radiometric errors in complex fourier transform spectrometry

Radiometric errors in complex Fourier transformspectrometry

Lawrence A. Sromovsky

A complex spectrum arises from the Fourier transform of an asymmetric interferogram. A rigorousderivation shows that the rms noise in the real part of that spectrum is indeed given by the commonlyused relation �R � 2X � NEP��A��N, where NEP is the delay-independent and uncorrelateddetector noise-equivalent power per unit bandwidth, X is the delay range measured with N samplesaveraging for a time � per sample, � is the system optical efficiency, and A� is the system throughput.A real spectrum produced by complex calibration with two complex reference spectra �Appl. Opt. 27, 3210�1988� has a variance �L

2 � �R2 �c

2�Lh � Ls2��Lh � Lc

2 �h2�Ls � Lc

2��Lh � Lc2, valid for �R, �c,

and �h small compared with Lh � Lc, where Ls, Lh, and Lc are scene, hot reference, and cold referencespectra, respectively, and �c and �h are the respective combined uncertainties in knowledge and mea-surement of the hot and cold reference spectra. © 2003 Optical Society of America

OCIS codes: 120.0120, 120.6200.

1. Introduction

The use of complex spectra in Fourier transform spec-trometry is a convenient way to characterize devia-tions from ideal behavior. Asymmetries in theinterferogram between positive and negative delayscans are related to phase errors that have a naturalcomplex representation. Unfortunately, a numberof books that treat interferometry present confusing,and sometimes erroneous, expositions on how a mea-sured real interferogram is related to a complex spec-trum. The first purpose of this paper is to clearlydefine that relationship. My second purpose is toprovide a clear and unambiguous derivation of thefundamental equation for the noise-equivalent spec-tral radiance �NESR of a Fourier-transform spec-trometer. The third purpose is to show how complexcalibration removes the effects of instrumental arti-facts, including uncompensated phase errors, beam-splitter emissions and other offsets, and how thenoise present in calibrated spectra is related to themeasurement noise in complex raw scene spectra andthe measurement noise and knowledge uncertainty

L. A. Sromovsky �[email protected] is with theSpace Science and Engineering Center, University of Wisconsin,Madison, 1225 West Dayton Street, Madison, Wisconsin 53706.

Received 29 March 2002; revised manuscript received 23 Octo-ber 2002.

0003-6935�03�101779-09$15.00�0© 2003 Optical Society of America

associated with the reference spectra used in calibra-tion.

2. Basic Equations of Interferometry

I begin with the basic equation of interferometry forthe balanced output of a Michelson interferometer,which relates a real spectral radiance L�� to a mea-sured interferogram radiance Im�x through a cosinetransform:1

Im� x � �0

�

��m��1L��d�

� �0

�

�� L��cos�2��xd�, (1)

where � is the wave number and x is the optical delaydifference between the first and second beams. Theinterferometer separates the incoming beam into twobeams, delays the second beam by a distance x, thenrecombines them to obtain the interferogram. Thefactor �� is the overall system efficiency factor,which can be written as

�� 12

�b��o��, (2)

where �b�� 4RT is the nominal beam-splitter ef-ficiency �equaling 1 for a perfect beam splitter whereR and T are the beam-splitter reflectivity and trans-

1 April 2003 � Vol. 42, No. 10 � APPLIED OPTICS 1779

mission and �o�� is the remaining optical efficiency,which would be unity if all optical components hadperfect transmissions, alignment, and flatness. Onefactor included in �o�� is the modulation efficiency�m��, which, for the balanced output, is the ratio ofthe zero-delay modulation amplitude to the baselineinterferogram level obtained when we average overthe modulations. That factor does not affect the firstterm and is thus removed by division. The factor of1�2 comes about because half of the incoming radia-tion is reflected back toward the source �for a one-portinterferometer, which is the type we are consideringhere. Note that, whereas the interferogram I�x isdefined for both positive and negative x, the spectrumis defined only for positive wave numbers. It shouldalso be noted that for dielectric beam splitters, asused in the Planetary Imaging Fourier TransformSpectrometer2 and the Atmospheric-Emitted Radi-ance Interferometer, the interferogram has a nega-tive modulation term. For these instruments, one ofthe recombined beams has an internal beam-splitterreflection, whereas the other beam has an externalreflection, resulting in a relative 180° phase shiftbetween them that produces a negative modulation.In the analysis of Brault1 the recombined beams eachhave one external reflection, leading to positive mod-ulation. My conclusions in this paper also apply fornegative modulations.

To simplify the subsequent discussion, we definetwo new functions:

B�� L��, I� x � Im� x � �0

�

�m��1B��d�.

(3)

These allow Eq. �1 to be written in the more compactform

I� x � �0

�

B��cos�2��xd�. (4)

For a real interferometer, the measured interfero-gram is generally not well described by Eq. �1 �or Eq.�4�, where it is assumed that the optical delay isindependent of wavelength. Imperfect compensa-tion of the beam-splitter optical delay, coupled withdispersion, can lead to an extra wavelength-dependent phase shift between the two interferingbeams. For example, in the Planetary Imaging Fou-rier Transform Spectrometer2 one of the two interfer-ing beams passes through the compensator threetimes and once through the beam-splitter substrate,whereas the other beam passes through the beam-splitter substrate three times and once through thecompensator. If the thickness of the beam splitterand the compensator are different, then because onebeam passes through the compensator two timesmore than the other, while the other passes through

the beam splitter two times more than the other,these elements will introduce a differential opticaldelay �x proportional to n��t, where n�� is the re-fractive index and �t is the thickness difference.The interferogram equation then becomes

I� x � �0

�

B��cos�2��x � ��d� (5)

or

I� x � �0

�

B��cos��cos�2��xd�

� �0

�

B��sin��sin�2��xd�, (6)

where �� x is the wave-number-dependent phaseshift. For a compensator and beam splitter with awave-number-independent refractive index, thephase shift has a linear wave-number dependenceand is equivalent to a shift in the origin of the delaycoordinate system and thus is of no practical impor-tance, except that small errors in locating zero delaywill produce a linear variation in phase versus wavenumber. If the refractive index itself varies linearlywith wave number, then the phase shift will have aterm with quadratic dependence, which can be im-portant. Because this phase shift does not dependon mirror position, and thus does not reverse signwhen x reverses, Eq. �6 implies that the interfero-gram is not symmetric. To invert Eq. �6 so that B��can be expressed as an explicit function of I�x, weneed first to create functions with specific symmetryproperties.

To create even and odd functions out of B��cos��and B��sin��, which are defined only for positive �,we need to extend their definitions to negative wavenumbers. The extended definitions are as follows:

C�� B��cos�� for � � 00 for � � 0 , (7)

S�� B��sin�� for � � 00 for � � 0 .

We can then rewrite Eq. �6 in terms of integrals overall �:

I� x � ��

�

C��cos�2��xd� � ��

�

S��sin�2��xd�.

(8)

In this form it is clear that I�x depends only on theeven part of C�� and the odd part of S��. The even

1780 APPLIED OPTICS � Vol. 42, No. 10 � 1 April 2003

and odd components of these functions can be con-structed as follows:

C�� Ce � Co, Ce�� 12

�C�� C��,

Co�� 12

�C�� C��, (9)

S�� Se � So, Se�� 12

�S�� S��,

So�� 12

�S�� S��. (10)

This provides the necessary background to introducethe complex Fourier transform needed to invert Eq.�6.

3. Complex Notation and the Complex Spectrum

Because two-sided cosine transforms of odd functionsare zero, and two-sided sine transforms of even func-tions are zero, we can rewrite Eq. �8 as

I� x � ��

�

�Ce�� iSo��cos�2��xd� � i ��

�

�Ce��

� iSo��sin�2��xd�, (11)

where we made the functions to be transformed thesame for both transforms �within a factor of i byadding iSo � Co to the cosine transform integrandand �iCe � Se to the sine transform integrand.These additions do not alter the equality becausethey both have zero integrals. Now that the func-tions to be transformed have the same form, it isuseful to define a complex spectrum

G�� Ce�� iSo��, (12)

which then allows us to simplify Eq. �11 to the form

I� x � ��

�

G��exp��2�i�xd�. (13)

We can then write the complex spectrum as the com-plex transform of the real interferogram:

G��

�

I� xexp� 2�i�xdx, (14)

which provides the desired inversion of Eq. �6. Toretrieve the physical spectrum, we substitute defini-tions of Ce�� and So�� into the definition of G toobtain

G�� 12

B��exp�i�� for � � 0

12

B��exp��i�� for � � 0� , (15)

thus

L�� 1B��

�2

��exp��i��

��

�

I� xexp� 2�i�xdx.

(16)

Recall that L�� and B�� are real and defined only forpositive �.

4. Complex Calibration

Deriving a real physical spectrum from a measuredasymmetric interferogram in the presence of complexoffsets is the aim of complex calibrations.4 The cal-ibration is performed in the spectral domain andmust correct for the uncompensated phase spectrumas well as remove extraneous contributions arisingfrom the instrument, including emission from beamsplitters4,5 and apertures, and the detector and aftoptics, some of which contribute dc offsets and someof which produce modulated signals. In general, themeasured interferogram will be a linear combinationof the desired interferogram with a superposition ofextraneous interferograms. Likewise the transformof that interferogram will be a linear combination ofthe form

��

�

I� xexp�2�i�xdx � G�� O��, (17)

where O�� denotes the extraneous spectrum of theinstrument, which is actually itself a linear combina-tion of form ¥k �kLk exp�i��,k where Lk is the kthextraneous source spectral component and �k and��,k are the respective optical efficiency and phase,which differ from � and � because different opticalpaths are involved. Because the instrument spec-trum is independent of external source variations, itcan be removed by subtraction.

If the interferogram is measured by a detector witha spectral responsivity of R��, then the raw signalspectrum can be written as V�� Rd��G�� O��.Two known reference sources, e.g., hot and coldblackbodies, are used to generate spectra Gh and Gc,which combine with the background spectrum O toyield raw spectral signals:

Vh � Rd�Gh � O

� �Rd

�

2�Lh exp�i� � Rd O �hot, (18)

Vc � Rd�Gc � O

� �Rd

�

2�Lc exp�i� � Rd O �cold, (19)

Vs � Rd�Gs � O

� �Rd

�

2�Ls exp�i� � Rd O �scene. (20)


In Eqs. �18–�20, for positive wave numbers, all fac-tors have an implicit dependence on �. Equations�3 and �15 were used to introduce spectral radiances�L. The real proportionality constant R � Rd��2 isthe system responsivity, which is to be determined bythe calibration process. Assuming that the twoblackbody radiances Lh and Lc are known, the firsttwo equations can be solved for both the backgroundspectrum and the proportionality constant, which canthen be used in the third equation to obtain the de-sired scene spectrum Ls:

R exp�i� �Vh � Vc

Lh � Lc, R �

�Vh � Vc�Lh � Lc

, (21)

Rd O � Vc � RLc exp i�, (22)

Ls � Lc � Re�Vs � Vc

Vh � Vc��Lh � Lc. (23)

The expression for the calibrated scene radiance �Eq.�23� is equivalent to Eq. �12 of Revercomb et al.4For noise-free measurements, that ratio is inherentlyreal. However, in a practical calibration, measure-ment noise will create a spurious imaginary compo-nent that is explicitly rejected by taking the real part.Revercomb et al.4 show that Eq. �23 eliminates thecontribution of anomalous phase associated withbeam-splitter emission. As shown here the calibra-tion is effective at removing any complex offset.

The expressions for the system responsivity R andfor the offset RdO, which depends on R, are especiallysusceptible to noise perturbations in regions of lowsignal, either because of low radiance or low respon-sivity. Because responsivity is a slowly varying func-tion of wave number, one might want to smooth theresponsivity in wave number to improve its accuracy inthe low signal regions. But because squared-errorterms appear in a magnitude, there will tend to be apositive bias introduced that smoothing will not re-move, which is demonstrated in Subsection 6.C. Abetter approach would be for one to smooth the differ-ence Vh � Vc before taking its absolute value. Notethat there is no such bias problem in the expressionsfor the calibrated radiances themselves.

5. Discrete Transforms

To treat the transform of noise in discrete interfero-gram measurements we need to make use of the prop-erties of discrete Fourier transforms. As describedin Section 12.1 of Numerical Recipes,6 the discreteFourier transform is a mapping of one set of complexnumbers hk, k � 0, 1, . . . , N � 1 into a second set ofcomplex numbers Hn, n � 0, 1, . . . , N � 1. Theforward and inverse transforms of these into eachother are given by

Hn � �k�0

N�1

hk exp�2�ikn�N, (24)

hk �1N �

n�0

N�1

Hn exp��2�ikn�N. (25)

Note that the inverse transform has a negative expo-nential argument. This terminology agrees withthat used by Brault,1 Press et al.6 and Mathews andWalker,7 but although the same formulas are used byInteractive Data Language �IDL, the IDL manual8refers to our inverse transform as the forward trans-form and our forward transform as the IDL inversetransform. The inverse transform given above fol-lows from the forward transform and the fact that

�k�0

N�1

exp�2�ik�n � m�N� � N�nm, (26)

�given by Kaplan9 where �nm is the Kronecker deltafunction. Note that there is no particular physicalsignificance associated with these transforms.

The connection with physics begins with the ap-proximation of the continuous transform integral ofEq. �13 by a finite sum. We first truncate the inte-gral to a finite delay range ��X, X�, which defines anew function

H�� X

X

I� xexp�2�i�xdx. (27)

This can also be written as the convolution of thecomplete spectrum with a sinc function:

H�� G�� 2X sinc�2X�, (28)

where sinc�x � sin��x��x. This is a consequenceof the sinc function being the transform of the boxcarfunction that truncates the infinite integral.

We next approximate the truncated continuous in-tegral as a finite sum, i.e.,

H�� k��N�2

N�2�1

I� xkexp�2�i�xk�, (29)

where � � 2X�N is the sample interval and N is thenumber of sample points xk, for k � �N�2, �N�2 1, . . . , N�2 � 1. If the spectrum G�� is band lim-ited to �� N��4X, then this sampling provides atleast two delay samples per wavelength of the high-est frequency and is thus adequate to fully define thespectrum.6 The same number of spectral samples isalso sufficient, so that we need compute only thetransform for N wave numbers:

�n � n��2X,

n � �N�2, �N�2 � 1, . . . , N�2 � 1. (30)

This results in the following expression for the sam-pled spectral values:

H��n �2XN �

k��N�2

N�2�1

I� xkexp�2�ink�N. (31)

Note that the exponential factor is now in the formgiven by Eq. �24 but the summation is over a differ-ent set of k values. This can be fixed when we make


use of two periodicities. First note that the exponen-tial factor in Eqs. �24 and �31 is periodic in k �and nwith period N. We can also make I�xk periodic bycreating an extended range of the definition of I suchthat I�x 2X � I�x, which is allowed because valuesin the extended range play no role in the originaldefining equation for H��. The periodicity of bothI�x and the exponential factor allows us write

�k��N�2

�1

I� xkexp�2�ink�N � �k��N�2 N

N�1

� I� xk�Nexp�2�ink�N,(32)

and thus we finally obtain the form

H��n �2XN �

k�0

N�1

Is� xkexp�2�ink�N, (33)

where we define a new function Is�xk in terms of theoriginal interferogram �prior to the range extension:

so that hk in Eq. �24 can be set equal to �2X�NIs�xk.Equation �33 is close to the form of Eq. �24, exceptthat, in Eq. �24, the index n varies from 0 to N � 1,whereas in Eq. �33 it varies from �N�2 to N�2 � 1.Using similar periodicity arguments, we find that thecorrect relationship between the Hn in Eq. �24 andour function H��n is as follows:

The wave-number values are thus ordered much likethe delay samples. The first N�2 values of Hn are forwave numbers from 0 to N��4X, and the second N�2values are for wave numbers from �N��4X to 0.

6. Noise-Equivalent Spectral Radiance

A. Derivation of the Noise-Equivalent Spectral RadianceEquation

The quantity I�x, defined in Eqs. �1–�4 as the mod-ulated part of the interferogram, is actually a radi-ance, and the power per unit area incident at thedetector that is due to the modulated part of theinterferogram is given by the product of I�x and the

optical system etendue �A� or throughput, which isa conserved quantity in the optical system.10 Thiscan be represented as the solid angle of view timesthe collecting area of the optics, or the area of thedetector times the solid angle of the beam convergingon the detector. Thus we have a power incident onthe detector of

PS� x � A�I� x, (36)

with typical units of watts. Let a detector noise per-turbation be defined as an equivalent incident powerperturbation �P�x for a given interferogram sampleat xk and for an averaging time �. This is a detectornoise perturbation measured in units of incidentpower. It can also be written as noise voltage di-vided by detector responsivity. The interferogramradiance perturbation yielding the same detectorpower change is obtained from Eq. �36, i.e.,

�I� x ��P� x

A�. (37)

Transforming �I�x the same way as I�x, usingEq. �33, we then obtain the resulting noise in H��nas

�H��n �2XN �

k�0

N�1

�I� xkexp�2�ink�N, (38)

which represents how one specific set of random noiseperturbations for N-specific detector readings trans-lates into a set of N perturbations in the complexspectral function H��n. In terms of the correspond-ing real scene spectral function D��n � 2��exp��i�H��n, we have the relations

�D��ncos �� 2�

Re��H��n�,

�D��nsin �� 2�

Im��H��n�. (39)

Is� xk � �I�xk � I�2XN

k� for k � 0, 1, . . . , N�2 � 1,

I� xk�N � I�2XN

k � X� for k � N�2, . . . , N � 1,(34)

Hn � �H��n � H� n2X� for n � 0, 1, . . . , N�2 � 1,

H��n�N � H� n2X

�N2X� for n � N�2, . . . , N � 1.

(35)


The expected variance in the real spectral componentD��ncos��n is then just the expectation value

�R2 � E��Dn cos �n u

2

�4�2�2X

N �2

E�k�0

N�1

�I� xkcos�2�nk�N

� �k��0

N�1

�I� xk�cos�2�nk��N� , (40)

which collapses to the form

�R2 �

4�2�2X

N �2

�k�0

N�1

E��I� xk2�cos2�2�nk�N (41)

when we substitute the relation

E��I� xk�I� xk�� E��I� xk2��k,k�, (42)

which is valid for uncorrelated errors, i.e., for errorsthat are independent of delay position. That wouldnot be the case for detectors with 1�f noise, whichcould produce correlated variations over many inter-ferogram samples. Nevertheless, we proceed herewith the simplifying assumption of uncorrelated er-rors. Further simplification of the expression for�R

2 is obtained when we assume that the expectedinterferogram error variance be independent of thedelay sample, i.e., that the expected noise varianceover many trials is independent of the delay position.That is a good approximation for thermal detectors,or for photon detectors operating under background-limited conditions. But for a cooled photon detectorlooking at an optical system that is also cooled, thenoise variance might be dominated by shot noise ofthe signal, which would lead to a noise variance thatdid depend on the delay position. �Under these con-ditions the delay-independent background would alsoplay a significant role in determining the detectornoise. Ignoring these complications and proceedingwith the delay-independent assumption, we define�I

2 � E��I�xk2� and move the expectation value out-side the summation, reducing Eq. �41 to the form

�R � �4X�I��2N for n � 04X�I��N for n � 0

, (43)

in which we used an expression given by Tuma11 toevaluate the sum ¥k�0

N�1 cos2�2�nk�N, which equalsN�2 for n � 0 and N for n � 0. The expression forn � 0 seems to conflict with the equation on page 34of Brault’s chapter,1 which states that the spectralnoise should be a factor of 2X��2N times the inter-ferogram noise. That factor would need to be mul-tiplied by 2�� to agree with Eq. �43. The extrafactor appears to arise because the spectrum thatsatisfies the power theorem, from which Brault’sequation is derived, is not the physical spectrum, butinstead is G��, as given by Eq. �15, the magnitude ofwhich is ��2 times that of the physical spectrum.Thus the equivalent noise in the physical spectrum is2�� times the noise in the real part of G.

We make use of Eq. �37 to relate the interferogramnoise variance to the detector noise variance:

�I2 � E��I� xk

2� �1

� A�2 E��P� xk2� �

1� A�2

NEP2

2�,

(44)

where � is the averaging time for each delay sample,1��2� is the corresponding noise bandwidth, andNEP2 is the expected mean-square detector noisepower per unit noise bandwidth. This definition ofNEP2 is the same as used on page 238 of Hanel etal.,12 but must be multiplied by the noise bandwidthto match the definition on page 204 of Dereniak andBoreman.13 It should be noted here that there issome confusion in the literature concerning the ex-pression for the effective noise bandwidth when asignal is averaged over a time period T. Davis etal.14 used �f � 1�T in their equation for NESR onpage 23. Hanel et al.12 also seem to have made thisassumption in their Eq. 5.14.1 for the NESR of a filterradiometer, although it is not explicitly stated. Theactual relationship, given by Eq. 5.153, page 190, ofDereniak and Boreman,13 is �f � �2T�1.

Substituting Eq. �44 into Eq. �43 we obtain, forn � 0,

NESR � �R � 2X�2�

NEP

A��2�N�

NEP

�A��N,

(45)

where we substituted �� 2X�1 in the second ver-sion. For n � 0 the NESR is a factor of �2 larger.This can be understood by the fact that for n � 0 theimaginary term is zero, so that all the noise appearsin the real component rather than being equally dis-tributed, as it is for all other wave numbers. Equa-tion �45 is essentially equivalent to Eq. 5.8.14 ofHanel et al.,12 provided that their efficiency factorproduct �1�2 is the same as our � and if we interpretthe product �N as the total time to scan the interfero-gram, which would be the case were there no timegaps between averaging periods. Comparing thiswith Eq. 2.5 of Davis et al.,14 we can obtain agreementwith our result if their efficiency factor product �1�2is equal to � and if we divide their NESR by therather large factor of N1�2. Because neither of theabove-mentioned references precisely defined theirefficiency factors, it is not possible to make a moredetailed comparison. Recall that our overall effi-ciency factor �, as defined in Eq. �2, cannot exceed 0.5because it includes the 50% loss from the beam exit-ing the entrance aperture.

Using the definition of specific detectivity �page 221of Crowe et al.15 as the reciprocal of the noise-equivalent power per unit area per unit bandwidth,i.e.,

D* �Ad

1�2

NEP, (46)


where Ad is the detector area, and using the substi-tution �� 1��2X, we can rewrite Eq. �45 as

NESR ��2Ad

1�2

�A��D*�2�N�

1�A��D* �Ad

�N�1�2

.

(47)

B. Phase Issues

The above derivation is most relevant if the imagi-nary component is small, i.e., the phase is smallenough to make cos�� 1. If the phase is largethen the error in the physical spectrum computedfrom the real part of H�� is amplified by the factor1�cos��.

In principle, we can eliminate the dependence onphase by computing the magnitude spectrum. How-ever, in the presence of errors, the expected value ofthe measured magnitude spectrum is not the truespectrum. This can be seen when we compute theexpected value of the squared magnitude of the erro-neous version of D��nexp�i��n

, which is given by

D�,n � D��nexp�i��n �

2�

�H��n. (48)

The expected magnitude squared is then given by

E�D�,nD�,n* � ED2 � 4�H��n�H��n*

�2

� cross terms�� D2 � 2�R

2, (49)

where the cross terms drop out because �H has a zeromean and where the middle term was evaluated withEqs. �38 and �42. This demonstrates that the mag-nitude spectrum will, on average, be larger than thetrue spectrum, and the amount larger is proportionalto detector noise variance. The fractional effect willof course be greatest when the signal is smallest. Itis important to note that, if magnitude spectra arecoadded, it will not result in convergence to the truespectrum. That process will yield the result givenabove. The best magnitude spectrum is obtainedwhen real and imaginary parts are coadded sepa-rately; then, after noise levels are reduced, the mag-nitude is computed last. The problem of noisebiasing of magnitude spectra exists for calibratedspectra as well as for raw spectra, except that, forcomplex calibrated spectra, there is no need to com-pute a magnitude spectrum because the phase is cor-rected in the calibration process.

C. Noise in Calibrated Spectral Radiances

Adding an error term �x � 2�H,x�� to each measure-ment in Eqs. �18–�20 converts each complex spectralmeasurement Vx to the form R�exp�i�Lx �x� RdO. After substituting these expressions into Eq.

�23 and subtracting the exact scene radiance Ls, weobtain the error in the calibrated scene radiance as

�L � Re �Ls � Lc � ��s � �cexp��i�

�Lh � Lc � ��h � �cexp��i��Lh � Lc

� �Ls � Lc, (50)

where the subscripts h, c, and s refer, as before, tohot, cold, and scene spectra, respectively. Note that,although L denotes radiance in this expression and inthe original complex calibration equations, theseequations also implicitly refer to the convolved spec-tra resulting from the truncation of the interfero-gram. For NESR calculations, D was used todistinguish convolved spectra from the original spec-tra L. Here that distinction is not made.

For the case in which the error in the referencemeasurements can be ignored, i.e., when �h � �c � 0,Eq. �50 can be reduced to the form

�L � Re��s exp��i�� Re��scos � � Im��ssin �.(51)

The expectation value of the squared calibrated ra-diance error is then given by

�L2 � E��L

2

� E��Re��s�2 cos2 �� E��Im��s�

2 sin2 ��

�4�2 E��Re��H�2� � �R

2, (52)

where the cross term drops out because the real andimaginary parts of �s are uncorrelated. This simpli-fication also used the defining equations for �R,namely, Eqs. �39 and �40, and the fact that both realand imaginary parts of �H have equal variance. Thisalso shows that E�Re��s

2� � �R2. Equation �52 is

hardly a surprising result. The radiance error in acalibrated spectrum is the same as in the uncali-brated spectrum when calibration errors are negligi-ble. The significance of this result is that it appliesfor an arbitrary phase in the complex spectrum.Thus we no longer have to worry about the treatmentof radiometric errors in the presence of significantphase errors.

When calibration source errors themselves are sig-nificant, the rigorous analysis becomes more compli-cated. When such errors are large, numericalsimulations are required. However, for small er-rors, i.e., when � �� Lh � Lc, we can ignore terms ofsecond order and higher in ��Lh � Lc and simplifyEq. �50 to the form

�L Re��s � �cexp��i�� Ls � Lc

Lh � Lc�

� Re��h � �cexp��i��. (53)


We next square this approximate expression for �Land take the expectation value, which we simplifyusing the following relations:

E��Re��s � �ccos � � Im��s � �csin ��2�

� �R2 � �c

2, (54)

E��Re��h � �ccos � � Im��h � �csin ��2�

� �h2 � �c

2, (55)

E�Re��s��cRe��h��ccos2 �

� Im��s��cIm��h��csin2 �� c2. (56)

Here we used the notation �c2 � E��Re��c�

2� and�h

2 � E��Re��h�2� and the fact that E��Re��s�2� �

�R2. We proceed in the same manner as for Eq. �52.

We use the fact that expected variances of real andimaginary components are equal and that real andimaginary components are uncorrelated. The phasedrops out because it enters in the form of sin2 � cos2

�. The final result is given by

�L2 � E��L

2 � �R2 � �c

2�Lh � Ls

Lh � Lc�2

� �h2�Ls � Lc

Lh � Lc�2

,

(57)

which shows that the error in the calibrated radiancevaries with scene radiance level. When Ls � Lc, theerror is ��R

2 �c21�2, and the hot measurement

error has no effect. When Ls � Lh, the error is��R

2 �h21�2, and the cold reference measurement

has no effect. The minimum error occurs at sceneradiance levels between the two reference radiances.For an instrument such as the Atmospheric EmittedRadiance Interferometer,3 which uses an ambientblackbody as the cold reference and a blackbodyheated above ambient as the hot reference, most ofthe downwelling atmospheric radiance values will beless than the cold reference, and the lowest radiancewill have the largest error, which has the upper limitof ��R

2 �c2Lh

2��Lh � Lc2 �h

2Lc2��Lh � Lc

2�1�2.If a set of calibration spectra is taken for each scenespectrum and only those are used in the calibration,and if the source radiances are known accurately,then all three component variances will be approxi-mately the same. Thus, if the scene radiance rangedbetween the hot and cold reference spectra, the ex-pected error in the calibrated radiance would rangebetween �R�3�2 and �R�2, with best results forscene radiance in the middle of the reference range.

It is easy to show that, if we ignored errors inmeasuring the reference spectra and used �h

2 and �c2

to denote uncertainties in knowledge of the referenceradiances, the expression for �L

2 would have thesame form as Eq. �57. Thus �h

2 and �c2 can be

taken to be the sum of variances due to measurementerror and knowledge uncertainties in the referencespectra.

7. Summary

A rigorous definition of the complex spectrum wasexpressed in terms of the physical spectrum and theasymmetric interferogram that arises from an un-compensated phase spectrum. The precisely de-fined relationship between the discrete Fouriertransform and the physical observations of interfero-gram radiance was used to rigorously derive theequation for NESR for an interferometer radiometerin terms of detector noise-equivalent power per unitbandwidth, confirming the commonly used formNESR � 2X � NEP��A��N, which strictly ap-plies only to the product of the physical spectrum andthe cosine of the phase spectrum. The equations forcomplex calibration were derived, reproducing themain results of Revercomb et al.4 in more generalterms and defining error propagation characteristics.When calibration errors are negligible, the noise in areal spectrum derived from complex calibration is thesame as the noise in the real part of the complexspectrum prior to calibration. When calibration er-rors are significant but small compared with Lh � Lc,the variance in the real calibrated spectrum can beapproximated as �L

2 � �R2 �c

2�Lh � Ls2��Lh �

Lc2 �h

2�Ls � Lc2��Lh � Lc

2, where Ls, Lh, and Lcare scene, hot reference, and cold reference spectra,respectively, and �c and �h are the respective com-bined uncertainties in knowledge and measurementof the hot and cold reference spectra.

References1. J. W. Brault, “Fourier transform spectrometry,” in High Res-

olution in Astronomy, A. Benz, A. Huber, and M. Mayor, eds.�Geneva Observatory, Sauverny, Switzerland, 1985, pp.1–61.

2. H. E. Revercomb, L. A. Sromovsky, P. M. Fry, F. A. Best, andD. D. LaPorte, “Demonstration of imaging Fourier transformspectrometer �FTS performance for planetary and geostation-ary Earth observing,” in Hyperspectral Remote Sensing of theLand and Atmosphere, W. L. Smith and Y. Yasuoka, eds., Proc.SPIE 4151, 1–10 �2001.

3. H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best,R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz,J. F. Short, and H. B. Howell, “AERI—Atmospheric EmittedRadiance Interferometer,” in Eighth Conference on Atmo-spheric Radiation �American Meteorological Society, Boston,Mass., 1994, pp. 180–182.

4. H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L.Smith, and L. A. Sromovsky, “Radiometric calibration of IRFourier transform spectrometers: solution to a problem withthe High-Resolution Interferometer Sounder,” Appl. Opt. 27,3210–3218 �1988.

5. J.-M. Theriault, “Beam splitter layer emission in Fourier-transform infrared interferometers,” Appl. Opt. 37, 8348–8351 �1998.

6. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery, Numerical Recipes �Cambridge U. Press, Cam-bridge, UK, 1992.

7. J. Mathews and R. L. Walker, Mathematical Methods of Phys-ics �Benjamin, New York, 1965.

8. Research Systems, Inc., IDL Reference Guide �Research Sys-tems, Inc., Boulder, Colo., 1995, Vol. 1.

9. W. Kaplan, Advanced Mathematics for Engineers �Addison-Wesley, Reading, Mass. 1981, p. 764.


10. L. Mertz, Transformations in Optics �Wiley, New York, 1965,p. 15.

11. J. J. Tuma, Engineering Mathematics Handbook, 2nd ed.�McGraw-Hill, New York, 1979, p. 95.

12. R. A. Hanel, B. J. Conrath, D. E. Jennings, and R. E. Samuel-son, Exploration of the Solar System by Infrared Remote Sens-ing �Cambridge U. Press, Cambridge, UK, 1992.

13. E. L. Dereniak and G. D. Boreman, Infrared Detectors andSystems �Wiley, New York, 1996.

14. S. P. Davis, M. C. Abrams, and J. W. Brault, Fourier Trans-form Spectrometry �Academic, San Diego, Calif., 2001.

15. D. G. Crowe, P. R. Norton, T. Limperis, and J. Mudar, “Detec-tors,” in Electro Optical Components, W. D. Rogatto, ed.,�SPIE, Bellingham, Wash., 1993, pp. 175–283.


radiometric errors in complex fourier transform spectrometry

Documents