Signal Averager

Abstract

A special purpose computer is used to calculate the statistical T-test as a measure of the significance of the difference between two sample populations. The computer includes a four channel average response computer, squaring circuits, square root circuits, dividing circuits, differential circuits and summing circuits.

Parent Case Text

This application is a continuation-in-part application based on U.S. Pat. application Ser. No. 878,158 filed Nov. 19, 1969, now abandoned and entitled "Signal Averager."

Description

The present invention relates to computers and more particularly to a special purpose computer including a signal averager and a T-test device.

Signal averaging is a method of separating a signal from noise. The signal may be from any source, as long as it is repeated, and the noise may be of any type, as long as it is more random than the signal. For example, when the brain waves of a subject are evoked by some stimulus, for example, a flashing light, the waves may be undistinguishable from the other electrical activity of the brain, which is considered as being a form of noise in this context. On an oscilloscope, a single evoked response may be so buried in the noise as to be undistinguishable.

Noise may be of various types, depending upon the context. A photoelectric tube, even without activation by a photon, will exhibit some slight output due to the thermal release of electrons from its cathode. Some of the types of noise have been called "white noise," "gaussian noise," "shot noise" and "on-going electrical activity," depending upon their origin or context. However, all these types of noise are random, i.e., they do not present a steady repeated signal. Other types of unwanted signals, which are considered noise, may present a steady signal. For example, the 60-cycle hum from the power line which is amplified by a radio or record player is a type of noise.

The efforts to extract signals from their noisy background have been costly and, to some extent, rewarded. But every time a signal is rescued, it opens up the hope that an even fainter signal may be retrieved or found.

One approach of the prior art has been to use bandwidth filtering. When the noise is random, a proper selection of a filter to pass the signal will eliminate all the noise except the noise which is close to the frequency of the signal. If the signal is weak, even that noise may prove troublesome. However, if the noise has a frequency close to that of the signal, then both the noise and signal will pass the filter and not be separated.

The approach in signal averaging to the signal-noise problem is to take a number of samples of the signal plus its surrounding noise at a number of intervals in a repeated series. If the repeated series are each of time I, the series is I.sub.1, I.sub.2 . . . I.sub.N. If one takes sub-intervals of each time period I, which need not be uniform but usually are, the sub-intervals (sampling impulses) are i.sub.1, i.sub.2 . . . i.sub.n, etc. One then has for a period I.sub.5,for example, five samples I.sub.5 -1, I.sub.5 -2, I.sub.5 -3, I.sub.5 -4, I.sub.5 -5. At each time period I . . . I, the signal is repeated and the noise is random. For each example i.sub.1 . . . i.sub.n the signal contribution is the same in each interval I.sub.1 . . . I.sub.N, but the noise may add or subtract, or be the same, in a random fashion. It is necessary for the sample values, to be summed and averaged. The signal component of the sample is a constant for any single point, so its contribution to the stored sum will increase proportionally to the number of repetitions. The noise component will increase proportionally to the square root of the number of repetitions. Thus, the signal-to-noise ratio is improved by an amount equal to .nu.N. The averaging may take place at the end of the samples or after each sample. In either event, the instrument requires a memory system, for example, a magnetic memory system.

However, average response computers suffer from certain limitations. Two averages may be the same, although the distribution of values in the two samples was quite different. Furthermore, the significance of differences between two averages is impossible to assess accurately without knowledge of the appropriate use of measures of variance and statistical tests of significance.

The T-test is a statistical test which measures the significance of the difference between two sample populations, taking into account both the respective means and variances. For example, consider two averages X and Y, each based upon 10 measurements. For the difference between the two averages [X - Y] to be significant at the 0.001 level [i.e., a difference of that size would occur by chance only one time in a thousand], the value of T must be greater than 4.587. If the samples were based on 25 measurements, the same probability level would only require a "T" value of 3.725.

Preferably both the number of sweeps N and the level of significance may be varied on the T-test computer to set a predetermined T-test standard. For example, the tester may set the maximum number of sweeps N at 25 and the level on significance P at 0.001. As an example, in electroencephalography, for each stimulus group either the T-test of the evoked response (X values) compared to brain wave ongoing activity background (Y values) will exceed 3.725 (the predetermined T standard) or be less than 3.725. If the T-test comparison of X and Y is larger than the predetermined "T" standard, then there is only 1 in 1,000 chance that the difference is merely random. Occurrence of such a T-test result would provide an objective indication that the subject probably responded to the stimulus. This result could be recorded.

It is the objective of the present invention to provide a special purpose computer which, by signal averaging methods, is able to distinguish a signal from its noise environment and which will provide a measure of the statistical variance of the averaged signal.

It is a further objective of the present invention to provide such a computer which will provide an indication of the statistical significance of the difference between two signals derived from a noisy environment.

It is a further objective of the present invention to provide such a digital computer which will provide a method of process control.

In accordance with the present invention, a relatively small size and low-cost special purpose computer is provided. An adjustable clock controls the sampling rate. In this way the signal is measured as a function of time after the start.

The computation of the T-test involves obtaining from the average response computer the sum of the voltage values X at similar times i over a number of repeated cycles (.SIGMA.X(i) ) . The circuit provides (1) the sum of the squares of those voltages (.SIGMA.X.sup.2 (i)) , (2) the division of the sum by the number of repetitions N and the squaring of the result ( .SIGMA.X.sup.(i) /N).sup.2, and (3) the division of the sum of the squares by N, giving (.SIGMA.X.sup.2 (i) )/( N ). The variance .sigma..sup.2 is equal to (.SIGMA.X.sup.2 (i))/( N ) - (.SIGMA.X(i)).sup.2 /( N ) .

In order to compute the T-test, two averages, M.sub.1 and M.sub.2, and their respective variances .sigma..sub.1.sup.2 and .sigma..sub.2.sup.2 must first be computed. The T-test is then computed by dividing the difference between the two averages by the square root of the sum of the variances: "t"=(M.sub.1 - M.sub.2)/([.sigma..sub.1.sup.2 +.sigma..sub.2.sup.2 ].sup.1/2) .

This computation must be performed at every point i over the epoch for which the functions X.sub.1 (i) and X.sub.2 (i) are to be described. These computations may be performed using a binary code by digital devices or, alternatively, using analog devices.

Other objectives of the present invention will be apparent from the detailed description of a preferred embodiment which follows, taken in conjunction with the accompanying drawings, in which:

FIG. 1 is a flow sheet illustrating a closed-loop process control;

FIG. 2 is a block schematic diagram of the one embodiment of the digital computer of the present invention;

FIGS. 3-6 are block diagrams showing the computations performed in the t-test device;

FIG. 6A is a block circuit diagram of another embodiment of the T-test computer; and

FIGS. 7A, 7B, 8A, 8B, 9 and 10 are circuit diagrams of specific circuits used in the embodiment of the T-test computer of FIG. 6A.

Before explaining the accompanying drawings, the operation of the device will first be illustrated by three specific examples of its application. These examples are (1) the derivation of a hypothesized signal from its noisy environment and the testing of the credibility of that signal, (2) the testing of the effect of variations in conditions on a process, and (3) the quality control of articles whose testing involves the separation of the output signal from noise.

I. credibility of the Derived Signal

The average response computer, as it is generally used, operates from voltages produced by voltage pick-ups or transducers. The transducers, such as temperature or pressure-to-voltage instruments, are connected to the input of the average response computer. Generally, the quantity being measured is a sample X(i) of the total population. That is, only a few measurements are made compared to how many could be made. For example, a brain wave or a chemical process may be measured for 1 minute, but the brain wave or process continues much longer. The averages derived by the average response computer are therefore not true averages .mu.(i) of the population but rather averages of the sample, i.e., X(i). The t-ratio is the appropriate test for inferring characteristics of the population from this sample.

The T-test computer provides an indication of the credibility of the hypothesized, i.e., average or extracted signal. In electrical engineering, the signal-to-noise ratio,S/ N,is used to indicate the credibility of a signal. If the ratio is low, there is much noise and the signal may not be considered to be trustworthy. Generally, these evaluations are by an empirical basis. It is derived from experience depending upon the device, how much noise makes what appears as the signal unreliable. In the context of the average response computer, the average is hypothesized as reconstituting the signal. The hypothesis is that the extracted signal is the actual signal. After the signal is so derived, it may be compared with the raw data. The ratio of the derived signal to the original value at each point may be considered the signal-to-noise ratio at that moment. In the chart below, there are three events I.sub.1, I.sub.2 and I.sub.3, for example, three sequential brain waves. Each wave is sampled at time points T.sub.1, T.sub.2, T.sub.3 and T.sub.4, for a total of 12 values. The raw signal-to-noise ratios require a highly subjective analysis of the reliability of the ratio; for example, the ratios of 15:2 or 9:6 may be good or not and valid signals may be discarded. The T-test provides an objective statistical basis for evaluating the signal and, in addition, provides a standard weight for comparison purposes. The T-test formula is ##SPC1##

In the example given for T.sub.1 :

and for T.sub.2 :

for T.sub.1, with 2.degree. of freedom the level of significance (using a T-test table) is 1 - 21 that the values would happen by chance; but for T.sub.2 it is 60 - 70 of the time that the values would so occur. In other words T.sub.1 has a much higher reliability than T.sub.2 in the sense of the credibility of the component at T.sub.1 compared to T.sub.2.

The most probable shape for the process I(t) is thus: ##SPC2##

The T-test is needed when there has been obtained one set of sample averages X. It is then desired to check the likelihood that this set of sample averages came from a population that had a true average .mu.. That is, there is provided a test for the likelihood that X =.mu..

Ii. process Control

A second function of the present T-test computer is to test the likelihood that two separate sets of sample averages X.sub.1 and X.sub.2 come from the same parent population. In this case, the true characteristics of the present population are not tested. That is, one does not care what .mu. or.sigma. are. All that is necessary and desired is to know if there is a significant difference between the two sets of readings, or if they both come from the parent population.

In this case, the appropriate formula is:

where:

X.sub.1 = mean 1

X.sub.2 = mean 2

.sigma..sub.1.sup.2 = variance 1

.sigma..sub.2.sup.2= variance 2

Once "T" has been computed, standard statistical tables are used to establish the probability of obtaining a specified difference, by chance, between two means based on particular sample sizes.

Under standard conditions, the mean X.sub.1 and variance .sigma..sub.1.sup.2 of some phenomenon are obtained, using channels 1 and 2 of the computer. Some condition is then changed, perhaps as an experimental maneuver. A second means X.sub.2 and variance .sigma..sub.2.sup.2 are computed, using channels 3 and 4 of the computer. The significance of the difference between X.sub.1 and X.sub.2 is then assessed by computing "t".sub.12. The channels used to compute X.sub.2 and .sigma..sub.2.sup.2 are then erased. A second change of condition is imposed on the system being measured and a third mean X.sub.3 and variance .sigma..sub.3.sup.2 are computed. The significance of the difference between X.sub.1 and X.sub.3 is assessed by computing "t".sub.13.

Now, if "t".sub.13> "t".sub.12, the second change of condition or experimental maneuver had a greater effect on the system than the first. If "t".sub.12>" t".sub.13, the first maneuver was more effective than the second. In this way, the average response computer has the capability to control a system or a process so as to obtain optimum effects. An operator or automatic controls are in a servo-loop and are instructed as to the sequence of steps which will maximize or minimize the difference between the state of a system and some reference state. Note that the reference state can be either an initial measurement from the system itself or some arbitrarily selected configuration which is desired by the operator. The T-test computer can direct a process either to achieve or avoid some selected criterion, provided that the necessary variables are placed under operator or automatic control, and operator or automatic assessment of the value of "T" is appropriately linked to those variables. Process control by T-test would be organized as shown in FIG. 1.

Iii. quality Control

An electronic circuit or component is tested by providing it with an input voltage. The circuit produces a pulse, which is extracted from its noise by average response computer. Then T-test is applied to determine the quality of the circuit. In some instances, for example, breakdown voltage, a statistical quality control program is required as the testing of the circuit destroys or injures it. For example, in a population of 100 circuits, five are drawn at random (not periodically). These five are tested using an average response computer; for example, they may be automatically tested for their breakdown voltage. Their result is an average of 5 volts. However, the specification requires 6 volts. If the voltages of the five units are, respectively, 3, 6, 4, 5, 7 volts, then the T-test is:

for which 4.degree. of freedom is 10 - 5 level of significance, i.e., it is significant at the 4 level. In other words, there is a 95 chance that the entire population (the 100 units) is below the specification.

The preferred embodiment of the "t" test computer is shown in FIG. 6A. As shown, the computer has two inputs--an X input on line 100 and a Y input on line 101. The inputs 100 and 101 are to a two-channel sample and hold circuit 102. The purpose of the sample and hold circuit 102 is to sample the two signals X and Y and to hold them so that they become in phase. A suitable sample and hold circuit is shown in FIG. 7A. The output lines 103 and 104 of the sample and hold circuit 102 are each directly connected to one channel of a four-channel average response computer 105. In addition, the outputs 103 and 104 are connected to respective squaring circuits 106 and 107, the details of the squaring circuit being given in connection with FIG. 7B. The average response computer 105 gives a value of samples taken periodically in time divided by the number of samples, thereby providing a running average, that is, an average which changes with the additional samples. A suitable average response computer is described in Clynes U.S. Pat. No. 3,087,487. Suitable multi-channel average response computers are Mnemotron (Technical Measurement Corp.) Series 400 and the Hewlett-Packard Model HP 5480A, available from the Hewlett-Packard Company, Palo Alto, Calif. and described in the Hewlett-Packard Journal, April 1968, pages 8-13. The operator determines the number of samples N by the sampling rate which is set by the clock pulses produced by an internal clock, such as a crystal controlled oscillator whose output is divided, within the average response computer 105. The output of the first channel 108 is the average of the sum of the values of X, i.e., the sum of the voltages of each of the samples divided by the number of the samples N, which is the mean and may be expressed by the formula: .SIGMA.X/ N .sub.x = M.sub.1 . The output of the channel 109 of the average response computer 105 is the sum of the X values squared over the number of samples and may be expressed by the formula: .SIGMA.X/ N.sub.x =M.sub.1. The output of channel 110 is the sum of the Y values over the number of samples and may be expressed by the formula: .SIGMA.Y/ N.sub.Y = M.sub.2 and the output of channel 110' the sum of the Y values squared over the number of samples and may be expressed by the formula: .SIGMA.Y.sup.2/ N.sub.y .

Each of the channels is connected to a four-channel sample and hold circuit 111. The only purpose of the sample and hold circuit 111 is to eliminate time skewing errors. An alternative is to have a separate memory for each of the channels, in which case the sample and hold circuit 111 would not be necessary. The circuits of each of the four channels of the sample and hold circuit 111 are the same as the sample and hold circuit shown in FIG. 7A.

The output of channel 108, which is the mean, is then squared in a squaring circuit 112 and similarly the output of channel 110 is squared in a squaring circuit 113. Each of the squaring circuits is the same as shown in FIG. 7B. The output of the squaring circuit and the output of channel 108 are then combined in a differential amplifier 114. Similarly the outputs of the squaring circuit 113 and channel 110' are combined in differential amplifier 115. The detailed circuit of a suitable differential amplifier is shown in FIG. 8A. The formula for the computation which occurs in the differential amplifier 114 is:

and the formula for the mathematical computation which occurs in the differential amplifier 115 is

The outputs of the differential amplifiers are connected to the respective divide circuits 116 and 117, the details of which are shown in FIG. 9. The divide circuit 116 divides the variance .sigma..sub.x.sup.2 by the number of samples. The number of samples is obtained from a sweep counter of the type shown in U.S. Pat. No. 3,506,813. The output of the device circuits 116 and 117 are connected to summing amplifier (adder) 118 which performs the following mathematical computation:

a suitable circuit being shown in FIG. 8B. The output of the summing amplifier 118 is to a square root circuit 119, the details of which are given in FIG. 9. The output of the square root circuit is to the divide circuit 120, a suitable divide circuit being shown in FIG. 9. The second input to the divide circuit is from a differential amplifier 121 which may be of the type shown in FIG. 8A. The differential amplifier 121 provides the difference between the two means, that is, it accomplishes the mathematical computation as follows: (.SIGMA.X/ N.sub.x) -(.SIGMA. Y/ N.sub.Y) . The output of the divide circuit 120 is to the absolute value circuit 121, shown in FIG. 10, which provides the final result of the T-test.

All of the computers necessary for the T-test have been provided by the circuit of FIG. 6A and the "t" test result is taken at the output 123. The T-test computation performed by the circuit of FIG. 6 is as follows: ##SPC3##

A suitable squaring circuit, as shown FIG. 7B, uses three integrated circuits. The integrated circuits 150 and 151 are operational amplifiers and may be of the type Motorola No. MC 1556-G. That integrated circuit is a compensated and monolithic operational amplifier. The integrated circuit 152 is a multiplier which, suitably, may be Motorola Type 1594-L. The multiplier, as its two inputs 153 and 154 derived from a common line 155 which is the output of the operational amplifier 150, and acts to square the input from line 155; that is, its inputs are tied together. A suitable integrated circuit is a monolithic four-quadrant multiplier where the output voltages are a linear product of two input voltages. The Motorola 1594-L is a variable transconductance multiplier with internal level shift circuitry and voltage regulation. The scale factor is adjustable and preferably is set to be one-tenth of input. An operational amplifier 151 is used to complete the multiplier connections from the integrated circuit 152. Its output 156 provides a square of the input at 157. This type of multiplier connection is described in further detail in the specification sheet dated October 1970 DS-9163 from Motorola of Phoenix, Ariz., of their 1594-L integrated circuit.

A suitable sample and hold circuit is shown in FIG. 7A. It uses an operational amplifier 140. Preferably operational amplifier 140 is an integrated circuit, for example, of the type Motorola No. 1456G, described above.

A suitable differential amplifier circuit is shown in FIG. 8A. It uses an operational amplifier 160 having two inputs 161 and 162. Preferably the operational amplifier 160 is an integrated circuit. A suitable integrated circuit is Motorola No. 1456G described in the specification sheet DS9147R1 dated April 1970 as being epitaxial passivated and monolithic. It has a power supply voltage of +18V dc and -18V dc, a power bandwidth of 40KHz and power consumption of 45m W max.

The summing amplifier of FIG. 8B also uses an operational amplifier 165. The two inputs to be added are connected to one input of the amplifier 165. A suitable operational amplifier is the integrated circuit Motorola No. 1456G described above.

A suitable divider circuit is shown in FIG. 9. It uses a linear multiplier 170 and an operational amplifier 171. Preferably the multiplier 170 and the amplifier 171 are integrated circuits. A suitable integrated circuit for the multiplier 170 is Motorola No. 1594, described above, and for the amplifier Motorola No. 1456G, also described above. The inputs are 172 and 173 and the output at 174.

A suitable square root circuit is shown in FIG. 9. The square root circuit is a special case of a divider in which the two inputs to the multiplier are connected together Consequently the input line 173 and the input line 172 are connected together to form a common input line 175, shown in dashed line and the ground.

A suitable absolute value circuit is shown in FIG. 10. It uses two operational amplifiers 176 and 177. Preferably they are integrated circuits and may be of the type Motorola No. 1456G described above. The input 178 is to the minus input of amplifier 176 and the output 179 is from amplifier 177. The purpose of the circuit of FIG. 10 is to provide a positive quantity if the X or the Y terms are larger, the absolute value being the value regardless of the plus or minus sign of the quantity.

The block schematic diagram of another embodiment of the system of the present invention is shown in FIG. 2. The input is to line 10. A typical input would be an electrical connection with a continuous repeated signal buried in noise, such as an electroencephalographic signal. The input signal is filtered by band filters 11, which eliminate noise outside the frequency band of the signal. The filtered signal is then communicated to the analog-to-digital converter 12 which converts the signal, in the form of a wave of continuous voltage over time, into its digital representation, at some predetermined clock rate. Preferably the digital representation is an eight-bit binary code, so that it will be compatible with the computer code.

The digital code from converter 12 is communicated to the sequence switch 13 which may be a shift register. The switch 13 sends the digital representations of the signal to one of the sequence of the multi-channel memory units 14-18, which are the "address" of the average response computer. The memory units 14-18 are part of a high-speed memory 19 which may be a ferrite core magnetic matrix plane, or any other suitable memory.

The sequence switch 13 is controlled by the signal sync pulse generator 20, for example, an adjustable duty cycle free-running multivibrator. The sync pulse generator 20 is set to indicate the start of each repeated wave, i.e., the I.sub.1 . . . I.sub.n periods. The generator 20 also controls the external stimulator 21, which is electroencephalography may be a flashable light. The sequence switch switches back to the first memory unit 14 when it receives a pulse from generator 20. The sequence switch 13 is also controlled by the sample pulse generator 28 which generates the sample pulses i.sub.1 . . . i.sub.n. For example, 10 pulses may be generated for each period I.sub.1 . . . I.sub.n within which the wave is repeated, in which case 10 memory units would be required. The sample pulse generator controls the switch 13 so that, at each sample pulse, the digital representation is sent to the next memory address 14-18 in sequence. The sequence switch 22, at the conclusion of each series I.sub.1 . . . I.sub.1, or, if desired, during the series, communicates the digital representation of each memory unit in sequence to the digital sums and averages device 23. Device 23 adds up the digital representation of each sample and divided by the number of samples, i.e., each address content is added (summed) and its average (arithmetic mean) determined. The averaged output of device 23 may, by means of switch 29, be communicated directly to the digital-to-analog converter. The output of converter 25 is displayed on display 26, which may be, for example, a meter or an oscilloscope.

In accordance with the present invention, the sums and averages of the device 23 are communicated to the remainder of the T-tester device 24. It should be understood, however, that the same register (memory addresses 14-18) may be used for certain functions in the T-test, in which case the output of the sums and averages device 23 and the T-test device 24 would be communicated back to the register-memory via the sequence switch 13 to store the sums.

Consider a four-channel average response computer, such as is described in the Clynes patent, in which the following alternatives are provided:

All four channels average in the conventional way, together or sequentially. However, an option exists so that

Channel 1 computes .SIGMA.X (t)

Channel 2 computes .SIGMA.X.sup.2 (t)

Channel 3 computes .SIGMA.Y (t)

Channel 4 computes .SIGMA.Y.sup.2 (t)

X and Y can be two different processes measured at the same time, or the same process measured at two different times. The measurement is made at each ordinate i over an analysis epoch from 0 to T. 0 is the time at which a perturbation (stimulus) is applied to the system, and T is a time period selected by the operator. T is then divided into a number of equal intervals, determined by the number of ordinates provided in the memory.

The digital representations stored in each channel can be added or subtracted from the representations in any other channel, and can be erased independently, resetting all registers to zero sequentially or simultaneously. The first two steps of computations are shown in FIG. 3, which shows the use of a four-channel memory computer and two counters.

In one implementation, the computation of .sigma.X.sup.2 (t) for each ordinate of X(t) can be carried out dynamically using repetitive sweep of the D-A system through the set of memory registers. This is shown in Step 2 of FIG. 3. In a second implementation, the computation of .sigma.X.sup.2 (t) is static. For subsequent T-test computation, the static version is preferred.

In the static version, .sigma. X.sup.2 is developed for i.sub.1, or ordinate 1, as shown in step 2 of FIG. 3. The .sigma..sub.X.sup.2 (i.sub.1) is stored as a voltage V ].sigma..sub.X.sup.2 (i.sub.1) ] in a sample and hold circuit; the registers holding .SIGMA.X.sup.2 (.sub.1) are reset to zero; the voltage V [.sigma..sub.X.sup.2 (i.sub.1)] is applied to the A-D; and the value of .sigma..sub.X.sup.2 (i.sub.1) is stored permanently in the register corresponding to ordinate 1. This sequence is then repeated for each ordinate, developing .sigma..sub.X.sup.2 (t) from 0 to T.

Then, as Step 3, there occurs a computation of .sigma..sub.Y.sup.2, using the procedure shown in Step 2 of FIG. 3. This is followed by the computation of "t" which is Step 4 of this procedure, as shown in FIG. 5, which shows use of an analog device and differential amplifier. Alternatively, the indicated computation should be performed digitally. After Step 3, quantities of FIG. 4 are stored which is followed by "t" computation shown in FIG. 5. After completion of Step 4, we have the quantities stored in the computer shown in FIG. 6.

The value of "t" at some particular time can be obtained digitally by interrogation of ordinate i, or the total or average value of "t" can be automatically calculated, or the full set of values of "t" from i.sub.o to T can be appropriately recorded in various ways. Then, counter 2 and channels 3 and 4 are erased and a new sample of data Z(i) brought into the computer. M.sub.z (i), .sigma.Z.sup.2 (i), and "t" x Z are computed. Comparison of "t".sub.xy with "t".sub.XZ at any ordinate of interest or over the full epoch from O to T then permits evaluation of the data.

The T-test device 24 may have its own long-term memory which, for example, provides a complete t-Distribution table (Student's or Gosset's distribution), or which stores sets of values of T-instead of storing them in channel 4.

In FIG. 1, the analog display 26 may be sent to display the results of the T-test computation in analog form, by means of the digital-to-analog converter 25 or alternatively the display may be by means of the digital display 27. The digital display may be a series of numerical display tubes, such as tubes, or a print-out device, such as a teleprinter.

The T-test computer has been described as a complete instrument. However, it will be understood that a T-test computer module may be sold as a unit and adapted to be connected to an average response computer. Having regard to FIG. 6 the average response computer 105 may be already in the customer's possession. He will then add to it the T-test module consisting of the squaring circuits 106 and 107 and the circuitry starting with the squaring circuits 112 and 113 and including the circuits to the right of circuits 112 and 113. As explained above, the sample and hold circuits 102 and 111 are optional. The connections of the T-test module to the average response computer would be the output lines from squaring circuits 106 and 107 and the input lines to the module of lines 108, 109, 110 and 110'.

Claims

We claim:

1. A special purpose computer for the computation of the "t" test, including

a first squaring means to provide the squares of a sequence of voltage values X and Y to provide their squared values X.sup.2 and Y.sup.2;

an average response computer connected to the output of said first squaring means and having a first summing means which provides the sum of the voltage values X over repeated cycles providing .SIGMA.X and the sum of a different set of voltage values Y over repeated cycles providing .SIGMA.Y and the sums of the squared values .SIGMA.X.sup.2 and .SIGMA.Y.sup.2, counter means for counting the repeated cycles to provide N.sub.x and N.sub.y, and a first division means connected to the output of said summing means and to the counter means to divide each of said sums by the number of; the repeated cycles N.sub.x and N.sub.y;

a second squaring means connected to the output of said average response computer to provide the square of the sums (.SIGMA.X/N.sub.x).sup.2 and (.SIGMA.Y/N.sub.y).sup.2;

a first differential means connected to the output of said second squaring means and to said average response computer to generate the quantities

(.SIGMA.X.sup.2/ N.sub.x) - (.SIGMA.X/N.sub.x).sup.2 =.sigma..sub.x.sup.2 and (.SIGMA.Y.sup.2/ N.sub.y) - (.SIGMA.Y/N.sub.y).sup.2 = .sigma..sub.y.sup.2;

a second divider means connected to the output of said differential means and to the counter means to divide the respective variances by the number of repeated cycles to provide averaged variances .sigma..sub.x.sup.2/ N.sub.x and.sigma. .sub.y.sup.2/ N.sub.y;

addition means connected to the output of said second divider means to add said averaged variances;

a square root means connected to the output of said addition means to provide the square root of said added variances;

a second differential means connected to said average response computer to provide the subtraction .SIGMA.X/N.sub.x -.SIGMA. Y/N.sub.y ; and

a third divider means connected to the outputs of said second differential means and to said square root means to provide the signal corresponding to the T-test result of .SIGMA.X/N.sub.x -.SIGMA. Y/N.sub.y / (.sigma.x.sup.2/ N.sub.x +.sigma. y.sup.2/ N.sub.y).sup.1/2.

2. A special purpose computer including the special purpose computer for the computation of the T-test as claimed in claim 1 and further including storage means responsive to the signal corresponding to the said T-test result, said storage responsive means storing the signals corresponding to the T-test results of a first and a subsequent second analysis of a process, comparison means comparing the said two stored signals, and means producing a signal which corresponds to the said comparison.

3. A special purpose computer including the special purpose computer for the computation of the T-test as claimed in claim 1 and further including comparison means which is responsive to the signal corresponding to the said T-test result and compares that signal with a predetermined standard, and means producing a signal which corresponds to the said comparison.

4. A special purpose computer module for the computation of the T-test, said module connectable to an average response computer having four channels, said average response computer having a first summing means which provides the sum of the voltage values X over repeated cycles providing .SIGMA.X and the sum of a different set of voltage values Y over repeated cycles providing .SIGMA.Y and the sums of the squared values .SIGMA.X.sup.2 and .SIGMA.Y.sup.2, counter means for counting the repeated cycles to provide N.sub.x and N.sub.y,

and a first division means connected to the output of said summing means and to the counter means to divide each said sums by the number of the repeated cycles N.sub.x and N.sub.y;

said computer module including:

a first squaring means having an input and an output, the input connectable to receive a sequence of voltage values X and Y and the squaring means providing at its output the squared values X.sup.2 and Y.sup.2, said squaring means having its output connectable to the input of said average response computer;

a second squaring means connectable to the output of said average response computer to provide the square of the sums (.SIGMA.X/N.sub.x).sup.2 and (.SIGMA.Y/N.sub.y).sup.2;

a first differential means connected to output of said second squaring means and connectable to said average response computer to generate the quantities (.SIGMA.X.sup.2/ N.sub.x) - (.SIGMA.X/N.sub.x).sup.2 = .sigma..sub.x.sup.2 and (.SIGMA.Y.sup.2/ N.sub.y) - (.SIGMA.Y/N.sub.y).sup.2 =.sigma..sub.y.sup.2

a second divider means connected to the output of said differential means and connectable to the counter means to divide the respective variances by the number of repeated cycles to provide averaged variances .sigma.X.sup.2/ N.sub.x and .sigma.Y.sup.2/ N.sub.y;

addition means connected to the output of said second divider means to add said averaged variances;

a square root means connected to the output of said addition means to provide the square root of said added variances;

a second differential means connectable to said average response computer to provide the subtraction .SIGMA.X/N.sub.x -.SIGMA. Y/N.sub.y ; and

a third divider means connected to the outputs of said second differential means and to said square root means to provide the signal corresponding to the T-test result of .SIGMA.X/N.sub.x -.SIGMA. Y/N.sub.y / (.sigma.X.sup.2/ N.sub.x + .sigma..sub.y.sup.2/ N.sub.y).sup.1/2.

5. A special purpose computer including the special purpose computer module for the computation of the T-test as claimed in claim 4 and further including storage means responsive to the signal corresponding to the said " t" test result, said storage responsive means storing the signals corresponding to the T-test results of a first and a subsequent second analysis of a process, comparison means comparing the said two stored signals, and means producing a signal which corresponds to the said comparison.

6. A special purpose computer including the special purpose computer module for the computation of the T-test as claimed in claim 4 and further including comparison means which is responsive to the signal corresponding to the said T-test result and compares that signal with a predetermined standard, and means producing a signal which corresponds to the said comparison.