Information-theoretic approach to Fouriertransform spectrometry
Alessandro Barducci
Consiglio Nazionale delle Ricerche – Istituto di Fisica Applicata “Nello Carrara,”,Via Madonna del Piano 10, I50019 Sesto Fiorentino (Fi), Italy ([email protected])
Received September 14, 2010; revised December 28, 2010; accepted December 28, 2010;posted January 5, 2011 (Doc. ID 134951); published March 8, 2011
1. INTRODUCTIONFourier transform spectrometry (FTS) and Hadamard trans-form spectrometry (HTS) are the leading implementations ofmultiplex spectrometry, their main difference being that Four-ier transform (FT) spectrometers realize interferometricamplitude multiplexing while multiplex dispersive (MD) spec-trometers put into operation intensity multiplexing. As a com-mon point, any multiplex spectrometers measure not thespectrum itself, but a complex transformation of it, thus re-quiring specific data preprocessing for transforming backthe observed data into the desired spectral estimations. Be-cause of this preprocessing, the observation of narrow spec-tral regions requires a complete multiplex measurement,lowering the overall efficiency of all types of multiplex spec-trometry for many applications. For a long time, multiplexspectrometry has been believed to be able to improve the ex-perimental signal-to-noise ratio (SNR) over traditional non-multiplexing spectrometry [1–3]. However, recent works[4,5] have demonstrated the existence of heavy radiometricdisadvantages of FTS and MD spectrometry (MDS). Despitethese radiometric drawbacks, FTS and MDS together repre-sent the widest field of spectrometry with an increasing num-ber of applications.
In this work, we describe a novel (to the best of our knowl-edge) approach to FTS that is able to mitigate its radiometricdrawbacks, expanding its application to the observation ofnarrowband sources. This new approach, called wavelettransform spectrometry (WTS), is founded on the adoptionof the bandpass sampling scheme for performing interfero-metric measurements of a source in narrow spectral intervals,such as those resulting from the utilization of a narrow spec-tral filter. Different from the WTS discussed here, previous at-tempts to optimize interferogram sampling in FTS have beenmainly directed to improve the resolution achieved in spectralestimations [6,7].
The theory of the noisy communication channel [8] is ap-plied to an instrument in order to investigate the performanceof various spectroscopic architectures in terms of the trans-mitted signal entropy. In such a way, we are able to show thatan ideal nonmultiplexing dispersive spectrometer providesthe best information content and no multiplex scheme adopt-ing the same input port can improve this level of performance.This analysis confirms previous findings [4,5] concerning thetrue disadvantage of the multiplex effect (Fellgett’s advan-tage), while validating potential benefits coming from thewell-known étendue (Jaquinot) advantage. We also show thatWTS overcomes the radiometric and SNR limitations of FTS,constituting a viable spectroscopic architecture for visible andinfrared measurements.
The paper is organized as follows. In Section 2 we give anoverview of the mathematical framework of standard FTS.This mathematical structure is expanded in Section 3, wherethe theory of bandpass sampling is applied to the interfero-metric observation of narrowband sources. In Section 4 weprovide experimental evidence of the functionality of this no-vel spectroscopic technique (WTS), using simulation out-comes and interferometric data collected with a Sagnactriangular interferometer. Section 5 is devoted to assessingthe performance of FTS, WTS, MDS, and traditional nonmul-tiplexing spectrometry by means of the theory of information.Finally, Section 6 summarizes open problems and draws someconclusions.
2. BACKGROUND AND MATHEMATICALMETHODSThe traditional approach to FTS can be summarized by con-sidering the application of the sampling theory to the interfer-ogram of a two-beam spectrometer over a finite range ofoptical phase difference (OPD). In such a way, it is possibleto write a system of equations that thoroughly describes themain features of this kind of spectrometry. The following
Alessandro Barducci Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 637
equation is the theoretical modeling for the analog interfero-gram [1,4,9,10] observed at the spatial coordinate x of themeasurement domain:
IðxÞ ¼ FTfiðκÞg þ FTfiðκÞgjx¼0
2; ð1Þ
where IðxÞ is the analog interferogram made of the constantterm FTfiðκÞgjx¼0 and the autocorrelation functions of the in-coming electromagnetic intensity. Using the complex FT inlieu of the basic cosine transform (CT) simply requires extend-ing the source spectrum iðκÞ to negative wavenumbers κ ¼1=λ assuming even symmetry ið−κÞ ¼ iðκÞ. Correcting thesampled interferogram ISðxÞ for the continuous dc termFTfiðκÞgjx¼0, and applying the inverse FT, the following esti-mate ~iðκÞ of the source spectrum is obtained [4]:
~iðκÞ ¼ FT−1fISðxÞg
¼� Xm¼þ∞
m¼−∞
iðκ −mκSÞsinc½ðκ −mκSÞdOPD��
� sincð2κOPDmaxÞ; ð2Þwhere dOPD is the OPD range of a sample (pixel pitch),OPDmax is the maximum digitized OPD, and sincðxÞ ¼sinðπxÞ=ðπxÞ. The theory of ideal sampling provides a sys-tematic mathematical interpretation of the above relationship,in which FT−1fISðxÞg is the sum of infinite aliases iðκ −mκSÞm ¼ �1;�2;… of the original source spectrum iðκÞ asmodulated by the term sinc½ðκ −mκSÞdOPD� and smoothedby the factor sincð2κOPDmaxÞ that limits the spectral resolu-tion of the instrument [4,9,10]. This concept is illustrated inFig. 1, where the natural spectrum iðκÞ is shown together withits replicas iðκ −mκSÞ and the dc term FTfiðκÞgjx¼0. When thesampling frequency κS is too low κS < 2κmax, adjacent spec-trum replicas overlap each other, giving rise to huge errorsin every spectral estimation. This phenomenon is known asaliasing. The minimal sampling frequency that does not incurin aliasing effects corresponds to Nyquist’s frequency, asspecified in the next relationship:
κS ≥ 2κmax; ð3Þwhere κmax is the maximumwavenumber limit of the observedspectrum, i.e., iðκÞ ¼ 0∀κ > κmax. Figure 1 shows an exampleof interferogram sampling performed at the minimal admittedfrequency (only the lower order aliases are shown). Very si-milar reasoning has been pointed out by a number of authorsin the past [4,9,10].
We notice that the minimum wavelength that can be recon-structed from an interferogram measurement λmin ¼ 1=κmax
completely determines the interferogram sampling frequency,while the sampled OPD range depends on the requested spec-tral resolution δκ ¼ 1=OPDmax, which is assumed to be the fullwidth null to null (FWNN) of spectral estimations. In otherwords, these two application parameters λmin and δκ fix thenumber of interferogram samples M to be collected and theirsampling step. As is known, an identical numberM of spectralestimates covering the wavenumber interval ½0; κmax� can bededuced from a single-sided interferogram, provided thatthe employed detector can sense this broad spectral range½λmin;þ∞�. For a two-sided interferogram, half of the spectraldata points are traded for information related to the complexvalued spectral phase. This behavior enlightens a typical
drawback of traditional FTS, linked to the lack of detectorsable to sense almost any electromagnetic frequency. There-fore, traditional FTS may have a poor sampling efficiency,meaning that the number of acquired samples M is usuallygreater than the number of interpolated spectral channelsK . The situation is even worse for those applications that re-quire the observation of narrowband sources or of smallwavelength ranges of a broadband source, i.e., the observa-tion of selected spectral lines as in many spectroscopic inves-tigations of stellar atmospheres [11] or Sun-inducedfluorescence of vegetation [12]. It is useful to measure thesampling efficiency ηS of a FT spectrometer as
ηS ¼ KM
: ð4Þ
We note that a dispersive, nonmultiplexing instrument canalways be designed so as to have unitary sampling efficiency,while MDS incurs in sampling efficiency losses similar tothose of FTS, at least when narrowband spectral measure-ments are performed.
Recently, Barducci et al. [4] noted that the interferogram al-ways contains a constant signal component FTfiðκÞgjx¼0=2which does not convey information, while holding most ofthe interferogram power. Because of Plancherel’s theorem,the informative signal component FTfiðκÞg=2 has half the am-plitude of the source spectrum iðκÞ=2 but it is subjected to thesame noise as the entire interferogram IðxÞ ¼ ½FTfiðκÞgþFTfiðκÞgjx¼0�=2. As a consequence, the SNRof the effective sig-nal (informative) of an interferometer is always far below theSNR of its physical counterpart.
The most important result depicted in [4] regards the radio-metric precision δFTSiðκÞ requested to a FT spectrometer forreaching a given spectral resolution δκ when observing asource having full bandwidth B. These authors demonstratedin lemma 1 of their paper that
δFTSiðκÞ ≅AFTS
2FTfiðκÞgjOPD¼0
�δκB
�lþ1
¼ AFTS
2FTfiðκÞgjOPD¼0
1Klþ1 l ≥ 0: ð5Þ
Fig. 1. (Color online) Spectral estimation computed by inverse FT ofthe sampled interferogram of a source with a broadband spectrum.The orange curves are the replicas (aliases) introduced by sampling,and they can be separated from the baseband as long as the samplingfrequency κS is greater than Nyquist’s limit 2κmax. The pulse at the zerowavenumber in the baseband spectrum represents the dc contributionFTfiðκÞgjx¼0.
638 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Alessandro Barducci
Equation (5) has been deduced by applying the Riemann–Lebesgue lemma to the FT of the lth iðκÞ derivative that is as-sumed to be continuous and Lebesgue integrable [4]. The AFTS
term is the unknown amplitude of the asymptotic limit of theanalog interferogram, i.e., the FT of the source spectrum iðκÞ.Equation (5) has been obtained stemming from the derivativeproperty of the FT operator: FTfdliðκÞ=dκlg ¼ ðB · OPDÞl·IðB · OPDÞ. We can apply the Riemann–Lebesgue lemma tothis relationship, obtaining a sufficient condition forFTfdliðκÞ=dκlg to be continuous and Lebesgue integrable:
limOPD→þ∞
ðB · OPDÞlþ1IðB · OPDÞ ¼ 0: ð6Þ
This equation can easily be turned into a condition on theasymptotic limit of the interferogram:
limOPDmax→þ∞
IðB · OPDmaxÞ ≤AFTSFTfiðκÞgjOPD¼0
21
ðB · OPDmaxÞlþ1
¼ AFTS
2FTfiðκÞgjOPD¼0
�δκB
�lþ1
: ð7Þ
It is worth noting that the Riemann–Lebesgue lemma states asufficient condition for the existence of direct and inversetransforms (Lebesgue integrable spectra and interferograms),but it is not at all necessary. The rectðÞ and sincðÞ are exam-ples of conjugated functions nonintegrable in the sense ofLebesgue that admit convergent Fourier integrals, i.e., a gen-eralized FT. These two functions do not represent likely radia-tion sources, while Gaussian and Lorentz profiles are typicalexamples of source spectra that also are Lebesgue integrablewith their derivatives. For a realistic source spectrum iðκÞEqs. (5)–(7) are usually obeyed, posing overwhelming boundsto the spectral resolution that can be achieved by FTS. Theasymptotic behavior depicted in Eqs. (5)–(7) clearly showsthat in the limit of a fine enough spectral resolution (largeOPDmax ≈ 1=δκ), any FT spectrometer is disadvantageous withrespect to a traditional dispersive instrument [4], the re-quested radiometric accuracy of which is directly propor-tional to the assigned spectral resolution δDISPiðκÞ ≅ iðκÞδκ .
3. THEORY OF WAVELET TRANSFORMSPECTROMETRYIn this section, we describe a new configuration and opera-tional procedure of FTS that is able to carry out high-spectral-resolution measurements of narrowband andbroadband sourceswith high accuracy and improved samplingefficiency. In order to explain this approach, we consider anarrowband source whose spectrum vanishes outside a tinyspectral interval centered around κc.
This type of spectrum is qualitatively illustrated in Fig. 2(blue curve), and it can be obtained filtering a broadbandsource with a suitable bandpass filter having a bandwidthΔκ ¼ κmax − κmin. For the sake of simplicity, we will assumethe filtered spectrum to be symmetrical around its centralwavenumber κc or, equivalently, that κc is the geometrical mid-point of its band ends κc ¼ ðκmax þ κminÞ=2. It is known that thelarge portion of space embedded within the two-sided band-pass spectrum can be reliably utilized for undersampling thesignal without originating aliasing. This kind of technique islargely employed in telecommunication engineering [13],where it is called bandpass sampling or simply undersampling.Sampling performed by means of this technique is subject to
some constraints that avoid alias to overlap each other, asstated by the following equation:
κs ≥ Δκ ¼ κmax − κmin
− κmin þ nκS ≤ κmin
− κmax þ ðnþ 1Þκs ≥ κmax: ð8ÞThe first relationship in Eq. (8) guarantees that replicas ori-
ginated by spectral components at positive (or negative)wavenumbers do not overlap with themselves. However, thiscondition is not able to prevent overlapping between the re-plicas of the section at negative wavenumbers and the base-band source spectrum component at positive wavenumbers.This possibility is precluded also when the last two relation-ships in Eq. (8) are obeyed. In these equations, aliases ofgeneric order n and nþ 1 [the value of index m in Eq. (2)]are adjacent to the signal baseband centered at κc without ori-ginating superposition. This concept, together with the posi-tion of spectral aliases in bandpass sampling, is depicted inFig. 2. Figure 2 shows the natural bandpass spectrum ofthe observed source with the aliases originated by the spec-trum element at negative wavenumbers in the neighbor of thebaseband element at positive wavenumbers. Recalling thatκc ¼ ðκmax þ κminÞ=2, simple mathematical steps allow us to re-formulate Eq. (8) as
κs ≥ Δκ ¼ κmax − κmin κs ≤2κmin
n¼ 2κc −Δκ
n
κs ≥2κmax
nþ 1¼ 2κc þΔκ
nþ 1: ð9Þ
We note that the last two relationships in Eq. (9) requiresthe sampling frequency κs to be greater than twice the band-widthΔκ, making the first relationship unnecessary. A supple-mentary property that can be graphically deduced (other,from Fig. 2) is that the optimal sampling frequency κs is ob-tained imposing that the center of the replicas n and nþ 1have equal distances from the center κc of the original
Fig. 2. (Color online) Spectral estimation computed from the inter-ferogram of a source with narrowband spectrum after bandpass sam-pling. Aliases introduced by sampling do not overlap the baseband(blue curve) as long as the sampling frequency κS is matched tothe source spectral characteristics. It is assumed that the constantterm FTfiðκÞgjx¼0 has been removed before spectral estimation areperformed. When this contribution is not completely rejected, possi-ble disturbances can affect spectral estimations utilizing the bandpasssampling scheme.
Alessandro Barducci Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 639
spectrum:
κc − ð−κc þ nκsÞ ¼ ½−κc þ ðnþ 1Þκs� − κc: ð10ÞWhen this condition holds true, we have
κs ¼4κc
2nþ 1κs ≥ 2Δκ: ð11Þ
Equation (11) defines the bandpass sampling frequencythat is most robust against aliasing; i.e., the sampling fre-quency that minimizes the probability of aliasing because itplaces both the two nearest aliases at the maximum distancefrom the central spectrum. Evidently, the parameter n can beconsidered as a sampling gain achieved by this particularapproach.
Possible solutions of Eq. (11) are represented graphically inFig. 3, where the hyperbolic curve is the plot of the first re-lationship in Eq. (11) and the horizontal line is the lowerbound κs ≥ 2Δκ. The blue part of the hyperbolic curve isthe ensemble of well-behaved solutions of Eq. (11), and highern s corresponding to sampling frequencies below the horizon-tal line give rise to aliasing. We can show that the values of nthat do not produce aliasing obey the following equation:
2nþ 1 ≤ 2κcΔκ : ð12Þ
The above equation specifies that the most favorablesampling gain n depends on the “geometry” of the sourcespectrum as specified by the central wavenumber κc to wave-band Δκ ratio. We also note that this ratio precisely defineswhat we mean by the phrase “narrowband source.” The higherthis ratio, the more favorable the sampling gain allowed bybandpass sampling will be. It is worth noting that the probabil-ity of aliasing can be further reduced by overestimating thebandwidth Δκ in Eqs. (8)–(12) with respect to the true band-width (W) of the input spectrum.
Adopting in Eqs. (11) and (12) a bandwidth Δκ slightlygreater than necessary (Δκ ≥ W) has the effect of addingan extra guard band that protects the sampling procedurefrom aliasing due to minor imperfections in system implemen-tation. Otherwise, when working at the minimal admitted
sampling frequency, these possible implementation troubleswould become critical. The guard band can be controlledusing the simple relationship reported below:
Δκ ¼ ð1þ aÞW; ð13Þwhere a is the relative measure of the guard band width asexpressed in units of W . We point out that the input narrow-band spectrum can be obtained filtering a broadband sourcewith a suitable filter. The adoption of a tunable monochroma-tor, such as a Fabry–Perot filter, would allow the sequentialobservation of large spectral regions with successive mea-surements of many narrow spectral intervals. In each mea-surement, the number of collected interferogram samples isonly slightly greater than (or equal to) the number of interpo-lated spectral samples due to a possibly nonzero guard band.If Hðκ; κc; WÞ is the filter transfer function tuned at the centralwavenumber κc and the dc term is disregarded, the analoginterferogram can be written as
IðxÞ ¼ FTfHðκ; κc; WÞiðκÞg
¼Z þ∞
−∞
Hðκ; κc; WÞiðκÞ expð−2πjκxÞdx: ð14Þ
The equation above closely resembles the definitions of theshort-term FT [14], and the wavelet transform (WT) [15]. Thus,we designate the approach described in this section as wave-let transform spectrometry (WTS).
4. WAVELET TRANSFORMSPECTROMETRY: PROOF OF CONCEPTIn this section, we give a simple proof of concept of the WTS,using simulations and interferometric measurements col-lected with a Sagnac common-path interferometer.
A. SimulationsInterferogram bandpass sampling with variable number ofsamples have been simulated according to Eqs. (1)–(13) usingGaussian sources clipped in a narrowband spectral interval.The resulting raw interferograms have been degraded by add-ing uniform white noise for reaching a linear SNR around 300,which is typical of many experimental setups. As shown inSection 2, this SNR is characteristic of the physical signal only[4], while the informative part of the interferogram has a muchlower SNR. According to our experience in this field, and con-sidering that we are observing narrowband sources, we as-sume that the effective SNR (SNReff ) should be between 50and 100. Because the WTS performs the undersampling ofthe FTS interferogram, it also extracts a subseries from theoriginal noise sequence, affecting the simulated FTS measure-ment. Extracting a subseries from a white noise sequence ofsamples originates a new white noise sequence that is inde-pendent of the original one. Therefore, the difference betweenFTS and WTS spectral estimates would be affected by a noiseof higher amplitude (relative noise gain of
ffiffiffi2
p). Large guard
bands have been utilized in order to make discernible the pos-sible effects of aliasing in spectral estimations extended tovery broad spectral intervals (0:5–2:5 μm−1), where basebandreplicas generated by bandpass sampling are evident.OPDmaxs in the range from 50 to 1500 μm have been adopted,originating raw interferograms composed by from 4000 up to15,000 independent samples. The agreement between WTSand FTS spectral estimations has been assessed, calculating
Fig. 3. (Color online) Determination of the optimal sampling fre-quency κS . The horizontal line in the lower part of the figure repre-sents the inferior limit for the sampling frequency, which is twicethe full bandwidth of the source. The hyperbolic curve representsthe first relationship in Eq. (11), which is the basic constraint to avoidaliasing. The sampling frequency in this equation is a function of thesampling gain n. The purpose of bandpass sampling is adopting the nthat determines the lowest sampling frequency greater than 2Δκ. Thevertical axis is in logarithmic scale.
640 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Alessandro Barducci
their correlation in the considered spectral interval and therelative difference of the area subtended by the reconstructedsource peak. We note that the area of the estimated sourcepeak should be greater for the WTS estimation due to possiblealiasing from adjacent replicas. The simulation results aresummarized in Fig. 4.
Figure 4(a) plots the raw and decimated interferogram ver-sus OPD for a Gaussian source centered at 800:0 nm with15:0 nm of standard deviation and OPDmax ¼ 200 μm. Thesetwo interferograms give rise to the spectral estimations de-picted in Fig. 4(b), where the wide spectral interval allowsthe investigation of the principal effects connected with band-pass sampling scheme. The area difference between WTS andFTS spectral estimations amounts to 1.2% of the area of theFTS estimation when these parameters are averaged in theinterval (768–832nm). In the same interval, the correlationbetween the two estimates is R2 ¼ 0:9995, meaning the twoarchitectures nearly obtain the same spectral shape. Eachbaseband replica visible in this figure shows the characteristicringing due to the convolution with the sincð2κOPDmaxÞ ofEq. (2). The far wings of this undulation clearly perturb adja-cent aliases (long-range aliasing), representing a possiblesource of errors for WTS estimations with low spectral reso-lution (moderate OPDmax). However, the area of the peak re-trieved by FTS is, in this case, larger than that obtained byWTS, indicating negligible effects of adjacent replicas in theWTS estimation. This behavior is further investigated bymeans of the simulation reported in Fig. 4(c), where low-re-solution (OPDmax ¼ 50 μm) FTS and WTS spectral estimationsare shown for a Gaussian source clipped to the region680:0 nm–720:0 nm with 700 nm of central wavelength, and5:0nm of standard deviation. WTS estimation has been ob-tained with 20 interferogram samples (1024 samples for theFTS simulation) and a 50% guard band interval (a ¼ 1:5).The lower signal in the guard band range between adjacentreplicas proves the secondary effect of the far wings of thesincð2κOPDmaxÞ function, demonstrating that theWTS schemeis able to efficiently control aliasing. The area of the WTS peakis 2.6% smaller than the peak area obtained by the FTS esti-mation, confirming the small effect of aliasing also for this si-mulation. The two methods reconstruct nearly the samespectral shape, characterized by a mutual correlation ofR2 ¼ 0:9992. Figure 4(d) shows an example of FTS andWTS estimations at higher spectral resolution (OPDmax ¼1500 μm), adopting a central wavelength of 1000nm anda ¼ 2:0. It is worth noting that the WTS is able to performits estimation using only 93 of the 14,000 samples employedby the FTS, obtaining an area difference of 2.2% with respectto the standard FTS estimate. Also, in this case, the area of thesource peak computed by FTS exceeds theWTS one, probablybecause of its greater height. The correlation between the twoestimations is as high as usual R2 ¼ 0:9989, confirming thegood agreement between the spectral estimations obtainedwith the FTS and the WTS schemes.
B. MeasurementsThe experimental setup has been made as smooth as possible,thus we avoided the introduction of a tunable narrowband fil-ter coupled with a wideband source, adopting a simpler con-figuration constituted by narrowband laser sources. Twotypes of sources have been employed: a He–Ne and various
Fig. 4. (Color online) Examples of simulation of interferogram band-pass sampling [Eqs. (8)–(10)] with variable number of samples. Green(blue) curves and markers indicate bandpass (standard) sampled in-terferograms and estimated spectra. Aliases generated by bandpasssampling in a wide spectral interval are evident. (a) Raw (blue curve)and bandpass sampling (green markers) interferogram of a Gaussiansource centered at 800:0 nm with 15nm of standard deviation andclipped to the 780–820nm interval (OPDmax ¼ 200 μm, a ¼ 3).(b) Spectral estimations obtained from interferograms in (a). Let usnote the evident ringing that limits the spectral resolution of the dataand may degrade the spectral estimations obtained with WTS due tolong-range aliasing. (c) FTS and WTS spectral estimations for a simu-lated Gaussian source (700nm central wavelength with 5:0 nm ofstandard deviation) clipped to the region 680:0–720:0 nm withOPDmax ¼ 50 μm. WTS estimation has been obtained with 20 interfer-ogram samples (1024 samples for the FTS simulation) and 50% guardband interval (a ¼ 1:5). (d) FTS and WTS spectral estimations for asimulated Gaussian source (1000nm central wavelength with5:0 nm of standard deviation) clipped to the region 993:0–1007:0 nmwith OPDmax ¼ 1500 μm. WTS estimation has been obtained with93 interferogram samples (15,000 samples for the FTS simulation)and 100% guard band interval (a ¼ 2).
Alessandro Barducci Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 641
solid-state lasers. The adopted instrument was the ALISEOSagnac imaging interferometer, a detailed description ofwhich can be found in [16]. Using this FT spectrometer, fullrange raw interferograms have been gathered with an SNRaround 100, and then bandpass sampling has been obtainedby decimation according to Eqs. (10)–(13).
Figure 5 shows an example of a raw interferogram (He–Nelaser at 632 nm of central wavelength) acquired with theALISEO and its optimal decimation for bandpass sampling.Raw interferograms always are made up of 1024 digital sam-ples, while the decimated interferogram in Fig. 5 contains 14samples only. The two interferograms in Fig. 5 have been pro-cessed in order to remove the constant dc offset FTfiðκÞgjx¼0typical of the interferometric signal and the distortion intro-duced by optical vignetting.
Four examples of spectral estimation, although not radio-metrically calibrated, are shown in Fig. 6. Figure 6(a) showsthe spectral estimation versus the wavelength obtained fromboth the raw interferogram and its bandpass sampled versionwhen observing a laser diode source with central wavelengthof 780nm. In this case, the spectral estimation has been ex-tended from 500 to 1000 nm in order to highlight the manyaliases that limit the free spectral range of the bandpass sam-pling procedure. We note the presence of aliases originated byboth the baseband source spectrum at positive wavenumbersand its mirror-symmetric satellite at negative wavenumbers.The two origins of the replicas are easily recognized due tothe slight asymmetry in the baseband spectrum of this so-lid-state laser as observed by ALISEO. Let us note that thealiases’ positions are not exactly symmetrical around thebaseband spectrum: a behavior due to the aforementionedasymmetry and to the selection of an undersampling fre-quency that approximately matches Eq. (11). In fact, choosingan undersampling step integer multiple of the raw samplingstep avoids signal interpolation that is otherwise necessary.Spectral estimations in Fig. 6(a) have been obtained as themodulus of the inverse FT of the concerned interferogram,an algorithm that helps to minimize possible phase errorsdue to a misknowledge of the true position of the interfero-gram center [4]. We also note that the tiny secondary peaksbetween the aliases are probably due to the convolution withthe sincð2κOPDmaxÞ term in Eq. (2). Because of the moderateOPDmax digitized by ALISEO, this filtering term is significantly
broad and is not well suited for the WTS sampling. We pointout that interferometric acquisitions at mild OPDmax do notallow the reconstruction of a high number of in-band spectralsamples, significantly lowering the sampling efficiency ofWTS. The area difference between FTS and WTS estimationsamounts to 6.5%, with the peak reconstructed by the WTSmethod being larger than that obtained by FTS. Also the cor-relation between the two spectra is significantly worse thanthose achieved with simulations R2 ¼ 0:9453. In this case,the area of the peak retrieved by the WTS is greater than thatobtained by FTS. Figure 6(b) shows the spectral estimationfor three additional laser sources centered at 808 (laserdiode), 530 (laser diode), and 632nm (He–Ne laser), respec-tively. The dark crosses shown in this figure depict theabsolute difference between the FTS and WTS estimations.An independent measurement has been executed for each
Fig. 5. (Color online) Example of interferogram acquired with theALISEO instrument and its bandpass sampling after removing vignet-ting effects and the dc offset: solid line, FTS interferogram (1024samples) of a He–Ne laser source; and filled rhombuses, WTS inter-ferogram of a He–Ne source after bandpass sampling (12 samples).
Fig. 6. (Color online) Examples of spectral estimations obtainedfrom interferograms acquired with the ALISEO instrument usingFTS and WTS. Blue curves always represent the FTS spectral esti-mate, which always equals the WTS estimate (green curves). (a) Spec-trum of a laser diode (780nm central wavelength) computed from thecomplete interferogram (1024 samples) and the corresponding band-pass sampling estimate (16 samples). Estimation has been executedtaking the modulus of the inverse FT, extended to a broad spectralinterval in order to show the aliases originated by controlled interfer-ogram undersampling. The decreasing alias amplitude is a combinedeffect of the OPD spectral dispersion and the 1=λ2 spectral densityfactor typical of the wavenumber domain. (b) Spectra of three lasersources at 530 (diode), 632 (He–Ne), and 808nm (diode) of centralwavelength, respectively. FTS spectra have been inferred from1024 samples interferograms, while WTS estimates of the samesources have been obtained considering interferograms made up of40 (530), 12 (632), and 14 samples (808nm). All spectra have beencalculated as the inverse DCT without any radiometric calibrationof the resulting signal; hence, they contain large contributions fromphase error. Dark markers (crosses) represent the absolute FTS-WTSdifference.
642 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Alessandro Barducci
of the three sources, and then the inverse digital CT (DCT) hasbeen employed as a spectral estimator for both FTS and WTS.The WTS versus FTS area difference and correlation are 8.8%and R2 ¼ 0:9827 for the source centered at 530, 0.4% and R2 ¼0:9944 for the laser at 632, 0.4% and R2 ¼ 0:9960 at 808 nm. Forall these examples of FTS and WTS spectral estimations, thepeak revealed by FTS has shown a greater area and heightthan that interpolated by WTS. Because the spectra have beenobtained using the DCT estimator, the comparison in Fig. 6incorporates possible differential effects of the phase errorbetween the two methods here considered in addition to un-correlated noise. This circumstance confirms the good agree-ment between WTS and FTS spectral estimations.
The coarse spectral resolution δκ shown in all plots of Fig. 6is largely due to the moderate OPDmax digitized by the inter-ferometer in these measurements. For instance, the laser at530 nm has been sampled up to 29:45 μm of OPDmax, which,in turn, bounds the experimental spectral resolution δλ to9:61nm FWNN. Considering that the original laser diodehas an intrinsic bandwidth around 5 nm FWHM, it can be con-cluded that the apparent bandwidth of 12nm FWHM for thesource portrayed in Fig. 6(b) is coherent with the interfero-meter configuration. As already noted, the coarse spectralresolution of ALISEO is suboptimal for performing WTS mea-surements; hence, data in Figs. 5 and 6 should be retained forproof of concept only. In the examples of Fig. 6, the convolu-tion of the source baseband spectrum with the broadsincð2κOPDmaxÞ instrument response function originates spur-ious signals in the guard band frequency intervals betweenadjacent replicas and possible long-range aliasing in WTS es-timates. This phenomenon is clearly shown in Fig. 6(a).
We note the good agreement between spectral estimationsobtained with FTS and WTS. The difference between spectraobtained with these interferential methods has the same am-plitude as the noise affecting the informative signal compo-nent of the interferogram, which, for these measurements,should have an effective SNR between 30 and 60. This agree-ment clearly demonstrates the functionality of bandpass sam-pling for observing narrowband sources. The more generalapplication of this sampling scheme to interferential spectro-metry leads to WTS by the interposition of a tunable bandpassfilter (e.g., a Fabry–Perot device) between the two-beam inter-ferometer and the examined source. It is remarkable that twosources in Fig. 6 have been observed with 12 and 14 interfer-ogram samples, allowing the retrieval of their spectrum in thedesigned spectral band. We point out that WTS can be an es-sential tool for optimizing the performance of FT soundersand spectrometers without imaging capabilities. For interfe-rometers adopting a fixed FTS sampling scheme (e.g., usuallyimaging interferometers such as ALISEO), the bandpass sam-pling approach can be selected to lighten the data processingburden typical of FT spectrometry.
5. COMPARING WAVELET TRANSFORMSPECTROMETRY WITH OTHERSPECTROMETRY ARCHITECTURESThe WTS approach has significant advantages over standardFTS in terms of both sampling efficiency and requested radio-metric accuracy. If no guard band is added to the bandpasssampling, WTS achieves unit sampling efficiency like nonmul-tiplexing dispersive spectrometers. Whenever some guard
band is added [a > 0 in Eq. (13)] and the minimal samplingfrequency κs ¼ 2Δκ is attained, the WTS reaches the followingsampling efficiency:�
M ¼ κsOPDmax ¼ 2ð1þ aÞWOPDmax
K ¼ 2Wδκ ¼ 2WOPDmax
⇒ ηS ¼ 11þ a
; ð15Þ
meaning that the WTS can achieve a sampling efficiency arbi-trarily close to unity.
The behavior depicted in Eq. (5) also limits the radiometricperformance of WTS, but in this circumstance, the actualbandwidthΔκ is smaller than the source bandwidth (Δκ ≪ B)because the entire source spectrum B is measured observing asequence of adjacent narrow spectral intervals. Hence, ineach measurement, the desired spectral resolution δκ givesrise to a smaller number of in-band spectral channels andthe radiometric accuracy δWTS
iðκÞ requested to a WT spectro-meter is a less critical parameter in instrument design andsource estimation. This property can also be stated as a directconsequence of the uncertainty principle. Equation (16)recaps this important radiometric advantage of WTS overstandard FTS:8><>:
δFTSiðκÞ ≅AFTS2 FTfiðκÞgjOPD¼0
�δκB
�lþ1
δWTSiðκÞ ≅
AFTS2 FTfiðκÞgjOPD¼0
�δκΔκ
�lþ1 ⇒
δWTSiðκÞδFTSiðκÞ
≅
�BΔκ
�lþ1
:
ð16ÞIt should be noted that FTS and MDS of wideband sources
are subject to large photonic noise [4,5] introduced by thepredominant noninformative signal component. This contri-bution has lower amplitude in WTS due to the stronglyreduced spectral interval Δκ, over which signals are inte-grated (lowering the dc term amplitude). If FTS and MDSadopt the same source prefiltering of WTS, an equal levelof photonic noise would result, but in this case, the samplingefficiency of FTS and MDS would be extremely poor. In otherwords, FTS and MDS can match only one aspect of the WTSperformance: they can achieve the same sampling efficiencybut with a larger photonic noise or the same photonic noise atthe price of a reduced sampling efficiency.
A. Information Theory and SpectrometryA useful approach for investigating the efficiency of differenttypes of spectrometers is to consider the information entropyof the data they gather. Speculations about the entropy in adataset began with the pioneering work of Shannon [8],who pointed out the rate at which bits can be communicated(transmitted and received) through a noisy transmission chan-nel with arbitrarily small frequency of error. The fundamentalresult of his work is summarized in the following equation:
h ¼ B log2P þ NN
; ð17Þ
where h is the maximum information entropy, or informationrate (often measured in bits per second or bits per word), thatcan be transmitted by a channel having a bandwidth B, apower P for the available signal, and N for the noise. Thismaximum entropy h is also known as channel capacity andis usually indicated by the symbol C [8].
When the observed signal is the specific intensity iðκÞ of theradiation field, the available signal sðκÞ is proportional to iðκÞthrough a measurement constant C accounting for the
Alessandro Barducci Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 643
combined effect of a finite integration time τ, effective field ofview Ω, integration area A, and spectral resolution δκ . The ra-diation field specific intensity iðκÞ is called “spectral radiance”in the remote sensing literature, representing the amount ofradiant power in a specific spectral interval that is transportedacross an element of area within an element of solid angle.While iðκÞ can be thought as a continuous electromagnetic ra-diation field, the signal sðκÞ is proportional to the integer num-ber of photons collected by the telescope aperture in time τ,and, as such, it is subject to the random variability originatedby the photonic noise that obeys the Poisson statistics withstandard deviation σsðκÞ. In this case, information entropycan be associated to the signal of the observed source evenbefore its measurement performed by the detector. For thispurpose, B can be interpreted as the maximum frequencyin the signal sðκÞ, which determines its most favorable sam-pling frequency f s ¼ 2B. This minimal sampling frequency,also known as the Nyquist limit, has the dimension of the in-verse of the wavenumber κ. The only relevant difference withrespect to the standard theory of communication is that thesignal and the related information flow are dispersed overthe wavenumber κ rather than the time t. Hence, the telescopeacts as a noisy communication channel with an average powerlimitation, which is bounded by the intensity of the observedsource. This channel inputs the ideal signal iðκÞ and outputsthe noisy signal sðκÞ. Adopting the symbol E½·� for the ensem-ble average operator, theorem 17 in Shannon’s work [8] allowsus to write the capacity hs of the telescope as
≅ f s log2 SNR E½sðκÞ� ¼ CiðκÞ C ¼ τAΩδκ; ð18Þwhere the approximation holds true for large SNRs. The ca-pacity hs represents the maximum number of error-free bitsconveyed by the signal sðκÞ per unit of spectral interval. Itis important to realize that hs is not necessarily the actualamount of the source information content of the signal sðκÞbut an upper bound to this quantity. As an example, sðκÞ couldsimply encode a source of constant intensity, an unrealisticinstance in which the intrinsic information content is vanish-ing. Unfortunately, there is no simple way to provide a satis-factory definition of the effective source information content,independent of the mathematical model adopted for its repre-sentation (e.g., a lossless compression algorithm). However, abasic property of it can be drawn. If γs indicates the actualsource information content transmitted by the channel (tele-scope) and held in the signal sðκÞ, we can write
γs ≤ hs; ð19Þwhere the equal holds true for a source that transmits infor-mation enough to saturate the capacity hs permitted by thecommunication channel (the telescope). The result held inEq. (19) is a direct consequence of theorem 11 in Shannon’swork [8], and it is due to the equivocation introduced in thesignal sðκÞ by the finite capacity of the telescope. The circum-stance γs ¼ hs can be assumed without loss of generality forthe spectrum of a natural source. Let us note that because theinformation content γs largely depends on the source itself, itcannot be considered as a figure of merit of the instru-ment alone.
Nowwe turn our attention to the measurement process thatwill be considered as a succession of steps regarding thetransmission of the input signal to the successive measure-ment devices, as sketched in Fig. 7. Here, the whole instru-ment is represented as a sequence of three blocks: thetelescope, the spectrometer, and the detector. Each elementconstituting the employed instrument is considered as a com-munication channel, which transmits to the next element theinformation received at its input end. The signal sðκÞ is theoutput of the foreoptics block, which feeds the spectrometerthat outputs the signal yðxÞ. At the end we have a detector thatprovides the digital signal oðκÞ (the measurement), which isthe input quantity for the final spectral estimation ~sðκÞ. Everyoutput signal defines, through theorem 17 in [8], the capacity hof the related communication channel that is assumed to belimited in power, and it is connected to the associated sourceentropy γ. For the sake of simplicity, we suppose that the γfactors describe the information content pertaining to the ob-served source only, while in real situations, the employed in-strument may add some information contribution to thisquantity (e.g., spectral attenuation). This sequence of commu-nication channels must obey the theory developed byShannon [8], so that γ ≤ h for any block in Fig. 7. In summary,the symbol h is reserved for indicating the capacities of thechannels in Fig. 7 as measured in bits per spectral interval(density of entropy), while γ will be used for the source infor-mation content expressed in the same units.
Let us note that the signal at the spectrometer exit portmight be dispersed over a nonspectral domain, as in the caseof FTS, where yðxÞ is an interferogram dispersed over an OPDrange. Without a loss of generality, the signal yðxÞ can be de-fined as some type of integral transform of sðκÞ, which reducesto the identity operator for a nonmultiplexing spectrometer. Inorder to account for any kinds of spectrometer, we refer toyðxÞ as a datagram, and we indicate x as the measurementdomain. With such a naming convention, γy is a datagram den-sity of entropy measured in bits per measurement interval.The γy characteristic of being a datagram density of informa-tion might give rise to some uncertainty when comparing it tothe information content provided by the telescope γs, which isinstead a spectral density of information. Possible troubles at
Fig. 7. Scheme of the instrument and the measurement processusing the information theory. Each element necessary for the spectralestimation is considered as a communication channel that obeys theShannon theory. The figure indicates the signals at the I/O ports ofevery channel (elements), their capacity, and the information entropypertaining to the observed source.
644 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Alessandro Barducci
this level can be avoided when the integrated information con-tent is considered. We therefore introduce for each block inFig. 7 two additional integral quantities H and Γ that will beuseful for comparison and performance investigation. The fol-lowing equations recap the definition of the principal entropyparameters for the spectrometer and the telescope channels:
8>>>>>><>>>>>>:
hs ¼ f s2 log2
�1þ s2ðκÞ
σ2s ðκÞ�
Hs ¼RDκ
hsðκÞdκγs ¼ hsΓs ¼ Hs
Φs ¼ Γsτ
8>>>>>>>>><>>>>>>>>>:
hy ¼ κs2 log2
�1þ y2ðxÞ
σ2yðxÞ�
Hy ¼ RDx
hyðxÞdxΓy ¼ R
DxγyðxÞdx
Φy ¼ Γy
τγy ≤ hyΓy ≤ Hy
: ð20Þ
In the equations above, the ensemble average operator hasbeen subtended, and the channel parameters Γ andH quantifythe integrated information content (Γ) and the channel capa-city (H) are measured in bits. Evidently, Dx and Dκ representthe measurement and the spectral intervals for the variables xand κ, respectively. In such a way, Γs½Hs� can be directly com-pared to Γx½Hx�. Additionally, we have introduced in Eq. (20)the overall flux of information Φ measured in bits per secondfor the various channels in Fig. 7. All the parameters inEq. (20) can be defined in a similar way in terms of the signalsoðκÞ and ~sðκÞ. We provide the following definition for a causalinstrument.
Definition 1. An instrument is said to be causal ifΓs ≥ Γy ≥ Γo ≥ Γ~s, the equality being true for ideal equipmentonly.
When Γy > Γs;Γo > Γs;Γo > Γy;Γ~s > Γs;…, we have acommunication channel (device) that adds to its output rele-vant source information that has not been conveyed by its in-put signal. Obviously, such a system violates the cause–effectprinciple and is not realizable in practice. Therefore, theabove definition reflects an axiomatic truth: a noncausal in-strument would guess the source spectrum without the needof measurement. From the above definition and Eq. (20), thefollowing corollary results.
Corollary 1 [or optical data processing inequality (ODPI)].For any causal instruments, the following inequality must holdtrue: Φ~s ≤ Φo ≤ Φy ≤ Φs, the equality being true for an idealinstrument only.
The property depicted by corollary 1 can also be inter-preted by relying on the wider theoretical framework of infor-mation theory. It is evident that the signal yðxÞ of a multiplexspectrometer springs from an optical processing of the inputsignal sðκÞ and that the corresponding source entropy Γy canbe considered as the mutual information between these twosignals. The data processing inequality [17,18] requires thismutual information Γy to be less than or equal to the informa-tion Γs ¼ Hs carried by the input signal sðκÞ. Therefore, cor-ollary 1 is an extension to the optics and spectroscopy of thedata processing inequality detailed in information theory. Thephysical meaning of this principle is that no optical processingexists that can increase the information content of a signal.
Now, utilizing the previous conceptual framework, we com-pare various multiplexing and nonmultiplexing types of spec-trometers. In the comparison, the possible spectral low-passfiltering performed by the spectrometers will be disregardedand we will assume that the considered spectrometers adoptthe same input port. This last point is coherent with instru-
ments having imaging capability, where spatial integrationof spectral measurements is limited by the desired image re-solution. Let us note that some multiplex devices, such as FTspectrometers, do not have specific requirements for the inputslit; hence, they can achieve essential radiometric and SNRadvantages connected to the utilization of a larger input port(the well-known Jaquinot’s advantage). In spite of its rele-vance, this subject is not addressed here.
B. Traditional and Multiplex SpectrometryMultiplexing can be interferometric as in FTS or dispersive asin HTS, but in both cases the output yðxÞ is far above its inputsignal: yðxÞ ≈ E½sðκÞ�M=2, M being, as usual, the number ofgathered samples [1–4,19–24]. On the other hand, the avail-ability of a large amplitude signal yðxÞ makes the channel ca-pacity hy greater than hs. It is important to recall that yðxÞ isthe signal at the exit port of the spectrometer, and, as such, itonly holds photonic noise. The SNR of such a signal, there-fore, is roughly proportional to the square root of thesignal itself. We can show that
hy ≫ hs Hy ≈Hs þM2log2
M2≫ Hs: ð21Þ
In this perspective, multiplex spectrometers can be consid-ered as the result of a technical effort devoted to set up a high-capacity communication channel (Hy ≫ Hs) for transmittinga flow of source information that, according to definition 1and corollary 1, has to be much more tiny Γs ≥ Γy:
Hy ≫ Hs ¼ Γs ≥ Γy ⇒ Hy ≫ Γy Φs ≥ Φy: ð22ÞOn the other hand, a dispersive nonmultiplexing spectro-
meter free from optical losses produces at its exit port exactlythe same signal received in input yðxÞ ¼ sðκÞ and x ¼ κ.Because of this signal identity, the channel entropy as wellas the source information content and its temporal flux mustbe identical to those output by the telescope. Thus, for anideal nonmultiplexing spectrometer, we have
Hnonmulty ¼ Hs Γnonmult
y ¼ Γs Φnonmulty ¼ Φs: ð23Þ
In other words, a loss-free nonmultiplexing dispersive in-strument achieves the maximum allowed temporal informa-tion flux Φnonmult
y ¼ Φs that cannot be matched by anyother causal spectrometer adopting the same input port. Theabove reasoning can be condensed in the following lemma.
Lemma 1. When possible optical losses can be disregarded,a traditional nonmultiplexing dispersive spectrometer has thehighest possible temporal flow of information, higher than anyother causal instrument that adopts the same input portΦnonmult
y ≥ Φmulty .
The demonstration of this lemma is quite easy and relies onEq. (22), definition 1, and its corollary 1. Considering that wecan neglect any optical losses, it follows that Φnonmult
y ¼ Φs
[see Eq. (23)]. If a multiplexing spectrometer has a temporalflux higher than that exhibited by the nonmultiplexing deviceΦmult
y > Φnonmulty , then its time information flux must also be
larger than the temporal flux Φs at its input: Φmulty > Φs. But
this outcome violates corollary 1; hence, this multiplexingspectrometer would not be causal.
Lemma 1 can be easily reformulated in order to account forimaging systems without spectroscopic discrimination ability(e.g., panchromatic imagers). This possible extension can be
Alessandro Barducci Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 645
fruitfully applied to the domain of coded aperture (multiplex)imaging [25,26], where an optical processing of the input sig-nal is applied before the measurement in order to magnify itsSNR. Despite these expectations of improved radiometricproperties, the lemma extension shows that this optical pro-cessing of the input signal (coded aperture imaging) cannotoriginate any advantage in terms of retrievable source infor-mation over straight (traditional) imaging, which always col-lects the greatest information amount. We remark that lemma1 has direct implications on emerging optical technologiessuch as compressive sensing (sampling), which requires aper-ture coded systems (imagers or spectrometers). The perfor-mance of instruments such as those discussed in [27–29] isheavily affected by lemma 1 and the ODPI.
In view of Eqs. (18) and (21), the huge channel capacity hyof multiplex spectrometers must be connected with the pre-sence of a noninformative signal component ynoninfðxÞ, whichincreases the physical SNR of the multiplexing spectrometerwithout improving the information flow it conveys γy, as sta-ted by the following equations:
yðxÞ ¼ yeffðxÞ þ ynoninfðxÞ ynoninfðxÞ ≫ yeffðxÞγy ≅ κs log2ðyeffðxÞ=σyðxÞÞ: ð24ÞThis result, which does not include the potential benefit
from a wider input port, has been pointed out by Barducciet al. for both types of multiplexing: interferential [4] (FTS)and dispersive [5] (MTS). The theory discussed here [e.g.,Eq. (22)] implicitly means that any attempts to increase theexperimental SNR by a specific optical configuration (e.g.,multiplexing) fed by an assigned input port will unavoidablyresult in the generation of a noninformative signal component.The SNR of the informative signal component determines theinformation flow γy at the exit port of the spectrometer, ac-cording to Eq. (24). This effective SNR (the amplitude ofthe informative signal yeffðxÞ) cannot be expanded withoutbounds, unless the information conveyed by a multiplex in-strument would exceed the information flow at its input port,violating the ODPI and lemma 1. The generation of a physicalsignal of higher SNR, as in Eq. (24), must be therefore con-nected with the introduction of a constant signal, the onlyknown type of signal which does not carry information. Thisnoninformative signal component brings, however, a highphotonic noise amplitude that dims the informative part ofthe signal yeffðxÞ. The noninformative component of the signalwill beautify the physical SNR, reducing the true informationcontent delivered by the instrument.
It is worth noting that the comparison of multiplexing andnonmultiplexing spectrometry has been made on the assump-tion of an equal integration time τ for the two instruments. Thehigher physical signal outputted by a multiplexing instrumentdoes not allow any integration time reduction because of thefollowing reasons:
• The integration time must be set taking into accountonly the informative component of the signal yðxÞ producedby a multiplexing spectrometer, so losing the gain of physicalsignal amplitude it obtains with respect to a nonmultiplexingdevice.
• Reducing τ for a multiplexing spectrometer would resultin a lower input information content and a lower inputtemporal flux Φs.
Previous papers on this subject erroneously stated that thehigher physical signal should give rise to a higher level of in-formation (higher SNR), or to a possible reduction of integra-tion time maintaining the information collected by anonmultiplexing spectrometer. Lemma 1 clearly shows thatthe bit rate (source information) at the output end of a non-multiplexing spectrometer always is greater than or equal tothat originated by a multiplex instrument. Therefore, no ad-vantage in terms of information collected can be gained byadopting a multiplex technique with the exception of Jaqui-not’s advantage connected to the lack of input slit. It can easilybe shown that the gain of source information γy introduced byJaquinot’s advantage at the exit port of an FT spectrometer isγyðno-slitÞ ¼ γyðslitÞ þ ½κs log2ðAno-slit=AslitÞ�=2, Aslit and Ano-slit
being the input port’s areas of an FT spectrometer with andwithout the slit, respectively. The following observations areessential to any kind of spectroscopic approach.
The limit imposed by the ODPI on the radiometric perfor-mance of multiplex spectrometers is quite general and appliesto all types of multiplexing, including multiplex imaging. Pos-sible data processing performed after the detector block is un-essential because, due to the data processing inequality, it canonly lower the information received from the spectrometer.
According to Eq. (18), the only viable strategy for enrichingthe actual information content pertaining to the source an in-strument can gather is to expand the telescope aperture or theallowed detector integration time. Both these actions increasethe constant C in Eq. (18), hence strengthening the entropy ofthe input signal and the source information flux Φs trans-mitted by the whole instrument. Astronomers have pursuedthis option for centuries.
C. Efficiency of SpectrometersA generic spectrometer can be characterized using the follow-ing two parameters, which define its cost ξ and efficiency η:
ξ ¼ Hy
Hs; η ¼ Γy
Γs: ð25Þ
The interpretation of the above efficiency η is quite simple,being the fraction of the input source information the spectro-meter delivers to its output port. Because of definition 1, theefficiency of any spectrometers must be less than the unit. Thespectrometer cost ξ instead is a positive measure of the “en-gineering effort” necessary for the instrument implementa-tion; it is not related to its economic value. For a multiplexspectrometer, the cost ξ is far above the unit (ξ ≫ 1), whereasthe maximum efficiency is less than one. In view of theorem11 in [8] (Γ ≤ H) and recalling that for a natural source weexpect that Γs ¼ Hs, it results in
η ≤ ξ: ð26Þ
The above property holds true for any spectrometers, and itis useful to determine the optimal (minimal) cost of an instru-ment designed to have a given efficiency η:
ξopt ¼ η: ð27ÞIt can be shown, in fact, that ξopt is the minimum cost of a
spectrometer that transmits the source information Γy ¼ ηΓs
to its output, provided that such source entropy is conveyed toits input port. Under the assumption of a constant SNR, the
646 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Alessandro Barducci
efficiency-to-cost ratio of a spectrometer can be stated as
ηξ ¼
log2ð1þ SNR2effÞ
log2ð1þ SNR2Þ ; SNR2eff ¼
y2effðxÞσ2yðxÞ
; ð28Þ
where for a multiplex spectrometer SNReff is far below itsphysical SNR. The overall information content Γ~s retrievedfrom the spectral estimation ~sðκÞ is shown in the followingequation:8<:
γ~s ¼ f ~s2 log2ð1þ SNR2
~sÞΓ~s ¼
RDκ
γ~sðκÞdκ ¼ K2 log2ð1þ SNR2
~sÞ ¼ ηs M2 log2ð1þ SNR2
~sÞf ~s ≤ f s;
;
ð29Þwhere the effect of the finite sampling efficiency has beenmade evident. We remark that the high sampling efficiencyachieved by nonmultiplexing spectrometry andWTS improvesthe efficiency-to-cost ratio η=ξ and the source informationconveyed by the measurements.
6. CONCLUSIONSIn this paper we have discussed a new technique of multiplexspectrometry, WTS. This new spectroscopic approach relieson the bandpass sampling scheme as applied to the FTS ofa prefiltered spectral source. Our analysis has proved thatWTS has a high sampling efficiency and improved radiometricperformance if compared with standard FTS. The enhancedsampling and radiometric efficiency make the WTS a practicaltool for observing narrowband sources, as in spectroscopicapplications where single absorption/emission lines are inves-tigated. The use of a tunable spectral filter can permit us toextend the application of WTS to the measurement of broad-band sources, maintaining the advantages stated above. Wehave demonstrated the utilization of this experimental ap-proach by means of numerical simulations and interfero-metric measurements of various laser sources, obtaininggood agreement between FTS and WTS measurements. Thecorrelation between WTS and FTS spectral estimates rangedfrom 0.9453 up to 0.9999 for the several experiments and si-mulations so far considered, with average values of 0.9992 forthe simulations and 0.9796 for the experimental results. TheWTS-FTS difference of the area subtended by the main peak inthe reconstructed source spectrum has always been in therange 0.4%–8.8%, with an average amount of 4.025% for theexperimental data and 2% for the simulations. For the experi-mental data, the absolute value of the relative WTS-FTS dif-ference ranges from 0.01% near the peak centers up toabout 10% in the far wings. This difference is coherent withthe noise affecting the measurements, although its correlatedspectrum suggests that it would be partially ascribed to pos-sible phase distortion.
Using information theory, we have been able to demon-strate that multiplex spectrometry, including WTS, cannotbring a SNR or radiometry advantage over nonmultiplexingdispersive instruments with the exception of Jaquinot’s advan-tage. This demonstration has been condensed in lemma 1; wehave also derived the ODPI, the equivalent of the data proces-sing inequality of the information theory. We have argued thatthese findings (ODPI and lemma 1) would also bound the SNRand radiometric performance of coded aperture imaging sys-tems, demonstrating that the direct observation of the signalof interest is always preferable in terms of collected source
information. The ODPI and lemma 1 also limit the radiometricperformance of compressive sampling as long as it adoptscoded aperture instruments.
ACKNOWLEDGMENTSThis work was partially supported by the Italian SpaceAgency. The author is indebted to Cinzia Lastri and DonatellaGuzzi for having executed the measurements with the ALISEOinstrument and for their help with the subsequent data proces-sing. The author is obliged to Paolo Marcoionni, VanniNardino, and Ivan Pippi for their support and the valuable dis-cussions on the theory of bandpass sampling applied to FTS.This instrument was partially developed under a contract ofthe Italian Space Agency. Thanks are due to two anonymousreferees who provided important suggestions for improvingthis paper.
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