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 510.84 
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Digitized by the Internet Archive 
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t ■ 07 
 
 REPORT NO. 3^9 
 
 COO-lU6y-oiU5 
 
 "MODEL T" 
 A DEMONSTRATION OF IMAGE MULTIPLICATION 
 USDJG STOCHASTIC SEQUENCES 
 
 i A 
 
 by 
 
 O. E. MARVEL 
 
 & 
 
 \\ 
 
 August, 1969 
 
REPORT NO. 3^9 
 
 "MODEL T" 
 
 A DEMONSTRATION OF IMAGE MULTIPLICATION 
 
 USING STOCHASTIC SEQUENCES* 
 
 by 
 
 0. E. MARVEL 
 
 August, 1969 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 61801 
 
 ^Supported in part by Atomic Energy Commision Contract No, 
 AEC AT(ll-l) 1^69. 
 
ii 
 
 TABLE OF CONTENTS 
 
 Page 
 
 1. INTRODUCTION 1 
 
 2. STOCHASTIC SEQUENCE REPRESENTATION 5 
 
 3. TIMING 8 
 
 k. SYSTEM OPERATION 11 
 
 5. CIRCUITS Ik 
 
 6. CONCLUSIONS 22 
 
 BIBLIOGRAPHY 23 
 
3S<f 
 
 LIST OF FIGURES 
 
 Figure Page 
 
 1. Model T 2 
 
 2. "Model T" Inputs k 
 
 3. Timing Chart 10 
 
 k. Block Diagram of "Model T" 13 
 
 5. Stochastic Analog Staircase Generator 15 
 
 6. Input Processors 17 
 
 7- Output Processor 20 
 
 8. Timing and Gating Generation 21 
 
 in 
 
1. INTRODUCTION 
 
 "Model T", shown in Figure 1, demonstrates a new method of 
 electronically multiplying pictures. Since the corresponding points in 
 the two pictures are multiplied at the same time (in parallel), a large 
 number of inexpensive multipliers must be found. Assuming that the 
 intensity of each point on the picture can be conveniently converted to a 
 stochastic sequence, a stochastic multiplier -- "And" gate -- costing 
 about twenty-five cents is ideal. Since stochastic sequences 
 represent the analog intensities of the picture, computations and 
 processing are done with digital circuits which are fast and reliable. 
 
 Stochastic process multiplication produces with a digital "And" 
 gate the product of two analog variables represented as the probability 
 of occurrence of a standardized pulse occurring in a fixed time slot. 
 When two independent stochastic pulse trains with duty cycles u, and u 
 are fed into an "And" gate, as shown in the figure below, the output of 
 the "And" gate is given by 
 
 T 
 Vil U 3 (t)dt 
 
 = P{u_(t) has a pulse in a time slot} 
 
 = P{u_ (t) and u p (t) both have pulses in a time slot} 
 
 = P{u, (t) has a pulse in a time slot} x 
 
 P{u p (t) has a pulse in a time slot} 
 
 = U l * U 2 
 
 This computer was developed to experimentally verify 
 electronic circuits to be used in the TRANSFORMATRIX system and was 
 guided by Professor W. J. Poppelbaum. 
 
MODEL. T 
 
 A Demonstration of 
 the World's Fastest 
 Stochastic Multipliers 
 
 Each Operatesat 
 30,000 Multiplications / 
 with a IO Million Rulse / 
 
 Clock Rate 
 
 econ 
 ► econd 
 
 Figure 1. Model T 
 
cm 
 
 Q 
 
 -» o 
 
 •»**•*« 
 
 H e 
 
 U,(;c) 
 
 ♦ — i — t- 
 
 » 7 * I * 9 2 
 
 *-— I • 
 
 In "Model T", two h by h input pictures are multiplied and 
 displayed as a k by h output picture, with 20 gray levels, on a Heathkit 
 Model 10-18 oscilloscope, Figure 2. The output picture is produced by 
 first converting the intensities of each input point with a photo conductor 
 and operational amplifier to a voltage. This voltage is then compared 
 with a Stochastic Analog Staircase (SAS) to form the Clocked Analog 
 Stochastic Sequence (CASS) signal. Upon arrival of the proper gating 
 pulse, the corresponding CASS signals from the two input pictures are 
 multiplied and integrated for 32.552 microseconds. The resulting 
 integrand, thus, represents the intensity of the output point and after 
 being converted to an analog voltage, drives the z-axis, intensity, of 
 the oscilloscope. Of course, at the same time the z-axis voltage arrives, 
 the beam of the oscilloscope is correctly positioned. We have, there- 
 fore, produced a picture whose point by point brightness depends on the 
 product of the point by point brightness from two pictures. 
 
SLOT FOR 
 SLIDE 
 
 PHOTO 
 CONDUCTORS 
 
 LIGHT BULB 
 
 XA m , 
 
 XB 
 
 mn 
 
 mn 
 
 Figure 2. "Model T" Input;; 
 
2. STOCHASTIC SEQUENCE REPRESENTATION 
 
 The probability that a pulse from a random pulse generator is 
 present in a fixed time slot is the stochastic sequence representation 
 of a number from to 1. The above definition, although conceptually 
 simple, is almost impossible to measure. We may, however, estimate the 
 value of the stochastic sequence over a number of time slots. Thus, 
 we are representing an analog variable from to 1 by the number of time 
 slots that contain a logical "1" divided by the total number of possible 
 time slots, and this quantity may be determined by a digital counter or 
 integrating voltmeter. 
 
 A voltage is converted to a Clocked Analog Stochastic Sequence 
 (CASS) by comparing it with a Stochastic Analog Staircase (SAS) — a 
 voltage waveform that randomly takes on a number of quantized amplitude 
 levels with equal probabilities. Assuming the voltage is between to 5 
 volts and the SAS has 32 levels with 5/31 of a volt separating adjacent 
 levels, then each level of the SAS has a probability of 1/32 of occurring. 
 CASS is the output of a voltage comparitor with the inputs arranged so 
 that when the input voltage is greater than the SAS the output is a logical 
 1, see below. 
 
 i"V 
 
 »- — » ' i I l l oV 
 
 t t ♦ a a i 
 
 n 
 
 i—* 
 
 4 r * 3 • • 
 
The accuracy of this conversion process is determined by: 
 
 (A) The voltage comparitor offset. In "Model T", a 5 
 millivolt worst case offset gives us 3.1% of the 
 level separation as an error. 
 
 (B) The number of quantized levels in SAS. In "Model T", 
 32 levels gives us a worst case quantization accuracy 
 of 3.1%. 
 
 (C) The statistical properties of the Pseudo-Random-Noise 
 Generator. "Model T" uses a 2>h bit feedback shift 
 register which has a worst case probability error 
 which is very small. 
 
 After the stochastic sequences have been processed, they are 
 converted into a voltage by an averaging circuit. The accuracy of this 
 conversion process can be obtained from the probability theory of 
 stochastic sequences. If the probability of a pulse occurring in a 
 time slot is p, then the probability of no pulse is 1 - p. The mean or 
 expected value of the stochastic sequence is obtained from: 
 
 m = E(x) = Z{ (probability of value) (value) } 
 = (P)(D + (1-P)(0) -p 
 
 and the variance obtained from: 
 
 a 2 = E[(x-m) 2 ] = E(x 2 ) - [E(x)] 2 = E(x 2 ) - m 2 
 a 2 = p(l) 2 + (l-p)(0) 2 - p 2 = p - p 2 = p(l-p) 
 
 Since there are N possible time slots, the mean is Np and the variance 
 Np(l-p). 
 
The accuracy to be specified here is a percentage of full scale 
 reading as in a voltmeter. Since each output point is calculated and 
 displayed 30 times per second, while the human eye and oscilloscope phosphor 
 act as averagers, a 95% confidence level per display seems reasonable. 
 Thus the 2a -standard deviation error is given by 
 
 Error = 
 
 Y * 100* 
 
 200 
 
 N 
 
 [Np(l-p)] 
 
 1 
 
 200 r /. v-,2 
 j[p(l-p)] 
 
 2 
 (N) 
 
 For "Model T" W is 325 giving 
 
 Error = ll.l[p(l-p)] 
 
 ERROR t 
 (%) 
 
 .2r 
 
 .r* 
 
 ir 
 
3. TIMING 
 
 The basic timing is dictated by the following three 
 requirements . 
 
 (A) The output display frame rate must be the same as the 
 frame rate of a conventional TV receiver^ - 
 
 30 frames /second. 
 
 (B) The fastest economical digital circuits in integrated 
 circuit form are transistor-transistor logic circuits 
 with a maximum practical pulse repetition rate of ten 
 megahertz. 
 
 (C) The circuits when proven in "Model T" will be used in the 
 TRANSFORMATRIX system which will display a 32 x 32 output 
 picture . 
 
 Since TRANSFORMATRIX will display 1,02U output points at a 
 rate of 30 frames /second, a new output point will be calculated every 
 32.552 microseconds or at a rate of 30.72 kilohertz. This gives us a 
 master clock frequency of 9- 9°^ megahertz and 325 time slots per stochastic 
 process. 
 
 Figure 3 shows the timing and gating pulses which are derived 
 from a 9.98^ megahertz multivibrator -- Master Clock. A counter then 
 produces an output pulse -- Point Clock -- for every 325 Master Clock 
 pulses. 
 
 2 
 
 In this display there is no interlacing, thus the frame and 
 
 field rates are equal, 
 
Since "Model T" only displays 16 intensities per frame, it 
 processes data for only .52 milliseconds while it rests for 32.81 milli- 
 seconds every frame. A counter selecting 16 of the point clock pulses 
 out of every 102U produces the Input Clock. Monostable multivibrators 
 are used to produce Shift Clock, Reset Clock, and the blanking and rest 
 pulses from Input Clock. 
 
10 
 
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 Q. 
 
 
 2 
 
 U 
 
 t 
 
 I 
 
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 UJ o 
 
 o z 
 
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 ^ X 
 
 <»■ PO "1 
 
 Figure 3. Timing Chart 
 
11 
 
 h. SYSTEM OPERATION 
 
 The block diagram for "Model T" is shown in Figure h. The basic 
 timing and gating pulses as discussed in the last section are produced in 
 the clock control and gating control units. Because the oscilloscope 
 is displaying the intensity of an output point while the next output 
 points' intensity is being calculated, the k bit X-Y counter must produce 
 the present X-Y position as X and Y analog voltages for the oscilloscope 
 and the next X-Y position as digital commands for multiplexing the CASS 
 signals. 
 
 A 3^- bit feedback shift register produces two 5 bit digital noise 
 signals which are converted to the Stochastic Analog Staircase - SAS - by 
 digital to analog converters. Since the two digital noise signals are 
 independent, the two SAS signals are independent also. Each SAS goes to 
 a set of sixteen comparitors whose other inputs are the voltages representing 
 the light intensities from the two input pictures. Thus, we are generating 
 sixteen pairs of stochastic sequences, each pair representing one of the 
 input pictures . 
 
 The input processors shown in Figure h perform two operations 
 on the stochastic sequences. First, the independent stochastic sequences, 
 CASS, representing the same grid location on each picture must be multiplied, 
 Then, because the oscilloscope only displays one product at a time as an 
 output intensity, the 16 CASS products must be multiplexed or funneled one 
 at a time to the integrator. 
 
12 
 
 The integrator produces a binary representation of the output 
 intensity by integrating for 32.552 microseconds the CASS. During the 
 next 32.552 microseconds period, the binary number is stored and connected 
 to a voltage which is applied to the z axis of the oscilloscope through 
 the driver switching unit. Two other signals are applied to the driver 
 switching circuit: The first is a blanking pulse which turns off the 
 beam during the time it is moving from point to point. The second is the 
 rest pulse which cuts off the beam during the 32.81 millisecond rest 
 period. 
 
 The driver switching unit allows either the positive or negative 
 product of the input pictures to be displayed. 
 
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1U 
 
 5 . CIRCUITS 
 
 To keep cost and size down, integrated circuits are used 
 jughout "Model T". All digital circuits are positive logic TTL 
 integrated circuits. 
 
 Figure 5 shows the syncronous random pulse generator and the 
 two stochastic analog staircase - SAS - generators. The syncronous 
 random pulse generator is a 3^+ hit feedback shift register where the 
 bit to be entered and shifted is given by Q = Q , © Q © Q, © . 
 Thus, the shift register contains a continuously changing - at a 10 MHz 
 rate - binary word in which the probability of any bit being a one is 
 1/2. Also, at any instant of time, i.e. within a fixed word, the value 
 of any bit is independent of the value of any other bit. 
 
 The outputs of 5 stages of the shift register go to one high 
 speed D/A converter to produce SAS #1, and the outputs of 5 different 
 stages go to another D/A converter to produce SAS #2. The digital to 
 analog converter consists of a high speed voltage switch with weighted 
 resistor decoder and high speed operational amplifier. 
 
 Figure 6 shows 16 identical input processor units which perform 
 four functions. First the light intensity to voltage transfer function 
 must be standardized. The transfer function for the photoconductor is 
 given by 
 
 R - R (D- 8U 
 
 the resistance in ohms, R is the resistance at one foot 
 candle which may vary 4 33%, and I is the intensity in foot candles. 
 
Figure 5. Stochastic Analog Staircase Generator 
 
16 
 
 With the photo conductor as the input resistor and a trimpot R^ as the 
 feedback resistor of an operational amplifier, the output voltage of the 
 op amp is 
 
 INT " R Q U; V S 
 
 th V a standardized voltage, the variation in R can be cancelled 
 
 o U 
 
 by adjusting R^ to make all photo processors have the same transfer 
 functions . 
 
 The CASS signals are produced by comparing the V s against 
 the SAS signal with a 710 voltage comparitor which produces a logical 
 one whenever V is greater than the SAS signal. Since the comparitors 
 have a maximum differential input voltage of 5 volts and the SAS has 32 
 levels, to obtain the greatest noise margin the photo processor stage 
 (R„) is adjusted to give V a 5 volt swing as the light varies from 
 .1 to 1.0 foot candles. 
 
 A four input NAND gate, when enabled by the proper I and J 
 gating signals, produces the logical inverse of the product of two 
 CASS input signals. 
 
 Figure 7 shows the output intensity processing circuitry which 
 performs 16 identical cycles per displayed frame. After the 16 lines from 
 the input processor have been inverted and OR'ed, the CASS product is 
 AND'ed with a k MHz square wave to produce a pulse when CASS is a 
 logical one, except for three periods when the AND gate is inhibited and 
 no pulses are produced. A ripple counter then counts the number of 
 . "34 Mhz pulses in a 32.552 microsecond period. At the end of this 
 period, while the input to the counter is inhibited, the 5 MSB's of the 
 
17 
 
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 Figure 6. Input Processors 
 
18 
 
 counter are shifted into a 5 bit storage register and the counter reset. 
 During the next cycle the binary number in the storage register is 
 converted to an analog voltage by a D/A converter. This analog voltage 
 is applied to the z axis of our oscilloscope through a + 25 volt driver 
 stage which has the ability to amplify the applied voltage - display the 
 picture - or invert and amplify the applied voltage - display the negative 
 of the picture. 
 
 Simultaneous with the signal flow shown in the last three 
 figures, timing and gating signals are generated as shown schematically 
 in Figure 8. Master clock is a 96OI multivibrator wired as a 9.98k MHz 
 astable multivibrator. Ripple counters and coincidence circuits are used 
 to produce point clock and input clock. 
 
 The 2 bit X and Y counters, Figure 8, are decoded to produce 
 the I to I and J to J gating signals. The outputs of the counters 
 are also delayed one cycle and converted to M and N, the horizontal and 
 vertical drivers. 
 
 Two problems were encountered and solved after "Model T" was 
 built. The first was the long rest time built into "Model T" which 
 caused a bright spot on the oscilloscope at X Q ,Y . This problem was 
 eliminated by connecting a retriggerable monostable multivibrator, with 
 its period set slightly longer than the period of input clock - 32.552 
 microseconds, to input clock. Thus the inverted output of the multi- 
 vibrator is zero while we have the 16 pulses of input clock and +k volts 
 which is used to blank the oscilloscope when there are no input clock 
 pulses. The second problem was the capacitive coupling of the z axis 
 which causes intensity level standardization problems. This problem 
 
19 
 
 will be eliminated in TRANSFORMATRIX by using an oscilloscope with DC 
 coupling. But, in "Model T", the irnpedence of the grid circuit of the 
 oscilloscope tube as well as the input capacitance were increased, giving 
 a more satisfactory time constant for the z axis grid drive. 
 
20 
 
 Figure 7. Output Processor 
 
21 
 
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22 
 
 6. CONCLUSIONS 
 
 Because of the ease and low expense of multiplying and gating 
 analog quantities in stochastic sequence form, large para ll el processors 
 may be economically built using stochastic elements . "Model T" by using 
 these stochastic techniques has shown the feasibility and problems that 
 must be overcome to process pictures at standard television rates. 
 
23 
 
 BIBLIOGRAPHY 
 
 1. Afuso, C, "Analog Computation with Random- Pulse Sequences", 
 
 Department of Computer Science Report #255, University 
 of Illinois, Urbana, Illinois, February, 1968. 
 
 2. Esch, J., "A Display for Demonstrating Analog Computations with Random 
 
 Pulse Sequences," Department of Computer Science Report #312, 
 University of Illinois, Urbana, Illinois, March, I969. 
 
 3. Esch, J., "Rascel - A Programmable Analog Computer Based on a Regular 
 
 Array of Stochastic Computing Element Logic," Department of 
 Computer Science Report #332, University of Illinois, Urbana, 
 Illinois, June, 1969. 
 
 h. Korn, G., Random- Process Simulation and Measurement, McGraw-Hill, 1966. 
 
orm AEC-427 
 
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 "MODEL T" - A DEMONSTRATION OF IMAGE MULTIPLICATION 
 USING STOCHASTIC SEQUENCES 
 
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 Orin E. Marvel, Research Assistant 
 
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 Department of Computer Science 
 
 University of Illinois 
 
 Urbana. Illinois 6l801 
 
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