Abstract:
In order to deliver an input signal of each operation cycle after a desired time delay, there are disposed data memory means for storing the input signal, counter means for appointing write addresses of the data memory means, and address memory means for appointing read addresses of the data memory means. The address memory means is divided into partial memory areas equal in number to time delay elements, whereupon while sampling the input signal at a predetermined sampling period and changing the count value of the counter means one by one for each of the desired time delay elements at each sampling point, the variations of the input signal in a sampling interval between the particular sampling point and the adjacent sampling point are successively written into the memory means. Further, while changing the contents of the partial memory areas corresponding to the desired time delay element to the number of the time delay elements in each sampling interval, the variations in a sampling interval preceding a predetermined sampling number to the particular sampling interval are successively read out from the memory means. The input signal of each operation cycle in the preceding sampling interval is presumed by an interpolation operation based on the variation and the sampling period, and the result is used as an output signal of the desired time delay element.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates to a method of generating a time delay in a digital differential analyzer (hereinbelow, abbreviated to &#34;DDA&#34;) etc. 
     In the description of the principles of this invention, a case where the method of this invention to generate a time delay is applied to a DDA will be taken as an example. The invention, however, is also applicable to apparatuses other than a DDA, which will be described in detail in connection with the preferred embodiments. 
     In the case of employing a DDA for, e.g., the analysis of a process control system or the calculation of a correlation coefficient, a time delay T expressed by the following equation needs to be generated: 
     
         e.sub.o (t)=e.sub.in (t-T)                                 (1) 
    
     where e in  denotes an input signal, and e o  an output signal. 
     2. Description of the Prior Art 
     In general, as methods of generating time delays, there are (a) a method which utilizes approximation by a transfer function (refer to &#34;On the analog simulation of a pure time delay&#34;, Simulation, 1972, 18, (5), pp. 161-170), and (b) a method which utilizes a time delay arithmetic unit based on a storage mode. 
     Regarding the former, Pade&#39;s approximate formulas are famous. However, they can generate only short time delays of ωT&lt;2[rad] with the second approximate formula and ωT&lt;6[rad] with the fourth approximate formula, ω in the inequalities being representative of the angular frequency of an input signal. Another disadvantage is that an output signal becomes oscillatory for indicial response. Moreover, a large number of DDA arithmetical elements (for example, ten or more elements for Pade&#39;s fourth approximate formula) are required. In view of such drawbacks, it can be said that it is unpractical to apply to the DDA the method which employs approximation by a transfer function. 
     On the other hand, a kind of storage mode of the latter wherein an input signal is sampled at a certain period h, the sampled signal is stored into a digital memory and the stored signal is read out from the memory and then reconstructed and delivered after the lapse of a certain time delay, is a method suited to the DDA which handles all signals as digital signals. It has also the feature that a long time delay can be realized merely by increasing the capacity of the memory. 
     In the case of applying this method to the DDA, however, several problems as stated hereunder are involved. 
     The first problem is that the sampling period h must be made small in order to compute a time delay of high precision, so a large memory capacity is required in order to generate a long time delay. FIGS. 1(A)-1(D) illustrate the principle of the storage mode. An input signal e in  being a continuous function as shown in FIG. 1(A) is sampled at a sampling period h as shown in FIG. 1(B). Sampling values e in  (p) (p=0, 1, . . . ) in the order of the No. of the sampling points are successively stored in sequence into addresses of a digital memory as shown in FIG. 1(C). Upon lapse of a time delay T, the sampling values are read out from the digital memory in the order of e in  (0), e in  (1), . . . and then reconstructed and delivered. Accordingly, a time delay output signal e o  becomes a stepped waveform as shown in FIG. 1(D). Here, the error ε of the time delay output signal becomes as follows: ##EQU1## 
     Accordingly, the error ε for p·h≦t&lt;(p+1)·h becomes zero at the sampling point p, and it increases more as the next sampling point (p+1) comes nearer. By way of example, in the case where the time delay has been generated by the method of FIGS. 1(A)-1(D) as to: 
     
         e.sub.in =A sin x                                          (3) 
    
     the absolute error ε becomes: ##EQU2## where 0≦τ&lt;h 
     The maximum error develops in the vicinity of the maximum gradient of Equation (3), that is, in the vicinity of x=0. The maximum error in the vicinity of x=0 can therefore be approximately evaluated by substituting p=0 and τ=h into Equation (4), as follows: 
     
         ε.sub.max ≅A sin h                       (5) 
    
     It is understood from Equation (5) that, in order to limit the calculation error of the time delay to 0.1% of the full scale A, the sampling period h must be made 0.001 [rad]. In other words, about 6,280 (≃2/0.001) points must be sampled within one cycle. Now, let&#39;s consider a case where a sine wave having an angular frequency ω is sampled at a sampling period h over a delay time T. Letting B (bytes) denote the length of one word of a memory for expressing a sampling value, the capacity W of the memory required per time delay element becomes: ##EQU3## Assuming by way of example ω=10 rad/s, T=5 sec, h=0.001 rad and B=4 bytes, the capacity W becomes as large as 0.2M bytes. 
     The second problem is attributed to the calculation method of the DDA. 
     The DDA adopts the calculation method in which each of an input variable, an output variable and an integral independent variable is rounded off into 1 bit or several bits and then transmitted (refer to Japanese Patent Application Publication No. 50-25148 and Japanese Patent Application Publication No. 50-32849). This leads to the problem that also the output signal of the time delay must be converted into an increment of 1 bit or several bits. 
     SUMMARY OF THE INVENTION 
     An object of this invention is to provide a method of generating a time delay as based on the digital memory storage mode, which method is free from the problems described above, is inexpensive and has the function of generating a long time delay at high precision. 
     In order to accomplish the object, this invention is contrived as stated below. At the first feature, two sorts of digital memories A and B to generate a time delay are provided. An input signal of a time delay element expressed in an incremental format of 1 bit or several bits is accumulated every operation cycle (hereinbelow, termed &#34;iteration&#34;) of a DDA. A sampling period h is set to be an integral number of times of the iteration, and the accumulative values or the increment values of the input signal in 1 sampling interval are successively stored into the memory A every sampling period. After the storage processing has advanced to an address which is appointed by a quotient u (integer value) with the time delay T divided by the sampling period h, the increment values of the input signal are read out every sampling period h from those addresses of the memory A which precede at most (u-1), interpolation operations of an incremental format as described later are performed in the respective iterations of the DDA, and the results of the interpolation operations are stored into the memory B. Thereafter, the contents of those addresses of the memory B which precede at most a residue v (integer value) resulting from the division of the time delay T by the sampling period h are read out, and are provided as an output signal of the time delay. 
     As the second feature, address counters ADCA and ADCB which manage the addresses of the memories A and B to store data thereinto are respectively provided for the memories A and B. The ADCA has 1 (one) added every sampling period H and every time delay element, and the sampling value is stored into the resulting address of the memory A. Simultaneously, the ADCB has 1 (one) added every iteration and every time delay element, and the interpolated value is stored into the resulting address of the memory B. In this way, it is permitted to write data into both the digital memories continuously in a ring shape without limitation. 
     As the third feature, address memories ADMA and ADMB with manage the addresses of the memories A and B to read out data therefrom are respectively provided for the memories A and B, and each address memory ADMA or ADMB includes storage locations equal in number to the time delay elements. The content of that memory within the ADMA which corresponds to the time delay element noted has the total number of the time delay elements added every sampling period h, and the data is read out from the resulting address of the memory A. Simultaneously, the content of that memory within the ADMB which corresponds to the time delay element noted has the total number of the time delay elements added every iteration, and the data is read out from the resulting address of the memory B. In this way, it is permitted to read out ring-shaped data formed on the memories A and B, continuously and unlimitedly. 
     Hereunder, the principles of this invention will be described in detail. 
     First, in order to sharply reduce the capacity of a memory for storing sampling values, this invention executes the operation of interpolation indicated by Equation (7) and obtains a time delay output signal e o  (t) at time t (t=T+p.h+π) after the lapse of the time delay T, where o≦π&lt;h. ##EQU4## 
     FIG. 2 shows the time delay output signal delivered in such a way that the input signal e in  shown in FIG. 1(A) is sampled at the same sampling period h as in FIG. 1(B) and that after the lapse of the time delay T, the sampling values are subjected to the interpolation operation of Equation (7). Merely by comparing FIG. 1(D) and FIG. 2, it is understood that the present method attains a higher accuracy. An error ε in the interpolation of Equation (7) becomes: ##EQU5## Here, by taking as an example a case where the input signal expressed by Equation (3) is operated by the method of Equation (7), the error at the generation of the time delay in the present method will be compared with that in the method of FIGS. 1(A)-1(D). The maximum value of the error expressed by Equation (8) develops when the conditions of dε/dτ=0 and d 2  ε/dτ 2  &lt;0 hold. When the conditions causing the maximum error are obtained by substituting Equation (3) into Equation (8), it is understood that the maximum error develops in a state illustrated in FIG. 3. Accordingly, the maximum error ε max  becomes: ##EQU6## As seen from Equation (9), in order to limit the error at the generation of the time delay to 0.1% of the full scale A, the sampling period h may be made about 0.08945 [rad]. In other words, one cycle may be sampled at about 70 points. Since the method of FIGS. 1(A)-1(D) requires about 6,280 sampling points in order to attain the same accuracy, the time delay generating method based on the interpolation mode in accordance with this invention suffices with about 1/90 of the sampling points in the method of FIGS. 1(A)-1(D) and can remarkably reduce the capacity of the digital memory. 
     A problem to be solved next is how the output signal of the time delay obtained by executing the aforecited interpolation operation in combination with the sampling value-storing digital memory is converted into an incremental format and then delivered by the use of simple means. An operating method therefor is illustrated in FIG. 4 and FIGS. 5(A)-5(E). 
     First, Equation (7) is converted into the incremental format of the following equation: ##EQU7## Operating steps indicated by Equation (11) are executed by means of the DDA by setting τ at Δt which is the fine increment of the fundamental integral independent variable t of the DDA, and the interpolation operation in which the resulting output increment ΔZ i  is used as the increment Δe o  of the output signal of the time delay element is executed. ##EQU8## Here, letter i represents the iteration No. of the DDA included between the sampling points p and (p+1), symbol Y i  represents the content of a Y register at the i-th iteration or the gradient of the aforecited Δe o , symbol R i  represents an integral residue at the i-th iteration, and symbol R i-1  represents an integral residue at the (i-1)-th iteration. Further, all these variables shall be handled with floating-point numbers. 
     In actuality, however, it is not advantageous to operate Equation (11) as it is. This is because a divider circuit is required for the computation of the gradient Y i . In accordance with this invention, therefore, the sampling period h is set at an integral number of times m of Δt and the integer m is selected to be the power of 2, as illustrated in FIG. 4, whereby the gradient Y i  is obtained by only addition and subtraction. This will now be described. 
     By making m the power of 2, the sampling period h becomes as follows: ##EQU9## 
     Subsequently, when the increment of the input signal e in  between the sampling points p and (p+1) is put as: ##EQU10## the gradient Y i  in Equation (11) becomes: ##EQU11## Further, when Δt is set at 2 k  (k=integer) and when the mantissa of the increment SDY p+1  is denoted by SDY p+1  (M) and the exponent thereof by SDY p+1  (E), the gradient Y i  becomes as follows: ##EQU12## 
     Thus, it can be said that the gradient Y i  can be evaluated by only the addition and subtraction of the exponent. 
     In the above, the interpolation operation method of this invention for use in the generation of the time delay has been described. There is another problem which must be solved. It concerns the write and read control method for the digital memory used for generating the time delay, and the structure of the memory. Hereunder, a method of the present invention in this connection will be explained. 
     Here, FIG. 4 will be taken as an example. In case where the output signal having the time delay T as shown by e o  in FIG. 4 is to be generated from the input signal e in , the sampling period h is selected at m·Δt as stated before. The increment SDY of the input signal e in  in the sampling interval as represented by Equation (13) is sampled every sampling period h, and it is stored into the digital memory A. This is illustrated in FIGS. 5(A)-5(E). FIG. 5(A) shows a block diagram of a time delay element (e -ST  : S indicates the Laplace operator). The increment of e in  in the sampling interval is evaluated by operation steps given in Equation (16): ##EQU13## 
     Here, ΔY i ,j denotes the j-th input signal at the i-th iteration, l the number of inputs of the time delay element, and SDY i  the accumulative value of the input signal at and after the sampling point (p-1). 
     In this way, the incremental values of the sampling interval are stored into the digital memory A (hereinbelow, termed &#34;memory A&#34;) in succession as shown in FIG. 5(B) (phase I). Subsequently, the incremental value of the input signal sampled the time delay T earlier is read out from the memory A, and it is used as an input signal Y in FIG. 5(C) so as to compute the output signal of the time delay (phase II). Since, however, m is made 2 b  in this invention as described before, the integral number of times of the sampling period h does not always agree with the time delay T, and a residue or remainder remains for a time interval T&#34; (=q·Δt) as shown in FIG. 4. In accordance with this invention, therefore, the method to be described below is performed. Values n and q which satisfy the relation of the following equation are found: 
     
         T=T&#39;+T&#34;=n·h+q·Δt (0≦q·Δt&lt;h) (17) 
    
     When a sampling point of p=n has been reached in FIG. 5(B), the increment value SDY p-n+1  of the input signal having been stored at a sampling point (p-n+1) is read out from the memory A, and it is used to obtain the gradient Y i  with Equation (15). Using this value Y i  as the input signal in FIG. 5(C), the computation of Equation (11) is performed. Thus, the output signal ΔZ&#39; of the time delay at t&#39;=n·h is evaluated every Δt as illustrated as the intermediate output signal e o  &#39; of the time delay element in FIG. 4. As shown in FIG. 5(D), the output signal ΔZ&#39; is stored into a memory B every Δt. In order to generate the time delay T&#34; (=q·Δt) which becomes the odds, a value ΔZ&#39; i-q  stored in the address of the memory B preceding q addresses is read out as the output signal ΔZ i  of the desired time delay. This output signal is made the output signal e o  of the time delay element in FIG. 4. In addition, FIG. 5(E) illustrates the time relationship between the write signal into the memory B and the read signal from the memory B. 
     Now, the method of controlling the write and read operations of the memories A and B will be described with reference to FIGS. 6 to 8. 
     In general, the lengths of time delays are not fixed, but time delays having various lengths exist within a single system. The angular frequencies ω of the input signals of the system are also various. Accordingly, the optimum sampling period satisfying a computation accuracy becomes different. From the standpoint of reducing to the utmost the capacities of the memories A and B used for the computation of the time delay, it is desirable to sample the input signal at the sampling period which is the most suitable for each particular time delay. In this case, however, the write and read control for which the memories A and B are possessed for each time delay element must be conducted, and the control of the interpolation operation must also be conducted for each time delay element. These result in the disadvantage of complicated circuitry. This invention therefore adopts a method according to which the input signals of all the time delay elements are sampled at a fixed sampling period which is common to all the time delay elements. 
     While the DDA according to this invention will be described in detail later, it does not possess a special-purpose arithmetic unit for each arithmetical element. By preparing a single general-purpose arithmetic unit, data required for the corresponding element are loaded from the memories into the arithmetic unit in accordance with control instructions within a control memory, and the operations of several sorts of arithmetical elements including an integrator, a potentiometer, a time delay element etc. are performed in accordance with a time chart of FIG. 6. FIG. 6 illustrates an example in the case where four arithmetical elements of No. 1-No. 4 are comprised. ELC indicates the operation timing signal of the arithmetical elements, and ITE an iteration signal. Here, the arithmetical elements No. 1 and No. 4 are the time delay elements, and the sampling period h is set at 2·Δt by supposing m=2. In the figure, TA indicates the write and read timing signal for the memory A in FIG. 5(B), and this signal is generated by the AND logic between a flag DLYF appointing the time delay elements and a sampling signal SAMP. On the other hand, TB indicates the write and read timing signal for the memory B in FIG. 5(D). 
     The write operation into the memories A and B is performed by a method illustrated in FIG. 7. Numeral 71 in the figure designates an address counter (ADC) which indicates write addresses into the memories A and B, and which can appoint address V (=2 w ) at the maximum. Both the memories A and B have addresses O-V assigned thereto. In the write into the memories A and B, the content of the ADC 71 or the write addresses is/are made addresses 0 by a reset signal (RESET) in the intial state. During an operation, the content of the ADC 71 is incremented +1 in accordance with an increment signal UP, and the increment value SDY i  in the sampling interval and the increment ΔZ&#39; of the interpolated time delay output signal as previously stated are respectively written into the addresses of the memories A and B indicated by the ADC 71. As the increment signal UP, the timing signal TA is applied to the memory A, and the timing signal TB to the memory B. That is, the memory A has its write address renewed by the timing signal TA, and the memory B by the timing signal TB. 
     On the other hand, when the content of the ADC 71 has exceeded the value V, it becomes zero again. Accordingly, each memory A or B can be constructed into a memory 72 having a ring structure as shown in FIG. 7. With such construction, data can be written into the memories A and B unlimitedly in accordance with the increment signal UP. In this case, when address 0 is returned to beyond the maximum address V, data having been stored in address 0 till then is erased, and new data is written therein. The same applies to address 1 and the subsequent addresses. Now, letting p denote the sampling point and M denote the number of the time delay elements, data at the p (p=0, 1, . . . , n-1)-th sampling point concerning the j (j=1, 2, . . . , M)-th time delay element is written into address (M×p+j) of the memory 72. 
     Subsequently, the control of the read operation from the memories A and B is performed by a method illustrated in FIG. 8. In case of the read operation, address memories 80 which manage the addresses of the memories A and B to read out data therefrom are disposed to the number of the elements. The managing memories are respectively denoted by RA 0 , RA 1 , . . . and RA M-1 , and they are generally termed an &#34;ADM 81&#34;. The memories equal in number to the elements are needed because the respective time delays have different intervals to be read out. 
     An initial address INIT is set into the ADM 81 from a host computer (not shown) via a multiplexer (MPX) 82, for each time delay and before its operation. Such initial addresses are different between the memory A and the memory B, and are computed by the following steps in advance: ##EQU14## 
     Here, 
     IA j  ; the initial address of the memory A concerning the j-th time delay element, 
     IB j  ; the initial address of the memory B concerning the j-th time delay element, p1 VA; the maximum value of the addresses of the memory A, 
     VB; the maximum value of the addresses of the memory B, 
     M; the total number of the time delay elements, 
     N j  ; n in FIG. 4 concerning the j-th time delay element, 
     Q j  ; q in FIG. 4 concerning the j-th time delay element. 
     During the operation for generating the time delay, a counter (C1) 83 is incremented +1 by the increment signal UP after the counter 83 has been reset by the signal RESET in accordance with the iteration signal ITE in FIG. 6. Further, the contents of the addresses of the ADM 81 appointed by the counter 83 are read out and are added by an adder circuit 84 to the total number of the time delay elements in accordance with the increment signal UP. The added result is written again into the address of the ADM 81 appointed by the counter 83. Data are read out from the addresses of the memories A and B consequently appointed. The increment signal UP is the signal TA in FIG. 6 for the memory A, and the signal TB for the memory B. 
     The ADM 81 can handle the same range of addresses as that of the ADC 71 in FIG. 7, and can appoint address V at the maximum. Besides, each memory A or B is constructed as the foregoing memory 72. With the above method, therefore, data can be simply read out from the memories A and B unlimitedly by renewing the read addresses of the memories A and B for each time delay element and continuously in time. The maximum value VA of the addresses of the memory A and the maximum value VB of the addresses of the memory B must satisfy the following inequalities: ##EQU15## 
     As the memory A and the memory B, individual memory means may be disposed, or single memory means may well be divided into two memory areas to be used as a memory A area and a memory B area. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIGS. 1(A) to 1(D) are diagrams showing a conventional method of generating a time delay as employs a digital memory, 
     FIG. 2 is a diagram showing an output signal of a time delay based on an interpolation mode, 
     FIG. 3 is a diagram showing the state in which the maximum error develops in the interpolation mode, 
     FIG. 4 and FIGS. 5(A) to 5(E) are diagrams for explaining the principles of this invention, 
     FIG. 6 is a time chart of basic control signals in the execution of an operation by a DDA which adopts this invention, 
     FIG. 7 is a diagram showing a method of controlling write into a digital memory in this invention, 
     FIG. 8 is a diagram showing a method of controlling read from the digital memory in this invention, 
     FIGS. 9(A) to 9(C) are diagrams showing the circuit arrangement of an embodiment of a DDA which executes operations for generating a time delay in accordance with this invention, 
     FIG. 10 is a time chart of a pipeline control in the DDA shown in FIGS. 9(A) to 9(C), 
     FIG. 11 is a diagram showing the bit construction of an operation control instruction in the DDA of FIGS. 9(A) to 9(C), 
     FIG. 12 is a diagram showing the structures of floating-point numerical value systems which are used in the operations of the DDA of FIGS. 9(A) to 9(C), and 
     FIG. 13 is a diagram showing the circuit arrangement of an embodiment in the case where this invention is applied to operations other than those of the DDA. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     FIGS. 9(A)-9(C) show the block arrangement of a DDA which adopts this invention. In order to achieve a high operating speed, the operations of the DDA are carried out under a pipeline control as shown in FIG. 10 (refer to Japanese Patent Application Publication No. 54-15389). 
     An integrating operation in each iteration of the DDA is broadly classified into and is executed by the following three operation phases: 
      ○1  P phase; Pickup phase 
     The fine increments ΔY i ,j of the j-th input variable in the i-th iteration are totaled for j=1 to l, to obtain an increment ΔY i . ##EQU16## 
      ○2  Y phase; Update phase 
     The addition between the increment ΔY i  and the content (denoted by Y i-1 ) of the Y register of an integrator in a period preceding one iteration is performed, and the added result is made the content of the Y register at the i-th iteration. That is, the following operation is executed. 
     
         Y.sub.i :=Y.sub.i-1 +ΔY.sub.i                        (20) 
    
      ○3  I phase; Integration phase 
     The addition between the aforecited Y i  and the content (R i-1 ) of an R register is performed, 1 bit or several bits with an overflow component included is/are delivered as ΔZ i  from the added result (denoted by R i ), and the result with ΔZ i  removed from R i  is set in the R register. That is, the following operation is executed. 
     
         R.sub.i :=R.sub.i-1 +Y.sub.i ·ΔX.sub.i -ΔZ.sub.i(21) 
    
     Here, ΔX i  denotes the fine increment of the integral independent variable in the i-th iteration. 
     FIGS. 9(A) to 9(C) correspond to the respective operation phases. FIG. 9(A) shows the block arrangement of the DDA for executing the P phase, FIG. 9(B) that for executing the Y phase, and FIG. 9(C) that for executing the I phase. 
     Latches 9121, 9122, 9123, 9124, 9126, 9127, 9128 and 9129 in these figures are buffers which are used for performing the operations of the pipeline control. 
     The time delay generating computation of this invention consists of the two parts of the interpolation operation of Equation (11) and the operation of Equation (16) for evaluating the variation of the input signal (variable) between samplings. The former corresponds to the I phase of the inegrating operation, while the latter corresponds to an operation with the P phase and the Y phase combined. However, the contents of the Y register are different between the former and the latter. Further, when the write and read times of the memories A and B, etc. are considered, the Y phase and the I phase become longer than ordinary integrating operations by the DDA. Therefore, the operating periods of time of the respective phases required for the operation for generating the time delay cannot be confined within 1 iteration of the DDA consisting of all the operation phases (P, Y and I phases). In the embodiment, therefore, the operation for generating the time delay is performed in operation cycles corresponding to two arithmetical elements. Referring to FIG. 10, the N-th and (N+1)-th cycles are used for one time delay element. The operations of Equation (11) are carried out in phases Y N  and I N , while the operations of Equation (16) are carried out in phases Y N+1  and I N-1 . 
     In FIGS. 9(A) to 9(C), a computer 901 transmits respective initial values to a ΔZ memory 903 for storing the output increment ΔZ i  in the operation of Equation (21), a Y memory 904 for storing the operated result of Equation (20), an ADMA memory 905 for storing the initial address IA j  in Equation (17), an R memory 906 for storing the operated result of Equation (21), and an ADMB memory 907 for storing the initial address IB j   in Equation (17), via a common bus 902 and multiplexers 9021, 9022, 9023, 9024 and 9025. In addition, the computer 901 transmits a DDA operation control instruction to a control memory 908. 
     The computer 901 transmits a predetermined value to a program counter 909 for appointing the address of the control memory 908, and also transmits the exponent in Equation (15) to a BEKI register 910. Further, it functions to start and stop a controller 911 which generates a series of timing signals necessary for executing the operation of the DDA. 
     The operation of the DDA is performed in such a way that the operation control instruction read out from the address PC of the control memory 908 appointed by the program counter 909 is executed in accordance with the series of timing signals generated by the controller 911. 
     Here, the operation control instruction kept stored in the address appointed by the program counter 909 has a bit structure which is adapted to appoint the kind of an arithmetic unit to be used for an operation in a predetermined iteration (for example, i-th iteration), an operation mode associated therewith, etc. 
     FIG. 11 shows an example of the bit structure of the operation control instruction. 
     Among various parts in FIG. 11, EL indicates the kind of the arithmetic unit which is to be used for executing a desired operation, and ΔX A  indicates that address of the ΔZ memory 903 in which the fine increment ΔX i  of the integral independent variable in Equation (21) is stored. DT indicates a flag of 1 bit for indicating whether or not the aforecited ΔX i  is the time increment Δt, and P x  indicates the polarity of the aforecited ΔX i . ΔY A1 , ΔY A2  and ΔY A3  (in this example, the number of inputs is assumed to be three) represent those addresses of the ΔZ memory 903 in which ΔY i ,1, ΔY i ,2 and ΔY i ,3 in Equations (16) and (19) are stored, respectively. P 1 , P 2  and P 3  represent 1-bit flags for controlling the polarities of the aforecited ΔY i ,1, ΔY i ,2 and ΔY i ,3, respectively. 
     The functions of the circuit arrangement in FIGS. 9(A)-9(C) will now be described more in detail by taking as an example a case where the operation of this invention for generating the time delay is executed on the basis of the operation control instruction having the bit structure of FIG. 11. 
     In the example of FIG. 11, the N-th arithmetic unit is assumed an arithmetic unit DLY I  which executes the interpolation operation of Equation (11) in the operation phases Y N  and I N  shown in FIG. 10, while the (N+1)-th arithmetic unit is assumed an arithmetic unit DLY II  which executes the operation of obtaining the variation of the input signal in the sampling interval as given in Equation (16), in the operation phases P N+1  and Y N+1  in FIG. 10. 
     When the address appointed by the program counter 909 is N, the operation control instruction of the arithmetic unit DLY I  is read out from the control memory 908. The operation control instruction read out is decoded by a decoder 913, and the decoded signals of the respective parts of the instruction are fed to the corresponding circuit portions. 
     The decoded result of the part EL indicative of the kind of the arithmetic unit to be used is fed to the controller 911. Upon receiving this result, the controller 911 creates the timing signals for executing the appointed operation for generating the time delay as based on the pipeline control and transmits it to the predetermined circuit portions. In case of the arithmetic unit DLY I , in order to perform the interpolation operation of Equation (11), the controller 911 transmits an enable signal (ENABLE) to the terminals E of the ΔZ memory 903, the Y memory 904 and the ADMA memory 905 and also transmits the series of timing signals necessary for the operation of Equation (11). In FIGS. 9(A)-9(C), lines which connect the controller 911 and such terminals are omitted for the sake of brevity. 
     In the P N  phase of the arithmetic unit DLY I , the decoded results (ΔY A1 ), (ΔY A2 ) and (ΔY A3 ) of the respective addresses ΔY A1 , ΔY A2  and ΔY A3  are applied to the address terminal A of the ΔZ memory 903 as in case of other arithmetic unit, and the values ΔY i ,j (j=1, 2 and 3) are successively read out from the appointed addresses and set into a ΔY register 914. 
     The value ΔY i ,1 set in the ΔY register 914, and the content (SDY i ,0 =0) of a SDY register 915 reset by the operation timing signal ELC in FIG. 6 prior to the i-th iteration are applied to a floating-point added FADD 916, which performs the following floating-point operation: 
     
         SDY.sub.i,0 +ΔY.sub.i,1 =ΔY.sub.i,1            (22) 
    
     The result ΔY i ,1 is set into the SDY register 915 as SDY i ,1. 
     Here, the numerical value systems of data handled in the DDA of this invention will be briefly described. FIG. 12 shows the numerical value systems of the data handled in this invention. Variables to be used for operations are floating-point numbers which assume the numerical value system of either a data format A or B. Variables belonging to the data format A are the integrand Y, the integral residue R, the summation SDY of the input variable, etc. which do not employ any incremental format. On the other hand, variables belonging to the data format B are the fine increment ΔY of the input variable, the fine increment ΔX of the integral independent variable, the fine increment ΔZ of the output variable, etc. which have the incremental format. 
     After the operation of Equation (22), the value ΔY i ,2 set in the ΔY register 914 and the content (SDY i ,1 =ΔY i ,1) of the SDY register 915 are applied to the FADD 916, which performs the following operation: 
     
         SDY.sub.i,1 ΔY.sub.i,2 =ΔY.sub.i,1 +ΔY.sub.i,2(23) 
    
     The result is set into the SDY register 915 as SDY i ,2. 
     By repeating similar operations, ΔY i  of Equation (19) are obtained in the SDY register 915 as SDY i ,l =SDY i ,3. 
     In the above operations, the SDY register 915 and the FADD 916 correspond to accumulators. 
     In case where the decoded result (P 1 ), (P 2 ) or (P 3 ) of the polarity bit is the negative polarity, the addition between SDY i ,j and the 2&#39;s complement concerning the negative-polarity fine increment is executed in the FADD 916. For example, in case where ΔY i ,2 has become negative in polarity, the following operation is performed: 
     
         SDY.sub.i,1 -ΔY.sub.i,2 =ΔY.sub.i,1 -ΔY.sub.i,2(24) 
    
     Next, in case where the flag (DT) has indicated that ΔX i  read out from the address of the ΔZ memory 903 appointed by the decoded result (ΔX A ) of the address ΔX A  corresponds to the time increment Δt, the aforecited ΔX i  is set into a ΔX register 917. 
     In case of the arithmetic unit DLY I , the addresses ΔY A1  -ΔY A3  appoint such specific addresses that numerical values within the ΔZ memory 903 are zero, and the content of the SDY register 915 obtained in the P N  phase becomes zero. Although the operations of the P N  phase are performed in the same procedure as in the other arithmetic units, the values of SDY obtained are not used. 
     In the next Y N  phase, the values SDY, P c , ΔX and P x  obtained in the P N  phase are first latched into latches for buffers 9121-9124, respectively. Thereafter, the gradient Y i  which is stored in the Y memory 904 and which is used in the operation of Equation (11) is read out and is sent to the circuit of the next I N  phase. 
     The read of the gradient is carried out in such a way that the content of the buffer latch 9121 having latched the output of the program counter 909 is applied to the Y memory 904, whereupon the Y memory is enabled by the enable signal from the controller 911. The gradient Y i  is read out from the same address of the Y memory 904 as No. of the arithmetic unit, and is set into a Y register 925. Via an FADD 926 provided with an accumulator, it is fed to the circuit of FIG. 9(C) for executing the I N  phase. 
     In the I N  phase of the arithmetic unit DLY I , the values P x , Y, ΔX and P c  sent from the circuit of FIG. 9(B) are respectively latched into latches for buffers 9126-9129. Thereafter, the operation of Step 1 of Equation (11) or the interpolation operation for each time increment Δt and the conversion of the output signal into the incremental format are carried out. In addition, the time delay which corresponds to q·Δt shown in FIG. 4 is generated by writing the data into and reading the data from the memory B. Finally, the output signal having the time delay T=m·n·Δt+q·Δt is provided. 
     The interpolation operation of Equation (11) is started by multiplying the content Y i  of the buffer latch 9127 and the content ΔX i  of the buffer latch 9128 by means of a multiplier 927. The output Y i  ·ΔX i  of the multiplier 927 is added by an FADD 928 to the integral residue R i-1  preceding one iteration as read out from the R memory 906. The result passes from the FADD 928 and is applied to a decoder 929. This decoder 929 forms the increment ΔZ i  &#39; in which the mantissa of (m1+1)-m2 is composed of 1 bit to several bits as in the data format B of FIG. 12. The increment ΔZ i  &#39; is sent to an IDM memory 930. The residue R i  with the increment ΔZ i  &#39; removed from the integral value as obtained by the FADD 928 is stored via the multiplexer 9024 into the same address of the R memory 906 as No. (N) of the arithmetic unit DLY I  appointed by the buffer latch 9129. 
     On the other hand, the increment ΔZ i  &#39; sent to the IDM memory 930 is stored into the IDM memory 930 corresponding to the memory 72 in FIG. 7, by the method explained with reference to FIG. 7. That is, the IDM memory 930 and an ADCB counter 931 correspond to the memory B and the ADC counter 71, respectively. The content of the ADCB counter 931 is applied to the terminal A of the IDM memory 930 via a multiplexer 932, and it is stored into the address of an SDM memory 918 appointed by the output of the multiplexer 932, by the enable signal (E) from the controller 911. 
     The read of the output increment ΔZ i  having the time delay T, from the IDM memory 930 is executed as soon as the above write has ended. 
     The ADMB memory 907 and a counter (CI) 933 in the circuit of FIG. 9(C) correspond to the address memory ADM 81 and the counter (C1) 83 in FIG. 8, respectively. The counter (CI) 933 is reset every pulse of the iteration signal ITE (in FIG. 6) which is generated by the controller 911. The read of the increment ΔZ i  is started in such a way that the content of the counter (CI) 933 is applied to the A (address) terminal of the ADMB memory 907, and that the address J of the IDM memory 930 in which the desired ΔZ i  is stored is read out from the ADMB memory 907 by the enable signal (E) from the controller 911. Since a timing signal TB&#39; remains &#34;off&#34;, the address J read out has not the aforecited M added thereto in an adder circuit 934. It is applied to the A (address) terminal of the IDM memory 930 via the multiplexer 932. Thereafter, the desired output increment ΔZ i  is read out from the IDM memory 930 by the enable signal (E) from the controller 911. 
     The output increment ΔZ i  having the time delay T as thus read out is writted into the address (N) of the ΔZ memory 903 corresponding to the arithmetic unit DLY I , via a multiplexer 9240 and the multiplexer 9021 in FIG. 9(A). 
     After the desired time delay output has been obtained, the content of the address (the content of the counter 933) of the ADMB memory 907 corresponding to the arithmetic unit DLY I  has the aforecited M added thereto by the adder 934 and is again stored into the address of the ADMB memory 907 corresponding to the arithmetic unit DLY I  via the multiplexer 9025, in order to use the resulting content for the operation of the arithmetic unit DLY I  in the next iteration. The contents of the counter (CI) 933 and the ADCB 931 have one added thereto at the fall of the timing signal TB in FIG. 6. The &#34;add one&#34; operation by the adder circuit 934 is effected with the timing signal TB&#39; which is generated by the controller 911 and which falls somewhat earlier than the timing signal TB. 
     Thus, the series of operations of the arithmetic unit DLY I  are completed. There will now be described in detail the functions of the arithmetic unit DLY II  which executes operations for finding the variation (SDY P ) of the input signal of the time delay element in the sampling interval as given by Equation (16), in the time delay element of this invention. 
     First, in the operation phase P N+1  shown in FIG. 10, the summation of the input signal of the arithmetic unit DLY II  during the i-th iteration is obtained. This operation is executed by the circuit of FIG. 9(A), and is carried out in the same procedure as in the P N  phase of the arithmetic unit DLY I . 
     In the next Y N+1  phase, as in the Y N  phase of the arithmetic unit DLY I , the values SDY, PC, ΔX and P x  obtained in the P N+1  phase are respectively latched in the buffer latches 9121-9124. Thereafter, predetermined operations to be stated below are performed. 
     In the Y N+1  phase, the variations of the input signal in the sampling interval are accumulated at the respective iterations. To this end, Y i-1  preceding one iteration is read out from the Y memory 904 corresponding to the arithmetic unit DLY II  and is set into the Y register 925 via a multiplexer 924. The content Y i-1  of the Y register 925 and the content SDY i  of the buffer latch 9122 are applied to the FADD 926, in which the following operation is performed: 
     
         Y.sub.i :=Y.sub.i-1 +SDY.sub.i                             (25) 
    
     Y i  consequently obtained is transmitted to the multiplexer 924 and the SDM memory 918. 
     Next, whether or not the present i-th iteration is an iteration corresponding to a sampling point (refer to FIG. 4) is examined. When it is not the iteration corresponding to the sampling point, the value Y i  obtained by the FADD 926 is stored via the multiplexers 924 and 9022 into the address of the Y memory corresponding to the arithmetic unit DLY II . On the other hand, when the present iteration corresponds to the sampling point, the above Y i  is written into the SDM memory 918 by the method illustrated in FIG. 7. Simultaneously therewith, the variation SDY of the input signal of the arithmetic unit DLY II  in the sampling interval as has been stored in the (p-n+1)-th iteration is read out from the SDM memory 918 by the method illustrated in FIG. 8, it is used to operate the exponent of Equation (15) so as to obtain the gradient Y i  for use in Equation (11), and the gradient Y i  is stored into the address of the Y memory 904 corresponding to the arithmetic unit DLY I . 
     Which of the processings is performed is decided by the controller 911. The timing signal TA in FIG. 6 consequently issued by the controller 911 is transmitted to a counter (CS) 919 and an address counter (ADCA) 920 in FIG. 9(B), whereby the above processings are changed-over. 
     Hereunder, the latter processing will be described in detail. 
     The SDM memory 918 in FIG. 9(B) corresponds to the memory A, and the counter (CS) 919, the address memory (ADMA) 905 and the address counter (ADCA) 920 correspond respectively to the counter (C1) 83 and address memory (ADM) 81 in FIG. 8 and the counter (ADC) 71 in FIG. 7. 
     The write of the gradient Y i  into the SDM memory 918 is executed in such a way that the content of the counter (ADCA) 920 is sent to the A terminal of the SDM memory 918 via a multiplexer 921, and that it is written into the address appointed by the content of the counter (ADCA) 920, by the enable signal which the controller 911 issues in time with the timing signal TA. 
     Next, in case of reading out the above data from the SDM memory 918, that address of the ADMA memory 905 whose content is the content of the counter (CS) 919 or is the address of the SDM memory 918 storing the data to-be-read-out is applied to the A terminal of the ADMA memory 905. The enable signal issued by the controller 911 is applied to the E terminal of the ADMA memory 905. Then, the aforecited address is read out from the ADMA memory 905 and is applied to the A terminal of the SDM memory 918 via the multiplexer 921 without being subjected to the &#34;add one&#34; operation. 
     The desired data or the variation of the input signal of the arithmetic unit DLY II  in the sampling interval is read out by the enable signal from the controller 911. The data read out is separated by a decoder 922 into the exponent (1 to m1) and the mantissa (m1+1 to m2) as in the data format A shown in FIG. 12. The separated exponent is added by an adder 923 with the content of the register (BEKI) 910 which has been set from the computer 901 via the common bus 902 before the operation. The sum is transmitted to the Y memory 904 via the multiplexers 924 and 9022 together with the mantissa. 
     The gradient Y i  thus evaluated is written into the address of the Y memory 904 corresponding to the arithmetic unit DLY I . In addition, the numerical value zero is written via the multiplexer 9022 into the address of the Y memory 904 corresponding to the arithmetic unit DLY II . 
     The gradient Y i  for use in Equation (11) has been obtained by the above procedure, whereupon for the next sampling, the content of the address (the content of the counter (CS) 919) of the ADMA memory 905 corresponding to the arithmetic unit DLY II  has the foregoing value M added by an adder circuit 935, and the sum is again stored into the address of the ADMA memory 905 corresponding to the arithmetic unit DLY II  via the multiplexer 9023. The contents of the counter (CS) 919 and the ADCA 920 have one added at the fall of the timing signal TA in FIG. 6. The &#34;add&#34; operation by the adder circuit 935 is effected by the timing signal TA&#39; which is issued by the controller 911 and which fall somewhat earlier than the timing signal TA. 
     The counter (CS) 919 is reset every iteration signal (ITE in FIG. 6) issued by the controller 911. 
     In the above, the embodiment in which the time delay generating method of this invention is applied to the DDA has been described in detail. Now, an embodiment in the case of applying the invention to an apparatus other than the DDA will be briefly described. 
     FIG. 13 shows a circuit block diagram of the embodiment for the apparatus other than the DDA. A portion 1300 enclosed with a broken line is the circuit in the DDA corresponding to FIG. 9(A). It functions to evaluate the variation of the input signal in the sampling interval and to compute the gradient for use in the interpolation operation with the variation. Likewise, a portion 1301 enclosed with a broken line is the circuit in the DDA corresponding to FIG. 9(C). It functions to perform the interpolation operation by the use of the gradient evaluated by the portion 1300 every iteration, and to provide an output signal having a desired time delay. 
     The time delay generating circuit shown in FIG. 13 generate only the time delay alone. As its input signal, Y is applied from a terminal 131, and as its result, an output signal Z is delivered from a terminal 132. Unlike those of the DDA, the input and output signals are not handled in the incremental format, but they are represented by the data format A in FIG. 12. 
     Further, a memory A control circuit 1302 is the same circuit as the portion in FIG. 9(B) consisting of the multiplexer 9023, ADMA memory 905, counter (CS) 919, counter (ADCA) 920, adder circuit 935 and multiplexer 921. This circuit 1302 controls the addresses of an SDM memory 1303 to write the input signal e in  (Y) into or read it from the memory 1303. A memory B control circuit 1304 is the same circuit as the portion in FIG. 9(C) consisting of the multiplexer 9025, ADMB memory 907, counter (CI) 933, counter (ADCB) 931, adder circuit 934 and multiplexer 932. This circuit 1304 controls the addresses of an IDM memory 1305 to be written in or read out. 
     A common bus 1308 is a transmission line which transmits information from a computer (not shown). It sends the initial address IA j  in Equation (17) to be stored in the ADMA memory within the memory A control circuit, the initial address IB j  in Equation (17) to be stored in the ADMB memory within the memory B control circuit, and an index (=-b) for use in a gradient computation of Equation (26), referred to below, to the respective circuits. It also sends a control signal for starting or stopping a controller 1306 which generates a series of timing signals necessary for the operation of the time delay. 
     As compared with the embodiment for the DDA, the arithmetic circuit shown in FIG. 13 differs only in the portion for evaluating the gradient for use in the interpolation operation from the input signal and in the method of operating the interpolation. Since the write and read controls for the SDM memory 1303 and the IDM memory 1305 are made on the basis of the timing signals in FIG. 6 and by procedures similar to those in the case of the DDA, they will be briefly explained. 
     The operation for generating the time delay in the embodiment of FIG. 13 is started in such a way that a start signal transmitted from the computer via the common bus 1308 is sent to the controller 1306. When starting the operation, the controller 1306 supplies a reset signal RST to predetermined circuit parts in FIG. 13, to reset the contents thereof. Thereafter, it generates the series of timing signals required for the operation for generating the time delay. 
     The embodiment shown in FIG. 13 first performs operation contents corresponding to the arithmetic unit DLY I  of the DDA by means of the circuit of the portion 1301, and thereafter executed operation contents corresponding to the arithmetic unit DLY II  of the DDA by means of the circuit of the portion 1300. Then, the operations of one iteration are completed. 
     In the interpolation operation, a gradient G p  computed by the portion 1300 in the immediately preceding sampling interval and given below is read out from the same address of a Y memory 1307 as No. of a time delay element noted, and it is set into a Y register 13080. ##EQU17## 
     Here, the suffixes p and q in Equation (26) are identical to those in FIG. 4, and the variable b is identical to that in Equation (12). 
     Subsequently, the result Z&#39; i-1  of the interpolation operation preceding one iteration is read out from the same address of a Z memory 1309 as No. of the time delay element noted, and it is added with the gradient G p  being the content of the Y register 13080 by means of an adder FADD 1310 which is of the floating-point operation type. The result of the addition is denoted by Z&#39; i . 
     
         Z&#39;.sub.i :=Z&#39;.sub.i-1 +G.sub.p                             (27) 
    
     The result Z&#39; i  is stored into the same address of the Z memory 1309 as that of the aforecited value Z&#39; i-1 . Simultaneously therewith, a procedure similar to the procedure described as to the arithmetic circuit of the DDA is carried out by the memory B control circuit 1304, thereby to store the result Z&#39; i  into the IDM memory 1305. Using as an address the content of a counter within the memory B control circuit 1304 as corresponds to the counter ADCB in FIG. 9(C), the output signal Z i  having the desired time delay is read out from the IDM memory 1305. It is set into a buffer register (BR3) 1311 and then delivered to another arithmetic element from the output terminal 132. 
     The interpolation operation thus far described is carried out by employing as its basic signal the timing signal TB shown in FIG. 6. 
     Next, there will be explained the procedure for obtaining the gradient G p  by performing the operation indicated in Equation (26) by means of the portion 1300. 
     The gradient G p  is evaluated every sampling period h expressed by Equation (12). The input signal Y is written into the SDM memory 1303 on the basis of the timing signal TA in FIG. 6. The write control is executed by the memory A control circuit 1302, and the procedure thereof will not be explained here because it is the same as in the case of the DDA. After the sampling value Y p  of the input signal has been stored into the SDM memory 1303, the (p-q+1)-th sampling value Y p-q+1  of the time delay element noted is read out from the address of the SDM memory 1303 indicated by the memory A control circuit 1302 and is set into a buffer register (BR1) 1312. The content of the buffer register (BR1) 1312 and that of a buffer register (BR2) 1313 are added by an adder (FADD) 1314 of the floating-point operation type, and the variation of the input signal is evaluated by the following operation: 
     
         ΔY.sub.p-q+1 :=Y.sub.p-q+1 -Y.sub.p-q                (28) 
    
     The variation ΔY p-q+1  is separated into a mantissa and an exponent by a decoder 1315. The exponent is added by an adder 1316 with a content (=-b) having been set in a register (BEKI) 1317 by the computer in advance, and the sum is again written into the address of the Y memory corresponding to the time delay element noted, along with the aforecited mantissa. Simultaneously therewith, the content of the register BR1 is set into the register BR2 so as to be used for the computation of the gradient in the next sampling interval. 
     The embodiment in the case where the time delay generating method of this invention is applied to any arithmetic circuit other than the DDA, is constructed and operated as thus far described. 
     The embodiments shown in FIGS. 9(A)-9(C) and FIG. 13 are the arithmetic circuits which handle data expressed by floating-point numbers. Fixed-point operations can be performed by the same block arrangement except the decoder and the adder which are used for the computation of the gradient. In this case, the decoder and the adder are replaced with a shifter, the mantissa of the gradient is shifted by the shifter to the amount of the exponent of the gradient, and only the resulting mantissa is used as the value of the gradient. 
     As set forth above, according to this invention, the capacity of a memory for use in the generation of a time delay is allowed to be about 1/90 as compared with that in a method which merely samples an input signal and read out and reconstructs it after lapse of the time delay. In addition, owing to the development of a simple memory control system, a long time delay can be generated at high speed and at high accuracy merely by increasing the memory capacity. In such manner, the invention is greatly effective.