JPRS ID: 8586 TRANSLATION ACOUSTOOPTIC DEVICES FOR SPECIAL AND CORRELATIVE ANALYSIS OF SIGNALS

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. ~ l~ _ FOR ~
ANO CORRELATIVE ANALYSIS OF SIGNALS
24 JULY i979 CFOUO~ i OF i
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JPRS L/8586
24 July 1979
Translation
- ACOUSTOOPTIC DEVICES FOR SPECTRAL -
AND C,ORR~LATIVE ANALYSIS OF SIGNALS
RY
S~ V~ KULAKOV -
FBIS FOREIGN BF~OADCAST INFORMATION SERVICE
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JPRS L/ 8586
24 ,7uly 1979 ~
ACOUSTO~PTIC DEVICES FOR SPECTRAL
AND CORRELATIVE ANALYSIS ~F SIGNALS
Leningrad AKU5TOOPTZCHESKSYE USTROYSTVA SPEKTRAL'NOGO T KORRE-
_ LYAT5i0NNOG0 ANALZZA SZGNALOV in Russ3.an 1978 signed to press
27 Jul 78 pp 24-29, 4~-55, 80-103, 138-143
~Exernts �rom book by S. V. Kulakov, "Nauks" Publishin,q House, �
144 pages, 2000 copies]
CONTENTS PAGE
1.4. The Acoustic Light M~odulator [ALMJ as an Element of an
Optical Signal Processing System 1
2.2. Multichannel Acoustooptic Spectral Devicea 6
Chapter 3. Correlative Acouatooptic Signal 16
3.1. Acoustooptic Convolvers and Correlatora 17 "
3.5. Dispersion Quadripoles Based on an Acoustooptic Device
with ~nlarged Image of the Reference LFM [Linear
- Frequency-Mndu].ated] Signal 10 -
3.6. Tim~ Scale Converters i~ased on Acoustooptical
- Parametric Quadripolea 23
3.7. Selection of th~ Version of the Acoustooptic Correlator..... 40
e
Bibliography 43
_ a _ [I - USSR - F FOUO]
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F'UK Ub'~'1G1AL US~; UNLY
~ . .
N
PUBLICATION DATA ~ �
Engliah CiCle : ACOUSTOOPTIC DEVICES FOR SPECTRAL AND
~ CORItELATIVE ANALYSIS OF 3IGNALS
ltuasian title : AKUSTOOPTICHESKIYE USTROYSTVA
- SPEKTRAL'NOGO I KORRELYATSIONNOGO
ANALIZA SIGNALOV
Author (s) ~ S. V. Kulakov
Editor (s) ;
Publishing Houae ~ Nauka
Place of Pvblication ~ Leningrad
Date of Publication ~ 1978
Signed to press ~ 27 Jul ?8
Copiea ~ 200U
_ COPYRIGHT ; Izdatel'~tvo "Nauka", 1978
- b -
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UDC 621.39~..14
ACOUSTOOPTIC DEVICES FOR SPECTRAL AND CORRELATIVE ANALYSIS OF SIGNALS '
Leningrad AKUSTOOPTICHESKIYE USTROYSTVA SPEKTRAL'NOGO I KORRELYATSIONNOGO
ANALIZA SIGNALOV in Ruasian 1978, Nauka pp 24-29, 41-51, 52-55, 80-103, 138-143
- [~xcerpCs from a book by 5. V. Kulakov)
1.4. The Acouatic Light Modulator [ALM] as an Element of an OpCical Signal
Processing System pp 24~29
When solving ~giy~sis and syntheais problema, optical information procesaing
systema take the form of a aet of individual elements with the correaponding
- couplings betweet, them. The ma~ority oF the elementa of an optical ayatem
(lenaea, free space layera, spatial band f ilters, and so on) are lin~ar with
respect to the ].ight wave transmieted through them. For these elements~ in
a number of papers [22, 77] both the transmitting functions and the reaponsea
to the correaponding d-effects are defined, that is, the characterietica
analogous to the characterietics of linear electric circuita are faund.
Such electrical analogiea turn out to be highly useful when calculating op-
tical aystema, for they permit the use of the methods and mathematical appara-
tus well developed for aaalysia and synthesis of electric circuita and systems.
The devicea used to input the processed information (signals) to the optical
systems the spatial light modulators are also characterized by the
_ transmission functions of the light wave incident oq them. These tranamis-
sion functions are sometimea called transmission coefficients, traneparency
functions, and so on.
' For the acoustic light modulator, along with the transmissinn coefficients
it is expedienC to find Che frequency-dependent coefficient which defines the -
linear transformati.on of the input electr:tc signal to an acoustical wave
packet propagated in the acoustooptic interaction medium. This coefficient
will be called the electroacouatical transmiasion coefficient. Let us also
find the electrooptical impulse reaponse of the modulator.
Let us define the spectrum of the spatial frequencies of the acoustic wave
- packet corresponding to the elFCtric signals s(t). To begin with, we shall
assume that f.n the acoustoopti.c interaction mediwn there is no damping of the
- 1
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elas~ic waves, and the electroopeical converter of the modulator has an in-
f inite pMS~ band.
_ Tl~e time-s~ace signal in the ~perture o� the acoustic 11ght modulator (the _
acouseic wave packet} correaponding to the input aignal e(r) can be written
in the following form;
1~(s~ vt),Qrfj)/(ue-s), (1.4.1)
where r(z) is Che weighC function defined by the ~aperture atop.
LeC ua f ind the Fourier eraneformation of the aignals (1.4.1):
~hoo ~l,o� .
, J ut) oxp (-lW.=) af ~ ~ r / (ut - exP (-lW.=) d=~ (1. y ~ 2)
-co
Here wz ~-W~~ ~ 2~~~audio is the apatial frequency [30, 77], w is the angu-
lar dynamic frequency~ -
_ ,
Substituting the variable in (1.4~2), we obtain
+W +m
- ~ l. I=: ut) e:P ds a eYp (-l~~,ut) ~ r(ut - U1 /(D) ~rP (IW.U) du� -
_m _m (1.4.3)
+m
The integral of type ~ r(vt U) f(y)axp (futrJ~dJ ie very aimilar with respect Co _
'-m
form to the integral defining the inatantaneoua spectrum [56]. Therefore
we shnll call
+W
' F"i (w� ut) = f r(ut - Y1 I~Y) exP l1W.Y)aU
(1.4.4)
the instantaneous spectrum of the apatial frequencies (ar the spatial instan-
taneous spectrum) of the mirror image of the eignal, f(y).
Let us rewrite expresaion (1.4.4) in the following form:
+m
~ ue) _ ~ r (t1J(ut ezP IIW. (ue -:)1 d:� (1.4. 5)
_m
Taking the inverse Fourier transformation of the left and right aidea of
(1.4.5), we obtain
+W
r(=)1(ut s). = 2R ~ Fi (W,, Jt) oxP I-!w. -=11 a~s�.
(1.4.6)
The expression (1.4.6) defines the spatial signal in the aperture of the
ALM as the set of harmonic waves of the type exp [-~wZ (vt-z)]. -
Let us express the spatial inatantaneous spectrum in terms of the apatial
_ spectra of the signal and the weight function defint~n by the -~perture stop. .
~
2
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Uging thc known Cheorem of the spQCerum of the product of two functione [12j~ ~
we 1�ve
:
_ Fi(w� ut)~la J ~lW~)~~W,-W;IexPf/~W,--W:)~'~la~:~ 1.4.~
-ao ~ )
* ~
where F(wZ) ia the complex-con~ugaee spati~l apecrrum of the signal, R(wZ)
is the spatial apectrum of the weight function.
xn ehe special case where '
1 for L L
- i 6s~-~- Z ~
r (s) ~
o foi� I=I> i ~ . (1~4.8)
expression (1.4.7) assumes the form
~ L
_ . ~ Afn (w, - wi) - v
~'L ~WI~ U~~ a ~ ~ `w~, 2 exp ~Wf W~) ut) d~;, ~1 � 4 � 9~ -
. w, -
. _
where L is Che size of the ALM aperture in the direction of propagaCion of
the elasCic wavea~
+m
Thus, r (=)1(~~ - = 2ic ~ exP ~'~W. (~t -:U dW, X -
+m ,
sin - L
2 (1.4.10)
x~ ~ W, e7CP I~ ~~r - w~) vtJ dw~.
f
~m ~
Expression (1.4.10) defines the time~space aignal in the aperture of the
ideal ALM.
Now let us take into account the distortions which are introduced by the
elecCroacoustical (most frequently piezoelectric) converter and the damping
effect of the elasCic waves in the acoustooptic. interacCion medium of the
modulator.
- '1'he piezoelectric converCer can be considered as a linear quadripole with in-
_ put elec.tric.signal and output electric signal, and a complex transmission
coefficient K~(w) can be assigned to it. This transndssion coefFicient also
takes into account the effect of the binding layer between the piezoconverter
and a acoustooptic interaction medium, and it can be determined experi-
mentally.
If the inpuC signal s(t) has the spectral density S(w), it is obvious that
- F' (~~,1= S' (~~.1 K': (1. 4.11)
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where * is Che rjign of complex con~ugat~,on~
Th~ actual acoustooptic interaction medium is characterized by the damping
, coefficienC a(t~). Consequenely, tha time~apace aignRl in the gperture of the
ALM muaC be represeneed by the set of damping elementiary waves of Che type
,Qxp,l-Ix ~w,)Ixl oxp I-~?w,(vt-z)),
Thus, the actual time-space aignal in the a~perture of the ALM can be repre-
aenred in the following form;
1 +m . ,
~ f:) i(vt~ =1= zi~ ~ 8!A I- I a(W,I I=~ aaP I-/W, (u~ - s)1 dW, X~ -
~Fm G
~ atn - Z
X K f S~ ~W;) Ka (w;l W~ _ Ws e:P U(~?, - w;) ut) d~;. (1. 4.12)
~
The presence of damping of the elastic waves in the acouatooptiic interacCion
medium requirea the introduction of correctiona Co the determination, of the
instantaneoua spectrwn of the spatial frequenciea.
By the insCanCaneoua spectrum of the apatial frequencies of the time-apace
signal in the rectangular apperture of the ALM we mean the apectrum defined
by the expr~ssion
~ +m
Ft (~n� s~ ut) ~ ezp a~~,~ ~=1 n f S' ~W:) Kn X
sin ~:1 2 (1.4.13) ~ .
, X _ W~ - e:P I/ ~W, - w;) ut J dW,. -
Then ' ~
+m
r(:1 a(v~, s) � 2a ~ Fi (W,~ s, vt) exP - s)1 dW,
_m (1.4.14) �
is the Cime-space aignal in the aperture of the real AI~M.
The frequency properties of the real ALM are completely defined by the pro-
duct
~ c~.. s~ ~ x, exp ~_i a cW,~~ c~. 4. ~s>
which we shall call the ALM electroacousCical transmission coefficient.
This coefficient establishes g unique relation between the spectrum of the
input electric signal and the instantaneous spectrum of the spatial frequen-
cies of the time~space signal in the aperture of a real acoustic light modu-
lator.
4
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~ On feeding a d pul~e to rhe ~,npue o� Che ALM ax the eime we call the time-
~ apace eignal in the a perture o� the AL,M Che electroacouetical impulse re-
- aponse. Substituting the apectral density S~(w) R exp (~~wT) in (1~4~12), for
Clie electroacouatical impulse response of the ALM we obtain
B~~'~~ ~t~ � 2a J exP ~ a~w,) X
. .~.oo � _
_ o~P ~`/W~ - a~~ dWS n ~ exP ~1W:~t) "~fa ~Wi~ x
_ -oo �
sin (w.'- w;) l
, X w~ _ W~ exP U(W,-~:) ut) dW;. (1.4.16)
Then the Cime-apace signal in the ~perture of the acousric light madulator
with aperture input signal can be found using Che known expression
r(:) s(ut, s) ~~+(nc) 8(~r~ uT, s) dns. (1. 4.17) .
~ It must be noted that the elecCroacoustical transmiasion coefficient and the
impulse response of the SLM are not relaCed to each o~ther by the Fourier
transformation. However, these characCeristics completely reflecC the
frequency and Cime charucteristics of the linear process of conversion of -
_ the input elecCric signal to the time-spsce signal in the ALM aperture,
As was indicated in item 1.1, various operating conditions of the diffraction
acoustic light modulator are distinguiahed. The modulation process is
most simply described under Raman-Nutt conditions when purely phase modula~
tion is assumed. We shall limit ourselves Co the investigation of this case ~
here.
Let the electric signal s(t) be fed to the input of the acoustic light
modulator operating under the Raman-Nutt conditions. The plane light wave
incident normally on the ALM
e' (t, z) = F,o eap I/ ~Wa.~ - k~~x)~
(1)
Key: 1~ light
is modulated by the time--space sigr.al, and at the output of the modulator
it can be represented by the expression
(t. ut) = Eo QxP /[Wer~ Ar I:1 i(vt, :)I, (1. 4~ 18)
where A is the proportionality coefficient. With a harmonic input signal it
has the meaning of the phase modulation index (the phase advance klightX
which is insignificant to the following discussion will be omitted just as
in item 1.3.) Thus, the ALM operating in the Raman-Nutt diffraction mode
has the following transmission coefficient of the light wave incident on it:
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T (ut, s).� exp (/Ar i (ut, :)j, (1.4 .19)
Here ehe expon~nC in (1.4.19) is determined by Che diffraction acCivity o�
~l~e ilC0U9C00~)C~.C inCeriiction medium and Che expresaions (1,4.14) or (1.4~17)
e~taUli~hing Clie relation beCween ehe inpuC elect,ric aignal s(t) and Che time-
space signal i.n the aperture of the ALM.
_ 'Ttie acousric lighr modulator ia a linear device with respecC to Che light
wave normally incidenC on it. At the same time Che nonliaear traneformation
of the input electric signal of Che type (1.4.19) satisf ies the generalized _
superposition principle [11]. This fact can be uaed both for analysis and
syntheais of ehe acoustooptic devicea and, obviously, for the solution of
the problem of linearization of their amplitude characteristics~
2~2, Multichannel Acoustooptic SpecCral Devices pp 41-51
In radio engineering the problem frequenCly arises of analyzing a multidimen-
sional signal made up of N elementary inlependent electric signals exiating
- in separaCe channel:s or separated in space, As a rule, these elementary ~
signals are distinguished by initial phases. Specialized optical compuCers
to wh3.ch the elemenCary signals are ~nput by means of multichannel ALM find ;
application in the processing of multidimensional signals [3, 83, 97, 104,
121]. 5ometimes these devices are not so precisely called multichannel ~
acoustooptic epectral analyzers, ~lthough the Cerm "~hultichannel analyzer"
presupposes the presence of N independent channels and, consequently, N out-
put (in ehe general case, two dimensiehal) signals. In the investigated de-
vice N elementary input signals (or one N-dimenaional input signal) forms one
output (in the general case, three dimensional) signal. Inasmuch as the ,
operating principle of tbis acouatooptic computer consists in formation of the
light wave spectrum modulated by the signals in the multichannel ALM, we shall
ca11 iC a multichannel acoustooptic spectral device. _
The most prospective is the application of the multichannel acoustooptic .
spectral devices for processing multidimensional electric signals of phased
antenna arrays.
In 1963, a multic~~annel acoustoopCic device was described in reference [83]
for simultaneous observation of many radar targets. In this device the
_ signals from the elements of the linear antenna array were fed after frequency
conversion and amplification to piezoquartz electroacoustic converters �
attached to one end of an acoustic polygon (a multichannel ucoustic light
modulator). Here the phases of the signals trans:aitted to the polygon repeated '
the phases of the si~nals reaching the elementa of the antenna array. The I-
distances between the electroacoustic converters on the scale determined by ~
the ratio of the lengths of the electromagnetic and elastic waves ccarreaponded .
to the distances between elements of the antenna array. Thus, the total
elastic wave created by the linear acoustic array was propagated in the
acoustic polygon at the angle to the acoustic array at which the electro-
magneCic wave was incident on the antenna array.
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, The acoustic polygon was i11um~.nAted by co1l~.maCed light~ and in rhe rear
focal plane o� the integxaCing ob,~ecC~,ve~ sex~,ea wexe f ormed, each pair of ,
which (wiCh small phase moduLation 3ndexes) correaponded to the radar CargeC
- observed at ehe correaponding angle~ By measuxing the angular po~ition of
rhe diffrnclion spot, it was possible to determine the angular coordinate
- of the radar target~ In apite of the quite similar description of a device
presentied in the indicated paper, it did not become widespread as a result of
_ technical difficulties connected with execuCion of iC.
In reference [97] a sCudy was made of the applicarion of the acoustooptic
spectral devices for proceasing the signals of a phased antenna array (PAA)~
Two meehods of processing the signals of the linear receiving antenna array
are described: 1) multiple time delay and 2) apatial mulCichannel nature.
The combinaCion of theae meChods makes it possible to create a device for
_ procesaing the signals of a planar PAA. Let us discuss them ~n more detail.
The functional diagram ~f Che signal processing device of the linear PAA which
uses the multiple time delay method is presented in Figure 13~ The pulse ~ -
~ signals induced in the elements of the linear PAA reach the frequency con-
verters; then they are delayed by the corresponding Cime and summed. The �
magnitudes of the time delay are selected so thaC the total aignal will con-
sist of N radial pulses (where N is the number of elements of the PAA), the
interval between which would be proportional to the angle of incidence ~ of
- a plane electromagnetic wave incident on the PAA. The total aignal is
fed to a single-channel acoustooptic.spectral analyzer. The position of the -
main lobe of the diffraction peak of the f irsC order is determined by the
average f requency of the total pulse and, consequently, depends on the angle
The functional diagram of the acousCooptic signal processing device of the
lin ear PAA executed by the method of spatial multichannelness is presented
in Figure 14. The signals from the elements of the PAA are fed after fre-
quency conversion and amplification to the corresponding electroacoustic
converters of the multiichannel ALM, which is illuminated by a plane monochro-
matic light wave I. A diffraction pattern is formed in the rear focal plane
of the integrating objective. Here the position of the main peak of the
first-order diffraction maximum reckoned along the wy axis is defined by the -
' angle A at which the plane electramagnetic wave is incident on the linear
PAA. By measuring Che coordinate wy, the angle 6 is determined.
- If the planar PAA has N columns and M rows, then an N-channel light modulator _
is used in the signal processing device of such an array~ and a bunch of N- ~
pulses is created in each channel by the multiple time delay method~ By
measuring the coordinates wy and t~Z of the main peak of the first-order
" diff raction maximum, the angular coordinates of the observed target are de-
termined.
In reference [97] the results are presented from experimental studies of an
acoustooptic device with multichannel ALM with the following parameters:
. ~ .
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. ~
I ~
y y ~ ~ I ?
� ~ I I
~ ~ . _
" t t t t ~ '
t i
� i_
3 3 3 3 3 I
4
. .
~ 3 6 ~ ~
. I
Fi�ure 13~ FuncCional diagram of the PAA signal processing I~
device using the method of multiple time delay~ 1--- mixer, ~
2~-� heterodyne, 3-- delay line, 4~~ adder, 5~~ spectral ana- i
lyzer, 6-~ display, I-- plane electromagnetic wave front.
i
average pass band frequency of the channel 20 megahertz, duration of the pro-
cessed signal in each channel 25 microseconds, wid~h of Che channel pass '
band 5 megahertz, number of channels 24. ,
The multichannel ALM is executed in the form of a glass cell filled with dis-
tilled water with an array of electroacoustic converters made from an x-cut ~
piezoquartz plate on which gold electrodes are applied. The electrode width
is 1.5 mm, the spacing beCween ad~acent electrodes, 3 mm. A 4 megawatt
helium-nec., laser was used as the light source. The optical system was made
of high-quality optical parts. Ths read-outs system is a photomultiplier
with narrow slits (1 micron) in front of a photocathode fast~ned to a ~aoving
_ platform. The signals from the photomultiplier were amplified by a logari-
thmic amplifier and fed to an automatic recorder. The experiments were per-
formed with specially developed PAA signal simulator. The angles of incidence
of a plane electromagnetic wave on a linear PAA amounting to zero and 19.5�
were simulated. For rhese angles the distribution curves of the light in- ~
tensity in the region of the first diffraction maximum were calculated.
Then the actual light distribution was measured using the read-out system.
The results of the measurements agree well with the calculation. The possi-
bility of suppressing the side lobes by the weighted processing method was
also checked out.
In [104, 121J it was proposed that a multichannel acoustooptic device be
used to process t:e radio telescope signals, in particular, the signals of ;
~
8
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~ ~Ok O~~I~IAL US~ OI~LY
9 B B 9 ~ ,
i
t t t 1 t
~ ~
Z .L.
1
~ exosei �
� V~t
.
~ "'y. ~
~ ~ I
1 Y~
_ y~ ~ / Y:
`
y ~
1
~
t~
~igure 14. Functiongl diagram nf the signal processing device
of the PAA by the method of spatial multi~hannelness~ 1~~
mixer, 2~- heterodyne~ 3-- mulCichennel ALM~ 4-~ ineggrating ~
ob~ective~
Key: a~ inputs
a radioheliograph. An original optical system wae developed~ Obviously this
is tY?e most proapective application of guch devicea.
In the papers devoted to multichannel acouatooptic apectral devices, as a
rule, studies are made of the factors determining the diatribution of the
light oacillations in the output plane of the optical computer for the simp-
- lest input signals. However, the procedure for calculating the output signals
for arbitrary input signals is left out of these papers. Inasmuch as the
investigated device belongs eo the linear spectral devices, it is expedient
to determine ita instrument funcCion which in the given case will be multi-
dimensional. When determining the instrument function we use the known
principles of the theory of linear multidimensional ayatems [13].
The superpasition integral for systems with N inputa and M outputs ia written
as follows:
r
~ (v) = J G (V. � s (z) a:~ (2 . 2 .1)
r.
9
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APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100074431-9
~on o~~zcY~w us~ orn~x
ahar~ ~(x) ~nd ~(y) gx~ ettp ~,npue gad oueput v~cCnr~ of ~he ~y~eem~ G(y~ x)
i~ eh~ pu1~~ m~t~ix~
I~ u~ingle pul~e S(y - x) ie fed eo the ieh inpue of Che eyeC~m~ Ch~ ouCputi
veceor of ehe ~yeeem h~~ eh~ form
B,r (u~ x)
gr (V~ s) ~ B,r (U~ 3) ~
~ 8xr rI (2 ~ 2. 2) ~
, where g~i(y, x) ig ehe pu1~~ reeponse of thp ~th ourpue excieQd by ~ eingle !
pulge ae the ith input~ Fep~;ing the d�effecC to the remaining inputs~ we ~
det~rmin~ the pulse matrix of the eyeeem ;
8u x) 8'~~ ~V~ t) 8~N (D~ s) ~ '
~ 8u tV ~ zI d~~ r) StN ~Y~ r) I
a (u~ ~ . . . . . . . . . . . . . . . . . (2, 2.3) ~
.
qXt t~~ sI Sx~,tV~ txrv (V~ ;
- for a system with N inpues and or.~e output the pulse matrix ie a row-maCrix ;
I
G ~Y~ x) ~ (B~t rY? 8i~ IY~ ~ _ ~
s�v(Y, 1], (2,2.4)
and here
ar(v. =)aB1fIV, (2.2.5)
L~t us define the pulae matrix of a multichannel acoustooptic device~ the
functional diagram of which is preaented .in Figure 14 on feeding d-inputs in
the frequency? region to its inputs. This pulse matrix, by analogy with the
instrument functiona of the single-channel spectral devicea, will be called
the instrument matrix of the mu~tichannel acoustooptic apectral device~
Here it is necessary to feed a harmonic oscillation to the inputa of the
spectral device .
t,: (t) ~ cos (2. 2.6)
(1) ~2~
Key: 1. input 2~ audio
T'hus, the inpuC signal in the frequency region can be written in the form
j~t ~w~ a(ra ~W T W~1~ T Tt ~W ` W~~~) SI for ~ f'i N~ \L ~ L~ I~
where ~i is the ith column of the unit matrix N x N~ that is,
10
FOR OFFICIAL US~ ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100070031-9

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100074431-9
~Ott 0~'FICIAL US~ ONLY
, d ,
i ~ -
0 '
~r f-? iCh row. (2, 2. g)
0
_ ~
_ ,
0
~ mh~ planp 11ght wav~ I incidenr c~n the multich~nn~l ALM (~Qe Fig~r~ 14) is
_ ph~ee-moduS.ated by h~rmoniC ~l~~ein w~veg.
In the caee of idenCical chgnnelg of the AY~M, the 11ght wav~ at its output
- can be described a~ followe: _ ;
. cns ~~~~~t - A co~ (w~~t - k~.:,li .@~ f or -
--~0.5N-(l--flla-(U,5(N-}-f)-t~6~ .
e'(~u =i~ Y~1~ :U~C"`(U.5N-t)a--(-n.5(N-{-f)-1~6,
U for -~-(U.SN--t)a-(0,5(N-}-f)-1Jb< ~2~2~9~
GU~~--(n,SN-t~a-(U,5(N-}-f)- ~
-(t-f)~b, 1Cf~N~
where a is the width of one channel of the ALM~ a+ b is the epacing between
ad~acent channels. ~
. '1'he analytical aignal which can be repreeented by the fol.loWing expresgion
corresponsla eo the narrow-band aignal (2.2.9) (w � w , A � 1):
lighe audio
, m~
~ sa= W~~,~~e:C?~~E_h= s~]
o P/ M) P/(~. -1~' ~ gr
�
for-~0.5N-(c-f))n-(0.5(N-}-f)-tJ6~
e'~~~ Yt~� 6Y~