166 OPTICS LETTERS / Vol. 8, No. 3 / March 1983

Acousto-optic processors for real-time generation of time-frequency representations

Ravindra A. Athale, John N. Lee, E. Larry Robinson,* and Harold H. Szu

Naval Research Laboratory, Washington, D.C. 20375

Received October 1, 1982

Acousto-optic processors for calculating different two-dimensional (2-D) time-frequency representations for one-dimensional temporal signals in real time are described. The various 2-D representations discussed in the litera-ture, such as the Wigner distribution and the ambiguity function, are shown to be obtainable through minor varia-tions in an acousto-optic processor consisting of two Bragg cells in a parallel configuration. Also obtained and dis-cussed are two new time-frequency representations that, for amplitude-modulated signals, correspond to mean-frequency-selective correlation and Doppler-frequency-selective convolution. Experimental results are presentedto highlight the special features of the different time-frequency representation.

Over the years several two-dimensional (2-D) time-frequency representations have been formulated (withinthe constraints of the uncertainty principle).1 2 Thesehave been formulated for signal-detection and charac-terization tasks in areas such as speech, radar, and sonarprocessing. 3-6 A theoretical overview of the variousrepresentations from a signal-processing viewpoint wasgiven recently by Claasen and Mecklenbrauker.7 Themost familiar representation is the ambiguity function,for which many optical schemes using 2-D transducers(e.g., film and liquid-crystal light valves) have beenproposed and implemented.-13 Other representationswere implemented recently using film.14 Optical pro-cessors using one-dimensional acousto-optic (A-0)devices instead of 2-D transducers have received con-siderable attention recently because of the superiorcharacteristics (rapid input rate, large dynamic range,high time-bandwidth product, etc.) of currently avail-able Bragg cells.15 -'7 A-O processors, however, wereprimarily directed toward producing the ambiguityfunction. In this Letter we describe how the basic A-Oprocessor designs presented in the literature can bemodified to calculate all the previously developed 2-Drepresentations and also two new time-frequency rep-resentations. These new representations are shown tocorrespond to carrier-frequency selective correlationand Doppler-frequency-selective convolution for thespecial case of amplitude-modulated (AM) input sig-nals. Experimental results are presented highlightingthe unique features of each representation.

We first briefly describe the various representationsthat were experimentally investigated.

1. The cross-ambiguity function is the time-fre-quency representation that will perform the task ofdetecting a signal with arbitrary time and frequencyshift. It may be written in either asymmetric,

Xls# 12 ( Sl (t)S2*(t - -0 exp(-j2xt)dt,

(1)

or symmetric,

A1 y(r T) = XSI (t +) S2* (t -)X exp(-j2irgt)dt, (2)

form. We note the relationship

A12(,M, 'r = exp(j7rgr)X12(A, - (3)

The 2-D ambiguity function displays cross correlationalong the X direction and the carrier-frequency differ-ence between the two signals along they gdirection. Aspecial case of the cross-ambiguity function is the au-toambiguity function, where SI(t) = S2(t) S(t).{Note: Here S(t) denotes the complex representationof a real signal, r(t); i.e., r(t) = Re[S(t)}.]

2. The instantaneous power spectrum (IPS) is the2-D Fourier transform of the asymmetric form of theautoambiguity function:

I(t, V) = s- x(u, '-) expU2w(at - vr)]drdg

= S(t)S*(v) exp(-j2irvt), (4)

where S(v) is the Fourier transform of S(t). I(t, iP) canbe interpreted as the complex energy-density functionin the time-frequency domain.4

3. The Wigner distribution is obtained from the 2-DFourier transform of the symmetric form of the ambi-guity function, A12 :

W12Nto, VD) s : A12(g, T)

X expU2irv(t 0 - vrJdAd-r

sl1 (to + ) S2* (to X exp(-j2wrv(r)d-. (5)

The cross-Wigner distribution gives cross convolutionalong the to direction and provides the mean carrierfrequency along the iPo direction. W1 2 expressed in Eq.(5) has a symmetric form. An asymmetric form of theWigner distribution also exists and is related to thesymmetric form through a relation similar to that stated

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March 1983 / Vol. 8, No. 3 / OPTICS LETTERS 167

in Eq. (3). The difference in phase distribution causesthe asymmetric Wigner distribution to be Fourier-transform related to the IPS rather than to x(t, r).

4. We have noted that, for the special case of AMmonofrequency signals, two new time-frequency dis-tributions can be defined. These correspond tomean-frequency-selective correlation (MFC) andDoppler-frequency selective convolution (DFC) [Eqs.(6) and (7) below] and are convenient to implementacousto-optically:

MFC: MI2 T, PO) = s 8S(t)St - T)

X exp(-j2wrvot)dt, (6)

DFC: N12(M, to) = Sdr)S2(to -'

X exp(-j27rgr)dT, (7)

where S(t) and S2(t) are complex representations ofthe real AM signals. Note the absence of conjugationin the integrals of Eqs. (6) and (7). By using Sl(t) =

a,(t) exp(j2,irvt) and S2(t) = a2(t) exp(j2rv2t) andapplying the principle of stationary phase, it is seen thatthe main contribution to M12(Tr, O0) occurs at P0 = pi +

'2 , whereas that to N1 2 (g, to) occurs at A = P2 - P1-

M(r, PO) at io = v1 + V2 gives cross correlation betweenal(t) and a2(t), and N(g, to) at yc = P2 - PI gives crossconvolution between a1 (t) and a2 (t). Thus it can beseen that MFC and DFC combine the properties of theambiguity function and the Wigner distribution in aninteresting fashion. In particular, the auto-MFC pro-vides a unique description of a multicomponent signalby displaying the autocorrelation functions of the var-ious components while simultaneously giving infor-mation about their carrier frequencies. This point iselaborated later through experimental results.

There have been basically two designs of space-in-tegrating A-O processors for calculating the ambiguityfunction. One design consisting of two Bragg cells ina crossed configuration was first proposed by Said andCooper1 5 and later refined by Cohen.16 This processorcalculates the symmetric version of the ambiguityfunction. By making minor modifications to the basicdesign, one can generate the symmetric versions of allthe time-frequency representations. The second de-sign is based on two Bragg cells in a parallel configura-tion and was invented by Tamura et al.17 This pro-cessor calculates the asymmetric version of the ambi-guity function. Here we describe only the asymmetricprocessor and discuss the modifications required togenerate the time-frequency representations mentionedabove. A similar study was also conducted for thecrossed Bragg cell processor, and all the five time-fre-quency representations were experimentally generated.Except for the phases of the outputs, these two A-Oprocessors gave identical results.

A schematic diagram of the parallel Bragg cell pro-cessor as discussed by Tamura et a!.1 7 is shown in Fig.1. The first Bragg cell receives the real rf signal r2(t).The -1 diffraction order from this cell [containingS2*(t)] is imaged telecentrically onto the second Braggcell after being filtered in the Fourier plane by a phasefilter of the form exp(j2rq4). The second Bragg cell

BRAGG CELL-1

y

BRAGG CELL-2

Fig. 1. Schematic diagram of an A-O processor for comput-ing the asymmetric ambiguity function. The light source Sis pulsed to freeze the acoustic-wave motion.

receives the real rf signal ri(t). Selecting the +1 orderfrom the second cell gives SJAt)S2 *(v)exp(-j27rvt).This is followed by a simple spherical lens in either animaging or a Fourier-transforming configuration. Thedesired time-frequency representations can now begenerated as described below.

1. The asymmetric ambiguity function results whenthe final spherical lens performs a 2-D Fourier trans-form on the doubly diffracted beam.

2. The asymmetric Wigner distribution can begenerated by reversing the direction of the acoustic-wave propagation in the first Bragg cell [thus effectivelyinverting the signal S2*(t) before shifting by r, resultingin convolution instead of correlation].

3. Since the light distribution on the doubly dif-fracted beam already corresponds to the IPS (includingthe correct phase), the final spherical lens simply imagesit to the output plane.

4. To obtain the MFC, the +1 diffracted order fromthe first cell is selected in a system that is otherwiseidentical with that for the ambiguity function, thusremoving the complex conjugation of S2(t - ¶1.

5. To obtain theDFC, the +1 diffracted order of thefirst Bragg cell is selected in a system that is otherwiseidentical with that for the Wigner distribution, againremoving the complex conjugation of S2(to- ').

It should be stressed again that the symmetric andasymmetric versions of the various distributions areidentical in magnitude. The only difference is in thephase factors associated with them, which give sub-stantially different results when 2-D Fourier transformsare performed on the complex amplitudes of the sym-metric and asymmetric versions. Thus, in applicationsrequiring postprocessing of the complex time-frequencyrepresentations, the version being produced opticallyshould be carefully matched to the postprocessing al-gorithm chosen. If the magnitude of the time-fre-quency representation is of interest, then the choicebetween the two A-O processors will be dictated bypractical considerations, such as the number of opticalcomponents required and ease of switching from onerepresentation to the other ones.

Figure 2 displays the experimental results obtainedwith the asymmetric A-O processor. Bragg cells madewith TeO 2 operating in the slow shear-wave mode wereused (center frequency, 90 MHz; bandwidth, 50 MHz;

X y

168 OPTICS LETTERS / Vol. 8, No. 3 / March 1983

=94

(a)

'c Iin p=o

l1}(C)

i- 0

'1v

003V

1V2

IbI

V~O 1/3 01 1`2

{d)

- 11 1( )3 IV2

(e)

Fig. 2. Outputs of the asymmetric A-O processor configuredto compute five different time-frequency representations fora multicomponent input signal. (a) ambiguity function, (b)mean-frequency-selective correlation, (c) Doppler-fre-quency-selective convolution, (d) Wigner distribution, (e)instantaneous power spectrum.

time aperture, 40 Asec). The frequency coverage in theoutput (Doppler bandwidth or the range of carrierfrequencies) is, however, limited by the height of theacoustic beam in the second Bragg cell to approximately25 MHz. The five different time-frequency repre-sentations were generated by inserting into both Braggcells the multicomponent signal consisting of a four-bitAM Barker code (1101) of 4-gsec duration on a 72-MHz(P,) carrier, a cw signal at 80 MHz (VA), and a 5-MHz-bandwidth white-noise signal centered at 70 MHz (V3).The results exhibit both auto terms and cross termsbetween different components. For example, the au-tocorrelations and cross correlations of the Barker code,the cw signal, and the random noise can be seen alongthe time axis at various frequency locations in the am-biguity function [Fig. 2(a)] and the MFC [Fig. 2(b)].Since the ambiguity function gives information aboutthe frequency difference between the input signals, allthe autocorrelations overlap at g = 0, whereas the crosscorrelations appear at p = L(I -'A), ±(v2 - vo), and+(PI - i'3), as seen in Fig. 2(a). The MPC, however,gives information about the mean frequency of theinput signals, thus displaying the autocorrelations of thecomponents at their respective carrier frequencies (v1,P2, and P3) while the cross correlations appear at (P, +v)/2, (v1 + v3 )/2, and ("2 + v3)/2 [Fig. 2(b)]. Thus in the

ambiguity function the autocorrelation terms (e.g., a1113111 amplitude pattern along the r axis at M = 0corresponding to the Barker code) overlap one anotherand the cross-correlation terms are spatially separated,whereas in the MFC the autocorrelation terms separatespatially and some of the cross-correlation terms over-lap, giving fewer cross terms. This feature of MFC canbe of great use in correlation detection of AM signalsburied in out-of-band noise components while simul-taneously giving the carrier frequency.

In the Wigner distribution and the DFC, the fre-quency location of the auto and cross terms is the sameas in the MFC and the ambiguity function, respectively.The difference in these cases is that a convolution op-

eration is performed along the time axis. This is veri-fied by observing the autoconvolution of the Barkercode (1212201 amplitude pattern) in Figs. 2(c) and 2(d).As in the MFC, the Wigner distribution spatially sep-arates the autoconvolution terms while partially over-lapping the cross-convolution terms, and the DFCoverlaps the autoconvolution terms while separating thecross-convolution terms spatially.

For the IPS [Fig. 2(c)], the multicomponent signalitself is displayed along the time axis, and the spectrumof the composite signal is seen along the frequency axis.The cross terms here correspond to combination of thetemporal behavior of one component associated withthe spectrum of another component [Eq. (4)). Crossterms are not separated unless the components aredisjoint in both time and frequency space.

In conclusion, A-O processors for calculating severaldifferent time-frequency representations were dis-cussed. Two new time-frequency representations wereintroduced for the special case of AM signals on a car-rier. These representations were found to combine thefeatures of the ambiguity function and the Wignerdistribution in unique fashions. Experimental resultsobtained with the A-O processors were also presented,bringing out the special features of the time-frequencyrepresentations, particularly for multicomponent sig-nals. These real-time A-O processors could be usefulin radar and sonar data processing.

* Permanent address, Department of Physics, AustinCollege, Sherman, Texas 75090.

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