The present invention relates to a parallel precoder circuit that processes parallel input information series, and outputs parallel output information series, and more particularly, to a parallel precoder circuit that is applied to a differential quadrature phase shift keying (DQPSK) system.

In the optical communication system, an optical duobinary modulation system and a differential phase shift keying (DPSK) system are investigated as a technique of increasing a relay distance and increasing a transmission speed. On the other hand, in recent years, a multiple-value technique using a differential quadrature phase shift keying (DQPSK) system calls attention.

Circuits of a transmission and reception system of the optical communication system using the DQPSK system have a precoder, an encoder, and a decoder. The precoder performs differential encoding operations of an input set of 2-bit information series including an in-phase component signal and a quadrature-phase component signal, and one-bit delay feedback information series obtained by delaying the own output information series by one bit, and outputs the operated result. The encoder executes a base band modulation to information series as the output of the precoder, thereby obtaining and outputting a DQPSK signal. The decoder modulates the DQPSK signal output from the encoder, and demodulates the signal, thereby restoring the input set of the information series input to the precoder.

In the optical communication system, a distributed feedback (DFB) laser and a Mac-Zehnder modulator achieve the function of the encoder, and a photodetector achieves the function of the decoder in many cases. In other words, individual optical elements achieve the functions of the encoder and the decoder in many cases. On the other hand, a logical circuit is usually used for the precoder.

A transmission speed F [Hz] in the optical communication is an ultra-high speed of 10 Gb/s and 40 Gb/s. Therefore, when a serial precoder circuit that processes a signal as serial data according to the transmission speed F is used, the logical circuit is required to operate at an ultra-high speed.

When the transmission speed F becomes an ultra-high speed, a clock unit time per one bit becomes short. Therefore, timing adjustment of a circuit that achieves a one-bit delay becomes difficult.

To solve this problem, various techniques are conventionally considered. For example, a technique concerning a serial precoder circuit that operates at a high speed without structuring a feedback route within a DQPSK precoder by using a toggle flip-flop (T-FF) circuit is disclosed in M. Serbay, C. Wree and W. Rosenkranz, "Implementation of differential precoder for high-speed optical DQPSK transmission," Electric Lett., vol. 40, no. 20, Sep. 2004.

The serial precoder circuit described in the above literature achieves the serial precoder circuit without structuring a feedback route within a DQPSK precoder. Therefore, a circuit that delays one bit is not necessary. Consequently, the difficulty of adjusting the timing of a circuit that achieves a one-bit delay cannot be avoided.

However, the serial precoder circuit described in the above literature does not solve strict requirement for the operation speed of logical circuits such as the T-FF circuit, a delay flip-flop circuit (D-FF) circuit, an inverting circuit, and AND circuit, an OR circuit, and an exclusive OR (EXOR) circuit. Therefore, it is difficult to achieve the serial precoder circuit according to the conventional technique described in the Nonpatent literature 1 by using a general-purpose application specific integrated circuit (ASIC), such as a framer, and a field-programmable gate array (FPGA).

In general, the requirement for the operation speed of the logical circuit can be lowered by achieving the function of the precoder, by serial-parallel converting the input information series, and processing the data with a parallel precoder circuit having plural serial precoders developed in parallel.

A parallel precoder is e.g. disclosed in US 2005/0068209 A.

For example, in the case of the parallel precoder circuit that has a serial precoder circuit simply developed to x (2≤x, where x is a positive integer) parallel precoders, parallel input signal of information series (a set of an in-phase component signal and a quadrature-phase component signal) transmitted in advance in the serial transfer, that is, a parallel input signal that is old in time series, is input to the serial precoder circuit. Output information series obtained from the serial precoder circuit are connected in cascade as a one-bit delayed feedback signal of the serial precoder circuit that processes the output information series adjacent in the parallel signals. In this configuration, a feedback route that passes through x serial precoder circuits is generated. Therefore, even when the serial precoder circuit is developed in parallel, the operation speed required for the serial precoder circuit cannot be mitigated.

Assume that a circuit is configured to directly generate a parallel output signal from a parallel input signal, without using the output information series of the serial precoder circuit to which a parallel input signal old in time series is input as a one-bit delayed feedback signal of the last information series. In this case, a delay in the feedback route can be decreased. However, a scale of the circuits increases enormously.

It is an object of the present invention to at least solve the problems in the conventional technology.

This is done by the subject-matter of the appended independent claims. Specific embodiments are defined in the dependent claims.

The above and other objects, features, advantages and technical and industrial significance of this invention will be better understood by reading the following detailed description of presently preferred embodiments of the invention, when considered in connection with the accompanying drawings.

• Fig. 1 is a configuration example of a parallel precoder circuit according to a first embodiment of the present invention;
• Fig. 2 is a configuration diagram of a general optical DQPSK transmission and reception system;
• Fig. 3 is a block diagram of a parallel precoder circuit that configures a precoder shown in Fig. 2;
• Fig. 4 is a configuration diagram of a parallel precoder circuit having a serial precoder circuit simply developed in parallel;
• Fig. 5 is an explanatory diagram of a conversion of a circuit configuration when an associative law is established in a differential encoding operation;
• Fig. 6 is an explanatory diagram of a conversion of the circuit configuration when the associative law is established in the differential encoding operation;
• Fig. 7 is a signal space diagram of the DQPSK system;
• Fig. 8 is an example of input and output relationships of the differential encoding operation based on a phase difference;
• Fig. 9 depicts phases shown in Fig. 8 expressed in binary data;
• Fig. 10 is a logical circuit diagram that depicts a configuration example of a differential encoding operation circuit shown in Fig. 1;
• Fig. 11 is a logical circuit diagram that depicts a configuration example of a differential encoding operation circuit shown in Fig. 1;
• Fig. 12 is a logical circuit diagram that depicts a configuration example of a differential encoding operation circuit shown in Fig. 1;
• Fig. 13 is a logical circuit diagram that depicts a configuration example of a differential encoding operation circuit shown in Fig. 1;
• Fig. 14 is a configuration example of a parallel precoder circuit according to a first embodiment of the present invention;
• Fig. 15 is a configuration example of a parallel precoder circuit according to a second embodiment of the present invention;
• Fig. 16 is a configuration example of a parallel precoder circuit according to a third embodiment of the present invention;
• Fig. 17 is a configuration example of a parallel precoder circuit according to a fourth embodiment of the present invention;
• Fig. 18 is a configuration example of a parallel precoder circuit according to a fifth embodiment of the present invention;
• Fig. 19 is a configuration diagram of a multiple-input differential encoding operation circuit and a delay circuit shown in Fig. 18;
• Fig. 20 is a configuration example of a parallel precoder circuit according to a sixth embodiment of the present invention;
• Fig. 21 is a block diagram of a configuration of a circuit of a DQPSK transmission system according to one system that is used in the optical communication system;
• Fig. 22 is a signal space diagram based on (0,π) and (0,π/2) modulations;
• Fig. 23 is an example of input and output relationships of a differential encoding operation based on a phase difference;
• Fig. 24 is a table of phases shown in Fig. 23 expressed in binary data; and
• Fig. 25 is an explanatory diagram of a parallel precoder circuit according to an eighth embodiment of the present invention.

Exemplary embodiments of the present invention are explained in detail below with reference to the accompanying drawings. Note that the present invention is not limited to the embodiments.

A parallel precoder circuit according to a first embodiment of the present invention is explained below with reference to Fig. 1 to Fig. 14. The parallel precoder circuit according to the first embodiment of the present invention is applied to the DQPSK system. The parallel precoder circuit calculates n-set (2≤n, where n is an integer) parallel output information series DO₁ (d_I,d_Q) to DO_n (d_I,d_Q) using 2-bit information series including a signal d_I of an in-phase component (an I system) and a signal d_Q of a quadrature-phase component (a Q system) as one set, from n-set parallel input information series B_I (b_I,b_Q) to B_n (b_I,b_Q) using 2-bit information series including a signal b_I of the I system and a signal b_Q of the Q system as one set. The parallel input information series B_I (b_I,b_Q) to B_n (b_I,b_Q) are hereinafter also simply referred to as parallel input information series BI to Bn, and the parallel output information series DO₁ (d_I,d_Q) to DO_n (d_I,d_Q) are hereinafter also simply referred to as parallel output information series DO₁ to DO_n. The 2-bit information series including signals of the I system and the Q system are also referred to as a set of signals.

The parallel precoder circuit according to the first embodiment includes differential encoding operation (hereinafter referred to as SP) circuits 1a ((1a-(i,j)) that perform a differential encoding operation derived from a differential encoding definition described later based on two sets of input signals, and delay circuits 1c (1c-1 to 1c-n) that are configured by flip-flops and output one set of input signals by delaying the signals by a unit time. The unit time is an inverse number of the operation speed of the parallel precoder circuit. When a transmission speed is F [Hz], the unit time becomes "n/F" [sec]. The SP circuits 1a and the delay circuits 1c are connected in a net configuration described later.

Each of the SP circuits 1a-(i,j) is disposed in an ith row and a jth column. The SP circuit in ith row is an SP circuit disposed in a signal route that connects between parallel input information series B_i and a delay circuit 1c-i. An SP circuit that is input with an output of the delay circuit 1c-n and parallel input information series B₁ is defined as an SP circuit in the zeroth column. A jth SP circuit from the parallel input information series B₁ in an even row (2≤i, where i is an even number) is defined as the SP circuit in the jth column. An SP circuit that is disposed at a pre-stage of the delay circuit 1c-i in an odd row other than one row is an SP circuit disposed in the last column. In the first embodiment, a last column h is a smallest integer equal to or above (log₂n)+1. For example, when the number of bits n is "7", the last column becomes "4".

Fig. 1 is a configuration diagram of the parallel precoder circuit according to the first embodiment when "n=8". In Fig. 1, the parallel precoder circuit includes encoding (SP) circuits 1a (1a-(1,0), 1a-(2,1), 1a-(4,1), 1a-(6,1),1a-(8,1),1a-(4,2), 1a-(6,2), 1a-(6,3), 1a-(8,2) 1a-(8,3), 1a-(3,4), 1a-(5,4), and 1a-(7,4)), and delay circuits (Ds in Fig. 1) 1c (1c-1 to 1c-8).

In Fig. 1, the SP circuit 1a-(1,0) is disposed in the first row and the zeroth column, the SP circuit 1a-(2,1) is disposed in the second row and the first column, the SP circuit 1a-(4,1) is disposed in the fourth row and the first column, the SP circuit 1a-(6,1) is disposed in the sixth row and the first column, the SP circuit 1a-(8,1) is disposed in the eighth row and the first column, the SP circuit 1a-(4,2) is disposed in the fourth row and the second column, the SP circuit la-(6,2) is disposed in the sixth row and the second column, the SP circuit 1a-(6,3) is disposed in the sixth row and the third column, the SP circuit 1a-(8,2) is disposed in the eighth row and the second column, the SP circuit 1a-(8,3) is disposed in the eighth row and the third column, the SP circuit 1a-(3,4) is disposed in the third row and the fourth column, the SP circuit 1a-(5,4) is disposed in the fifth row and the fourth column, and the SP circuit 1a-(7,4) is disposed in the seventh row and the fourth column. The fourth column is the last column.

Parallel input information series B_2k are connected to one input set and parallel input information series B_2k-1 are connected to the other input respectively of an SP circuit in an even row and the first column, that is the SP circuit 1a-(2k,1) in the (2k)th ((2k)th row, 1≤k≤n, where k is an integer) row and the first column. However, instead of the parallel input information series B₁, the output set of the SP circuit 1a-(1,0) in the first row and the zeroth column is connected to the other input set of the SP circuit 1a-(2,1) in the second row and the first column.

The output set of the SP circuit 1a-(2k-2^(m-1),m-1) is connected to one input set, and the output set of the SP circuit 1a-(2k,m-1) is connected to the other input respectively of the SP circuit 1a-(2k,m) in the (2k)th row and the mth (1≤m≤h-1) column. However, when no SP circuit is disposed in the (2k-2^(m-1))th row and the (m-1)th column, the output set of the SP circuit in the jth column having a largest column number j among SP circuits disposed in the (2k-2^(m-1))th row is connected to one input set of the SP circuit 1a-(2k,m). The operator "^" denotes power.

In Fig. 1, the output set of the SP circuit 1a-(2,1) is connected to one input set, and the output set of the SP circuit 1a-(6,2) is connected to the other input set respectively of the SP circuit 1a-(6,3) of k=3 and m=3.

When 2k-2^(m-1) is smaller than 1, no SP circuit is disposed. In Fig. 1, no SP circuit is disposed at the position of k=2 and m=3, that is, in the fourth row and the third column.

The output set of the parallel input information series B_2k-1 is connected to one input set of the SP circuit 1a-(2k-1,h) in the (2k-1)th row and the hth column disposed in the last column. The output set of the SP circuit in the jth column having a largest column number j among SP circuits disposed in the (2k-2)th row is connected to the other input set of the SP circuit 1a-(2k-1,h). However, no SP circuit is disposed in the first row in the last column. In Fig. 1, the last column is "4". Therefore, the parallel input information series B₅ are connected to one input set, and the output set of the SP circuit 1a-(4,2) is connected to the other input set respectively of the SP circuit 1a-(5,4) .

The output set of the SP circuit in the jth column having a largest column number j among SP circuits disposed in the (2k)th row is connected to the input set of the delay circuit 1c-2k in the (2k)th row (the even row). In Fig. 1, the output set of the SP circuit 1a-(4,2) is connected to the input set of the delay circuit 1c-4.

The output set of the SP circuit 1a-(2k-1,h) disposed in the last column, that is, in the (2k-1)th row in the hth column, is connected to the input set of the delay circuit 1c-(2k,1) in the (2k-1)th row (the odd row). However, the output set of the SP circuit 1a-(1,0) is connected to the input set of the delay circuit 1c-1 in the first row. The output set of the delay circuit 1c-i becomes a parallel output DO_i.

Fig. 2 is a block diagram of a configuration of circuits of a general optical DQPSK transmission and reception system. In Fig. 2, an optical DQPSK transmission and reception system 20 includes a precoder 21, an encoder 22, and a decoder 23. The precoder 21 and the encoder 22 are transmission systems, and the decoder 23 is a reception system.

The precoder 21 is a serial precoder circuit, and includes a differential encoding operation (SP) circuit 211 that perform a differential encoding operation derived from a differential encoding definition described later, and delay circuits (D in Fig. 2) 212 and 213. Input information series (IN_I,IN_Q) are connected to one input set, and outputs of the delay circuits 212 and 213 are connected to the other input set respectively of the SP circuit 21. In other words, the SP circuit 211 and the delay circuits 212 and 213 configure a feedback loop.

The delay circuits 212 and 213 include flip-flops that operate synchronously with the clock, and delay-adjusted delay lines. The delay circuit 212 and 213 delay the outputs d_I and d_Q of the SP circuit 211 by one clock unit time. For example, when a transmission speed of a transmission path 27 is F [Hz], one clock unit time becomes 1/F [sec].

The encoder 22 includes a distributed feedback (DFB) laser 221, a splitter 222, Mach-Zehnder modulators 223 and 224, a phase converter 225, and a combiner 226. The splitter 222 distributes light emitted by the DFB laser 221, and inputs the distributed light to the Mach-Zehnder modulators 223 and 224. The Mach-Zehnder modulators 223 and 224 modulate the distributed light based on the outputs of the precoder 21. That is, based on the output d_I and d_Q of the SP circuit 211. The combiner 226 multiplexes and outputs the output of the Mach-Zehnder modulators 223 shifted by (1/2) π by the phase converter 225, and the output of the Mach-Zehnder modulators 224. In other words, the encoder 22 generates $$\mathrm{\phi }\mathrm{I}=\left(\surd \mathrm{E}/\mathrm{T}\right)\cos \left(2\mathrm{\pi }\mathrm{fct}\right)$$ from the output d_I of the SP circuit 211, generates $$\mathrm{\phi }\mathrm{I}=-\left(\surd \mathrm{E}/\mathrm{T}\right)\sin \left(2\mathrm{\pi }\mathrm{fct}\right)$$ from the output d_Q of the SP circuit 211, multiplexes both outputs and generates a signal of φI+φ_Q.

The decoder circuit 23 includes a splitter 231, delay circuits 223 and 233, phase converters 234 and 235, and wave detectors 236 and 237. The splitter 231 distributes a signal received by the encoder 22, and inputs the distributed signals to the delay circuits 232 and 235 and the phase converters 234 and 235. The wave detector 236 detects an output signal OUT_I based on a signal one clock delayed by the delay circuit 232 and a signal (1/4)π shifted by the phase converter 234. The wave detector 237 detects an output signal OUT_Q based on a signal one clock delayed by the delay circuit 233 and a signal (-1/4)π shifted by the phase converter 235. In other words, after passing through the encoder 23, the input IN_I and IN_Q of the precoder 21 are restored as the output signals OUT_I and OUT_Q. There are various other configurations of the encoder 22 and the decoder 23 than those shown in Fig. 2. However, the processing that the precoder 21 should execute does not depend on the configurations of the encoder 22 and the decoder 23.

In Fig. 2, the precoder 21 is input with the input information series (IN_I,IN_Q), and outputs the output information series (d_I,d_Q). In the following, IN represents the input information series (IN_I,IN_Q), DO represents the output information series (d_I,d_Q), and "*" represents the operator of the differential encoding operation that the SP circuit 211 performs.

Assume that IN₁, IN₂, IN₃, IN₄, IN₅, IN₆, IN₇, and IN₈ are input as the input information series IN in a unit time cycle starting from time t1. The output information series DO can be expressed as follows $${\mathrm{DO}}_1={\mathrm{IN}}_1\ast \mathrm{DDO}$$ $${\mathrm{DO}}_2={\mathrm{IN}}_2\ast {\mathrm{DO}}_1$$ $${\mathrm{DO}}_3={\mathrm{IN}}_3\ast {\mathrm{DO}}_2$$ $${\mathrm{DO}}_4={\mathrm{IN}}_4\ast {\mathrm{DO}}_3$$ $${\mathrm{DO}}_5={\mathrm{IN}}_5\ast {\mathrm{DO}}_4$$ $${\mathrm{DO}}_6={\mathrm{IN}}_6\ast {\mathrm{DO}}_5$$ $${\mathrm{DO}}_7={\mathrm{IN}}_7\ast {\mathrm{DO}}_6$$ $${\mathrm{DO}}_8={\mathrm{IN}}_8\ast {\mathrm{DO}}_7$$ DDO represents output information series before time t1. From Equation (3-1) to Equation (3-8), the following Equations are obtained. $${\mathrm{DO}}_1={\mathrm{IN}}_1\ast \mathrm{DDO}$$ $${\mathrm{DO}}_2={\mathrm{IN}}_2\ast \left({\mathrm{IN}}_1\ast \mathrm{DDO}\right)$$ $${\mathrm{DO}}_3={\mathrm{IN}}_3\ast \left({\mathrm{IN}}_2\ast \left({\mathrm{IN}}_1\ast \mathrm{DDO}\right)\right)$$ $${\mathrm{DO}}_4={\mathrm{IN}}_4\ast \left({\mathrm{IN}}_3\ast \left({\mathrm{IN}}_2\ast \left({\mathrm{IN}}_1\ast \mathrm{DDO}\right)\right)\right)$$ $${\mathrm{DO}}_5={\mathrm{IN}}_5\ast \left({\mathrm{IN}}_4\ast \left({\mathrm{IN}}_3\ast \left({\mathrm{IN}}_2\ast \left({\mathrm{IN}}_1\ast \mathrm{DDO}\right)\right)\right)\right)$$ $${\mathrm{DO}}_6={\mathrm{IN}}_6\ast \left({\mathrm{IN}}_5\ast \left({\mathrm{IN}}_4\ast \left({\mathrm{IN}}_3\ast \left({\mathrm{IN}}_2\ast \left({\mathrm{IN}}_1\ast \mathrm{DDO}\right)\right)\right)\right)\right)$$ $${\mathrm{DO}}_7={\mathrm{IN}}_7\ast \left({\mathrm{IN}}_6\ast \left({\mathrm{IN}}_5\ast \left({\mathrm{IN}}_4\ast \left({\mathrm{IN}}_3\ast \left({\mathrm{IN}}_2\ast \left({\mathrm{IN}}_1\ast \mathrm{DDO}\right)\right)\right)\right)\right)\right)$$ $${\mathrm{DO}}_8={\mathrm{IN}}_8\ast \left({\mathrm{IN}}_7\ast \left({\mathrm{IN}}_6\ast \left({\mathrm{IN}}_5\ast \left({\mathrm{IN}}_4\ast \left({\mathrm{IN}}_3\ast \left({\mathrm{IN}}_2\ast \left({\mathrm{IN}}_1\ast \mathrm{DDO}\right)\right)\right)\right)\right)\right)\right)$$

The precoder 21 shown in Fig. 2 that is configured by a parallel precoder circuit is explained next. Fig. 3 is a block diagram of the configuration of the precoder 21 that is configured by the parallel precoder circuit. In Fig. 3, the precoder 21 includes a demultiplexing circuit 42 at a pre-stage of a parallel precoder circuit 41, and a multiplexing circuit 43 at a post-stage of the parallel precoder circuit 41. When an output of a processing circuit, such as a framer (not shown), disposed at the pre-stage of the parallel precoder circuit 41 and an input of an encoder (not shown) disposed at the post-stage of the parallel precoder circuit 41 are parallelized, the demultiplexing circuit 42 and the multiplexing circuit 43 do not need to be disposed.

The demultiplexing circuit 42 generates n-set parallel input information series B₁ to B_n by serial-parallel converting the input information series IN. The parallel precoder circuit 41 calculates the parallel output information series DO₁ to DO_n from the parallel input information series B₁ to B_n. The multiplexing circuit 43 outputs serial output information series O by parallel-serial converting the parallel output information series DO₁ to DO_n calculated by the parallel precoder circuit 41.

The parallel input information series B₁ to B_n are the data transmitted in time series in the order of the parallel input information series B₁, the parallel input information series B₂, ..., and the parallel input information series B_n, when the parallel input information series are serially transmitted. The parallel output information series DO₁ to DO_n are the data transmitted in time series in the order of the parallel output information series O₁, the parallel output information series O₂, ..., and the parallel output information series O_n, when the parallel output information series are serially transmitted.

The input information series IN and the output information series O are input and output respectively, synchronously with a clock of the transmission speed F [Hz], that is, a clock of I/F [sec] cycle. In other words, the parallel precoder circuit 41 operates in an operation clock T=F/n [Hz].

Fig. 4 is a block diagram of the configuration of the parallel precoder circuit that has the serial precoder circuit simply developed into eight parallel circuits. In Fig. 4, the parallel precoder circuit includes differential encoding operation (SP) circuits 101 to 108 that perform differential encoding operations, and delay circuits that delay the output sets of the SP circuits 101 to 108 by the operation clock T.

The parallel input information series B₁ are connected to one input set of the SP circuit 101, and the output set of the delay circuit 118, that is, the output information series DO₈, are connected to the other input set of the SP circuit 101. The parallel information series B₂ to B₈ are connected to one input set of the SP circuits 102 to 108, and the output sets of the SP circuits 101 to 107 are connected to the other input set of the SP circuits 102 to 108. In other words, the parallel precoder circuit shown in Fig. 4 uses the output information series of the SP circuit to which the parallel input information series old in time series is input, as the signal one-bit adjacent to the last information series. Therefore, the operation speed of the delay circuits 110 to 118 becomes F/8 [Hz]. However, because a feedback route that passes through the delay circuit 118 and the SP circuits 101 to 108 is generated, the operation speed required for the SP circuits 101 to 108 becomes the same as that when the precoder is configured by the serial precoder circuit.

When DOO₁ to DOO₈ represent the output sets of the SP circuits 101 to 108, and also when "*" represents the operator of the differential encoding operation achieved by the SP circuits 101 to 108, the parallel output information series DO₁ to DO₈ of the parallel precoder circuit can be expressed as follows. $${\mathrm{DO}}_1={\mathrm{B}}_1\ast {\mathrm{DO}}_8$$ $${\mathrm{DO}}_2={\mathrm{B}}_2\ast {\mathrm{DOO}}_1$$ $${\mathrm{DO}}_3={\mathrm{B}}_3\ast {\mathrm{DOO}}_2$$ $${\mathrm{DO}}_4={\mathrm{B}}_4\ast {\mathrm{DOO}}_3$$ $${\mathrm{DO}}_5={\mathrm{B}}_5\ast {\mathrm{DOO}}_4$$ $${\mathrm{DO}}_6={\mathrm{B}}_6\ast {\mathrm{DOO}}_5$$ $${\mathrm{DO}}_7={\mathrm{B}}_7\ast {\mathrm{DOO}}_6$$ $${\mathrm{DO}}_8={\mathrm{B}}_8\ast {\mathrm{DOO}}_7$$

The output sets DOO₁ to DOO₇ of the SP circuits 101 to 108 in Equation (5-1) to Equation (5-8) are equivalent to the parallel output information series DO₁ to DO₇. Therefore, Equation (5-1) to Equation (5-8) can be expressed as $${\mathrm{DO}}_1={\mathrm{B}}_1\ast {\mathrm{DO}}_8$$ $${\mathrm{DO}}_2={\mathrm{B}}_2\ast \left({\mathrm{B}}_1\ast {\mathrm{DO}}_8\right)$$ $${\mathrm{DO}}_3={\mathrm{B}}_3\ast \left({\mathrm{B}}_2\ast \left({\mathrm{B}}_1\ast {\mathrm{DO}}_8\right)\right)$$ $${\mathrm{DO}}_4={\mathrm{B}}_4\left({\mathrm{B}}_3\ast \left({\mathrm{B}}_2\ast \left({\mathrm{B}}_1\ast {\mathrm{DO}}_8\right)\right)\right)$$ $${\mathrm{DO}}_5={\mathrm{B}}_5\ast \left({\mathrm{B}}_4\ast \left({\mathrm{B}}_3\ast \left({\mathrm{B}}_2\ast \left({\mathrm{B}}_1\ast {\mathrm{DO}}_8\right)\right)\right)\right)$$ $${\mathrm{DO}}_6={\mathrm{B}}_6\ast \left({\mathrm{B}}_5\ast \left({\mathrm{B}}_4\ast \left({\mathrm{B}}_3\ast \left({\mathrm{B}}_2\ast \left({\mathrm{B}}_1\ast {\mathrm{DO}}_8\right)\right)\right)\right)\right)$$ $${\mathrm{DO}}_7={\mathrm{B}}_7\left({\mathrm{B}}_6\ast \left({\mathrm{B}}_5\ast \left({\mathrm{B}}_4\ast \left({\mathrm{B}}_3\ast \left({\mathrm{B}}_2\ast \left({\mathrm{B}}_1\ast {\mathrm{DO}}_8\right)\right)\right)\right)\right)\right)$$ $${\mathrm{DO}}_8={\mathrm{B}}_8\ast \left({\mathrm{B}}_7\ast \left({\mathrm{B}}_6\ast \left({\mathrm{B}}_5\ast \left({\mathrm{B}}_4\ast \left({\mathrm{B}}_3\ast \left({\mathrm{B}}_2\ast \left({\mathrm{B}}_1\ast {\mathrm{DO}}_8\right)\right)\right)\right)\right)\right)\right)$$

The parallel output information series DO₈ at the right side of Equation (6-1) to Equation (6-8) show the parallel output information series before the parallel input information series B₁. The parallel input information series B₁ to B₈ are the signals obtained by converting the input information series IN₁ to IN₈ into parallel data by the demultiplexing circuit 42. The parallel input information series B₁ corresponds to the input information series IN₁. The parallel input information series B₂ corresponds to the input information series IN₂. The parallel input information series B₃ corresponds to the input information series IN₃. The parallel input information series B₄ corresponds to the input information series IN₄. The parallel input information series B₅ corresponds to the input information series IN₅. The parallel input information series B₆ corresponds to the input information series IN₆. The parallel input information series B₇ corresponds to the input information series IN₇. The parallel input information series B₈ corresponds to the input information series IN₈. Therefore, Equation (6-1) to Equation (6-8) is equivalent to Equation (4-1) to Equation (4-8), respectively. In other words, the serial precoder circuit is equivalent to the parallel precoder circuit shown in Fig. 4.

The parallel precoder circuit according to the first embodiment of the present invention shown in Fig. 1 satisfies Equation (6-1) to Equation (6-8), and this is explained below.

The SP circuit 1a-(2k,1) in the first column outputs a result of the differential encoding operation of parallel input information series B_2k and parallel input information series B_2k-1. The SP circuit 1a-(2k,2) in the second column outputs a result of the differential encoding operation of the output sets of the SP circuit 1a-(2k-2,1) and the SP circuit 1a-(2k-1) therefore, the output of the SP circuit 1a-(2k,2) is equal to a result of the differential encoding operation of the input four sets of the parallel input information series B_2k to B_2k-3.

In the parallel precoder circuit shown in Fig. 1, when the output of the SP circuit 1a-(1,0) in the first row and the zeroth column is BB₁ and when the operator of the differential encoding operation is "*", an output SPO 1a-(2,1) of the SP circuit 1a-(2,1) in the second row and the first column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{2,1}\right)={\mathrm{BB}}_1\ast {\mathrm{B}}_2$$ An output SPO 1a-(4,1) of the SP circuit 1a-(4,1) in the fourth row and the first column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{4,1}\right)={\mathrm{B}}_3\ast {\mathrm{B}}_4$$ An output SPO 1a-(6,1) of the SP circuit 1a-(6,1) in the sixth row and the first column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{6,1}\right)={\mathrm{B}}_5\ast {\mathrm{B}}_6$$ An output SPO 1a-(8,1) of the SP circuit 1a-(8,1) in the eighth row and the first column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{8,1}\right)={\mathrm{B}}_7\ast {\mathrm{B}}_8$$ Equation (7-1) to Equation (7-4) is hereinafter also referred to as Equations (7).

An output SPO 1a-(4,2) of the SP circuit 1a-(4,2) in the fourth row and the second column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{4,2}\right)=\mathrm{SPO}1\mathrm{a}-\left(\mathrm{2,1}\right)\ast \mathrm{SPO}1\mathrm{a}-\left(\mathrm{4,1}\right)$$ An output SPO 1a-(6,2) of the SP circuit 1a-(6,2) in the sixth row and the second column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{6,2}\right)=\mathrm{SPO}1\mathrm{a}-\left(\mathrm{4,1}\right)\ast \mathrm{SPO}1\mathrm{a}-\left(\mathrm{6,1}\right)$$ An output SPO 1a-(8,2) of the SP circuit 1a-(8,2) in the eighth row and the second column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{8,2}\right)=\mathrm{SPO}1\mathrm{a}-\left(\mathrm{6,1}\right)\ast \mathrm{SPO}1\mathrm{a}-\left(\mathrm{8,1}\right)$$ Based on Equations (17-1) and (17-2), Equation (8-1) to Equation (8-3) become $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{4,2}\right)=\left({\mathrm{BB}}_1\ast {\mathrm{B}}_2\right)\ast \left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right)$$ $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{6,2}\right)=\left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right)\ast \left({\mathrm{B}}_5\ast {\mathrm{B}}_6\right)$$ $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{8,2}\right)=\left({\mathrm{B}}_5\ast {\mathrm{B}}_6\right)\ast \left({\mathrm{B}}_7\ast {\mathrm{B}}_8\right)$$ Equation (9-1) to Equation (9-3) is hereinafter also referred to as Equations (9).

The SP circuit 1a-(2k,3) in the third column outputs a result of the differential encoding operation of the parallel input information series B_2k and the parallel input information series B_2k-4. The output set of the SP circuit 1a-(2k,3) is equal to a result of the differential encoding operation of the parallel input information series B_2k to B_2k-7 (corresponding to the output set of the SP circuit 1a-(8,3) in Fig. 1). When 2k-7 is equal to or smaller than 0 (corresponding to the output set of the SP circuit 1a-(6,3) in Fig. 1), the output of the SP circuit 1a-(2k,3) is equal to a result of the differential encoding operation of the parallel input information series B₁ to B_2k-7.

In Fig. 1, the output SPO 1a-(6,3) of the SP circuit 1a-(6,3) in the sixth row and the third column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{6,3}\right)=\mathrm{SPO}1\mathrm{a}-\left(\mathrm{2,1}\right)\ast \mathrm{SPO}1\mathrm{a}-\left(\mathrm{6,2}\right)$$ The output SPO 1a-(8,3) of the SP circuit 1a-(8,3) in the eighth row and the third column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{8,3}\right)=\mathrm{SPO}1\mathrm{a}-\left(\mathrm{4,2}\right)\ast \mathrm{SPO}1\mathrm{a}-\left(\mathrm{8,2}\right)$$ Based on Equations (7) and Equations (9), Equation (10-1) and Equation (10-2) become $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{6,3}\right)=\left({\mathrm{BB}}_1\ast {\mathrm{B}}_2\right)\ast \left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right)\ast \left({\mathrm{B}}_5\ast {\mathrm{B}}_6\right)$$ $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{8,3}\right)=\left(\left({\mathrm{BB}}_1\ast {\mathrm{B}}_2\right)\ast \left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right)\right)\ast \left(\left({\mathrm{B}}_5\ast {\mathrm{B}}_6\right)\ast \left({\mathrm{B}}_7\ast {\mathrm{B}}_8\right)\right)$$ Equation (11-1) and Equation (11-2) are hereinafter also referred to as Equations (11).

In Fig. 1, the output SPO 1a-(3,4) of the SP circuit 1a-(3,4) in the third row and the fourth column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{3,4}\right)=\mathrm{SPO}1\mathrm{a}-\left(\mathrm{2,1}\right)\ast {\mathrm{B}}_3$$ The output SPO 1a-(5,4) of the SP circuit 1a-(5,4) in the fifth row and the fourth column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{5,4}\right)=\mathrm{SPO}1\mathrm{a}-\left(\mathrm{4,2}\right)\ast {\mathrm{B}}_5$$ The output SPO 1a-(7,4) of the SP circuit 1a-(7,4) in the seventh row and the fourth column can be given by $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{7,4}\right)=\mathrm{SPO}1\mathrm{a}-\left(\mathrm{6,3}\right)\ast {\mathrm{B}}_7$$ Based on Equations (7), Equations (9), and Equations (11), Equation (12-1) to Equation (12-3) become $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{3,4}\right)=\left({\mathrm{BB}}_1\ast {\mathrm{B}}_2\right)\ast {\mathrm{B}}_3$$ $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{5,4}\right)=\left(\left({\mathrm{BB}}_1\ast {\mathrm{B}}_2\right)\ast \left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right)\right)\ast {\mathrm{B}}_5$$ $$\mathrm{SPO}1\mathrm{a}-\left(\mathrm{7,4}\right)=\left(\left({\mathrm{BB}}_1\ast {\mathrm{B}}_2\right)\ast \left(\left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right)\ast \left({\mathrm{B}}_5\ast {\mathrm{B}}_6\right)\right)\right)\ast {\mathrm{B}}_7$$ Equation (13-1) to Equation (13-3) is hereinafter also referred to as Equations (13).

The outputs SPO 1a-(2,1), 1a-(4,2), la-(6,3), and 1a-(8,3) of the SP circuits 1a-(2,1), 1a-(4,2), 1a-(6,3), and la-(8,3) in the (2k)th row are input to the delay circuits 1c-2, 1c-4, 1c-6, and 1c-8, and are delayed by one clock, thereby becoming the parallel output information series DO₂, DO₄, DO₆, and DO₈. The SP circuit 1a-(2k-1,4) differentially encodes parallel input information series B_2k-1 and the input to the delay circuit 1c-2(k-1), and outputs a result of the differential encoding operation to a delay circuit 1c-(2k,1) in the odd row. The delay circuit 1c-(2k,1) delays the data by one clock, and outputs parallel output series DO3, DO5, and DO7.

The SP circuit 1a-(1,0) outputs BB₁ as a result of differentially encoding the parallel output information series DDO before the parallel input information series B₁ and the parallel input information series B₁. The output BB₁ can be expressed as $${\mathrm{BB}}_1={\mathrm{B}}_1\ast \mathrm{DDO}$$ The delay circuit 1c-1 delays the output BB₁ of the SP circuit 1a-(1,0) by one clock, and outputs the parallel output series DO₁.

Therefore, the parallel output information series DO₁ of the parallel precoder circuit shown in Fig. 1 can be given by $${\mathrm{DO}}_1={\mathrm{B}}_1\ast \mathrm{DDO}$$ From Equation (7-1) and Equation (14), the parallel output information series DO₂ can be given by $${\mathrm{DO}}_2=\left({\mathrm{B}}_1\ast \mathrm{DDO}\right)\ast {\mathrm{B}}_2$$ From Equation (13-1) and Equation (14), the parallel output information series DO₃ can be given by $${\mathrm{DO}}_3=\left(\left({\mathrm{B}}_1\ast \mathrm{DDO}\right)\ast {\mathrm{B}}_2\right)\ast {\mathrm{B}}_3$$ From Equation (9-1) and Equation (14), the parallel output information series DO₄ can be given by $${\mathrm{DO}}_4=\left(\left({\mathrm{B}}_1\ast \mathrm{DDO}\right)\ast {\mathrm{B}}_2\right)\ast \left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right)$$ From Equation (13-2) and Equation (14), the parallel output information series DO₅ can be given by $${\mathrm{DO}}_3=\left(\left({\mathrm{B}}_1\ast \mathrm{DDO}\right)\ast {\mathrm{B}}_2\right)\ast \left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right))\ast {\mathrm{B}}_5$$ From Equation (13-2) and Equation (14), the parallel output information series DO₆ can be given by $${\mathrm{DO}}_6=\left(\left({\mathrm{B}}_1\ast \mathrm{DDO}\right)\ast {\mathrm{B}}_2\right)\ast \left(\left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right)\ast \left({\mathrm{B}}_5\ast {\mathrm{B}}_6\right)\right)$$ From Equation (13-3) and Equation (14), the parallel output information series DO₇ can be given by $${\mathrm{DO}}_7=\left(\left({\mathrm{B}}_1\ast \mathrm{DDO}\right)\ast {\mathrm{B}}_2\right)\ast \left(\left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right)\ast \left({\mathrm{B}}_5\ast {\mathrm{B}}_6\right)\right))\ast {\mathrm{B}}_7$$ From Equation (11-2) and Equation (14), the parallel output information series DO₈ can be given by $${\mathrm{DO}}_7=\left(\left({\mathrm{B}}_1\ast \mathrm{DDO}\right)\ast {\mathrm{B}}_2\right)\ast \left({\mathrm{B}}_3\ast {\mathrm{B}}_4\right))\ast \left(\left({\mathrm{B}}_5\ast {\mathrm{B}}_6\right)\ast \left({\mathrm{B}}_7\ast {\mathrm{B}}_8\right)\right)$$

The order of differential encoding operations in Equation (15-1) to Equation (15-8) that show the outputs of the parallel output information series DO₁ to DO₈ of the parallel precoder circuit shown in Fig. 1 is different from the order of differential encoding operations in Equation (6-1) to Equation (6-8) that show the outputs of the parallel output information series DO₁ to DO₈ of the parallel precoder circuit shown in Fig. 4.

Therefore, when an associative law is established in the differential encoding operation, a circuit configuration that achieves "(X(X_I, X_Q)*Y(Y_I, Y_Q))*Z(Z_I, Z_Q)" as shown in Fig. 5 can be substituted by a circuit configuration that achieves "X(X_I, X_Q)*(Y(Y_I, Y_Q))*Z(Z_I, Z_Q))" as shown in Fig. 6.

First, logical Equations of the differential encoding operation that the SP circuit 1a shown in Fig. 1, the SP circuit 211 shown in Fig. 2, and the SP circuits 101 to 108 shown in Fig. 4 perform are explained. In the differential encoding, θ_n represents an information source of differential encoding at time nT, and Φ_n represents transmission phase information. The information source θ_n can be given by $${\mathrm{\theta }}_{\mathrm{n}}=\mathrm{mod}\left(\left({\mathrm{\Phi }}_{\mathrm{n}}-{\mathrm{\Phi }}_{\mathrm{n}-1}\right)\mathrm{,2}\mathrm{\pi }\right)$$ In other words, the information source θ_n becomes a remainder of a division of a subtraction of transmission phase information Φ_n-1 one clock before from the transmission phase information Φ_n by 2π.

From Equations (16-1) and (16-2), the transmission phase information Φ_n can be given by $${\mathrm{\Phi }}_{\mathrm{n}}=\mathrm{mod}\left(\left({\mathrm{\theta }}_{\mathrm{n}}+{\mathrm{\Phi }}_{\mathrm{n}-1}\right)\mathrm{,2}\mathrm{\pi }\right)$$

While logical Equations of the differential encoding operation are derived from Equation (17), the logical Equations are different depending on the information series and phases allocated to four signal points (1,1), (1,1), (1,1), and (1,1) in a signal space diagram shown in Fig. 7.

Fig. 8 is an example of input and output relationships of the differential encoding operation based on a phase difference. In Fig. 8, a phase (1/4)π is allocated to the signal point (1,1), a phase (3,4)π is allocated to the signal point (1,1), a phase (5,4)π is allocated to the signal point (1,1), and a phase (7,4)π is allocated to the signal point (1,1), in the signal space diagram shown in Fig. 7.

In Fig. 8, when the transmission phase information Φ_n-1 one T before is "(1/4)π" and also when the information source θ_n is "0", the transmission phase information Φ_n becomes "(1/4)π". When the information source θ_n is "(1/2)π", the transmission phase information Φ_n becomes "(3,4)π". When the information source θ_n is "π", the transmission phase information Φ_n becomes "(5,4)π". When the information source θ_n is "(3/2)π", the transmission phase information Φ_n becomes "(7,4)π".

When the transmission phase information Φ_n-1 one T before is "(3,4)π" and also when the information source θ_n is "0", the transmission phase information Φ_n becomes "(3,4)π". When the information source θ_n is "(1/2)π", the transmission phase information Φ_n becomes "(5,4)π". When the information source θ_n is "π", the transmission phase information Φ_n becomes "(7,4)π". When the information source θ_n is "(3/2)π", the transmission phase information Φ_n becomes "(1/4)π".

When the transmission phase information Φ_n-1 one T before is "(5,4)π" and also when the information source θ_n is "0", the transmission phase information Φ_n becomes "(5,4)π". When the information source θ_n is "(1/2)π", the transmission phase information Φ_n becomes "(7,4)π". When the information source θ_n is "π", the transmission phase information Φ_n becomes "(1/4)π". When the information source θ_n is "(3/2)π", the transmission phase information Φ_n becomes "(3,4)π".

When the transmission phase information Φ_n-1 one T before is "(7,4)π" and also when the information source θ_n is "0", the transmission phase information Φ_n becomes "(7,4)π". When the information source θ_n is "(1/2)π", the transmission phase information Φ_n becomes "(1/4)π". When the information source θ_n is "π", the transmission phase information Φ_n becomes "(3,4)π". When the information source θ_n is "(3/2)π", the transmission phase information Φ_n becomes "(5,4)π".

Fig. 9 depicts binary data of the phases shown in Fig. 8, that is, information series of the phases. In Fig. 9, (1,1) is allocated to (1/4)π of the transmission phase information Φ_n and Φ_n-1. (0,1) is allocated to (3,4)π of the transmission phase information Φ_n and Φ_n-1. (0,0) is allocated to (5,4)π of the transmission phase information Φ_n and Φ_n-1. (1,0) is allocated to (7,4)π of the transmission phase information Φ_n and Φ_n-1. (1,1) is allocated to (0) of the information source θ_n. (0,1) is allocated to (1/2)π of the information source θ_n. (0,0) is allocated to π of the information source θ_n. (1,0) is allocated to (3/2)π of the information source θ_n. The transmission phase information Φ_n corresponds to the information series DO (d_I,d_Q). The transmission phase information Φ_n-1 corresponds to the information series DO (d_I,d_Q)_n-1. The information source θ_n corresponds to the information series B (b_I,b_Q).

From the input and output relationships shown in Fig. 9, logical Equations of a main addition standard type differential encoding operation become $$\mathrm{DO}\left(\mathrm{n}\right)=\mathrm{B}\left(\mathrm{n}\right)\ast \mathrm{DO}\left(\mathrm{n}-1\right),$$ and the logical Equations can be given by $$\begin{array}{l}{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}\right)={\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\cdot {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)+{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\cdot {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\\ +\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\cdot \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}+\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\cdot {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\\ +\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)+\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\\ +{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\cdot {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)+{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\end{array}$$ $$\begin{array}{l}{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}\right)={\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\cdot {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)+{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\cdot \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\\ +\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\cdot {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)+\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\cdot {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\\ +\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}+\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\cdot {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\\ +{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)+{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\end{array}$$ Based on the theorem and the rule of the Boolean constant, the terms of Equation (16-1) and Equation (16-2) can be modified into Equation (17-1) and Equation (17-2), Equation (18-1) and Equation (18-2), Equation (19-1) and Equation (19-2), or Equation (20-1) and Equation (20-2), as follows $${\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}\right)={\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot {\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)+\overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}+{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\cdot {\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}$$ $${\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}\right)={\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\cdot {\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)+{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot \overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)+\overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}+\overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot {\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}$$ $${\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}\right)={\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\cdot {\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)+{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)+\overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}+\overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}$$ $${\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}\right)={\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\cdot {\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)+\overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\cdot {\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)+{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}+\overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\cdot \overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}$$ $${\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}\right)=\left(\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\oplus {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\right)\cdot \left(\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\oplus {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\right)+\left(\overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\oplus {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\right)\cdot \left({\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\oplus {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\right)$$ $${\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}\right)=\left(\overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\oplus {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\right)\cdot \left(\overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\oplus {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\right)+\left({\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\oplus {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\right)\cdot \left(\overline{{\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)}\oplus {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\right)$$ $${\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}\right)=\left({\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\oplus \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\right)\cdot \left({\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\oplus \overline{{\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)}\right)+\left({\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\oplus \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\right)\cdot \left({\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\oplus {\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)\right)$$ $${\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}\right)=\left({\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\oplus \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\right)\cdot \left({\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)\oplus \overline{{\mathrm{b}}_{\mathrm{Q}}\left(\mathrm{n}\right)}\right)+\left({\mathrm{d}}_{\mathrm{I}}\left(\mathrm{n}-1\right)\oplus {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\right)\cdot \left(\overline{{\mathrm{d}}_{\mathrm{Q}}\left(\mathrm{n}-1\right)}\oplus {\mathrm{b}}_{\mathrm{I}}\left(\mathrm{n}\right)\right)$$

When Equation (17-1) and Equation (17-2) are used as the logical Equations of the differential encoding operation performed by the SP circuit 1a, the SP circuit 211 and the SP circuits 101 to 108, a circuit configuration as shown in Fig. 10 is obtained. When Equation (18-1) and Equation (18-2) are used as the logical Equations of these SP circuits, a circuit configuration as shown in Fig. 11 is obtained. When Equation (19-1) and Equation (19-2) are used as the logical Equations of these SP circuits, a circuit configuration as shown in Fig. 12 is obtained. When Equation (20-1) and Equation (20-2) are used as the logical Equations of these SP circuits, a circuit configuration as shown in Fig. 13 is obtained.

It is explained below that an associative law is established in the differential encoding operation performed using Equation (17-1) and Equation (17-2), or Equation (18-1) and Equation (18-2) . When "(X(X_I, X_Q)*Y(Y_I, Y_Q))*Z(Z_I, Z_Q)" as shown in Fig. 5 is expressed by Equation (21), Equation (21) can be given by Equation (22) based on Equation (18-1) and Equation (18-2). In Equation (22), t_I is expressed by Equation (23) based on Equation (17-1) and Equation (17-2). In Equation (22), t_Q is expressed by Equation (24) based on Equation (17-1) and Equation (17-2). $$\left(\begin{array}{c}{\mathrm{t}}_{\mathrm{I}}\\ {\mathrm{t}}_{\mathrm{Q}}\end{array}\right)=\left(\left(\begin{array}{c}{\mathrm{z}}_{\mathrm{I}}\\ {\mathrm{z}}_{\mathrm{Q}}\end{array}\right)\ast \left(\begin{array}{c}{\mathrm{y}}_{\mathrm{I}}\\ {\mathrm{y}}_{\mathrm{Q}}\end{array}\right)\right)\ast \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{I}}\\ {\mathrm{x}}_{\mathrm{Q}}\end{array}\right)$$ $$\left(\begin{array}{c}{\mathrm{t}}_{\mathrm{I}}\\ {\mathrm{t}}_{\mathrm{Q}}\end{array}\right)=\left(\begin{array}{c}{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\\ {\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\end{array}\right)\ast \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{I}}\\ {\mathrm{x}}_{\mathrm{Q}}\end{array}\right)$$ $$\begin{array}{l}{\mathrm{t}}_{\mathrm{I}}=\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\cdot \overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\stackrel{‾}{{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\overline{\cdot {\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\stackrel{‾}{{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\overline{\cdot {\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ +\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\overline{\cdot {\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ \\ =\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+\cdot {\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\overline{\cdot {\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\left(\overline{{\mathrm{z}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{z}}_{\mathrm{I}}+\overline{{\mathrm{z}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\right)\cdot \left(\overline{{\mathrm{z}}_{\mathrm{I}}}+{\mathrm{z}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\right)\cdot \left({\mathrm{z}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{Q}}\right)\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\left(\overline{{\mathrm{z}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\right)\cdot \left(\overline{{\mathrm{z}}_{\mathrm{I}}}+{\mathrm{z}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{z}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\right)\cdot \left({\mathrm{z}}_{\mathrm{I}}+\overline{{\mathrm{z}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\right)\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ +\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ \\ =\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ +\left(\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{I}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{I}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}+{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ +\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\end{array}$$ $$\begin{array}{l}{\mathrm{t}}_{\mathrm{Q}}=\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\stackrel{‾}{{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ +\left(\stackrel{‾}{{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ \\ =\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\left(\overline{{\mathrm{z}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{z}}_{\mathrm{I}}+\overline{{\mathrm{z}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\right)\cdot \left(\overline{{\mathrm{z}}_{\mathrm{I}}}+{\mathrm{z}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\right)\cdot \left({\mathrm{z}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{Q}}\right)\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ +\left(\left(\overline{{\mathrm{z}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \left(\overline{{\mathrm{z}}_{\mathrm{I}}}+{\mathrm{z}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{z}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\right)\cdot \left({\mathrm{z}}_{\mathrm{I}}+\overline{{\mathrm{z}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\right)\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ \\ =\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ +\left(\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{I}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}+{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\end{array}$$

When "X(X_I, X_Q)*(Y(Y_I, Y_Q))*Z(Z_I, Z_Q))" as shown in Fig. 6 is expressed by Equation (25), Equation (25) can be given by Equation (26) based on Equation (17-1) and Equation (17-2) . $$\left(\begin{array}{c}{\mathrm{s}}_{\mathrm{I}}\\ {\mathrm{s}}_{\mathrm{Q}}\end{array}\right)=\left(\begin{array}{c}{\mathrm{z}}_{\mathrm{I}}\\ {\mathrm{z}}_{\mathrm{Q}}\end{array}\right)\ast \left(\left(\begin{array}{c}{\mathrm{y}}_{\mathrm{I}}\\ {\mathrm{y}}_{\mathrm{Q}}\end{array}\right)\ast \left(\begin{array}{c}{\mathrm{x}}_{\mathrm{I}}\\ {\mathrm{x}}_{\mathrm{Q}}\end{array}\right)\right)$$ $$\left(\begin{array}{c}{\mathrm{s}}_{\mathrm{I}}\\ {\mathrm{s}}_{\mathrm{Q}}\end{array}\right)=\left(\begin{array}{c}{\mathrm{z}}_{\mathrm{I}}\\ {\mathrm{z}}_{\mathrm{Q}}\end{array}\right)\ast \left(\begin{array}{c}{\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ {\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\end{array}\right)$$

In Equation (26), S_I is expressed by Equation (27) based on Equation (18-1) and Equation (18-2). $$\begin{array}{l}{\mathrm{s}}_{\mathrm{I}}={\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left({\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left({\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left(\stackrel{‾}{{\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left(\stackrel{‾}{{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}}\right)\\ \\ ={\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left({\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left({\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left(\left(\overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{x}}_{\mathrm{I}}}+\overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{y}}_{\mathrm{Q}}+{\mathrm{x}}_{\mathrm{I}}+\overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{y}}_{\mathrm{I}}+{\mathrm{x}}_{\mathrm{I}}+{\mathrm{x}}_{\mathrm{Q}}\right)\cdot \left(\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\right)\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left(\left(\overline{{\mathrm{y}}_{\mathrm{Q}}}+\overline{{\mathrm{x}}_{\mathrm{I}}}+\overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\cdot \left(\overline{{\mathrm{y}}_{\mathrm{I}}}+{\mathrm{x}}_{\mathrm{I}}+\overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{y}}_{\mathrm{Q}}+{\mathrm{x}}_{\mathrm{I}}+{\mathrm{x}}_{\mathrm{Q}}\right)\cdot \left({\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\right)\right)\\ \\ ={\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left({\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left({\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left(\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{Q}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left(\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{Q}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ \\ =\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{I}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}+{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ +\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}\end{array}$$

In Equation (27), S_Q is expressed by Equation (28) based on Equation (18-1) and Equation (18-2). $$\begin{array}{l}{\mathrm{s}}_{\mathrm{Q}}={\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left({\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left({\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left(\stackrel{‾}{{\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left(\stackrel{‾}{{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}}\right)\\ \\ ={\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left({\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left({\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left(\left(\overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{x}}_{\mathrm{I}}}+\overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{y}}_{\mathrm{Q}}+{\mathrm{x}}_{\mathrm{I}}+\overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{y}}_{\mathrm{I}}+{\mathrm{x}}_{\mathrm{I}}+{\mathrm{x}}_{\mathrm{Q}}\right)\cdot \left(\overline{{\mathrm{y}}_{\mathrm{Q}}}+\overline{{\mathrm{x}}_{\mathrm{I}}}+{\mathrm{x}}_{\mathrm{Q}}\right)\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left(\left(\overline{{\mathrm{y}}_{\mathrm{Q}}}+\overline{{\mathrm{x}}_{\mathrm{I}}}+\overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\cdot \left(\overline{{\mathrm{y}}_{\mathrm{I}}}+{\mathrm{x}}_{\mathrm{I}}+\overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\cdot \left({\mathrm{y}}_{\mathrm{Q}}+{\mathrm{x}}_{\mathrm{I}}+{\mathrm{x}}_{\mathrm{Q}}\right)\cdot \left({\mathrm{y}}_{\mathrm{I}}+\overline{{\mathrm{x}}_{\mathrm{I}}}+{\mathrm{x}}_{\mathrm{Q}}\right)\right)\\ \\ ={\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left({\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \left({\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left({\mathrm{y}}_{\mathrm{Q}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{Q}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ +\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \left(\overline{{\mathrm{y}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}+\overline{{\mathrm{y}}_{\mathrm{Q}}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}+{\mathrm{y}}_{\mathrm{Q}}\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}+{\mathrm{y}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\right)\\ \\ =\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left({\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot {\mathrm{x}}_{\mathrm{Q}}\\ +\left(\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{Q}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{I}}+{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}\right)\cdot \overline{{\mathrm{x}}_{\mathrm{I}}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\\ +\left(\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot {\mathrm{y}}_{\mathrm{I}}+\overline{{\mathrm{z}}_{\mathrm{I}}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot {\mathrm{y}}_{\mathrm{Q}}+{\mathrm{z}}_{\mathrm{I}}\cdot \overline{{\mathrm{z}}_{\mathrm{Q}}}\cdot \overline{{\mathrm{y}}_{\mathrm{Q}}}+{\mathrm{z}}_{\mathrm{I}}\cdot {\mathrm{z}}_{\mathrm{Q}}\cdot \overline{{\mathrm{y}}_{\mathrm{I}}}\right)\cdot {\mathrm{x}}_{\mathrm{I}}\cdot \overline{{\mathrm{x}}_{\mathrm{Q}}}\end{array}$$