Experimental SNR determination from ELODIE data-set.

Introduction

Main goal of this short branch is to discuss the concept for signal to noise ratio (SNR) measurement technique.

It is assumed that ELODIE includes 1959 spectra at three different resolutions:

Resolution	Code
R=42000	H
R=10000	L
Blue fract.	Elo1

[Resolutions considered]

By using the R=10000 resolution two different energy ranges were provided

Band	From (A)	To (A)	By (A)
Blue	3960	4567	0.2
Red	6437	6800	0.2

[Energy Bands considered]

Processing

## Loading required package: magic

## Loading required package: abind

The previously stored data are now reloaded,

The real SNR values have been provided as estimated from the array of sensors device.

After loading the different families of spectra, we calculate their SNR values as per element basis. Then we merged all the SNR estimations with the original one.

Our first estimation for the SNR was based only into the wavelength expression for the spectra with out taking care of any sub-spectra region.

We can depict some relationships depending on each data-set studied

plot(osnr[, 2], osnr[, 3], xlab = "Original SNR", ylab = "Measured SNR", main = "Relationship in case of Elo1 dataset")

plot of chunk unnamed-chunk-3

plot(osnr[, 2], osnr[, 5], xlab = "Original SNR", ylab = "Measured SNR", main = "Relationship in case of L dataset")

plot of chunk unnamed-chunk-4

plot(osnr[, 2], osnr[, 6], xlab = "Original SNR", ylab = "Measured SNR", main = "Relationship in case of H dataset")

plot of chunk unnamed-chunk-5

plot(osnr[, 2], osnr[, 7], xlab = "Original SNR", ylab = "Measured SNR", main = "Relationship in case of local H dataset")

plot of chunk unnamed-chunk-6

Preliminary Conclusions

The biggest conclusion is that it gets a wrong endpoint to associate a SNR value to a signal, to re-sample it and still considers this initial SNR as the good one.

In this way it is possible to realize how, for the biggest density (R=42000) there is a stronger relationship between some fraction of both spectra. Even when there is another set of spectra which are clearly uncorrelated.

osnr[1:20, 1:5]

##        THEORY D.SNR            B.SNR            R.SNR            L.SNR
## 1  00001.fits 163.8 26.2394481246712 26.2394481246712 67.5230554616693
## 2  00002.fits 288.8 361.398612567866 361.398612567866 792.303514579114
## 3  00003.fits 122.7  28.397618881473  28.397618881473 62.1796934990082
## 4  00004.fits 137.3 30.4157467625289 30.4157467625289 71.2549576135147
## 5  00005.fits 101.1 12.8878536655673 12.8878536655673 23.9785331052851
## 6  00006.fits 144.9 15.4119255615594 15.4119255615594 33.3450468407331
## 7  00007.fits 134.8 42.9762981206909 42.9762981206909 194.560573063463
## 8  00008.fits 105.1 45.7922086863789 45.7922086863789 181.996115257925
## 9  00009.fits  80.7 45.1509638488004 45.1509638488004 152.633122795621
## 10 00010.fits 105.5 27.5296749602775 27.5296749602775  56.471982313251
## 11 00011.fits 120.9 17.0562902075109 17.0562902075109 32.7831945448975
## 12 00012.fits  73.6 56.7717060857951 56.7717060857951 126.725619462166
## 13 00013.fits  74.8 21.8103401437975 21.8103401437975 41.5268746926864
## 14 00014.fits 376.0 12.0862434983147 12.0862434983147 18.6512256856873
## 15 00015.fits 224.6 12.0041337176951 12.0041337176951 18.7040553824664
## 16 00016.fits  89.2  81.792193976718  81.792193976718 232.658226130451
## 17 00017.fits  89.2 81.8429836737006 81.8429836737006 232.520209718436
## 18 00018.fits 187.2 22.4364059641126 22.4364059641126 48.3407376451479
## 19 00019.fits  72.1 17.8581165878733 17.8581165878733  36.554377031841
## 20 00020.fits 171.6 25.6097392978677 25.6097392978677 54.4687664192282

Error measurement by band around 5500A, by local FFT around that wavelehgth. Considered windows will be 256, 512, 1024, 2048, 4096

rslH <- 0.05
orgH <- 3900
psnr <- dsnr
for (dlt in seq(8, 12)) {
    cpos <- (5500 - orgH)/rslH
    pas <- 2^(dlt)
    hcsnr <- apply(rd_H[, floor(cpos - pas/2):floor(cpos + pas/2)], 1, fsnr)
    #
    hcsnrd <- as.data.frame(cbind(names(hcsnr), as.numeric(hcsnr)))
    text <- paste(dlt, ".HBND")
    tex2 <- paste(dlt, ".SNR")
    colnames(hcsnrd) <- c(text, tex2)
    psnr <- merge(psnr, hcsnrd, by.x = "THEORY", by.y = text, sort = TRUE, all = TRUE)
}
osnr[1:20, c(1:2, 6:7)]

##        THEORY D.SNR            H.SNR         HBND.SNR
## 1  00001.fits 163.8  80.387641787958 146.180826265927
## 2  00002.fits 288.8 269.266968402058 433.563487691221
## 3  00003.fits 122.7 74.5942053808666 107.186068345025
## 4  00004.fits 137.3 80.8961004913999  160.33534450115
## 5  00005.fits 101.1  27.052446993182 56.0259421167451
## 6  00006.fits 144.9 53.8588479708267 113.892388132789
## 7  00007.fits 134.8 109.360429963834 145.568290600405
## 8  00008.fits 105.1 89.2406366300405 133.000083401165
## 9  00009.fits  80.7 66.0652374631983 107.637620747169
## 10 00010.fits 105.5 63.8270265818283 120.021647122842
## 11 00011.fits 120.9 47.0704185352661 111.200855610754
## 12 00012.fits  73.6 59.7811856481925 99.9694170129399
## 13 00013.fits  74.8 42.4882034728498 74.4647799699079
## 14 00014.fits 376.0 25.1858452094172 61.5033164587391
## 15 00015.fits 224.6 24.9494558706771  59.322966977515
## 16 00016.fits  89.2 82.2784887816092 126.177342416592
## 17 00017.fits  89.2 82.2955728234692 126.147704066393
## 18 00018.fits 187.2 70.6073725324837 147.992434642949
## 19 00019.fits  72.1 36.5030519194865 66.0549595555163
## 20 00020.fits 171.6 76.5534664675584 141.344661262157

Now, it is time to have a look to the different periodograms at different sampling resolutions

psnr[1:20, 1:5]

##        THEORY D.SNR           8 .SNR           9 .SNR          10 .SNR
## 1  00001.fits 163.8 140.038319449606 121.369812987247 125.140809473659
## 2  00002.fits 288.8 418.273353293255 433.502502770795 400.223296517381
## 3  00003.fits 122.7  108.89236030774  108.70521349289 107.360640193766
## 4  00004.fits 137.3 140.517642808016 128.363945564501 133.133057536069
## 5  00005.fits 101.1 46.6842498069175  36.451119878017 40.7973343343739
## 6  00006.fits 144.9 96.5301855458177 85.4436498986788 85.2192022599505
## 7  00007.fits 134.8 144.635136323297  151.12315278166 151.464514561051
## 8  00008.fits 105.1 140.648906158605 141.635116376949 139.768910460531
## 9  00009.fits  80.7 115.220012244607 120.734860286675 115.539911041657
## 10 00010.fits 105.5 105.207116578212 96.7242177854302 97.2370793564668
## 11 00011.fits 120.9 93.3261196335231 66.5550071436699 72.4445408371857
## 12 00012.fits  73.6 102.764083730727 104.716935798403 100.307901539783
## 13 00013.fits  74.8 70.4828155583254 65.7714059377006  65.273153806047
## 14 00014.fits 376.0 52.3766366461237 36.3619128666981 36.5105702237634
## 15 00015.fits 224.6 49.6456688450098 36.0562418031433 35.4126156206947
## 16 00016.fits  89.2  133.14393160961 131.010693741214 128.685865623536
## 17 00017.fits  89.2 133.174868218658 131.019132336857 128.731172290053
## 18 00018.fits 187.2 137.311637403973 106.397121452758 115.595370868224
## 19 00019.fits  72.1 56.6672788103907 53.2359067557004 54.5816670396327
## 20 00020.fits 171.6 114.295874447645 102.871913131149  116.75194127855

and

psnr[1:20, c(1:2, 6:7)]

##        THEORY D.SNR          11 .SNR          12 .SNR
## 1  00001.fits 163.8 130.459662312782 121.410129390908
## 2  00002.fits 288.8 380.401105041159 364.395001835365
## 3  00003.fits 122.7 113.835403831215 109.056382075912
## 4  00004.fits 137.3 130.964080702413 121.056673510573
## 5  00005.fits 101.1 42.1264306140594  40.436507684288
## 6  00006.fits 144.9  86.503285724092 80.7471246809198
## 7  00007.fits 134.8 156.853727415829  153.20040374366
## 8  00008.fits 105.1 136.152210072654 128.463708588377
## 9  00009.fits  80.7 109.859060086462 100.049431730978
## 10 00010.fits 105.5  99.997897400918 94.1966128093806
## 11 00011.fits 120.9 77.4024562916032 75.0277282134922
## 12 00012.fits  73.6 94.2507827487855 88.7258992134475
## 13 00013.fits  74.8 67.5519116849584 64.8299125108343
## 14 00014.fits 376.0 36.0754361323104 36.6057213757243
## 15 00015.fits 224.6 35.7468925035846 36.1121758641171
## 16 00016.fits  89.2 131.116107674721 121.934857482697
## 17 00017.fits  89.2  131.23955951129 122.379451808434
## 18 00018.fits 187.2 121.164629652973 114.069774307889
## 19 00019.fits  72.1 56.6401551120061 52.9605617526288
## 20 00020.fits 171.6 127.847453299838 119.590226650698

Main conclusion here is that we are not able to predict the originally declared SNR So, let's work additionally with

Learning from how adding noise works

Let's start from one sampled spectrum like a blue one with index 1000. Its general picture looks like

plot(1:length(delo1[iddx, ]), delo1[iddx, ], type = "l", col = 3)

plot of chunk unnamed-chunk-11

with a spectral power graph like,

ta <- spectrum(data = as.numeric(delo1[iddx, ]))

## Error: argument "x" is missing, with no default

se <- spectrum(ta)

## Error: object 'ta' not found

if we have a look at the statistical parameters for this distribution we have 3.2938 × 10^-4 as mean and 8.0694 × 10^-5 as standard deviation.

Signal and Periodograms

Let us plot four plots per spectrum, which is its blue signal portion, its smoothed blue periodogram and then, periodograms for L and H resolutions.


for (i in 1:100) {
    linea <- sprintf("%12s %10s %10s %10s %10s", osnr[i, 1], osnr[i, 2], osnr[i, 
        3], osnr[i, 4], osnr[i, 5])

    par(mfrow = c(2, 2))
    tit <- paste(rownames(rd_B[i, ]), " B fraction")
    plot(1:length(rd_B[i, ]), rd_B[i, ], type = "l", main = tit, xlab = "Wavelength", 
        ylab = "Energy")
    plot(sp_B[[i]], main = tit)
    tit <- paste(rownames(rd_L[i, ]), " L fraction")
    plot(sp_L[[i]], main = tit)
    tit <- paste(rownames(rd_H[i, ]), " H fraction")
    plot(sp_H[[i]], main = tit)
    par(mfrow = c(1, 1))

    cat("\n --------------------------------------------------------\n")
    cat(paste("\n  NAME      ORIG_SNR     BLUE_SNR            L_SNR            H_SNR\n", 
        linea, "\n", sep = ""))

}