D irect P h a se C o rrectio n o f D ifferen tia l F T -IR S p ectra
M . S H A N E H U T S O N an d M A R K S . B R A IM A N *
D ep art m en t of B ioch em istry and B iop hysics P ro gra m , U niversit y o f V irg inia H ealth Sciences C enter #440 , C harlo ttesv ille,
V irg inia 22 908
S tep -sc an t ra n sien t F o u rier tran sf orm inf rared ( F T - IR ) d ifferen ce
sp ect ra a re of ten m ea su red in an ac-cou p led con ® gu rat ion . T h e resu ltin g d ifferen t ial int en sity sp ectra con ta in b oth p osit ive an d n eg ativ e b an d s. T h is con d ition p o se s p rob lem s f or d irect p h ase co rrection b y t h e st a n d a rd M ertz an d F o rm an m et h od s. R est rictin g th e
calcu lat ed p h a se a n g le to t h e ran ge [ 2 p /2, p /2] w as p rev iou sly
sh ow n to ® x so m e of th ese p ro b lem s, b u t w e sh ow th at t h e u se of a
red u ced - reso lu tio n p h a se sp ectru m can p ro d u ce o th er a rt ifact s. T h e
effect of red u ced reso lut ion is an alyzed f or a sim u lated n o ise-free
sp ect ru m a n d f or a m easu red tran si en t sp ectru m of a rea l p h o to ch em ica l sy st em , b a ct eriorh o d op sin . Exa m in ation o f th ese resu lts
rev ea ls th at t h e M ertz a n d M ert z S ign ed m et h od s ca n p ro d u ce sp ectral b an d s o f red u ced m a gn itu d e an d u n u su al b an d sh ap e, w ith
con sid erab le am o u n ts of in ten sit y rem a inin g alon g t h e im a gin a ry
axis af ter p h a se co rrect ion . H ow ever, t h ese errors can b e elim in at ed
b y se lf-con vo lut ion o f t h e m ea su red int erferogram , w h ich d o u b les
all p h ase a n g les, p rior t o sm o ot h ing . T h is p ro ced u re rem oves th e
p ot en tia l d isc o n tin u ities in t h e p h ase an gle d u e t o sig n ch an g es in
th e d ifferen tia l sp ect ru m . W ith b acteriorh o d o p sin , th is d ou b led - a n gle m et h od fo r d irect p h a se correctio n is ab le to p rod u ce a t ra n sien t
sp ect ru m w h ich close ly m a tch es t h a t p rod u ced b y u sin g a se p a rat ely
m easu red d c int erferog ram to calcu lat e th e p h ase an g le.
In dex H eadin g s: S t ep -sc an d ifferen ce sp ect rosc op y ; D ou b led an g le;
In terferog ram se lf-co n v o lut ion ; P h ase reso lut ion .
IN T R O D U C T IO N
In step -scan tran sien t F o urier transfo rm in frared (F TIR ) difference spectra of b io lo g ical sam ples, th e tim edep en den t in tensity chan g es are 2±3 o rders o f m ag n itu de
sm aller than the static in tensity. To m easu re these in tensity chan g es w ith m axim um sen sitiv ity, it is ad v an tageo u s to ac-co u p le th e detecto r. 1 T h e resu ltin g sp ectra
con tain b o th p o sitiv e and n egativ e in ten sities, w h ich are
no t h and led co rrectly b y th e stan d ard M ertz an d F o rm an
ph ase co rrectio n alg o rith m s. 2,3
W hen th e spectru m co ntain s reg ion s o f p ositive an d
neg ativ e in tensity, th ese p hase co rrection m eth o d s fail to
pro du ce the true sp ectru m . A neg ativ e in tensity can b e
in terp reted in eith er of tw o w ay s d u e to th e eq u ality :
( 2 A )e iu 5
A e i( u
1 p )
.
(1)
T h e M ertz and F o rm an alg orithm s im p licitly assu m e th at
m o st of th e in tensity in th e spectru m is p ositive. W h en
neg ativ e in tensities are p resen t, they are lik ely to be m isin terp reted as p o sitiv e in ten sities w ith a ph ase sh ift o f p
rad ians. H ow ev er, th e p h ase-an gle erro rs are n o t lim ited
to p sh ifts. A s dem on strated b elow , th e use o f reso lu tion red u cin g sm o o th in g p roced u res lead s to a m o re com plicated set o f erro rs.
A po p ular so lutio n to th is pro blem has been to m easure
a sep arate d c-co u p led in ter fero gram fro m w h ich th e
Received 2 9 D ecem b er 19 97; accepted 3 0 M ar ch 199 8.
* A uthor to w h om co rr esp o nd en ce sh o uld be sen t.
974
Volume 52, Number 7, 1998
ph ase in form atio n can b e ex tracted . 4,5 A d isad v antag e o f
th is ap p ro ach is th at care m u st be tak en to en su re instrum en t stab ility betw een these tw o m easu rem ents. A lternatively, on e can m easure th e tw o in ter ferog ram s co n curren tly w ith sp ecial hard w are. 6 ,7
To circu m v en t the need fo r a sep arate dc-co up led in ter ferog ram , M cC o y an d d e H aseth p rop o sed a m o d i® catio n o f th e M ertz alg o rithm fo r v ib ration al circu lar d ichro ism sp ectra, 8 w h ich also h av e po sitiv e an d n egativ e
peak s. T his p hase co rrectio n m etho d has b een in corp orated into so m e com m ercial so ftw are app licatio ns as the
``M ertz S ig n ed’ ’ m eth od .9 T h e b ro ad differen ce b and s o f
qu arter w av e p late/p olarizer sp ectra w ere sh o w n to be
pro perly p hase co rrected w ith this algo rith m . 8 H o w ev er,
as sh o w n b elo w , th is m eth od d oes no t w o rk nearly as
w ell o n sp ectra that hav e n arrow b an ds of altern atin g
po sitiv e an d neg ativ e inten sity. T h is situ atio n is o ften en cou n tered in tim e-reso lv ed step -scan sp ectro sco p y as w ell
as oth er step -scan m eth od s and v ib ratio n al circu lar dichro ism . 4 ±7
In ligh t o f these sho rtco m in g s, w e pro po se here a d ifferent m eth od fo r p h ase co rrecting sp ectra w ith alternatin g p o sitiv e and neg ativ e b and s, based o n do u b lin g th e
ph ase an gles p rio r to lo w erin g th e reso lu tion . T h e ad v an tag e o f d o ub ling th e ph ase an g les is that th e p h ase facto r
e i2 u is iden tical fo r u an d u 1 p . T h is b ene® t elim in ates
all th e po tential p ¯ ips in the ph ase. C o m p arison of this
m etho d w ith the M ertz an d M ertz S ig n ed m eth od s sho w s
th at o ur n ew app roach elim inates the p hase artifacts associated w ith the latter tw o m eth od s. T h e do u b led -an g le
ph ase co rrectio n m eth od is cap ab le of p ro d ucin g tim ereso lved step -scan sp ectra th at m atch th ose pro du ced w ith
th e d c S tored P hase m etho d , w ith o ut th e need for a sep arate ex p erim en tal m easu rem ent. T his n ew p hase co rrectio n m etho d m ay also ® n d utility in p h ase co rrectio n o f
oth er ty p es o f F T- IR sp ectra th at d isp lay b oth po sitive
and neg ativ e b and s.
TH EO R Y
O ptical and electro n ic factors can in tro du ce o d d (sin e)
com po n ents in to the id eally ev en (co sine) in terfero gram . 1
T h us, F o u rier tran sform atio n o f th e in ter fero gram p rodu ces a co m p lex resu lt. P hase co rrectio n alg orithm s are
desig ned to ro tate each com p lex v alu e in th is rep resen tatio n o f th e spectru m on to th e real ax is. W ith the M ertz
correction m eth o d, th e frequ en cy-d epen d ent p h ase ang le,
u 9 (n Å ), is calculated from a sho rt d ou b le-sid ed regio n of
th e in terfero gram arou n d the po sitio n o f zero path d ifference (Z P D ). T h e ph ase co rrectio n can be rep resen ted as
B (n Å ) 5
[b(n Å )e iu
(n Å )
] e2
i u 9 (n Å )
(2)
w h ere b (n Å )e i u (n Å ) is th e co m p lex sp ectru m o b tain ed b y F o urier tran sform atio n o f th e in terfero g ram . A s lon g as u (n Å )
0003-702 8 / 98 / 5207 -0 974$2.0 0 / 0
q 1998 S ociety for A pplied S pectros copy
APPLIED SPECTROSCOPY
is slo w ly var y in g , th en u 9 (n Å ) ù u (n Å ), an d th e M ertz ph ase
co rrectio n m eth o d p rod u ces th e tru e sp ectru m , w ith o n ly
n oise rem ain ing in th e im ag inar y part o f th e sp ectru m .
T h e assum p tio n that b(n Å ) . 0 w ith th e M ertz alg o rithm
m ean s that n eg ative in tensities can resu lt in erro n eou sly
calcu lated p h ase an g les o f u 9 (n Å ) 5 u (n Å ) 1 p . S u bstitutin g
into th e ab o v e equ ation ,
B (n Å ) 5
b (n Å )e i u
(n Å ) 2 i( u (n Å ) 1 p )
e
5
b(n Å )e 2
ip
5
2 b(n Å ).
(3 )
T h u s, peak s in so m e rang es of the p hase-corrected sp ectru m m ay b e inco rrectly sign ed becau se o f th e p rad ian s
p hase error.
M ertz S ig n ed A lg orith m . A m o d i® catio n of the M ertz
p hase co rrectio n alg orith m d esign ed to h and le the p o sitive an d n eg ativ e p eaks o f vib ratio nal circu lar dich ro ism
sp ectra w as p rop o sed by M cC o y an d de H aseth. 8 T h is
m eth o d lim its th e p o ssib le p h ase an gles to the rang e 2 p /
2 to p /2 . T hu s, th e p ¯ ip s in th e ph ase ang le cau sed b y
n egative inten sities are reversed by d isallo w in g p h ase an g les in the seco n d an d third q uad ran ts. O f co u rse, w ith a
p oo r ch o ice o f th e Z P D p o sitio n, p ro p er p h ase co rrectio n
w ou ld req u ire p hase an g les o u tsid e th e allo w ed ran ge.
H ow ev er, w ith th e lim ited p h ase an g le d isp eÂ
rsio n p rod uced b y carefu lly eng in eered in terfero m eters in m o d ern
F T- IR sp ectro m eters, th ere ex ists a pro p er ch o ice fo r the
Z P D th at allo w s th e M ertz S ign ed alg o rithm to m ap th e
p hase an g les of all n eg ativ e-in ten sity b and s to u 9 (n Å ) 5
( u (n Å ) 1 p ) 2 p , so th at the p hase co rrectio n p ro d uces th e
tru e spectrum .
N ev erth eless, as discu ssed b elow , low erin g th e reso lutio n o f th e p hase spectrum to im p ro v e the sign al-to n oise ratio can result in p h ase errors o f in term ediate v alu es; i.e., u (n Å ) , u 9 (n Å ) , u (n Å ) 1
p . In th ese cases, th e
M ertz S ig n ed m eth od d o es n ot p ro d uce th e tru e sp ectru m .
P h a se C o rrection of a S im u la ted In terfero g ra m . To
d em o nstrate th e erro rs cau sed by lo w ering th e resolu tio n
to calcu late th e p h ase sp ectru m , w e co n stru ct a sim u lated
d ifferen ce in terfero g ram I(x ), co rresp o nd ing to eig h t differen tial inten sity ban d pairs of var y in g spacin g, as fo llow s:
I (x) 5
O
8
j5 1
[
1
co s[2 p xn Å
u (n Å j 1
w here
nÅ j 5
j
1
1
u (n Å j )] 2
D n Å j )]
2
co s[2 p x ´(n Å
]
46 cm 2
u (n Å ) 5
p /2 ´(n Å 2
1
D n Å j)
(4 )
j ´(2 0 0 cm 2
D nÅ j 5
j
1
1
1
),
j ´(2 cm 2
(5 )
1
)
(6 )
an d
1 00 0 cm 2 1 ) 2 .
(7 )
W ith the u se o f H ap p±G enzel ap o d ization an d a ph ase
co rrectio n an gle u 9 (n Å ) tak en d irectly fro m th e ab ov e d efinition of u (n Å ), th e sp ectru m calcu lated from th e sim u lated in terferog ram is sh ow n in F ig . 1 A . T he d o ub le-sid ed
interferog ram u sed for th is calcu latio n co n tain s 1 0 00
p oin ts w ith a d iscrete sp acing o f 2 .5 m m , fo r a spectral
resolu tio n of 8 cm 2 1 o ver a b an d w id th o f 0 ±2 00 0 cm 2 1 .
C alculation o f th e sam e sp ectru m by u sing the M ertz
F IG . 1. P h ase corr ection of a sim ulated d ifference inter fer og ram , I( x)
[se e text f or th e de® nition of I(x)] . (A ) True sp ectru m , i.e., sp ectru m
obtain ed after F our ier tran sf orm ation and ph ase cor rection usin g the
pr ed e® ned ph ase angle: u (n Å ) 5 p /2 (n Å 2 10 00 cm 2 1 ) 2 . T h e sp acing b etw een each pair of oppo sitely sig ned d fu nction s is listed. ( B ) S p ectru m
pr oduced by usi ng M er tz ph ase co rrection w ith a p hase r eso lu tion o f
128 cm 2 1 . T he su p erim po se d dotted line r eprese nts the real p art of the
F ou rier transf orm at 128 cm 2 1 r eso lution. (C ) S pectrum pr o du ced b y
usin g M er tz S igned p hase cor rection w ith a p hase r eso lution of 1 28
cm 2 1 . A t this r eso lution the M er tz S ig ned m ethod is n ot ab le to p r od uce
the n eg ative bands w ithout err or. (D ±E ) S p ectra corr esp ond ing to B and
C , but w ith 6 4 cm 2 1 ph ase reso lution . W ith an increase in the p h ase
reso lution, th e M ertz S igned m ethod is n ow able to ph ase co rr ect the
negativ e b ands w ithou t err or.
p hase co rrectio n m etho d at a p hase reso lu tio n o f 1 2 8
cm 2 1 (in stead o f relyin g o n ou r a p riori kn o w led g e o f
the p h ase an g le) resu lts in th e characteristic ``re¯ ected
p eak s’ ’ pattern fo r so m e o f the n eg ativ e b an ds (F ig . 1 B ).
H ow ev er, th e n egativ e com po n ents at 2 4 8, 4 50 , an d 65 2
cm 2 1 rem ain u n re¯ ected . It is o nly th e m o re w id ely
sp aced d ifferen tial ban d p airs at hig h er frequ en cy that
ex hib it re¯ ectio n, w ith th e w eak er n egative co m p o nen t
b eco m in g m o re co m p letely re¯ ected as the spacing in creases.
U sin g th e M ertz S ig n ed m eth o d at 1 2 8 cm 2 1 p hase
resolu tio n (F ig. 1 C ) o rien ts th e p eaks in th e co rrect m an n er qu alitatively, b ut th e neg ativ e p eak s ab ov e 1 0 00 cm 2 1
are red u ced in m ag nitu de o r sh o w un u sual b and sh ap e.
In so m e cases, th e p eak s ev en b eco m e split. T h ese artifacts o ccu r w h en th e real p art o f th e F o urier tran sfo rm at
1 28 cm 2 1 reso lu tio n (d otted line su perim p osed o n F ig .
1 B ) crosses zero at a frequ en cy w ith in a neg ativ e b and
o f the p hase-corrected sp ectru m . W h en th e p h ase reso lutio n is ch an g ed to 64 cm 2 1 , all th e neg ativ e p eak s are
re¯ ected in the M ertz p h ase-co rrected sp ectru m (F ig.
1 D ). T h e real part o f the F o u rier transfo rm at 6 4 cm 2 1
(do tted lin e su p erim p o sed o n F ig . 1 D ) is , 0 in th e vicinity of all th e n egative p eaks; th us, th e M ertz S ig n ed
m eth o d is able to p ro p erly p hase co rrect th e sp ectru m at
this p hase reso lutio n (F ig . 1E ). H o w ev er, if th e spacin g
o f 1 an d 2 ban d s is redu ced to # 3 2 cm 2 1 , th e p hasin g
erro rs an d resu ltin g sp ectral artifacts return (n o t sho w n).
T h ese artifacts can b e b etter u n d ersto od b y th o ro u g hly
APPLIED SPECTROSCOPY
975
F IG . 2. E f fect of a b and o f intensity M 2 5 6 2M 1 on the phase calcu lation of a n ear by b and o f inten sity M 1 . I t is assu m ed that the true ph ase
angles of the b and s are u 1 5 p /8 an d u 2 5 5 p /32. T he calculated phase
angle is plotted as a fu nction of th e b an d se paration , D n Å , nor m alized by
the no m inal p h ase r eso lution , 1/x m ax . I f the intensi ties o f bo th bands ar e
posi tive (M 2 5 2 M 1 ), then w ith either the M ertz o r M er tz S igned algorithm , the calcu lated p hase angle ( p lu s sy m b ols) is bou nded by 5 p /
32 an d p /8. H ow ev er, if the intensities are o f o pp osite sig n ( M 2 5
2 2M 1 ) , then the ph ase an gle calcu lated w ith the M ertz m etho d ( d ot ted
line) is b ou nd ed o nly b y 5 p /32 an d 9 p /8. A pp lication o f the M ertz
S igned phase corr ection ( AT A N fu n ction) restricts the resu ltant ph ase
angle to th e 2 p /2 to p /2 rang e (d ash ed lin e) , b ut large er ro r s in the
phase angle persist n ear a disco ntin u ity at D n Å ´x m ax 5 0 .4 4. T he b and
se par ation n ear w hich these er ro r s ap pear dep en ds on the relative m agnitude of the tw o ban ds an d the apodization fu nction.
exam in in g th e p h ase calcu latio n. T h e ap o dized in terferog ram can be represented as th e in verse F o urier tran sfo rm o f th e sp ectru m b(n Å ) tim es th e in stru m en tally g en erated p hase factor e i u (n Å ) , all m ultiplied b y th e apo d izatio n
fu n ctio n A (x):
I(x) 5
A (x)´F2 1 { b (n Å )e i u
(n Å )
}.
(8)
F o u rier transfo rm atio n o f th is ap o dized in ter fero gram
yields th e con v olu tio n of the co m p lex sp ectrum w ith th e
lin e sh ap e fu n ctio n , L (n Å ) 5 F{ A (x)}:
B 9 (n Å ) 5
L (n Å ) J [b (n Å )e i u
(n Å )
].
(9)
R eg ardless o f th e ap od izatio n fun ction u sed , as th e resolu tio n o f th e p hase calculation is lo w ered, L (n Å ) b road en s.
T h erefore, all b an ds are bro aden ed ; i.e., th e co n trib ution
to the calcu lated p hase an g le fro m n eig h bo rin g p eaks is
in creased .
L et u s no w ex am in e in m o re d etail the effects o f th is
con trib utio n u sing a sim p li® ed sp ectrum con sistin g o f
tw o d fun ction s of m ag n itu des M 1 an d M 2 at n Å 0 an d n Å 0 2
D n Å , resp ectiv ely. A p ply in g th e co n v olu tio n d escribed
abo v e, o n e o b tain s
B 9 (n Å ) 5
L (n Å 2
n Å 0 )M 1e i u (n Å ) 1
0
L (n Å 2
nÅ
0
1
D n Å )M 2 e i u (n Å
0
2
D nÅ )
.
(1 0)
E v alu atio n of th is eq u atio n at the cen ter frequ en cy o f th e
ban d at n Å 0 g ives
B 9 (n Å 0 ) 5
L (0 )M 1 e i u
1
1
L ( D n Å )M 2 e i u
2
(1 1)
w h ere w e hav e d e® n ed u 1 5 u (n Å 0 ) an d u 2 5 u (n Å 0 2 D n Å ).
T h us, B 9 ev aluated at th e p osition o f o ne b an d is actu ally
a w eigh ted v ecto r su m w ith co n tribu tion s fro m b o th
ban d s.
F ig u re 2 sh o w s th e beh avio r o f the calcu lated p h ase
976
Volume 52, Number 7, 1998
ang le, u 9 , as a fu n ctio n of th e separatio n betw een th e tw o
ban d s, D n Å , n orm alized b y th e ph ase resolu tio n 1 / x m ax . If
M 1 an d M 2 are of th e sam e sign , th en th e calcu lated p h ase
ang le u 9 (n Å 0 ) at the center o f th e resolu tio n b ro aden ed line
is co nstrain ed to the ran ge of u 1 to u 2 (F ig. 2 , so lid line).
A ssu m in g th at th e tru e p h ase ang le varies slow ly (i.e., u 1
ø u 2 ), the p hase erro rs in trod u ced b y u sing a red u ced
reso lutio n are sm all fo r b an d s o f th e sam e sign .
H ow ev er, if M 1 an d M 2 are o f o p po site sign , as in the
sim u lated spectru m o f F ig. 1 , then u 9 m ay v ar y fro m u 2
to u 1 1 p (F ig . 2 , d o tted lin e). In th e ex am p le in F ig. 2 ,
th e relativ e in tensity o f th e neig hb o rin g ban d s is g iven
by M 2 5 6 2 M 1 . T hu s, the in tensity in th e w in g s of b an d
2 ex actly can cels th e cen ter in ten sity of ban d 1 w h en th e
ban d sep aratio n is equ al to th e h alf-w id th at h alf-h eig h t
(H W H H ) o f the ap od izatio n fu n ctio n. P assing th ro u g h
th is separatio n , the calcu lated p hase an g le at n 0 un d erg o es
a rapid tran sitio n of p rad ians. A t m u ch sm aller sep aratio n s, it app roach es u 2 ; at m u ch larg er sep aration s, it ap pro ach es u 1 1 p .
F or th e H ap p ±G en zel fu nction (th e ap o d ization fu nctio n u sed thro ug h o ut th is p aper), D n Å H W H H 5
0 .44 /x m ax . 1
W h en D n Å ´x m ax !
0 .4 4, th e calcu lated ph ase an gle at n Å 0 is
heav ily in ¯ u enced b y th e larg er n eigh b orin g ban d , so th at
u 9 . u 2 . T his is th e situ atio n also enco u ntered in F ig. 1 B
fo r th e b and s at 2 4 8, 4 5 0, an d 65 2 cm 2 1 . In th ese cases,
th e n eig h bo rin g p ositive b and is clo se en o ug h to th e n egativ e b an d of in terest to d om inate th e p h ase an g le calculatio n . T h u s, no p ¯ ip is en co un tered in the calculated
ph ase an g le, an d a n egative ban d is pro perly p ro du ced
by the M ertz p hase co rrection m eth od .
O n the o th er h an d, w hen D n Å ´x m ax & 0.44 , the p h ase
ang le calculatio n is in ¯ u en ced m o re b y th e n eg ative b an d
itself, so that u 9 . u 1 1 p . T h is p attern can b e seen in
F ig . 2 , w here th e d otted line ap p ro aches 9 p /8 at large
separatio n s. A ll the b an ds in F ig. 1 D also fall into th is
categ or y. T h e tw o o pp o sitely sig ned ban d s in each pair
are m uch farth er ap art th an th e half-w idth of the line
shap e fun ctio n intro du ced b y reso lu tio n red uction an d
apo d ization . T hu s, the in ¯ uen ce o f the po sitive b an d o n
th e n eig h bo rin g neg ativ e ban d is sm all, an d a p sh ift do es
occu r in th e calcu lated p hase ang le. In th is case, th e
M ertz alg o rith m p ro d u ces re¯ ected b an d sh ap es.
W ith th e add itio n o f a (seco n d) p sh ift w h en D n Å ´x m ax
. 0.44 , the M ertz S ig ned alg orith m can o ften p rod u ce a
pro perly ph ase-co rrected spectru m ev en w h en th e M ertz
alg o rith m fails (F ig . 2 , dash ed line). T his result can also
be seen b y co m p arin g traces D and E in F ig . 1. H o w ev er,
in the reg ion of D n Å ´x m ax ø 0 .44 , th e p hase ang le u 9 calculated w ith th e M ertz S ig n ed alg orithm still u n d erg o es
larg e sw ing s aw ay fro m the tru e p h ase an gle. T he so u rce
of these larg e sw in g s is ap p aren t fro m F ig . 2 . U se o f the
fu ll-rang e arctan gen t fu n ctio n , as in th e M ertz m eth od
(d o tted line), p ro d uces a sm oo th tran sitio n from u 9 . u 2
to u 9 . u 1 1 p . L im itin g th e p h ase ang le to 2 p /2 to p /
2 w ith the AT A N fun ctio n , as in th e M ertz S ig n ed algo rith m (dash ed lin e), p rod u ces a calcu lated ph ase an g le
in w hich larg e erro rs rem ain . T h ese larg e erro rs o ccur
w h en ev er the p h ase reso lutio n is ch osen in su ch a w ay
th at n eig h b oring ban d s o f u neq u al inten sity and o p p osite
sig n n early can cel on e ano th er n ear the center o f th e
w eak er b an d .
T h is is p recisely th e situ atio n en co un tered fo r th e
p eak s ab ov e 1 0 00 cm 2 1 in the sim u lated spectru m o f F ig .
1 w ith a ph ase reso lutio n o f 1 2 8 cm 2 1 (traces B and C ).
T h e spacin g b etw een th e p eaks at th is p h ase reso lu tio n
is ju st larg e en o ug h to en su re th at th e po sitiv e b an d can
sig ni® can tly in ¯ uen ce, bu t n ot do m in ate, the calculatio n
o f th e p h ase ang le fo r the neg ativ e b an d. T h is pattern can
b e seen in th e 1 2 8 cm 2 1 resolu tio n F ou rier tran sform o f
the interferog ram (d o tted trace sup erim po sed o n F ig . 1 B ),
w hich sh o w s the nearly com plete can cellation of inten sity
at th e centers of th ese n egative b and s. T h e in term ed iate
p hase ang les th at result are m an ifest as p artially re¯ ected
p eak s in th e M ertz-calcu lated sp ectru m (F ig . 1 B ). T h ese
erro rs can n ot b e pro perly co rrected b y ap plicatio n o f the
M ertz S ig n ed alg o rith m (F ig . 1 C ).
W h ile the precedin g d iscussio n fo cu ses o n ph ase co rrectio n erro rs w ith th e u se o f th e M ertz m eth o d , it is
ap plicable to th e F orm an m etho d as w ell. T h e p h ase errors resu ltin g from o p po sitely sig ned b and s o ccur d uring
the calcu latio n of th e p hase spectru m , e 2 iu 9 (n Å ) , a step th at
is co m m o n to the M ertz an d F o rm an m etho d s. 2 ,3
P h a se C orrectio n via S elf-C o n v o lu tio n o f th e In terfero g ram (D o u b led -A n g le M eth o d ). A s sh o w n ab ov e,
the pro blem s enco u ntered in ap p ly in g th e M ertz S ig n ed
algo rith m to ac-co u p led spectra can b e traced to th e
sm oo thin g o f a d iscon tinu o us fu n ctio n . H ow ev er, it is
p ossib le to elim in ate th e d iscon tin u ities an d th us resto re
the slo w ly v ar yin g natu re o f th e p hase an g le, sim ply b y
co nv o lv in g th e interferog ram w ith itself p rio r to carr y in g
o ut the ph ase co rrection . T h is con v olu tio n is equ iv alen t
in th e F ou rier d om ain to sq u arin g th e com p lex rep resen tatio n o f the sp ectru m , i.e.,
F{I
J I} 5
[b(n Å )e i u
] 5
(n Å ) 2
b 2 (n Å )e i2 u
(n Å )
.
(1 2 )
T h u s, th e self-co n v olv ed in terfero g ram I J I is the in v erse F ou rier tran sfo rm o f a com p lex sp ectru m con taining on ly p ositiv e in tensities, b 2 , and a slo w ly var y in g
p hase an g le, 2 u (n Å ). T h e do u b led p hase ang le, 2 u 9 (n Å ), can
easily b e calcu lated b y ap p lyin g th e AT A N 2 fun ction to
the real an d im ag in ar y parts of th e F o u rier tran sfo rm o f
I J I. (N o te th at the algo rith m used in the A rray B asic
p ro g ram pro vid ed in the A pp en dix d oes n ot actu ally em p lo y th e AT A N 2 fu n ctio n, b ut carries ou t m athem atically
eq uiv alen t o p eration s.) T h e ad v antag e o f per fo rm in g the
self-co nv o lu tion an d th us d ou b lin g th e ph ase an g les is
that the p h ase facto r e i2 u (n Å ) b eco m es id en tical fo r u an d u
1 p . T h is co nd itio n elim in ates all th e p ¯ ip s in th e ph ase.
It is th u s p ossib le to tru ncate and ap o dize the self-co nv olv ed in terfero g ram (I J I ) p rior to F ou rier tran sfo rm atio n, i.e., to red uce th e ph ase reso lu tion , w ith ou t in tro d u cin g an y larg e p hase ang le erro rs.
Ta k in g h alf of th e red u ced -reso lutio n d o u bled -p h asean gle g iv es th e b est sm oo th ed estim ate for th e p h ase o f
the o rig in al in ter ferog ram . H o w ev er, d eterm inin g th e
h alf-ang le still req uires a ch oice b etw een ½[2 u 9 (n Å )] and
½[2 u 9 (n Å )] 1 p . T h is am b igu ity can be reso lv ed by u sin g
the criterio n that the ph ase an g le m ust be a slo w ly v ar ying fu n ctio n of n Å . D u rin g th e calculatio n of the d iscrete
array o f ph ase an g les, each u 9 (n Å ) is sim p ly ch osen to b e
the v alue closer to th at o f th e p revio u s elem en t o f th e
array. T his cho ice is n o w easy becau se th e reso lution red uctio n h as su p pressed th e no ise w ith o u t intro du cing
p hase ang les b etw een u 9 (n Å ) an d u 9 (n Å ) 1 p .
W h en the en tire p hase array h as been pro du ced , a g lo bal cho ice rem ain s. A d din g p to the p h ase an g le at all
freq u en cies also satis® es th e criterio n o f slo w ly v ar y in g
p hase. T he o nly w ay to m ake th is ch oice is b y ex am in in g
the resulting p hase-co rrected spectru m . A d din g p to the
p hase an g le is eq u iv alent to m u ltip lyin g th e en tire sp ectru m b y 2 1. T h u s, w hile the algo rith m p ro d u ces the co rrect relativ e o rien tatio n of p o sitiv e and n egativ e p eak s,
so m e k n o w led ge o f th e ``co rrect’ ’ g lo bal orientation o f
the spectru m is req uired. W ith resp ect to th e sim u lated
sp ectru m o f F ig . 1 , a p riori k no w led g e o f th e sig n o f an y
o ne ban d allo w s p ro p er glo b al o rien tatio n of all of th em .
T h u s, w ith o ut an y o ther in form atio n, th e D ou b led -A n g le
m eth o d can co rrect the p h ases o f o ur sim u lated d ifferen tial in terfero gram , p rod u cin g a resu lt (no t sh ow n ) th at
is id entical to F ig . 1 A .
F o r the D o u bled -A n gle p h ase correctio n m eth o d to
w ork p ro p erly, a d o u ble-sid ed in terfero gram is n eeded .
To see w h y, co nsid er th e effects o f tru n catin g on e sid e
o f a d o u ble-sid ed differential in terferog ram p rio r to co n v olv ing it w ith itself. T h is o n e-sid ed tru n catio n intro du ces erron eo us p hase ang les in the F o u rier tran sfo rm o f th e
u nco n vo lv ed in terferog ram ; co nv o lutio n do u b les these
erro n eo u s ang les. R ed u cin g th e reso lutio n after co n v olutio n sm o o th s th e erro rs slig h tly b ut d oes n ot sig ni® cantly red uce th em . T hey can be av o ided o n ly by usin g
the full do u ble-sid ed in terfero gram to calculate th e co n v olu tio n .
Z P D S election . T h e interfero g ram sig n al at th e Z P D
is ap pro xim ately eq ual to the sp ectral in ten sity integ rated
o ver th e b and w id th . T hu s a sp ectru m co n tain ing p ositive
an d n eg ativ e in tensities o f com p arab le m ag nitu de w ill
p ro d uce an in terfero g ram w ith ou t a cen terb urst at the
Z P D . E v en in such cases, th e Z P D can o ften b e d e® n ed
fro m th e con v olu tio n o f the interferog ram w ith itself. T h e
integ ratio n of th e sq u ared in ten sities resu lts in a cen terb urst in th e self-co nv o lved interfero g ram . T h e p osition o f
this cen terb u rst can th en be u sed to lo cate the Z P D o f
the o rig in al in ter fero gram . If a m easu red in terferog ram
o f N p oin ts is rep resen ted as I(n) w h ere n 5 (0 , . . . , N
2 1 ), an d I(n ) is set to zero fo r n , 0 and n $ N , th en
the self-co nv o lu tion is d e® n ed as
I J I (m ) 5
O
N2 1
n5 0
I (n )I
1
N
2
2
1 2
n 1
2
m .
(1 3 )
(N o te: T his d e® n itio n o f co n v olu tio n is dictated by th e
A rray B asic pro gram m ing lan gu ag e th at w e u sed to im p lem ent ou r ph ase co rrectio n m etho d as d etailed in th e
A pp en dix ). A ccord in g to this d e® n itio n , if th e Z P D is at
p oin t N /2 2 1 1 D x o f th e o rig in al interferog ram , th en
the self-co n vo lutio n w ill sh ift it to p oin t N /2 2 1 1 2 D x.
If the cen terb u rst of th e self-co nv o lv ed in terfero g ram o ccu rs at a po in t w here 2 D x is o d d, it is ro u nd ed d ow n b y
o ne so th at D x can b e in tegral.
M A T E R IA L S A N D M E T H O D S
B acterio rh o do p sin (b R ) sam p les w ere p rep ared fro m
H . ha lob iu m as d escribed p rev io usly. 10 P u rp le m em bran e
p ellets w ere w ashed w ith d istilled w ater an d tran sferred
to a C aF 2 w in d o w . A secon d C aF 2 w in do w w as coated
arou n d its ed g e w ith v acu um grease an d p ressed ag ain st
the ® rst to seal th e b R sam ple.
APPLIED SPECTROSCOPY
977
F IG . 3. (A ) Transi ent ac-coupled inter fer ogram of bacter iorh o dopsin photop ro ducts averaged over th e tim e range 0 ±1 m s after ph otolysis (8 cm 2 1
reso lution w ith a band w idth of 0 to 1 970 cm 2 1 ; i.e., a total of 88 8 points) . (B ) S am e inter fer o gram after application o f a d igital h igh-p ass F o u rier
® lter w ith a cu toff fr eq u en cy o f 400 cm 2 1 . (C ) Co nv olution of the ® ltered inter f erogr am (B ) w ith itse lf. T his self- co nvolved inter fer ogr am h as an
enhanced inten sity at the Z P D (o r a n earb y po int; se e text fo r details) and is ther efore used to identify the Z P D of the or igin al inter f ero g r am . F o r
the tw o-sid ed inter fer og ram in A , th ere ar e 44 4 po ints to the lef t of the Z P D an d 443 to th e r ight. F or the se lf-con volution in C , the Z P D is sh ifted
slightly, as calcu lated b y ou r pr og r am , w ith 4 4 5 points to th e lef t and 44 2 to the righ t.
Interfero g ram s w ere co llected on a B ru ker IF S -6 6 F TIR sp ectro m eter in tim e-reso lved step scan m od e at 8
cm 2 1 reso lutio n b y u sin g a K o lm ar p h oto vo ltaic H gC d Te
detector (M o d el #K M P V 1 1-1 -L J2/2 3 9). T h is d etecto r’ s
pream p li® er h as d u al ac- an d d c-cou p led ou tpu ts. T he
m easu red b an dw id th w as lim ited to 0 ±1 9 70 cm 2 1 as a
resu lt o f red u ced sam p lin g of the in terfero g ram . A lo n gpass o ptical ® lter placed in fro n t o f th e d etecto r p reven ted
aliasing of o p tical sign als fro m o u tsid e th is ban d w id th.
A p ulsed , freq u ency -do u bled N d 1 :YA G laser (5 3 2 n m ,
10 m J cm 2 2 ) w as used to trig ger th e b R ph o tocy cle fo r
th e tran sient F T-IR m easu rem en ts, w h ich w ere m ad e w ith
th e d etecto r an d intern al d ig itizer ac-co u p led . Tran sien t
sig n als fro m 1 0 ¯ ash es, spaced ev er y 3 00 m s, w ere record ed at each m irro r po sitio n . T im e-reso lved interfero gram s correspo n din g to 1 0 tim e slices o f 10 0 m s each
w ere so rted an d sto red to d isk, b u t th e 10 interfero g ram s
w ere av erag ed to g eth er prior to ph ase calculation .
To o btain a dc-co up led step -scan interfero g ram fo r
ph ase correctio n w ith th e d c S to red P hase m eth od , w e
reco rd ed three tim e slices o f 10 m s each w ith the laser
off, w ith 10 coad d itio ns at each m irro r p o sitio n . T he three
tim e slices w ere th en av erag ed an d sto red as a sing le
in terfero g ram . T his in terfero gram w as m easu red im m ediately p rio r to collectio n o f th e ac-cou p led d ifferen tial
in terfero g ram s to w h ich th e sto red p hase w as to b e ap plied.
D ata p ro cessing o f th e in terfero gram s w as carried ou t
in A rray B asic ro utines o n G R A M S /3 2 so ftw are (G alactic In du stries, S alem , N H ). T h e M ertz an d d c S to red
P h ase m eth o d s fo r p hase correctio n w ere im p lem ented by
ru n nin g th e stan dard icompute.ab A rray B asic co de su p plied w ith th e so ftw are. T h e M ertz S ig n ed an d D ou b ledA n g le m eth od s w ere im plem en ted b y sim p le m o d i® catio n s o f icompute.ab , as d etailed in th e A pp en dix .
RESULTS
W e tested v ario u s ph ase co rrection m etho d s o n typ ical
tim e-resolv ed F T- IR d ata fro m a b iolo g ical sam p le, b acteriorh od o p sin . T h e tran sien t differential in terfero gram
978
Volume 52, Number 7, 1998
m easu red fro m a b R sam p le after ph o to ly sis is sh ow n in
F ig . 3A . T h ese data, an d th e p h ase co rrection results ob tain ed fro m th em , are rep resen tativ e o f m easu rem en ts on
fo u r d ifferen t sam p les o f b R .
D eterm in ation o f th e Z P D . T h ere is a slo p in g baselin e in the raw in ter fero gram in F ig . 3A ; su ch a b aselin e
drift is q u ite co m m on in step -scan m easu rem ents. T his
artifact m ust b e rem o v ed by th e ap plicatio n of a d igital
hig h -p ass F o urier ® lter (F ig . 3B ). F ailu re to rem o ve th is
slo p ing b aselin e resu lts in a self-co nv o lved in ter ferog ram
w ith lo w -frequ en cy featu res th at p ro h ib it selectio n o f th e
Z P D p o sitio n (n ot sh o w n ). T h e self-con v olu tio n o f th e
® ltered in terfero g ram h as a clear centerbu rst at its Z P D
(F ig. 3 C ). T h e p o sitio n o f th is centerbu rst is used to d e® n e the Z P D o f th e o rig in al interfero g ram as d iscu ssed
abo v e.
C o m p a riso n o f P h a se C o rrection M eth o d s. F ig u re 4
sho w s th e resu lts o f different p hase co rrectio n alg orith m s
app lied to a step -scan tim e-reso lv ed interfero g ram o f b R
ph o toly sis. T he sam e m easu red differential in terfero g ram
(F ig. 3A ) and Z P D po sitio n w ere u sed fo r all o f the ph ase
correction m etho d s. F o r the D o ub led-A ng le m etho d , the
on ly a prio ri kn o w led g e o f th e spectru m th at w as utilized
is the p ositive sig n o f the large in tensity chan g e at 1 5 27
cm 2 1 . 11 T his in fo rm ation is n eeded to m ak e the co rrect
glo b al ch o ice o f sign fo r the sp ectru m .
T h e sp ectra pro du ced w ith th e d c S tored P hase (A ) an d
D o u bled -A n gle m etho d s (D ) m atch ver y closely ov er the
entire 8 5 0 ±1 95 0 cm 2 1 reg io n, w h ile th e M ertz alg orithm
(B ) p rod u ces a p attern o f ``re¯ ected p eaks’ ’ o v er m uch
of th e sp ectru m . T h e M ertz S ig n ed alg o rith m (C ) co rrects
som e, bu t n o t all, o f these p eak s. T he b road area o f n egativ e inten sity fro m 1 25 0 to 1 45 0 cm 2 1 is pro perly p hase
corrected , b ut th e jux tapo sitio n o f p o sitiv e an d n eg ativ e
ban d s from 1 5 00 to 1 6 00 cm 2 1 cau ses th e M ertz S ig n ed
alg o rith m to p ro d u ce b an d s of m u ch red uced inten sity,
corresp o n din g ro ug h ly to a p /2 p h ase error.
A s a m o re strin gen t test o f th ese m eth o ds, the resid u al
spectra rem ainin g alon g th e im ag in ar y ax is after p h ase
correction w ere ex am in ed (F ig. 5). B o th th e d c S to red
F IG . 4. P hase corr ection m ethod s applied to calculating th e transient
ac-coupled sp ectrum o f bR, averaged over th e ® r st 1 m s af ter a photoly sis ¯ ash . A ll fo u r traces w ere co m p uted f ro m the sa m e transient
differ en ce inter f ero g r am , each usin g the indicated m eth od o f p hase co rrection. T he sp ectral r eso lu tion is 8 cm 2 1 over a band w idth of 0 to 1970
cm 2 1 w ith H app ±G en zel apodizatio n and 128 cm 2 1 phase reso lution.
A lth ough the M er tz S igned m eth od co rr ects so m e areas o f `` re¯ ected
peaks’ ’ (1 2 50 ±1 45 0 cm 2 1 ) se en w ith the M ertz phase co rr ection , it pro duces b an d s o f redu ced m ag nitude in oth er region s (1 500 ±1600 cm 2 1 ).
H ow ever, the D ou bled-A n gle an d d c S tored P hase m etho ds pr oduce
nearly identical sp ectra.
P hase and D ou b led -A n g le m eth o ds p ro d uce residu al
im ag inar y spectra w h ich app ear as ran do m n o ise. H o w ev er, th e residu als fro m th e M ertz an d M ertz S ig ned
m eth o ds h av e featu res m u ch larger th an th e n o ise at
1 20 1 , 1 5 27 , an d 1 56 0 cm 2 1 ; these corresp o n d to im pro perly p hase-corrected b an ds in the real sp ectra. N o te th at
in th e bro ad regio n o f neg ative in tensity b etw een 12 5 0
an d 1 45 0 cm 2 1 , all fo ur m etho d s g iv e resid u al (im agin ar y ) sp ectra sm aller th an th e n oise. A ltho u gh th e M ertz
algo rith m succeed s in ro tatin g th e d ifferen tial in tensity in
this regio n fu lly o n to th e real ax is, it in co rrectly assig n s
the sign o f that in ten sity.
T h e p hase sp ectra p rod u ced w ith th e fo u r m eth o d s are
sh ow n in F ig. 6 . T h e ph ase an g les fro m b o th th e d c
S to red P h ase and D o ub led-A ng le m eth od s are close to
zero an d v ar y slow ly acro ss th e spectrum . H o w ever, th e
p hase an gles calcu lated b y u sing the M ertz an d M ertz
S ig n ed alg o rith m s co v er th e en tire allo w ed ran g es ([ 2 p ,
p ] an d [ 2 p /2 , p /2], resp ectiv ely ). A s d iscu ssed in th e
T h eo r y sectio n , w ith a tru e p h ase an gle clo se to 0 , th e
M ertz alg o rith m app lied to a sp ectru m o f alternating p o sitiv e an d neg ativ e b an d s sho u ld ideally p ro d u ce a d isco ntin uo u s p hase sp ectru m w h ich ju m p s b etw een 0 an d
6 p radian s. T h ere are h ints o f su ch disco n tin u ities in th e
M ertz ph ase sp ectrum (F ig . 6 , d ot-dash lin e), b ut th ese
featu res are sm oo thed b y th e low (12 8 cm 2 1 ) reso lu tio n
o f the p hase calcu latio n. L ik ew ise, in th e reg io n fro m
1 50 0 to 16 0 0 cm 2 1 , in w h ich th e M ertz S ig ned alg orithm
w ork s least effectively, the ph ase an g le tak es an in term ed iate v alue of 2 p /2 in stead o f 0 o r 6 p rad ians.
E ffect o f V a ryin g th e P h ase R eso lu tio n . A s d iscussed
in the T h eo r y sectio n abo v e, the sp ectral artifacts p rod uced w ith the M ertz S ig ned alg o rith m are du e to
sm oo thin g of the p h ase sp ectru m . T h e p hase an g le is calcu lated fro m a tru ncated regio n o f th e in ter fero gram an d
F IG . 5. R esid ual sp ectra left along the im aginar y ax is af ter ph ase co rrection w ith each of th e m ethod s. T he sc ale is the sa m e as in F ig. 4 .
W ith either the M ertz or the M er tz S igned m ethod, the r esid u al sp ectral
inten sity is as lar ge as the real sp ectral intensity ( F ig. 4) fo r so m e b and s.
O n the o ther h an d , the r esid u al inten sities of the D o ubled -A n gle an d d c
S tored P hase m ethods are com p arable to the noise .
then in terp olated to m atch th e p o int sp acin g o f the sp ectru m fro m th e full in ter fero gram .
F igu re 7A sh ow s ho w tru n catin g th e interfero g ram at
d ifferen t reso lu tion s affects th e real p art of its F o u rier
tran sform (i.e., th e sp ectru m u sed fo r calcu latin g the
p hase ang les). To g ive accu rate p h ases w ith th e M ertz
S ig n ed m eth od , th is real p art of the F o urier tran sform
g enerally need s to faith fully rep rod u ce th e p osition s o f
the zero-crossin g s o f th e tru e in tensity spectru m . H o w ev er, as th e p hase reso lutio n is lo w ered fro m 8 to 3 2 cm 2 1
F IG . 6. P hase sp ectra calculated by u sin g each of th e f our m ethod s at
128 cm 2 1 p hase reso lution. I nterp olation to the f ull 8 cm 2 1 sp ectral
reso lution w as per fo r m ed by zero-® llin g the tru ncated inter fero g ram
pr ior to F ourier transfo r m ation. T h e M ertz m etho d d oes n o t sim p ly
jum p f ro m an angle near 0 fo r po sitive ban d s to an angle n ear 6 p f o r
negativ e bands. In stead, larg e r egions o f the sp ectru m h av e calcu lated
phase an gles of interm ediate v alues. R estriction of the ph ase an gle to
[ 2 p /2, p /2] w ith the M er tz S igned m ethod does n ot resto r e all p hase
an gles to near 0 b ecause of these interm ed iate angles. H ow ev er, the
D oubled-A ngle m eth od does pro duce ph ase ang les that v ar y slo w ly
w ith n Å and ar e close to 0 ever yw h er e.
APPLIED SPECTROSCOPY
979
F IG . 7. B eh avior, as the reso lution is decrease d , of the r eal p art of the F o urier transf orm of (A ) the original inter f ero gr am an d (B ) the se lfconvolution of the inter f erog r am . N o ph ase cor rection h as b een applied.
and then to 1 2 8 cm 2 1 , th e n u m b er o f zero -cro ssin gs betw een 85 0 and 1 95 0 cm 2 1 d ro p s from 17 to 1 0 to 4 , an d
th eir p o sitio ns are sh ifted . T h e larg e p ositive real b an d at
15 2 7 cm 2 1 in the h igh -reso lutio n F ou rier transfo rm (F ig.
7A , top trace) co rresp on d s to an area o f near zero in ten sity at 1 28 cm 2 1 reso lu tio n (F ig. 7 A , bo ttom trace). T h e
sam e is tru e fo r th e three p o sitiv e b an ds n ear 1 2 00 cm 2 1 .
T h e o nly co nsistent featu re w ith a sig n th at d oes no t d epen d stro n gly on sp ectral reso lutio n is th e b ro ad area o f
neg ativ e in ten sity b etw een 1 25 0 and 14 5 0 cm 2 1 .
In con trast, F ig . 7 B sh ow s th e real part o f th e sp ectru m
calcu lated fro m the F o u rier transfo rm o f th e self-con vo lv ed in ter fero gram . N early all in ten sities are no w
greater th an zero reg ard less o f th e resolu tio n used . A lth o u gh the b an ds are b road en ed at 1 2 8 cm 2 1 reso lution ,
th e p o sitio ns o f th e p eak s are co n sisten t w ith th e b an d s
ob ser ved at 8 cm 2 1 resolu tio n .
F or th e M ertz S ig n ed m eth od , sm o o thin g d ue to th e
red u ced reso lutio n of th e p hase sp ectru m resu lts in interm ed iate v alu es o f th e calcu lated p h ase an g le u 9 (n Å ) in
reg ion s o f sh arp altern atin g p o sitiv e an d n egative featu res
in th e sp ectru m (see F ig. 8). T h e zero -cro ssing s o f th e
real p art o f th e F o u rier tran sfo rm (F ig . 7 A ) o ccur at th e
sam e w av en um b er v alu es as the b ig jum ps in th e p h ase
ang le (F ig . 8 ). A t 8 cm 2 1 resolu tio n , after ap plicatio n o f
th e 2 p /2 to p /2 lim itation , the ph ase an gle u 9 (n Å ) is close
to zero at m o st freq u encies ex cep t fo r th ose d irectly adjacen t to zero -crossin g s. U n fo rtun ately, these are m ad e
m o re frequ en t b y spectral n o ise. A t 32 an d 1 2 8 cm 2 1
reso lutio n, th e no ise is p rog ressiv ely su pp ressed , b ut the
ph ase spectru m is also p ro g ressiv ely sub ject to th e system atic erro rs in du ced b y th e b road ened lin e sh ape (see
T h eor y section ).
T h e resu lts o f these sy stem atic errors can b e seen clearly in F ig. 8C . T he w orst case o ccu rs b etw een 15 0 0 and
17 5 0 cm 2 1 . H ere, th e p h ase an gle m ig h t b e ex p ected to
¯ ip betw een ; 0 an d ; ( 6 p ) fo r th e M ertz m etho d , and
to b e corrected to ; 0 ever y w h ere b y th e M ertz S ig n ed
m etho d . Instead th e p hase an g le tak es o n an in term ediate
valu e near 2 p /2 o v er a larg e sp ectral ran ge. T his v alue
is un affected by w h eth er th e M ertz o r M ertz S ig ned
m etho d is used . T h e resulting sp ectra (F ig . 4B , 4 C ) thu s
coin cid e in this w av enu m b er ran g e, b u t sh o w in co rrect
m ag nitu des o f b o th p ositive an d n eg ativ e ban d s.
T h e M ertz an d M ertz S ig n ed m etho d s d o n ot d iv erge
ov er th e 1 50 0 ±1 7 50 cm 2 1 sp ectral regio n u n til th e ph ase
reso lutio n is raised to 3 2 cm 2 1 (sp ectra n ot sh ow n ). E v en
at th is reso lu tion , h o w ev er, the neg ativ e ban d at 1 56 0
cm 2 1 still sh o w s a sy stem atic artifact du e to th e bro adenin g o f th e larg e p o sitiv e n eigh b orin g b and at 15 2 7
cm 2 1 , an alog o u s to the artifacts sh ow n in F ig . 1 C . O nly
at 8 cm 2 1 reso lu tion d o es th e M ertz S ig ned m eth od p ro p erly co rrect the ph ase in th e 1 50 0 ±1 7 50 cm 2 1 ran g e.
H o w ever, th e p o orer sig n al-to-n o ise ratio in o ther reg ion s
of th e sp ectru m p reclu des u se o f th is p h ase reso lu tio n for
correcting th e en tire sp ectral rang e.
F IG . 8. P hase sp ectra calcu lated w ith the M ertz S ig ned m ethod at various sp ectral r eso lu tion s. T h e in ter fero g ram w as truncated at the indicated r eso lu tion, and zero - ® lled pr ior to F ou r ier transfo r m atio n and
phase angle calcu lation u si ng the fo r m ula, u 9 (n Å ) 5 AT A N (Im (n Å ) /R e(n Å )) .
D IS C U S S IO N
980
Volume 52, Number 7, 1998
T h e p resen ce o f b oth po sitiv e an d n eg ativ e peak s in an
F T-IR d ifferen ce sp ectru m po ses a b inar y cho ice o f u o r
u 1
p fo r each p h ase ang le fo r any d irect m eth od o f
p hase-correctin g a m easu red d ifferen tial interfero g ram .
F or m ost F T-IR sp ectrom eter d esig n s, th ere are tw o ch aracteristics o f the p hase sp ectru m th at can b e u sed to m ak e
this ch oice: (1 ) th e ph ase an g le sh ou ld b e ever y w h ere
close to zero sim ultaneo u sly fo r an ap p ro p riately cho sen
Z P D ; and (2 ) the ph ase an gle sh o uld v ar y slo w ly as a
fun ctio n o f n Å .
T h e th ree d irect m etho d s o f p h ase co rrectio n evalu ated
h ereÐ M ertz, M ertz S ig ned , an d D ou b led-A ng leÐ each
u se differen t criteria fo r d eterm in in g the ph ase an g le. T h e
M ertz algo rith m assu m es th at th e p h ase an gle varies
slo w ly an d that th e in ten sity is alw ays p ositive ex cept fo r
n oise. T h e latter assu m p tio n is clearly inco rrect fo r acco up led d ifferen ce spectra. U se o f th e M ertz alg orithm
results in an erro r o f p rad ians fo r b road n egative b an ds,
an d an erro r n ear p /2 rad ians fo r reg io ns co n tain ing
closely spaced p o sitiv e an d n egative ban d s of sim ilar intensity.
T h e M ertz S ign ed m eth o d su bstitu tes th e assu m p tion
o f p ositivity w ith a m o re realistic criterion fo r d ifferen tial
sp ectra, w hich assu m es th at th e ph ase an gles sh o uld all
lie w ith in the in ter v al [ 2 p /2 , p /2 ]. P hase factors are th us
restricted to lie in th e ® rst and fo urth q uad ran ts of th e
co m p lex p lan e; p rad ian s are ad d ed to an y calcu lated
an gle th at falls o utsid e th ese lim its. T h e m eth od is successfu l at co rrecting th e re¯ ectio n of b ro ad neg ativ e-in tensity p eak s. H o w ev er, it fails to co rrect sp ectral reg ion s
co ntainin g sh arp altern atin g p o sitiv e an d n egative ban d s,
freq u en tly y ield ing a resu lt id entical to th e M ertz alg o rith m , i.e., stro n gly atten uated b an ds d u e to an erron eo us
p hase th at can no t b e co rrected b y the ad d itio n of an y
m u ltip le o f p rad ian s. A s w e sh o w ab o ve, th is erro n eou s
p hase resu lts from b lurrin g o f n eig hb o rin g po sitive an d
n egative b and s b y th e u se o f a lo w ered ph ase reso lu tion .
L ike th e M ertz and M ertz S ig n ed m eth o d s, th e D o ub led -A n g le m etho d p resen ted h ere utilizes a trun cated
(red u ced -reso lutio n an d red uced -no ise) in terfero g ram to
calcu late th e p h ase, th ereb y im plicitly assu m in g a slo w ly
v ar y ing p h ase an g le. H o w ev er, th e erro rs o btain ed w ith
the M ertz S ign ed m eth o d as a resu lt of b lu rring sh arp ly
altern atin g p ositive an d n egative b and s into each oth er
are elim inated b y d ou b lin g th eir p hase an gles p rio r to
b lu rrin g . T his ap pro ach allo w s nearby b and s w ith p h ases
that d iffer b y nearly p to b e sm oo thed o n to each oth er
w ith o ut introd u cin g erron eo us interm ed iate ph ase an gles,
sin ce u an d u 1 p g ive eq uiv alen t an g les w hen m ultiplied
b y 2. T h erefo re, th e slo w ly v ar y ing n atu re of th e ph ase
sp ectru m is m ain tain ed. T h ere is a b in ar y ch oice th at
m u st b e m ad e w h en th e origin al p h ase an gles are restored
b y div idin g by 2 . H o w ev er, th is ch o ice can b e m ade easily becau se th e D o ub led-A ng le m eth od su cceed s in allow in g reso lutio n redu ction to im pro ve the sign al-to n oise ratio in reg io ns o f altern atin g b an d s, w ith o ut in tro d ucin g p h ase artifacts.
E ffect of N o ise o n P h a se C orrectio n . T h e an o m alou s
p hase an g les p ro d uced w ith the M ertz an d M ertz S ig ned
m eth o ds in th e p resen ce o f p o sitiv e an d neg ativ e b an d s
o ccu r at th e zero -cro ssing s o f th e redu ced-resolu tio n
sp ectru m . It m ig ht th erefo re b e assum ed th at n o ise is th e
cu lp rit. T his assu m ptio n is tru e o n ly in directly. If at certain freq u encies the sp ectral m agn itud e d rop s b elo w the
n oise lev el, th en th e p h ase an gle, u 9 (n Å ) 5 arctan [Im (n Å )/
R e(n Å )], b eco m es in d eterm in ate. H o w ev er, as sh ow n in th e
T h eo r y sectio n ab o v e, ph ase erro rs n ear zero -cro ssing s
can o ccu r ev en in th e co m p lete ab sen ce o f no ise. T herefore, it is n o t co rrect to assert th at the p hase ang le erro rs
that occu r n ear the zero -cro ssin g s o f R e(n Å ) w ith the M ertz
S ig n ed m eth od are d u e to ran do m n oise.
It is tru e th at no ise-free sp ectra, su ch as th e sim ulated
sp ectru m o f F ig . 1 , can alw ay s be p rop erly p hase corrected b y usin g the M ertz S ig n ed m eth od , if th e p h ase
resolu tio n is in creased su f® ciently. H ow ev er, for m easu red sp ectra, increasin g th e ph ase reso lu tion alw ays in creases n o ise. T h us, at h igh p hase resolu tio n , n oise lim its
the accu racy of the p h ase an gle calcu latio n; at lo w p h ase
resolu tio n , o n th e oth er h and , line shap e sm o oth ing in tro d u ces p hase erro rs reg ardless o f th e am o u nt o f n o ise
p resen t.
T h e D ou b led -A n g le m etho d elim in ates th e p hase erro rs
p ro d uced w ith altern atin g sig n ed ban d s at redu ced p h ase
resolu tio n . H o w ev er, th e effect o f n o ise w ith th e D o u b led -A n g le m eth od can b e m ore sev ere than w ith th e
M ertz m eth od s, esp ecially at hig h reso lutio n. T h u s, th e
p hase co rrection w ith th e D o u bled -A n gle m etho d m u st
g enerally be p erform ed at lo w p hase reso lu tion , ty p ically
1 28 cm 2 1 .
D irect (D o u b led -A n gle) P h ase C o rrectio n v s. th e d c
S to red P h ase M eth o d . In term s o f spectral accu racy, o ur
D ou b led -A n g le p h ase co rrectio n m eth o d is co m p arab le
to, b u t d oes n o t o utp erform , the d c S to red P h ase m eth o d .
H ow ev er, b ecause it is a d irect m eth od , there is no n eed
to co llect a sep arate interferog ram fo r th e ph ase calcu latio n . P ossib le ben e® ts o f th e direct ap pro ach in clud e
d ecreased su scep tibility to in stru m ental d rift an d less to tal
tim e req uired to co llect th e d ata.
H o w ev er, these b ene® ts m ay b e p artially neg ated b y
the n eed to u se m ore co m p u ter m em o r y to co llect d o ub lesid ed in terfero gram s. T h is is th e on ly sig n i® can t d raw b ack o f the direct D ou b led -A n g le p h ase co rrection m eth o d. T h is req u irem en t has a co st in term s of d isk space
an d R A M utilization . In add itio n , the m easu rem ent o f
tran sient F T- IR step-scan sp ectra requ ires a m ean s to p erturb th e sam p le rev ersib ly at each o f ; 10 3 p osition s o f
the m ov in g m irro r. T h e n eed fo r d ou b le-sid ed in ter ferog ram s in creases th e n u m b er of p o sitio ns requ ired to co llect a sin gle in ter fero gram b y alm o st tw o fo ld, so that an
ev en g reater prem iu m is p lace o n th e reversibility of th e
ch an g es effected b y the sam p le perturb atio n. H ow ev er,
the sign al-to -n o ise ratio in th e ® nal result is im p ro v ed b y
co llectin g a tw o -sid ed interfero g ram . M o st step-scan
tim e-reso lved ex p erim en ts req u ire sign i® can t sig nal av erag in g in an y case. T h erefo re, co llectin g tw o -sid ed in terfero gram s actu ally costs u sefu l m easurem en t tim e on ly
in th ose rare cases w here the sam p le is n ot stab le fo r th e
m in im u m tim e p erio d (o r m in im u m n u m b er o f ¯ ash es)
n eed ed to co m plete a sin gle (d ou b le-sided ) m irror scan .
A b ene® t o f u sin g th e D ou b led -A n g le p h ase co rrection
m eth o d m ay b e a red u ctio n in the com p lex ity of th e hardw are n eed ed to m easu re interferog ram s. It is n o t n ecessar y to sw itch d etecto r p ream p li® ers an d m ain am pli® ers
fro m d c to ac cou p lin g in ord er to co llect a ph ase sp ectru m , th u s sim p lify ing th e in stru m en t co n ® g u ration . In
ad dition , th e req uirem en ts o f a larg e d yn am ic ran g e and
a hig h d eg ree o f lin earity in the am p li® catio n and d igitizatio n p ro cesses are relax ed . T h ese ad v antag es m ay also
APPLIED SPECTROSCOPY
981
be realized for a h ost of oth er techn iqu es th at p ro d u ce
spectra w ith p o sitiv e an d n eg ativ e ban d s, such as v ib ratio n al circular dich ro ism , as w ell as p h ase-m od u latio n
and lock -in step -scan m easurem en ts.
T h e co n vo lutio n in vo lv ed in th e D ou b led-A ng le p h ase
correction m etho d is co m p utation ally inten siv e. F o r tim ereso lved step -scan spectrosco p y, an interferog ram is collected fo r each tim e po int. T hu s, th ere m ay b e several
hu n dred in terfero g ram s fro m a sin gle ex p erim en t. C o n vo lv in g each of th ese in ter fero gram s w ith itself d u rin g
th e p h ase correctio n p ro cess w o u ld req u ire a sig n i® can t
am o un t o f tim e. A dd itio n ally, th e resu ltin g p h ase spectra
are lik ely to ex h ibit erro rs d ue to p o o r sig n al-to-n oise
ratio. In stead , it is far better to calcu late a sing le ph ase
spectrum from th e av erag e o f all th e tim e-reso lv ed in terferog ram s o r, better y et, fro m the ® rst p rin cip al co m p o nen t d eriv ed b y princip al com po n ent an alysis. T his p h ase
spectrum can b e sto red, an d th en u sed to co rrect th e
ph ases o f th e in div id u al in terfero g ram s.
T h e w eigh t o f th ese facto rs v s. th e red uced in stru m en tal req u irem ents of the D o u bled -A n gle alg orithm w ill d eterm in e th e relative ad van tages o f d irect ph ase correctio n
vs. use of a d c-co up led sto red p h ase. F o r in stan ces in
w h ich a d c-in terfero gram can easily b e collected , th e
D o u bled -A n gle and d c S tored P hase m etho d s can b e used
in co m b in atio n, each to d ou b le-ch eck the oth er, since th e
tw o m eth o ds are su sceptib le to d ifferent sou rces o f erro r.
A CK N O W L E D G M E N T S
T he autho r s w ou ld lik e to thank G alactic In d u str ies, S alem , N H , fo r
perm issio n to pu blish th e r elev an t p arts of the m odi® ed ic om p ute.ab A rray B asic cod e. T h is w or k w as su p por ted by N I H g rant G M 4 685 4.
M .S .H . w as su pp or ted b y N I H M olecular B io physics Training G r an t
G M 0 832 3.
1 . P. G rif® th s and J. d e H aseth , F ourier T ra nsfo rm In fra re d S p ectro m etry ( Jo hn W iley and S ons, N ew Yo r k , 198 6 ).
2 . L . M ertz, T ra n sfo rm ations in O p tics (Jo hn W iley and S o ns, N ew
Yo r k, 1 965) .
3 . M . F o rm an , W. S teel, and G . Va nasse , J. O pt. S oc. A m . 56 , 59
(1 96 6).
4 . P. M alon , R . K obrinsk aya, and T. K eider ling, Bio polym er s 27 , 7 33
(1 98 8).
5 . E . L ipp and L . N a® e, B io polym er s 2 4 , 799 (1 985).
6 . A . D ioum aev and M . Br aim an, J. P hy s. C hem . B 1 01 , 16 55 ( 1 99 7) .
7 . X . H u , H . F rei, and T. S p iro, B ioch em istr y 35, 13001 (1 9 9 6 ).
8 . C . M cC oy and J. de H ase th, A ppl. S pectro sc . 42, 33 6 ( 1 9 88 ) .
9 . B ru ker O pu s V ersi on 3.0 (B ru k er A nalytische M esste chn ik G m b H ,
B iller ica, M assa chuse tts, 1997 ).
10 . M . B raim an and R . M ath ies, B iochem istr y 19, 5421 (1 9 80 ).
11 . M . B r aim an, P. A hl, and K . Ro thsch ild, P ro c. N at. A cad . S ci.
U .S .A . 8 4 , 522 1 ( 1987) .
A P P E N D IX
M od i® cation s o f the A rray B asic ro u tin es sup p lied
w ith G R A M S /3 2 so ftw are (v. 4.02 ; G alactic Ind u stries,
S alem , N H ) w ere m ad e to im p lem ent the new m eth o d o f
ph ase correctio n. T h e ch ang es m ad e to th e icompute.ab
pro gram to im plem en t the M ertz S ig ned or D o u bled -A n gle p hase co rrection m eth o d are in d icated b elow . Italics
in d icate ad d ed lin es o f co d e. A few lin es of the surrou n din g pro gram are inclu d ed to m ark th e lo catio ns of th e
add itio n s. T h e un italicized lin es are u naltered fro m th e
origin al, ex cep t fo r th e con v ersio n of three lin es in to rem arks b y ad d itio n o f the no tatio n R E M . A rrow s Ýß m ark
ju m ps to d ifferen t section s o f th e p ro g ram .
M ertz S ig n ed M eth od
3 0 20 rfft ph 2 : p h 2(1 )5 0 ’ fft p hase array
cm plx 2 5 p h 2 : tran sp ose cm plx 2 ’ get 2 row s: real & im ag
m ag2 5 sqrt((cm p lx 2(0 )* cm p lx 2(0 ))1 (cm p lx2 (1)* cm p lx 2 (1 ))) ’ g et m ag n itu d e of v alu es
m ag2 5 div rev (m ag 2 ,1 ) ’ g et 1 /m ag nitud e
cm plx 2 (0 ) 5 cm p lx 2(0 )* m ag 2 : cm p lx 2 (1 ) 5 cm plx 2 (1 )*m ag2 ’ no rm alize
’ * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * **
cm p lx2 (1) 5 cm plx2 (1 )*( cm plx2 (0 )/a b s(cm p lx2 (0)) ) ’ if th e rea l p art is , 0 , then
cm p lx2 (0) 5 cm plx2 (0 )*( cm plx2 (0 )/a b s(cm p lx2 (0)) ) ’ m u ltip ly b o th p arts by 2 1
’ * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * **
cm plx 2 (1 ) 5 2 cm p lx2 (1) ’ g et com p lex co nju g ate
Ýß
3 0 70 rfft ph : p h (1)5 0 ’ fft ph ase array
cm plx 5 p h : tran spo se cm p lx ’ get 2 row s: real & im ag
m ag 5 sq rt((cm plx (0)* cm p lx (0))1 (cm p lx (1)* cm p lx(1 ))) ’ g et m ag nitu de o f v alues
m ag 5 d iv rev(m ag ,1 ) ’ get 1/m ag n itu d e
cm plx (0 ) 5 cm plx (0 )* m ag : cm plx (1) 5 cm p lx (1)* m ag ’ n o rm alize
’ * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * *
cm p lx(1 ) 5 cm p lx(1 )* ( cm p lx(0 )/a b s(cm p lx(0 )) ) ’ if th e rea l p a rt is , 0 , then
cm p lx(0 ) 5 cm p lx(0 )* ( cm p lx(0 )/a b s(cm p lx(0 )) ) ’ m u ltip ly bo th p arts by 2 1
’ * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * *
cm plx (1 ) 5 2 cm p lx (1 ) ’ get co m p lex con ju gate
D o u b led -A n g le M eth o d
’ * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** *
’ A C D ifferen ce In terfero g ram com p ute w ith zero ® ll, p h asin g a n d a po d izatio n.
’ D ou b led -A n gle P ha se C o rrectio n A lgo rith m
’ * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** *
’ C op y rig h t (c) 1 9 92 ±9 8 G alactic In du stries C o rp . C o p y o n ly for use w ith G R A M S /3 2 t.
982
Volume 52, Number 7, 1998
free
pau seo ff : o n p ain t 0
m o d e 5 0 ’ S tan d alo n e
’ ’ po rtou t 2 4 4, 2 1 : d ebu g 5 0
’ * * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * **
m 5 0 .1 ’ g et pa ra m eters for ® lter
dia log o n ’ ’ H igh -P a ss F ilter o f In terfero g ram ’ ’
dia log a sk m , 0 1 2 56 ,0 ,1 , ’ ’ B rea kp oin t:’ ’
dia log o ff
`* * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * **
10 0 dim res® le(25 6 ),tex t(2 5 6 ),tem p(2 56 ),tem p 1 (2 5 6),tem p 2 (2 5 6 ),tem p3 (25 6 )
Ýß
’ F ind Z P D and set u p array fo r apo d ization fun ction s
10 0 0 free apd fn : d im ap dfn (n p ts(# s)) ’ create ap od izatio n fu nc array
if getffp (). g et¯ p () th en x ¯ ip
if cty p e 5 1 th en #s 5 # s 2 sum (#s)/pts ’ A v erag e su btractio n
R E M zp d 5 ind ex 0(ab s(# s) 2 m ax(ab s(# s))) ’ lo cate Z P D po int
np 5 np ts(# s)
’ * * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** *
free preig ra m ,ig ra m ,un a p od 1 : dim p reigra m (n p ),igra m (n p ), un a po d 1 (n p )
if g etffp() . g et¯ p () th en x¯ ip ’ m a ke sure th a t zero is on the left
’ * * F ilter in terfero g ram to rem o ve low -freq uen cy d rift
preig ra m 5 # s ’ tra n sfer to a rra y
® lter p reigra m , 0 ,0 , m ,0 , m 1 0 .1 ,1 , 1,1 ’ rem o ve low freq uen cy no ise
ig ra m 5 preig ra m ’ use array lim its to za p th a t a dd ed
’ b y th e ® lterin g
’ * * Self-co n volu tio n o f in terfero g ram
free tem p 1 : d im tem p 1 (2 * np ) ’ A rra y fo r self-con vo lutio n
tem p 1 5 0
tem p 1(n p/2 ,n p /2 1 n p 2 1 ) 5 igra m (0 ,n p 2 1) ’ zero ® ll ig ram on each sid e
con vo lve tem p1 ,ig ra m ’ S elf-con vo lu tion
’ * * R em ove a p od iza tio n effect of self-co nvo lu tin g b oxca rs
free un a po d 1 ,u na p o d2 : d im un a po d 1 (n p ),un a po d 2(2 * np )
un a po d 1 5 1
un a po d 2 5 0
un a po d 2 (n p /2 ,n p /2 1 n p 2 1 ) 5 u na p od 1 (0 ,n p 2 1 )
con vo lve u n ap o d 2,un a po d 1
un a po d 1 5 u n ap o d2 /(m a x(u n a po d 2))
ig ra m 5 tem p 1 /u n ap o d 1
’ * * F in d the Z P D o f th e co nvo lu tio n
zpd pc 5 in d ex0 (ig ra m -m ax(igra m )) ’ F in d Z P D o f self-co n vo lu tion
zpd 5 in t ( (zp dpc 2 (n p/2 2 1 ))/2 1 (n p /2 2 1 ) ) ’ a nd u se to a ssign zpd of ig ra m
zpd pc 5 (zp d 2 (n p/2 2 1 ))* 2 1 (np /2 2 1) ’ rea ssig n zpd pc b ased on zp d
’ * * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * **
if (zp d . 5 (np /2 ) 2 5 0) th en d b lsid e 5 tru e else d b lsid e 5 false
® llbeg 2 (1 /(n p 2 zp d 2 1))*zp d ’ set u p ® lls
® llin c 1 /(np 2 zp d 2 1 )
go to 1 1 0 0 1 (ap o d* 1 0 0)
Ýß
’ L in ear interpo latio n o f p h ase
30 1 0 if p h int . 0 g oto 30 5 0
dim p h 2(p ),ap2 (p),cm plx 2 (p /2,2),m ag2 (p/2 )
R E M ph 2 (0 ,h ) 5 # s(#zp d ,# zpd 1 h):ph 2 (p 2 h,p 2 1 ) 5 #s(# zpd 2 h ,# zp d 2 1 ) ’ ro tate Ig ram into sm all ph ase array
’ * * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * **
ph 2 (0 ,h ) 5 ig ra m (zp d pc,zp d pc 1 h ):p h 2(p 2 h ,p 2 1 ) 5 igra m (zp d pc 2 h,zpd pc 2 1 ) ’ ro tate Igra m in to sm a ll p h ase arra y
’ * * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * **
Ýß
30 5 0 R E M p h(0 ,h ) 5 # s(#zp d ,# zpd 1 h ):ph (n 2 h ,n 2 1) 5 # s(# zp d 2 h,#zp d 2 1) ’ rotate Ig ram in to b ig ph ase array
’ * * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * **
ph (0,h) 5 igra m (zp d pc,zpd pc 1 h): p h (n 2 h ,n 2 1) 5 ig ram (zpd pc 2 h,zpd pc 2 1 ) ’ ro ta te Ig ra m into b ig p h ase a rray
’ * * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * **
Ýß
APPLIED SPECTROSCOPY
983
’ S to re p hase array if ’ ’ S torepn ew ’ ’
’ * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** *
3 1 00 free co sT , sin T , q u a d : d im co sT (n /2 ), sin T (n/2 ), q ua d (n /2 )
’ * *C o m p u te h alf-an g les
tra n spo se cm plx
cm p lx(1 ) 5 cm p lx(1 )* ( 2 1 ) ’ u n do com p lex co n ju g a te
q u ad 5 cm p lx(1 ) , 0 ’ 5 0 if cm p lx(1 ) . 0 a n d 5 2 1 if cm p lx(1 ) , 0
co sT 5 sqrt (((cm plx(0))*(1 ) 1 1)/2) ’ h a lf-an g le fo rm ula s for co s
sin T 5 sq rt (((cm p lx(0 ))* ( 2 1 ) 1 1)/2 ) ’ a n d sin
sin T 5 sin T * (qu a d* 2 1 1 ) ’ m u ltip ly by 1 o r 2 1 to ch oo se p ro per qu a dra n t
’ * *T est for ¯ ip s in the ph a se a rra y a nd co rrect
fo r j 5 0 to n/2 2 2
test1 5 cosT (j)* co sT (j 1 1 )
test2 5 sinT (j)*sin T (j1 1 )
test 5 test1 1 test2 ’ test is co s(T [j] 2 T [j 1 1 ])
if test , 0 then ¯ ip 5 2 1 else ¯ ip 5 1 ’ if , 0 then T h as ju m p ed b y . pi/2
co sT (j 1 1 ) 5 ¯ ip * cosT (j 1 1)
sin T (j 1 1 ) 5 ¯ ip* sin T (j 1 1) ’ co rrect fo r ¯ ip
n ext j
cm p lx(0 ) 5 co sT ’ reassig n in to p h ase arra y
cm p lx(1 ) 5 sin T
cm p lx(1 ) 5 cm p lx(1 )* ( 2 1 ) ’ red o com p lex co n jug a te
tra n spo se cm plx
’ * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** * * ** * ** * ** * * ** * ** * * ** * ** *
if p h str , . 2 go to 3 2 00
n ew spc zq zq (n )
# s 5 cm plx
984
Volume 52, Number 7, 1998