OPTICAL REVIEW Vol. 6, No. 5 (1999) 424-432

Optical Digital Fast Fourier Transform System Masatomo SHIRAKAWA and Junji OHTSUBO* Faculty of Engileeerileg. Shizuoka Uleiversity, 3-5-1. Johoku. Hamamatsu, 432-8561 Japan

(Received March 29, 1999; Accepted May 16, 1999)

A system for optical digital fast Fourier transform (FFT) is proposed. Data exchange, adder, and multiplier circuits required for the optical FFT system are designed. Each circuit contains the same basic element of an optical beam switch composed of a polarization rotator and a birefringent plate. In the FFT algorithm, butterfly switching for data exchange is important and the method of optical switching networks is very

suitable for such operations. FFT calculation for one-dimensional data of a sinusoidal signal is experimentally demonstrated using the optical system. The throughput of the proposed system is estimated and its future perspective is discussed. Key words: optical digital computing, fast Fourier transform, optical interconnects, free=space switching, polarization switch

1. Introduction Digital electronic technology is essential in massive in-

formation processing and is now a key technology in worldwide multimedia networks. This technology makes it possible to develop very compact and low-cost informa-

tion equipment. To handle and communicate massive amount of data, integrated parallel signal processing sys-

tems using digital signal processing (DSP) chips have been introduced in this field. The capability of information processing realized by silicon chips is growing at a high rate year by year. However, in spite of the efforts to develop fast integrated circuits based on parallel data

processing, digital electronic systems have a drawback because of their interconnection problem since they are more suitable for serial data handling. In the future, more data must be treated and faster device speed is required in information processing systems. One possible solution to this problem is optical technology in which

to guarantee the accuracy of the calculation. An important part of the fast Fourier transform (FFT) algorithm which is used as a common algorithm for the fast calculation is data exchange. Fortunately, optical switching networks developed in various fields, especially in optical interconnects, are universally applicable to calcula-

tion and networking of optical signals. The switching networks are very appropriate for the implementation of optical FFT. DSP chips in electronic technology are also used to realize FFT, but the speed of the calculation increases linearly with increase of data number. In an optics case, it is expected that the calculation of FFT is

only limited by the switching speed of two-dimensional optical spatial switches due to the intrinsic parallelism in optics .14,18)

In this paper, an optical digital FFT system is proposed and its implementation for calculation and exchange of data is explained. The basic element of the sys-

the interconnection problem is greatly reduced.1-n) Opti-

tem is a switching circuit for input data beams which consists of a polarization switch and a birefringent

cal interconnection is a promising technology to avoid

plate.2,13-15,17-20) Three optical circuits required for the

the bottleneck of large-scale data exchange operations in digital electronic systems. Various types of optical interconnects already proposed are not only suited for interconnections in communication networks but also applicable to optical computing.12-17) Currently, performance of optical computing systems is limited by the available optical devices, but future development of these devices will

FFT calculation, namely, data exchange, adder, and multiplier circuits, are designed. In the experiments, FFT

result in great improvement of such systems.

calculation for one-dimensional 16-input data of a sinusoidal signal is accomplished using the optical system . The throughput of the experimental system is compared with a digital electronic one employing DSP chips

and the Luture perspective of the system is also discussed.

One of the important and widely used signal proc-

2. Design of Optical FFT Systems

essing techniques is Fourier transform by which we can analyze and characterize spectral components of a signal.

2.1 Algorithm

In optics, a lens is frequently used as a conventional Fourier transform device and two dimensional Fourier transform is very easily and quickly performed by a single lens. The result calculated from analogue Fourier transform by a lens, however is generally less accurate. Therefore, digital optical Fourier transform is believed

The FFT algorithm used in this paper is a type of 2radix of decimation-in frequency. We briefly summarize the method for later use. Number of data dealt with in a 2-radix algorithm is a power of two. The discrete Fourier

transform in the decimation-in-frequency algorithm is performed by divisions of data at each stage of the FFT process and multiplications with weights for the calculated results. For a given set f= { fk } with a data number

*E-mail: [email protected]

424

OPTICAL REVIEW Vol. 6, No. 5 (1999)

N, the discrete Fourier transform F= { Fk} is defined by

(N/2)-1 Fk= N-1 ~ f~ W"k= ~] {9~ W"k+h~ W(^+(N12))k}

~=0 ^=0

(N/2)-1

=~ {9~+h^ W(N/2)k}Wnk (1) n=0 where k is an integer number and O ~ k ~N- 1, gk and hk are the following two data sequences generated by the original data,

M. SHIRAKAWA & J. OHTSUBO 425 fO

Fo

fl

F4

f2

F2

f3

F6

f4

F1

f5

F5

f6

F3

f7

F7

g~=f^ O~n~ 2N

l^ -f O~n~ 2N1. (2)

(a)

X

+

yt* ~J ~ + (N/2)

The rotator in the Fourier transform is defined by

W(N12)k exp (-ilTk) 1 _1k=2m k=2m+1 (3) m is also an integer number and O ~m~N/2-1. In general, a discrete Fourier transform requires N2 multiplications, while the number of multiplications in Eq. (1) is reduced to (N/2)2. From Eq. (1), even and odd terms of Fk are separately rewritten by (N/ 2) - 1

F2~= ~ {9~+h.}W2^~ (4a) ~=0 (N/2)-1

F2~+1= ~ [{9^-h^}W"]W2~~ (4b) ^=0

+

X=x+y

+ Wn Y = (x - y)W"

y (b )

Fig. l. (a) Signal flow of the FFT algorithm of decimation-in-frequency for eight inputs. (b) Basic butterfly operation for two inputs.

tend it to a two-dimensional space. Namely, n-th data signal is displayed as a bar with a uniform brightness (in the

following, we define the uniform brightness as unity) whose length corresponds to its value. The row in the two-dimensional space represents the data number and the length of the column denotes the value of the data. Using this representation, parallel calculations are opti-

The expression is very simple and suitable for optical implementation. Based on the above logic, the data flow of

cally performed in the proposed system. Three optical circuits are considered based on Eq. (5); data exchange

the FFT of 2-radix of decimation-in frequency is shown in Fig. I for the number of data N= 8. FFT for N= 8 data is performed by three stages of the same operations, namely, exchange of data, calculations for addition and subtraction, and multiplications. The sequence of the

basic elements of the circuits for the FFT system are a polarization switch and a birefringent plate. Figure 2 shows an optical realization of a basic element and some

final results is not in descending order such as shown in Fig. 1(a), therefore a bit-reverse operation is needed to obtain the final result. The basis of the calculation for each stage in the algorithm consists of two-input (x, y) and two-output (X, Y) operations as shown in Fig. 1(b). The logic is written by

X=x + y, (5a) Y= (x - y ) W" . (5b) The calculation of FFT is reduced to a very simple basic procedure. In the following, we design circuits for the optical digital FFT based on Eqs. (4) and (5). 2.2 Design of optical circuits The circuits required for implementation of the optical FFT system are described in this section . The system for

the FFT algorithm is composed of instruction, control, and calculation circuits for numerical data; we mainly

circuit, optical parallel adder, and optical multiplier. The

of its operations. In the basic unit, the polarization of in-

put optical beam is changed by a polarization switch such as a ferroelectric or twisted nematic liquid crystal device,

and then the position of the beam is shifted in space by passing through the birefringent plate. Figure 2 shows possible switching patterns used in the following optical circuits. Using these optical components, positions of optical beams are controlled and switched and various logical operations are performed. The details of the optical circuits are explained in the following. 2.2. I Data exchange circuit Data for the input ports in each stage in the FFT al-

gorithm must be switched appropriately for successive operations as shown in the signal flow in Fig. 1. Therefore, data exchange is requisite to put the data as appropriate pairs and order in each step of the flow and to pass them to the following operation. In N= 8 FFT algorithm, data exchange is realized by three stages of net-

concerned here with the optical circuits for calculations

working circuits. The method discussed here can easily

and switching. There are many possible methods to

be extended to a system with a number of input data more than 2M (M> 3), so that we explain data exchange

represent the data format in the FFT algorithm. We consider a one-dimensional signal to be transformed and ex-

circuits for a particular case of N= 8. Three stages of in-

M. SHIRAKAWA & J.OHTSUBO

426 OPTICAL REVIEW Vol. 6, No. 5 (1999) ( a)

(a) Bar (O)

PS BP

(1)

oe

oe

p

p

(2)

(O)

(3)

(1) (2)

(3)

Optic Axis (b)

(b) Shift

(7)

oc e

oe

(6) (5)

Pe

(4)

(4)

(5)

p

(6) (7)

(c) Multiplex Fig. 3. Data exchange circuit for the first stage of the FFT flow. The switching channels (a) and (b) are placed spatially parallel. oe

p

( A)

(A)

(B)

(B)

(d) Cross

(a)

oe

p

(D) (C)

(C) Fig. 2 . Unit of polarization switch and four types of switching operations. PS, Polarization switch; BP, Birefringent plate. Double

(D)

circle and arrow represents s and p polarization states. Polarization

rotates 90'for hatched state at PS. (b )

terconnections for data exchange must be designed

Fig. 4. Data exchange circuit flow. The switching channels

f or

(a)

the second stage of and (b) are placed

the FFT s patially

along with the switching algorithm . However, the last stage performs only data exchange between two adjacent inputs, so that no special exchange circuit is necessary for the last stage in our method. The proposed data ex-

parallel .

change circuit is polarization-based optical interconnec-

change. The numbers in parentheses on the left hand

tions composed of those similar to the basic elements

side in Fig. 3 correspond to the input ports of the first stage in Fig. I . Each rectangle denotes a switching unit

used in the optical adder and the multiplier in the following discussion.

Figure 3 shows a schematic of the optical data exchange circuit for the flrst stage of the input in Fig. 1.

The operation of the circuit is equivalent to a shuffle ex-

consisting of a polarization switch and a birefringent plate as shown in Fig. 3. The input arrangement in Fig. 1 is L0lded with respect to the center of the row. The rear-

OPTICAL REVIEW Vol. 6, No. 5 (1999)

M SHIRAKAWA & J OHTSUBO 427 beam intensity and the sign of the addition is determined from the polarization direction of the final output. Figure 5 shows the fundamental circuit for optical adder for a pair of inputs. Four possible cases of data addition with

ranged inputs are used for the data exchange circuit. The switching channels (a) and (b) in Fig. 3 are placed spatially parallel with each other. For example, the output port (O, 4) is one of the appropriate pairs for the next oper-

ation of the adder as shown in Fig. 1. Two types of

sign bit are considered and the inputs x and y are assumed to be positive numbers in the figure. According to a calculation pattern, the appropriate switching for the

switching units are used. At the switching operations, single and double shifts for beams are employed in Fig. 3(a),

while only single shifts in Fig. 3(b). As a result, data

beams is given by the control of the polarization switches as a kernel pattern. In (a) x+y and (b) -x-y operations, the beam intensity at each spatial position in the output plane is always O or 1. A bright intensity is assigned a value of '1,' while a value of 'O' has no light. Thus the

beams are properly switched and shufiiedand aligned suitable for the next calculation of the data. In actuality,

two sets of the switching circuit as shown in Fig. 3 are used for the first stage of the networking in Fig. I . The pattern for the polarization switch is a kind of kernel and is given in advance as a control program for a particular

total number of brightness of '1' integrated along bars

input data number. Figure 4 shows a schematic of the data exchange circuit for the second stage of the input in Fig. 1. The switching network is a simple butterfly exchange. The

between the operations (a) and (b) is the polarization direction of the optical beams. Namely, if the polariza-

represents the final result from the adder. The difference

tion of the output beam is an s state (indicated by double circles), the sign of the addition is positive, while it is negative for a p state (indicated by arrows). In (c) x -y

letters in parentheses of the left side in Fig. 4 correspond to the input ports of the second stage in Fig. 1. Each rec-

and (d) -x+y operations, the output beams are spatially

tangle also represents a switching unit consisting of a polarization switch and a birefringent plate. The switch-

added, so that the intensity of each spatial position is O,

1, or 2. The intensity levels for I are taken as results from the adder and the other levels are discarded; there-

ing circuits (a) and (b) in Fig. 4 are also placed spatially

parallel with each other. For example, the output port (A, O is one of the appropriate pairs for the adder as

after, the integrated intensity represents the absolute

also given as a kernel in advance. We only present a

value of the addition. The sign of the calculation is also derived from the direction of polarization for the intensity levels of 1. Namely, if the polarization of the output

three-stage data exchange circuit for N= 8, but the exten-

beam is an s state, the sign is positive, and for a p state it

sion of the design for larger data number N= 2M (M> 3)

is negative. The relations between the resultant signs

shown in Fig. 1. The pattern for polarization switching is

is straightforward.

and the polarization directions are the same as those for

2.2.2 Parallel adder The important part of the operations for addition and

the previous two cases. Thus by taking into consideration the polarization states, the addition and subtraction operations are correctly performed with sign. In an optical sense, a complete adder with a sign is real-

subtraction of input data are performed by shift and addition of optical beams. The circuit consists of two units of switches and each optical axis of the birefringent plate in the switch aligns in the opposite direction. A resultant beam intensity has either a value of O, 1, or 2 with s, p, or mixed state of polarization. The absolute value of the out-

ized by the proposed method, but an optical device that performs both thresholding and detection of polarization direction at the same time is currently not available. The discrimination of light intensity level is easily done using

put from the adder is obtained by thresholding of the

an optical phase spatial light modulator such as a parallel

(a) x+y

(c) x-y

y

(y)

y

x

(x)

x

(b) -x-y

(x)

(d) -x+y

(y)

y

(y)

y

x

(x)

x

Fig. 5.

Operations of optical parallel adder.

M. SHIRAKAWA & J. OHTSUBO

428 OPTICAL REVIEW Vol. 6, No. 5 (1999) LC D Input a Control circuit

r~~T~l~~ L~I~L~JCCC

PBS2

OA-SLM

OS

Input b

e e e

e

Read-beam

a a WR

o

e

HM

PBS 1

M C C Output lbl C C

CC lal

Fig. 6. Operations of optical multiplier. LCD, Liquid crystal device; OS, Optical switch; OA-SLM, Optically addressed spatial light modulator; PBS, Polarization beam splitter; HM, Half mirror; M, Mirror.

For example, the sign of the addition is calculated by

two input data. In this example, we treat only the product of integer numbers, but the procedure can be expanded to a fixed-point arithmetic by changing the

two steps as follows. The output pattern in Fig. 5 is divid-

numerical weights for the operands. Also, a product of

aligned liquid crystal cell. At present, however, the sign of

a final result of the adder must be calculated separately.

ed into two patterns by a beam splitter. From one of

complex numbers is easily implemented by optical

them, the absolute value of the calculation is obtained using an optical phase spatial light modulator as a thresholding device. From the other, the sign of the addition is de-

parallelism .

termined from the direction of the beam polarization. In

The fundamental circuit of the FFT calculation in the experiments is shown in Fig. 7. Polarization switching

that calculation, the final result of the absolute value is

3. Experimental Results

used as a mask which defines the area for determination of the polarization direction. Though the example here is for the operation of integer number, the method is easily expanded to fixed-point arithmetic by software interpre-

and beam shifts required for the calculations of the main FFT operation are optically performed but the detection of the calculated results and the control for the switching

tation.

ments. Of course, some of the digital electronic parts can be replaced by appropriate optical devices. In the experi-

2.2.3 Multiplier Final operation of each stage is multiplication of weights to the data. Figure 6 shows a schematic of the multiplier. The main elements are also composed of a polarization switch and a birefringent plate . An optically

addressed spatial light modulator (OA-SLM) is used as an optical memory and a thresholding device (which outputs a value of '1' for a non-zero input intensity). In Fig. 6, a simple example of a product 2 x 3 is also demonstrated. Each square at the input plane represents a component of a bright data bar to be calculated. One of the input data b in Fig. 6 is shifted down and fed back optically

are done employing digital electronics in the experi-

ments, 16 input data (N= 24) are aligned spatially parallel at the input plane which is displayed as a bar pat-

tern on a twisted nematic liquid crystal device (TN-LCD) panel used for a liquid crystal television (LCTV). LCD panels are also used as segmented polarization switches and the polarizations of optical beams are switched by ap-

plying the control voltage to the LCD. The LCD panel used in the experiments has 220 x 320 pixels each with an element size of 90 x 80 pm. The beam separation between adjacent input pixels is I mm and beams can be shifted by the birefringent plate for a unit of I mm with

to the input plane for the number of times equal to the value of the input data a and is duplicated at each row.

an appropriate combination of this plate and the axis

Then the result of the product is calculated in the output plane by integrating with the bright area. The sign of the product is easily obtained by monitoring sign bits of the

ments for the data exchange, the adder, and the multi-

direction of the polarization switch. Since the basic eleplier in the FFT system are the same, all calculations and

operations are performed using the same optical circuit

M. SHlRAI~WA & J.OHTSUBO42 9

OPTICAL REVIEW Vol. 6, No. 5 (1999) LCDl

LCD2 BPI LCD3 BP2LCD4 BP3

CCD

Encoded input

S ort data

(Optical exchange circuit) Data or Control signal

Fig. 7. Optical setup for a single stage of FFT. PL, Polarizer.

Add/Sub (Optical adder)

in Fig. 7. Some of the operations in the FFT algorithm are done by digital electronics in the current experiments. However, by replacing the CCD which is used as

Decoding

a detector for the output by a spatial light modulator, all optical circuits are constructed and the optical implementation is realized by cascade connections of the proposed

circuits. For example, the OA-SLM and feedback loop for the next shift of data in the multiplier in Fig . 6 are

Weights

replaced by the CCD and electronic feedback in the ex(Optical multiplier)

periment. Furthermore, only two LCDS (LCDI and LCD2) are used for the displays of the data and the control pattern in the multiplier operations, and the other

Decoding

LCDS (LCD3 and LCD4) are non-activated (namely, data pass through the LCDS without changing the states) . For data with N= 1 6 , the FFT calculation is performed in four stages including data exchange, addition or subtraction, and multiplication. Figure 8 shows a block dia-

Fig. 8. Block diagram of the operation algorithm for each stage of

gram of the operation algorithm for each stage . First, the data are sorted into proper order as data exchange suitable for the following calculation. Addition or subtraction

much improved by increasing the switching speed of the

for the sorted data is performed by the adder. Then the weight for the result is applied by the multiplier . This process is repeated four times and thus the result of the FFT is finally obtained. The flnal result is not a descending order of the outputs and it requires bit inversion to align the data in the proper order.

Figure 9 shows examples of the experimental results. The input is a sinusoidal signal and digitized as an integer type as shown in (a). Figures 9(b) and (c) are the results from the adder and the multiplier of the first stage, respectively; (d) is the sorted input for the second stage, and (e) is the output from the fourth stage, namely, the final output of the FFT operation. The output pattern is not bit inverted. After swapping the data, we obtain a

delta-1ike function and the spectrum in (e) is easily proved to be coincident with the expected FFT result for a sinusoidal signal.

We evaluated the throughput of the optical FFT system as a currently attainable performance and compared it with that of an electronic FFT circuit fabricated by commercially available DSP chips. The throughput is shown in Table 1. In the demonstrated optical system, the speed of the operations was limited to a video frame rate since we used electronically addressed spatial light modulators . The throughput of the optical system can be

the FFT flow.

device. Figure 10 shows a comparison of the capability of the FFT calculation for the proposed optical system and an electronic system as a function of data number based on the normalized calculation step . The normalized calcu-

lation steps defined here include the entire processes such as encoding and decoding data for the FFT calculation. As long as an optical device can handle input data in optically parallel at once in each step and the number of

data is within the maximum area of the display device, the total calculation time remains unchanged for increase of input data number . But the calculation step increases linearly with increase of input data number in an electronic case and it also depends on the resolution. Actual

speed for the calculation is easily estimated by multiplying the clock or switching time by the number of steps. Therefore , high switching speed of optical devices promises fast calculation of the FFT by the proposed sys-

tem. The simulated overhead due to the electronic controls in the current experiments inevitably merged with the process when a large number of input data was handled. So we also estimated the performance for the optical-hybrid FFT system used in this experiment. Deviations from the ideal optical operations at higher data number due to such effects are plotted as open circles and triangles in the figure.

Some parts of the FFT calculation and the data con-

430

OHTSUBO M. SHIRAKAWA & J.

6, No. 5 (1999) OPTICAL REVIEW Vol.

(a) Input

(b) Output of adder

(d) Sorted result

(e) Final results

Fig. 9. Experimental result for 16 input optical FFT. (a) Input signal, (b) result from the adder at the first stage, (c) result from the

multiplier at the first stage, (d) sorted pattern for the second stage,

and (e) final FFT output pattern.

(c) Weighted result

M. SHIRAKAWA & J. OHTSUBO 431

OPTICAL REVIEW Vol. 6, No. 5 (1999)

an all optical FFT system. The main circuits are the data exchange, adder, and multiplier discussed in this paper.

trols were performed by digital electronic circuits in the

present experiments. But the most important parts can be optically implemented by using optically addressed spatial light modulators. Figure 11 shows appearance of

The module for each stage of the FFT calculation is shown Fig. 11(a) and a block diagram of the total FFT system for a data number of N= 16 (namely, four stages of the FFT calculation) is shown in Fig. 11(b).

Table I . Throughput of FFT calculation.

DSP Through put

240 MOPS

Resolution

A one-dimensional digital optical FFT system has

1.44 KOPS

(80 MHz) 32

Number of data

4. Conclusions

O ptical

been proposed. An optical unit composed of a polariza-

(Video-rate)

tion switch and a birefringent plate was used as an

16

elementary beam switching device . Circuits for data exchange, adder, and multiplier were built in for the calculation of the optical FFT. The technique of optical networking is applicable to butterfly operations in the FFT algorithm, since the optics is quite appropriate for such

8 bit (256 pixels)

24 bit

107

switching networks. In the experiments, we employed a

106

hybrid system of optical and electronic circuits and some of the important parts of the calculations were performed by the electronics. These electronic parts can be easily replaced by optical devices, however, and the main parts of the FFT calculation can be implemented optically. In the current demonstration, data of optical inputs formed a one-dimensional bar pattern and only a binary value of the light intensity was used. Optics intrinsically has the

~s

~:2 10 co c:

o

~=104 ~! s

~!} (:i

o 103 ~) (D

N (e

E 10 -o z

advantage of handling two-dimensional information with a continuous intensity level, so that future studies are

.e

.- -~! _ ~ ~•OI 'A

10

.....:,'_"~,i.....

expected to extend the complete two-dimensional FFT calculation based on optical networking.

i

10

103

2

10

Data number

104

Acknowledglnent The authors would like thank to T. Takemori for valuabie discus-

Fig. 10. Comparison of the performance between optical and

sions.

electronic FFT calculations. A, hybrid 16-bit system; O , hybrid 128-bit system; ~ , electronic 16-bit system; I , electronic 128-bit system; A, optical 16-bit system; o , optical 128-bit system.

(a)

LCD2

LCD 1

~~~~~~i (b)

~>~~~1*

el htl

~~~ll~~~l ~~lgn~~ht2

~~]

~~~il~1

<~~ escrete Fourier s ectrum

Fig. 1 1. Single stage (a) and total schematic of all optical FFT systems for N= 16 (b). Each gray rectangle in (b) is the same optical circuit as

shown in (a).

432

OPTICAL REVIEW Vol. 6, No. 5 (1999)

M. SHlRAKAWA & J.OHTSUBO 10) H. Yamazaki, M. Yamaguchi and K. Hirabayashi: Appl. Opt.

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