Noise-Equivalent Change in Radiance for Sampling Noise in a Double-Sided Interferogram

Noise-equivalent change in radiance for samplingnoise in a double-sided interferogram

Douglas L. Cohen

The formula for the noise-equivalent change in radiance �NEdN� for sampling noise �Appl. Opt. 38, 139�1999�� can work well when applied to the double-sided interferograms of radiance spectra dominated byisolated emission lines, but it does not work well when applied to broad, slowly varying radiance spectrasuch as a Planck blackbody curve. The modified formula for the sampling-noise NEdN works well whenapplied to the double-sided interferograms of both types of radiance spectra. © 2003 Optical Society ofAmerica

OCIS codes: 070.2590, 070.4790, 070.6020, 300.6300, 300.6340, 120.3180.

1. Introduction

Fourier transform infrared spectrometers and othercomputerized systems that use Michelson inter-ferometers measure spectral radiances by calculationof the Fourier transforms of interference signals, andfor these calculations to be accurate the interferencesignals must be accurately sampled. There havebeen more than a few analyses of the effect of sam-pling errors on interferometer performance,1–8 withat least one previous article relating the shape of thepower spectrum of the random error in the samplingposition to the shape of the noise-equivalent changein radiance �NEdN� associated with the measuredspectrum.1 Here we use the same notation as in Ref.1, and the material presented below can be read as acontinuation and modification of the results reportedin Ref. 1.

2. Summary of Notation Used to Describe MichelsonSystems

Figure 1 is a diagram of a standard Michelson-basedFourier transform infrared spectrometer. The sam-pling noise that contaminates the spectrum occurs at

D. L. Cohen �[email protected]� is with the Communications Di-vision, ITT Aerospace, P.O. Box 3700, 1919 West Cook Road, FortWayne, Indiana 46801-3700.

Received 6 August 2002; revised manuscript received 9 January2003.

0003-6935�03�132289-12$15.00�0© 2003 Optical Society of America

point B. The signal at point A is, by use of the samenotation as in Ref. 1,

Z� x� � ��

�

Z��exp�2i�x�d�, (1a)

where

Z�� Zs�� Zb��, (1b)

with

Zs�� 1�4� Adsc�� R��a��f�� Bsc��,(1c)

Zb�� 1�4� Ad�� R��a��f Bf��

� a Ba��. (1d)

The parameters in Eqs. �1a�–�1d� are defined as

x is the optical-path difference �OPD� in centi-meters as shown in Fig. 1;

� is the wave number in inverse centimeters;Ad is the detector area in squared centimeters;

sc is the solid angle in steradians subtended bythe input scene when viewed from the centerof the detector;

f is the solid angle in steradians subtended bythe fore optics when viewed from the centerof the detector;

a is the solid angle in steradians subtended bythe aft optics as viewed by reflection off theinterferometer from the center of the detector;

�� is the interferometer’s beam-splitter effi-ciency �a dimensionless number betweenzero and one� as a function of wave number;

1 May 2003 � Vol. 42, No. 13 � APPLIED OPTICS 2289

R�� is the detector’s responsivity in units of de-tector signal �either electrical current or volt-age� per unit optical power as a function ofwave number;

�a�� is the aft-optics transmission �a dimension-less number between zero and one� as a func-tion of wave number;

�f�� is the fore-optics transmission �a dimension-less number between zero and one� as a func-tion of wave number;

Bsc�� is the scene spectral radiance, in units ofpower per unit area per unit solid angle perunit wave-number interval, as a function ofwave number;

Bf�� is the background spectral radiance from thefore optics, in units of power per unit area perunit solid angle per unit wave-number inter-val, as a function of wave number;

Ba�� is the background spectral radiance from theaft optics, in units of power per unit area perunit solid angle per unit wave-number inter-val, as a function of wave number. (2)

We use function G�x� to represent the signal atpoint B in Fig. 1. The formula for G�x� is

G� x� � ��

� dx

uh�x � x

u �Z� x �,

where h is the impulse-response function of the de-tector circuit �which contains an antialiasing filter�and u is the OPD velocity, that is, the time rate ofchange of the OPD. This is just the standard con-volution formula for an electrical signal that passesthrough a linear circuit converted into an integralover the OPD. In a double-sided interferogram thesignal recorded at B is given by

�� x, L�G� x�,

where

�� x, L� � �1 for �x� � L0 for �x� � L . (3a)

The � function shows that the G�x� signal is recordedover only a finite range of the OPD:

�L � x � L. (3b)

The signal contaminated by the sampling noise canbe written as

G� x � r� x�� G� x� � r� x�dGdx

, (3c)

where r�x� is taken to be a wide-sense stationarystochastic process �or random function� that repre-sents the random error in the sampling position thatoccurs when the system attempts to sample G�x� at

Fig. 1. Signal diagram showing the flow of information from the input scene radiance that enters the Michelson interferometer to thedigital signal that leaves the Michelson interferometer. Although the Michelson interferometer is drawn with flat return mirrors, theequations derived still hold true when corner cubes are used to return the signal to the beam splitter.

2290 APPLIED OPTICS � Vol. 42, No. 13 � 1 May 2003

an OPD value of x. The average or expected value ofr at any x is zero:

E�r� x�� 0. (3d)

Because r is a wide-sense stationary random func-tion, its autocorrelation function can be written as

E�r� x�r� x �� Rr� x � x�. (3e)

Since E�r�x�r�x �� E�r�x �r�x��, it must be true that

Rr� x � x � � Rr� x � x�.

Defining a new OPD variable x� � x � x, we see thatRr is even and, of course, real:

Rr��x�� Rr� x��, (3f)

Im�Rr� x�� 0. (3g)

The noise-power spectrum Sr of r and the autocorre-lation function of r from a Fourier transform pair:

Sr��

�

Rr� x�exp��2i�x�dx, (3h)

Rr� x� � ��

�

Sr��exp�2i�x�d�. (3i)

According to Eqs. �3f � and �3g�, Rr is real and even, soits Fourier transform Sr must also be real and even:

Sr�� Sr��, (3j)

Im�Sr�� 0. (3k)

When we set the sampling error to zero, the unc-alibrated measured spectrum of the interferometersignal at point B in Fig. 1 can be written as

Gm��

�

�� x, L�G� x�exp��2i�x�dx,

where, according to Ref. 1, in a well-designed inter-ferometer measurement

Gm�� G�� H�u�� Z��. (4a)

Here H� f � is the complex transfer function of thedetector circuit, which is, of course, also the Fouriertransform of h�t�, the impulse-response function thatconnects the detector to the analog-to-digital con-verter, and argument f is the frequency in hertz of thesignal that travels through that circuit. �Since rela-tion �4a� is written in terms of the wave number �,argument f in H� f � is replaced by u�, the product ofthe OPD velocity u and wave number �.� The valueof L is chosen large enough, and the OPD interval �xbetween samples is chosen small enough that neitheraliasing nor the finite length of the signal specified by

relation �3b� introduces a significant difference be-tween

Gm��

�

�� x, L�G� x�exp��2i�x�dx

� ��L

L

G� x�exp��2i�x�dx, (4b)

and

G��

�

G� x�exp��2i�x�dx. (4c)

Reintroducing the sampling noise by taking r � 0gives, according to Ref. 1, the uncalibrated measuredspectrum contaminated by sampling noise Gm�� ns��, where the spectral noise that is due to sam-pling error is given by

ns��

�

�� x, L�r� x�dGdx

exp��2i�x�dx. (5a)

Substitution from relation �4a� gives

Gm�� ns�� H�u�� Z�� ns��. (5b)

Following the same path as in Ref. 1, we expect the nsspectral noise to be low enough in frequency not to besignificantly aliased by the sampling at point B inFig. 1. From Eqs. �5a� and �3d� we get

E�ns��

�

�� x, L� E�r� x��dGdx

exp��2i�x�dx

� 0. (5c)

This shows that the expected or average value of thespectral noise at any wave number � is zero. Re-versing the Fourier transform in Eq. �4c�, we notethat

dGdx

�d

dx ��

�

G��exp�2i�x�d�

� 2i ��

�

�G��exp�2i�x�d�.

Relation �4a� can now be substituted into the formuladG�dx to obtain

dGdx

� 2i ��

�

�H�u��Z��exp�2i�x�d�. (5d)

3. Deriving the Noise-Equivalent Change in Radiance

There are several ways to derive a NEdN formula byuse of the formalism set out above; perhaps the moststraightforward is to apply the Revercomb complexcalibration algorithm.9 Equation �5c� reminds usthat the sampling noise in the uncalibrated spectrumcan be reduced to insignificant levels if we average


together measurements of the same spectrum. In atypical setup calibration data are acquired with greatcare, reducing to insignificant levels all forms of noisethat are susceptible to being averaged away. Hencewhen we apply the Revercomb algorithm it makessense to assume noise-free calibration data. Thismeans the formula for the noise-contaminated spec-tral measurement from the Revercomb algorithm is

spectral radiancecontaminatedby sampling noise

� �G�� ns�� G1��

G2�� G1�� B2�� B1�� B1��,

(6a)

where G1�� and G2�� are the noise-free, complex,and uncalibrated signal spectra obtained at point Bin Fig. 1 when the interferometer measures theknown spectral radiances B1�� and B2��. Follow-ing the pattern of relation �4a� and Eqs. �1b�–�1d�, wesee that

G1�� H�u�� Zs1�� Zb��, (6b)

G2�� H�u�� Zs2�� Zb��, (6c)

with Zb�� given by Eq. �1d� and

Zs1,2�� 1�4� Adsc�� R��

� �a��f�� B1,2��. (6d)

Substituting Eqs. �1b�–�1d� and �6b�–�6d� into Eq.�6a� gives

spectral radiancecontaminated bysampling noise

�

Bsc�� 4ns��

AdscH�u�� R��a��f��. (7a)

At first glance it looks like

4ns��

AdscH�u�� R��a��f��

is the spectral noise that is due to sampling error inthe spectral radiance measured by the interferome-ter. We note, however, that although the measuredradiance Bsc is real, the same cannot be said aboutthe ratio ns��H�u��, which is almost guaranteed tohave a nonzero imaginary component. Becausespectroscopists know that Bsc must be real, the stan-dard way of handling this situation is to take the realpart of the noise-contaminated spectral radiance toget

sampling noisein measurement �

4Re�ns��H�u��

Adsc�� R��a��f��.

We call �� the complex phase angle or argument ofH�u�� so that

H�u�� H�u��exp�i��. (7b)

This gives

sampling noisein measurement �

4Re�exp��i��ns��

Adsc�� R��a��f��H�u��. (7c)

According to Eq. �5c� the average or expectation valueof ns�� is zero at any wave number �, so we knowthat the average or expectation value of the samplingnoise in Eq. �7c� is also zero at any wave number �.The sampling-error NEdN is then just the standarddeviation of this random quantity:

NEdNsamp �4{E[(Re�exp��i��ns��)2]}1�2

Adsc�H�u�� R��a��f��.

(7d)

The difficult part here is evaluation of

E[(Re�exp��i��ns��)2].

The approach adopted in Ref. 1 is equivalent to cal-culation of

E��exp��i��ns��2� � E��ns��2� (8a)

and then assuming that

E[(Re�exp��i��ns��)2] �12

E��ns��2�. (8b)

This is a standard approximation that works well foranalysis of high-frequency, additive noise such as de-tector or photon noise. When, however, it is used toanalyze low-frequency multiplicative noise, as wasdone for the sampling noise in Ref. 1 or the mirror-misalignment noise in Ref. 10, it could turn out to bea relatively poor approximation. To determine whythis is so, we first evaluate

E[(Re�exp��i��ns��)2]

exactly in the Appendix to get

E[(Re�exp��i��ns��)2] � ��, (8c)

where, according to Eq. �A7b� in the Appendix,

�� 2 ��

�

Sr�� exp��i��

� H�u�� Z��

� �� exp�i��

� �H�u�� *Z�� 2d� . (8d)


The sampling-error NEdN in Eq. �7d� now becomes

NEdNsamp �4��

Adsc�H�u�� R��a��f��.

(8e)

If we use the approximation in relation �8b� to calcu-late the sampling-error NEdN, then we are just stat-ing that


�12

E��ns��2� � ��. (8f)

Consulting Eq. �A6f � in the Appendix, we see that the� function can also be written as

�� 12

E��ns��2� � 22 Re(exp��2i��

� ��

�

Sr�� H�u��

� Z�� H�u��

� Z�� d� ). (8g)

When the integral

��

�

Sr�� H�u�� Z��

� �� H�u�� Z�� d� (8h)

is negligible or zero, the exact formula for � reducesto the formula in relation �8f �, which means thatrelation �8b� becomes a reasonable approximation.

4. Application of the Noise-Equivalent Change inRadiance Formula

There are many situations in which integral � isnegligible or zero, making relations �8b� and �8f � rea-sonable approximations. Sampling noise is oftennoted when isolated emission lines produce nearbyghost lines. Figure 2�a� and 2�b� specify the basicsetup required for the sampling noise to produceghost lines. In Fig. 2�a� the Bsc spectrum being mea-sured has an isolated emission line at wave number� � �e, which is significantly different from zero only

for wave numbers between �e � �w. The powerspectrum Sr straddles this emission line with its in-ner edges �see Fig. 2�b�� occurring at wave numbers� � ��c. In an ideal case, we can suppose that theinterferometer has negligible background radiance,so that

Ba�� Bf�� 0 (9a)

in Eqs. �1b�–�1d�. Only those regions of the � axiswhere Sr is nonnegligible can make a significant con-tribution to integral � in Eq. �8h�, so for power spec-trum Sr shown in Fig. 2�b� we can write

When Ba�� Bf�� 0, the �Z�� Z�� product in these integrals is, according to Eqs. �1b�–�1d�, proportional to �Bsc�� Bsc�� . Theemission line in Fig. 2�a� has a half-width of �w, so it isimpossible for both Bsc�� and Bsc�� to besignificantly different from zero when �c � �w. Yetunless �c � �w, the ghost lines cannot be easily seenbecause they are not well separated from the emissionline. Hence in this situation—isolated emission lines,a power spectrum acting to produce well-formedghosts, and negligible background radiance—integral� is small and relations �8b� and �8f � are acceptableapproximations. This is in fact the type of test caseexamined in Ref. 1, which is why the formulas derivedby use of relation �8b� worked well there.

Fig. 2. Scaled so that the x axes cover the same spread of wavenumbers. The zero wave-number position of �b� is aligned justbelow the peak value of the emission line in �a� to show how thepower spectrum in �b� brackets the emission line in �a�. Eachpeak in �b� has a wave-number width of �M.

�� C��M�

��C

Sr�� H�u�� Z�� H�u�� Z�� d�

� ��C

�C��M

Sr�� H�u�� Z�� H�u�� Z�� d� . (9b)


The approximation might not work well, however,when the interferometer is used to measure a broadradiance spectrum such as a Planck blackbody curve.To show why, we apply the passband white noisepower spectrum in Fig. 3—which, although idealized,has the same basic shape as the power spectrum inFig. 2�b�—to another type of test case: a simulatedinterferometer that measures the spectral radiancefrom a 400 K blackbody in the long-wave IR from 650to 1250 cm�1. The double-sided interferogram issampled 8192 times over an OPD between �L withL � 1.28 cm. The rms error in the sampling positionis 1.58 � 10�6 cm or approximately 0.5% of the OPDspacing between interferogram samples. �This sizeerror makes the spectral effects relatively easy tosee.� We again assume the interferometer’s back-ground radiance is negligible, with Ba�� Bf �� 0in Eqs. �1b�–�1d�; and we specify an ideal optical sys-tem in perfect alignment with

�a�� f�� 1, (10a)

and a detector responsivity of

R�� 1 amp s�s. (10b)

The blackbody radiances used to calibrate the simu-lated interferometer have temperatures of 30 and 500K. The antialiasing filter contained in the detectorcircuit of Fig. 1 is a three-pole, low-pass Butterworthfilter whose cutoff frequency divided by the OPD ve-locity u � 5 cm�s specifies a cutoff wave number of1600 cm�1, and the etendue in Eq. �1c� is taken to be

Adsc � 3.07 � 10�3 cm2 s. (10c)

Figure 4 contains ten simulated and noise-contaminated radiance measurements from the in-terferometer system specified in the previousparagraph. In Fig. 4, and only in Fig. 4, the nominalsampling noise is multiplied by 20 before being addedto the true radiance to make clear the noise shape

with respect to the true radiance spectrum �it is as ifwe had set the rms sampling error to 10% rather than0.5% of the sample spacing�. Because the 400 Kspectral radiance being measured is smooth, the sam-pling noise produces a family of smooth, Planck-likecurves. Figure 5 goes back to the use of a nominalrms sampling error �which is 0.5% of the sample spac-ing� specified by the noise-power spectrum in Fig. 3.The expected sampling-noise NEdN is calculated inFig. 5 as given by the formula in Ref. 1, discarding asinsignificant the contribution of integral � in relation

Fig. 3. Noise-power spectrum S4�� used to generate random er-rors in the sampling position for the simulated interferometerdiscussed. The width of each block is 10 cm�1, and the top of eachblock corresponds to a spectral power of ��10�11�4�cm2��2 � 10cm�1�. Consequently the rms error in the sampling position is�10�11�2�2� cm � 1.58 � 10�6 cm.

Fig. 4. Graph of ten simulated measurements of a 400 K black-body spectrum contaminated by the sampling noise in Fig. 3. Thenoise was increased by a factor of 20 over the size specified by thespectrum in Fig. 3 to make it easier to see.

Fig. 5. Top curve, sampling-noise NEdN from the formula in Ref.1; bottom curve, sampling-noise NEdN from Eq. �8e�. The crossesthat mark the calculated standard deviations of the spectral errorfollow the bottom curve. The standard deviations are from 30noise-contaminated measurements of a 400 K Planck curve, withthe sampling noise obeying the noise-power spectrum in Fig. 3.


�8f �. Figure 5 also calculates the sampling-noiseNEdN as given by the formula in Eq. �8e�, retainingthe contribution of integral � in relation �8f �. Al-though both formulas predict approximately thesame NEdN for situations such as the one shown inFigs. 2�a� and 2�b�, we see that the formula in Eq. �8e�predicts a much lower sampling-noise NEdN for thesmooth 400 K Planck radiance. The sampling-errorNEdN is just the standard deviation of the measure-ment error that is due to the sampling noise, so it iseasy to determine how well the lower curve matchesthe measurement errors: we just simulate a largenumber of measurements of the 400 K Planck curveand then use them to calculate the standard devia-tion of the measurement error as a function of wavenumber. The crosses in Fig. 5 give the calculatedstandard deviation over the range of wave numberscovered by the NEdN curves; the close match be-tween the crosses and the lower NEdN curve showsthat the formula in Eq. �8e� does a good job of esti-mating the sampling error.

Only a moderate amount of analysis is needed toshow why the formula in Eq. �8e� predicts a verysmall NEdN value near � � 1031 cm�1 in Fig. 5.The phase angle �� of the three-pole Butterworthlow-pass filter used in our example is to a good ap-proximation linear in wave number; that is,

�� K� (11a)

for some real constant K. �An approximately lineardependence of phase on frequency is a characteristicshared by many types of low-pass filter.� Substitut-ing relation �11a� into Eq. �8d� and consulting Eq. �7b�gives

�� 2 ��

�


� Z�� H�u��

� Z�� 2d� . (11b)

Returning to the definition of Z�� in Eqs. �1b�–�1d�and remembering that in our example

Ba�� Bf�� 0 and �a�� f��

� 1 with R�� 1 amp s�s,

we see that Eq. �8e� reduces to

NEdNsamp �

�H�u��

�

Sr��

� �H�u�� Z��

� �H�u�� Z�� 2d� 1�2

(11c)

for the curve plotted in Fig. 5. Over the � values forwhich NEdNsamp is plotted, Figs. 3 and 5 show that�� when Sr�� 0. The H�u�� transfer func-

tion varies slowly with respect to �, and in Eq. �11c�function Bsc�� is the Planck spectral radiance at 400K, which also varies slowly with respect to wave num-ber for the � values in Fig. 5. Hence the product

g�� H�u��Bsc�� (11d)

can be approximated as

g�� g�� dgd�

�at �

� (11e)

inside the integral of Eq. �11c� to get

��

�

Sr�� g��

� g�� 2d� � 4�g��

� �dgd�

�at �

�2

��

�

� 2Sr�� d� .

Substituting this into Eq. �11c� gives

NEdNsamp �2

�H�u�� g��

dgd��

at �

��

��

�

� 2Sr�� d� �1�2

. (11f)

The formula in relation �11f � shows that, when

�� g�� dgd�

�at �

(11g)

is zero, then NEdNsamp is small �it need not be exactlyzero because relation �11e� is only an approximation,not an exact equality�. Figure 6 plots � as a functionof � for the case we analyze, and it indeed crosses zeroat � � 1030.5 cm�1. This is very close to the esti-mated 1031-cm�1 wave-number value where the

Fig. 6. Plot of the �� curve showing where it crosses zero on thewave-number axis.


NEdNsamp curve �that is, the lower curve in Fig. 5�has its minimum.

5. Simplification of the Formula for theSampling-Noise Noise-Equivalent Change in Radiance

To get a simple approximation for the sampling-noiseNEdN, we again assume that the sampling noise isrelatively low frequency �see Ref. 1 and discussionfollowing relation �5b� above� and note that nowSr�� is significantly different from zero only for rel-atively small values of � . Hence only these small � values contribute significantly to the integral in Eq.�8d�. In most well-designed interferometers H�u��,R��, �a��, �f ��, and �� are slowly varying func-tions of � over the range of � being measured. So

H�u�� H�u��, (12a)

R�� R��, (12b)

�a,f�� a,f��, (12c)

�� (12d)

inside the integral over � . We still assume negligi-ble background radiance

Ba�� Bf�� 0. (12e)

Using Eqs. �1b�–�1d� and �7b� to expand the expres-sion for �� in Eq. �8d� yields

�� 2�Adsc

4 �2

�H�u��2R��2�a��2�f��2��2

� ��

�

Sr�� Bsc��

� �� Bsc�� 2d� . (12f)

The �� limits on the integral should be regarded asrequiring that the integration over d� covers allwave numbers for which Sr�� is significantly differ-ent from zero. Relation �12f � can now be substitutedinto Eq. �8e� to get, for � positive and lying inside theband of wave numbers measured by the interferom-eter.

NEdNsamp � ��

�

Sr�� Bsc��

� �� Bsc�� 2d� 1�2

. (12g)

For the situation shown in Figs. 2�a� and 2�b� anddiscussed following Eqs. �9a� and �9b� above, that is,for the situation needed to produce distinct sampling-noise ghost lines, we can assume that

��

�

Sr�� Bsc��

� Bsc�� d� � 0,

which reduces relation �12g� to

NEdNsamp � ��

�

Sr�� Bsc�� 2

� d� � ��

�

Sr��

� Bsc�� 2d� 1�2

. (12h)

Working with the first integral within the braces, wechange the variable of integration to �� andconsult Eq. �3j� to get

��

�

Sr�� Bsc�� 2d�

� ��

�

Sr�� Bsc�� 2d��.

Relation �12h� can now be rewritten as

NEdNsampghost

� 2��

�

Sr��

� Bsc�� 2d� 1�2

� 2�Sr�� 2Bsc��2��1�2,

with the � representing the convolution over �.

6. Conclusion

All the formulas given above for the sampling-noiseNEdN assume that the sampling-noise power spec-trum is dominated by low frequencies so that thealiasing introduced by sampling the interference sig-nal has no significant effect. When integral � in Eq.�8h� is negligible or zero, the formulas for thesampling-noise NEdN in Ref. 1 hold true. This oftenoccurs when the sampling-noise power spectrumstraddles isolated emission lines, as shown in Figs.2�a� and 2�b�. When the spectral radiance beingmeasured is not dominated by isolated emissionlines, the formula in relation �12g� produces, as longas the approximations in relations �12a�–�12d� andEq. �12e� hold true, a much better estimate of thesampling-noise NEdN. Relation �12g� also gives thecorrect sampling-noise NEdN when the power spec-trum straddles isolated emission lines as shown inFigs. 2�a� and 2�b�—again as long as the approxima-tions in relations �12a�–�12d� and Eq. �12e� holdtrue—because it is a more general formula and worksfor any spectral shape. When the approximations inrelations �12a�–�12d� and Eq. �12e� cannot be justifiedand when there is no desire to compare the shape ofthe radiance spectrum with the shape of thesampling-noise power spectrum, Eq. �8e�, the mostgeneral formula presented here can be used to calcu-late the sampling-noise NEdN.


Appendix

As a notational convenience I define the Fouriertransform of function f �x� to be

F��i�x�� f � x��

�

f � x�exp��2i�x�dx. (A1a)

Using Eq. �5d� I next specify function W�x� to be

W� x� �dGdx

� 2i ��

�

�H�u��Z��exp�2i�x�d�

� F�i�x��2i�H�u��Z��. (A1b)

We have to evaluate three terms, T1, T2, and T3, toget the formula we need:

T1 � E(�F��i�x�� x, L�r� x�W� x��2), (A2a)

T2 � E(�F�i�x�� x, L�r� x�W� x��2), (A2b)

T3 � E�F��i�x�� x, L�r� x�W� x��

� F�i�x �� x , L�r� x �W� x ��. (A2c)

Note that, since ��x, L�, r�x�, and W�x� are all strictlyreal, the second term is the complex conjugate of thefirst:

T2 � T1*. (A2d)

Working first with term T1, we see that, accordingto Eqs. �A1a�, �A2a�, and �3e�,

T1 � E��

�

dx�� x, L�W� x�exp��2i�x�

� ��

�

dx �� x , L�W� x �

� exp��2i�x �r� x�r� x ��

��

�


� ��

�

dx �� x , L�W� x �

� exp��2i�x � E�r� x�r� x ��

� ��

�


� ��

�

dx �� x , L�W� x �

� exp��2i�x � Rr� x � x�. (A2e)

We know that

��

�

�� x, L�exp��2i�x�dx � F��i�x�� x, L��

� 2L sinc�2�L�

(A3a)

for sinc�x� � sin�x��x and, by reversing the Fouriertransform in Eq. �A1b�,

��

�

W� x�exp��2i�x�dx � F��i�x��W� x��

� 2i�H�u��Z��. (A3b)

Now the Fourier convolution theorem, which statesthat the Fourier transform of the product of twofunctions equals the convolution of the Fouriertransform of each individual function, can be usedto write

��

�

�� x, L�W� x�exp��2i�x�dx

� �2L sinc�2�L�� 2i�H�u��Z��. (A3c)

The � on the right-hand side of Eq. �A3c� representsa convolution over wave number �. In a good in-terferometer measurement the value of L in Eq.�A3c� is chosen large enough to introduce only min-imal distortion into the measured spectral radianceBsc; and the interferometer’s background radiancesBa and Bf are slowly varying Planck functions. Inaddition, well-built interferometers are designed tohave functions �a, �f, �, and R �see definitions in �2�above� that vary slowly compared with the radiancebeing measured; and the detector-circuit �antialias-ing� transfer function H also varies slowly com-pared with Bsc. Consequently the sinc function inEq. �A3c� is narrow compared with ��HZ�, actingmuch like a delta function. Hence Eq. �A3c� can beapproximated as

��

�

�� x, L�W� x�exp��2i�x�dx

� �2i�H�u��Z��. (A3d)

Substituting relation �A3d� and Eq. �3i� into Eq. �A2e�gives

T1 � �42 � ��

�

Sr�� H�u�� Z�� H�u�� Z�� d� . (A3e)


According to Eq. �A2d�, parameter T2 is the com-plex conjugate of T1. Hence, knowing that onlythe transfer function H in Eq. �A3e� is complex, wehave

The first few steps of evaluating parameter T3 �whichis defined in Eq. �A2c�� are the same as those used toevaluate T1 in Eq. �A2e�. Following the same pro-cedure as in Eq. �A2e�, we get

The only difference between the final result in Eq.�A2e� and the right-hand side of Eq. �A5a� is that inEq. �A5a� the complex exponential in the integralover dx is exp�2i�x � instead of exp��2i�x �.Equation �3f � in the text lets us write

Rr� x � x� �12

Rr� x � x� �12

Rr� x � x �.

Substituting this into Eq. �A5a� gives

Since � and W are real,

��

�

�� x, L�W� x�exp��2i�x�dx�*

� ��

�

�� x, L�W� x�exp��2i�x�dx, (A5b)

and by use of relation �A3d� the formula for T3 can bewritten as

T3 � 22 � (��

�


� Z�� H�u�� *

� Z�� d� � ��

�

Sr��

� H�u�� Z��

� H�u�� *Z�� d� ) . (A5c)

Having found formulas for T1, T2, and T3, we canevaluate �� using the definition

�� E[(Re�exp��i��ns��)2]. (A6a)

Expanding the right-hand side of Eq. �A6a� gives

�� E(�12

exp��i��ns��

�12

exp�i��ns��*2)�

14

exp��2i��E�ns��2�

�14

exp�2i��E�ns��*2�

�12

E��ns��2�. (A6b)

According to Eqs. �5a�, �A1a�, and �A1b�,

ns�� F��i�x�� x, L�r� x�W� x��. (A6c)

Since �, r, and W are real,

ns��* � F�i�x�� x, L�r� x�W� x��. (A6d)

Substituting Eqs. �A6c� and �A6d� into the first twoterms on the right-hand side of Eq. �A6b�, after con-sulting Eqs. �A2a� and �A2b�, gives

�� T1

4exp��2i��

T2

4exp�2i��

�12

E��ns��2�. (A6e)

T2 � �42 � ��

�

Sr�� H�u�� *Z�� H�u�� *Z�� d� . (A4)

T3 � ��

�

dx�� x, L�W� x�exp��2i�x� ��

�

dx �� x , L�W� x �exp�2i�x � Rr� x � x�. (A5a)

T3 �12 �

��

�

dx�� x, L�W� x�exp��2i�x� ��

�

dx �� x , L�W� x �exp�2i�x � Rr� x � x�

�12 �

��

�

dx�� x, L�W� x�exp��2i�x� ��

�

dx �� x , L�W� x �exp�2i�x � Rr� x � x �.


Equation �A2d� can be used to write

�� 12

Re�T1 exp��2i�� 12

E��ns��2�,

and after substituting from Eq. �A3e� we get

Another way of handling the right-hand side of Eq.�A6e� is to apply Eqs. �A2c�, �A6c�, and �A6d� to thethird term to get

�� T1

4exp��2i��

T2

4exp�2i��

T3

2.

(A7a)

Substituting Eqs. �A3c�, �A4�, and �A5c� into Eq. �A7a�leads to

�� 2 ��

�

Sr�� (��exp��i��

� H�u�� Z��

� �exp��i�� H�u��

� Z�� exp�i��

� H�u�� *Z��

� �exp�i�� H�u�� *

� Z�� exp��i��

� H�u�� Z��

� �exp�i�� H�u�� *

� Z�� exp��i��

� H�u�� Z��

� �exp�i�� H�u�� *

� Z�� )d� .

If we define

a � exp��i�� H�u�� Z�� ,

b � exp�i�� H�u�� *Z�� ,

then the term within the braces inside the integralover d� becomes

�ab* � ba* � aa* � bb* � �a � b��a* � b*�

� �a � b�2.

Consequently the formula for �� can now be writtenas

�� 2 ��

�

Sr�� exp��i��

� H�u�� Z��

� exp�i��

� H�u�� *Z�� 2d� , (A7b)

which, after consulting Eq. �A6a�, becomes

Neither the power spectrum Sr nor the magnitudesquared of a complex number can ever be negative, sothe integral over d� must also be nonnegative. Thismakes sense because


can never be negative, since it is the expected oraverage value of a squared real number.

References1. D. Cohen, “Performance degradation of a Michelson inter-

�� 22 Re(exp��2i��

��

�

Sr�� H�u�� Z�� H�u�� Z�� d� )�

12

E��ns��2�. (A6f)


� 2 ��

�

Sr�� exp��i�� H�u�� Z��

� exp�i�� H�u�� *Z�� 2d� . (A7c)


ferometer due to random sampling errors,” Appl. Opt. 38, 139–151 �1999�.

2. D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman,J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D.Pillsbury, and D. E. Weidler, “POES high-resolution sounderstudy final report,” Project Report NOAA-1 �Lincoln Labora-tory, MIT, Cambridge, Mass., 1994�.

3. D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J.Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, and D. P.Ryan-Howard, “POES advanced sounder study �Phase II�,”Project Report NOAA-7 �Lincoln Laboratory, MIT, Cambridge,Mass., 1994�.

4. W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein,D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard,W. J. Scouler, and D. E. Weidler, “GOES high-resolution in-terferometer study,” Project Report NOAA-12 �Lincoln Labo-ratory, MIT, Cambridge, Mass., 1995�.

5. P. Haschberger, “Impact of the sinusoidal drive on the instru-

ment line shape function of a Michelson interferometer withrotating retroreflector,” Appl. Spectrosc. 48, 307–315 �1994�.

6. A. S. Zachor, “Drive nonlinearities: their effects in Fourierspectroscopy,” Appl. Opt. 16, 1412–1424 �1977�.

7. A. S. Zachor and S. M. Aaronson, “Delay compensation: itseffect in reducing sampling errors in Fourier transform spec-troscopy,” Appl. Opt. 18, 68–75 �1979�.

8. E. E. Bell and R. B. Sanderson, “Spectral errors resulting fromrandom sampling-position errors in Fourier transform spec-troscopy,” Appl. Opt. 11, 688–689 �1972�.

9. 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. 27,3201–3218 �1988�.

10. D. Cohen, “Performance degradation of a Michelson inter-ferometer when its misalignment angle is a rapidly varyingrandom time series,” Appl. Opt. 36, 4034–4042 �1997�.


Documents

Noise-Equivalent Change in Radiance for Sampling Noise in a Double-Sided Interferogram