards where the noise is apparent. A critical limit is where a spike in the signal appears strictly from random noise. In this example, a 99.9% critical limit, the red line above, means that in 1000 samples of 0.6 AR-1 red noise data sets of this specific size, 1 had its largest Fourier spike at or above this line, and 999 were below.

For this same data, the first image at the beginning of this topic shows the 50, 90, 95, 99, 99.9 critical limit lines for white noise. Note that there is an equal probability at every frequency of having a given level exceeded by random chance. Also note that the uppermost frequencies have lower Fourier thresholds. This should give you some sense for why the estimation of noise can be less than straightforward when this independence between signal and noise is absent.

An overall estimate of white noise in the time domain and the noise floor in the Fourier domain are two different ways to view the noise in spectral data. They should only agree if the white noise (uncorrelated, frequency independent) noise assumption holds.

Compare

The Compare button will open a Fourier plot where multiple data sets can be easily compared with one another. You can select the Use Mouse or Use Legend items to highlight either a curve or legend item to identify the different spectra.

Averages

The dialog will display the averages of multiple data sets as shown in the Numeric option.

Copy An Individual FFT to Clipboard

You can right click any of the Fourier plots and select Copy this FFT to Clipboard to place the a Fourier table on the clipboard.

Save An Individual FFT to Disk

You can right click any of the Fourier plots and select Save this FFT to File to write the Fourier table to disk.

View an Individual FFT

You can right click any of the Fourier plots and select View this FFT in a Numeric Table to open a window containing the Fourier table to disk.

Chnl       Frequency       Magnitude       Amplitude       Wavelength       Phase       dB Norm       PSD TISA

0      0.0000000000      30233.900261      187.78820038      0.0000000000      1.5707963268      0.0000000000      5677569.7204

1      0.0062111801      8071.8225520      100.27108760      161.00000000      4.2741443871      -11.47045140      809370.42622

2      0.0124223602      2956.8014353      36.730452612      80.500000000      5.6054297607      -20.19344031      108604.65500

3      0.0186335404      875.37300637      10.874198837      53.666666667      5.7429105065      -30.76602052      9518.9801278

4      0.0248447205      589.17337853      7.3189239569      40.250000000      6.0473059769      -34.20502122      4312.1151549

5      0.0310559006      391.23181422      4.8600225369      32.200000000      0.1400609986      -37.76120025      1901.3954343

6      0.0372670807      219.23615971      2.7234305555      26.833333333      6.0959520813      -42.79163980      597.07445623

7      0.0434782609      255.43358705      3.1730880378      23.000000000      6.1172136056      -41.46432348      810.51325951

8      0.0496894410      207.38381953      2.5761965159      20.125000000      6.2712379888      -43.27438614      534.26147332

9      0.0559006211      189.73983355      2.3570165659      17.888888889      0.0449992849      -44.04671322      447.21993090

10      0.0621118012      178.43017108      2.2165238643      16.100000000      0.0763644023      -44.58051768      395.49473231

...

Sinusoidal Modeling

You can right click any of the Fourier plots and select Sinusoidal Modeling of this FFT's Peaks to see a linear multiple sinusoid fit for the peaks in the Fourier spectrum. This is mainly of value for modeling data channels which contain oscillatory components in the Fourier domain. It will not generally be of any value with peak-type data. The following data consists of 7 sinusoids (4 prominent and 3 minor) plus noise. In order to fit this data to seven sinusoids, the sig value must be set to 7: Fourier Frequency Spectrum

Data: 7Cos1001

X: Frequency

Y: dB Norm

Data Source: C:\PeakLab\Config\CLIPBRD.PRN

Date: Oct 5, 2022

Algorithm: Best Exact N

Window: cs4 BHarris 4

Data Size: 1001

FFT Size: 1001 (7*11*13)

SSA Power: 2000.1851

MSA Power: 1.9981869

TISA Power: 0.20001851

Signal Count: 7

Interpolated Spectral Peaks

Frequency       dB Norm

1005.2100405      0.1894634337

1505.2708156      -39.88661372

2005.3347201      0.3216786483

2505.7508840      -49.70352675

3005.3422157      0.1489492234

3505.6778225      -59.32619748

4005.2042014      0.0696866871

Frequency Analysis

Frequency       Amplitude       Phase       Power       %       Rel %

1005.2100405      0.9872804354      1.2744816397      0.0487848690      24.711366704      97.718227673

1505.2708156      0.0099391605      1.1947412433      4.944285e-06      0.0025044658      0.0099036190

2005.3347201      0.9908999333      1.1542377462      0.0491432280      24.892888975      98.436036397

2505.7508840      0.0032527060      1.0766978791      5.295338e-07      0.0002682287      0.0010606794

3005.3422157      0.9951038027      1.1783240928      0.0495610905      25.104551990      99.273033189

3505.6778225      0.0010776440      1.1488934521      5.812389e-08      2.944193e-05      0.0001164247

4005.2042014      0.9987406811      1.3147948684      0.0499240216      25.288390194      100.00000000

0.1974187410      100.00000000

Sine Component Fit (Suboptimal)

Frequency       Amplitude       Phase       Power       %       Rel %

1005.2100405      0.9907648074      1.5051935769      0.0490909709      24.994640094      99.719623852

1505.2708156      0.0016254778      1.9196484092      1.321242e-07      6.727096e-05      0.0002683869

2005.3347201      0.9900905630      1.4665705491      0.0490220871      24.959567932      99.579698540

2505.7508840      0.0048723790      4.0752508628      1.187242e-06      0.0006044836      0.0024116721

3005.3422157      0.9905060370      1.4642413224      0.0490606534      24.979203924      99.658039087

3505.6778225      0.0062674530      4.2288279926      1.964364e-06      0.0010001549      0.0039902582

4005.2042014      0.9922347950      1.5071058985      0.0492289973      25.064916140      100.00000000

0.1964059925      100.00000000

r2       DOF r2       Std Err       F-stat

0.9895620642      0.9893381657      0.1459578896      4645.4147860

Data Power       Model Power       Error Power       Ratio

0.2000168605      0.1979307490      0.0020877631      94.805174507

Sine Component 1

Parm      Value      Std Error      t-value      95% Confidence Limits      P>|t|

Ampl      0.9907648074      0.0065210659      151.93295458      0.9779679485      1.0035616663      0.00000

Fare      1005.2100405      0.0363350403      27665.031675      1005.1387370      1005.2813439      0.00000

Phase      1.5051935769      0.0131801147      114.20185747      1.4793290831      1.5310580706      0.00000

Sine Component 2

Parm      Value      Std Error      t-value      95% Confidence Limits      P>|t|

Ampl      0.0016254778      0.0065225857      0.2492075786      -0.011174364      0.0144253192      0.80325

Fare      1505.2708156      22.133679366      68.008160357      1461.8359574      1548.7056738      0.00000

Phase      1.9196484092      8.0270229665      0.2391482393      -13.83248202      17.671778834      0.81104

Sine Component 3

Parm      Value      Std Error      t-value      95% Confidence Limits      P>|t|

Ampl      0.9900905630      0.0065216114      151.81686024      0.9772926336      1.0028884924      0.00000

Freq      2005.3347201      0.0363568006      55157.073415      2005.2633739      2005.4060662      0.00000

Phase      1.4665705491      0.0131881844      111.20336966      1.4406902193      1.4924508788      0.00000

Sine Component 4

Parm      Value      Std Error      t-value      95% Confidence Limits      P>|t|

Ampl      0.0048723790      0.0065229871      0.7469551836      -0.007928250      0.0176730080      0.45527

Freq      2505.7508840      7.3831289762      339.38874590      2491.2623232      2520.2394449      0.00000

Phase      4.0752508628      2.6778180686      1.5218550172      -1.179666140      9.3301678659      0.12837

Sine Component 5

Parm      Value      Std Error      t-value      95% Confidence Limits      P>|t|

Ampl      0.9905060370      0.0065221492      151.86804322      0.9777070522      1.0033050218      0.00000

Freq      3005.3422157      0.0363339738      82714.382831      3005.2709143      3005.4135170      0.00000

Phase      1.4642413224      0.0131803789      111.09250610      1.4383763101      1.4901063346      0.00000

Sine Component 6

Parm      Value      Std Error      t-value      95% Confidence Limits      P>|t|

Ampl      0.0062674530      0.0065234375      0.9607592606      -0.006534060      0.0190689661      0.33691

Freq      3505.6778225      5.7387550343      610.87776034      3494.4161607      3516.9394843      0.00000

Phase      4.2288279926      2.0811058025      2.0320100918      0.1448917482      8.3127642371      0.04242

Sine Component 7

Parm      Value      Std Error      t-value      95% Confidence Limits      P>|t|

Ampl      0.9922347950      0.0065224128      152.12695333      0.9794352929      1.0050342972      0.00000

Freq      4005.2042014      0.0362615354      110453.24347      4005.1330422      4005.2753606      0.00000

Phase      1.5071058985      0.0131546773      114.56806326      1.4812913228      1.5329204743      0.00000

Estimating Spectroscopic Resolution in the Fourier Domain

You can also use this Fourier S/N Estimation procedure with the D2 (2nd derivative) Savitzky-Golay spectra of data to get an approximation of spectral resolution.

The third title of the graphs will display the average values from the first sig count of peaks in the spectra by x value. For this specific analysis, the value of sig specifies the count of peaks starting with the first in the x-sequence. In all other instances, sig is the count of the most prominent magnitude peaks for the purpose of labeling the peaks, and for performing a sinusoidal fit. 