Abstract:
A modulating device including: a first convertor configured to generate a converted analog signal by analog conversion on a input digital signal, a modulator configured to generate a modulated signal by quadrature modulation, a phase shifter configured to generate a phase shift signal by phase rotation, a demodulator configured to generate a demodulated signal by quadrature demodulation, a second convertor to generate a converted digital signal by digital conversion, a calculating circuit configured to estimate a the first direct current offset based on the input digital signal and the converted digital signal, the first direct current offset being a noise of digital current component generated between the input digital signal inputted to the first convertor and the output signal inputted to the demodulator, and a correcting circuit configured to correct at least one among from the input digital signal to the output signal based on the first direct current offset.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application is based upon and claims the benefit of priority from the prior Japanese Patent Application No. 2013-060479 filed on Mar. 22, 2013, the entire contents of which are incorporated herein by reference. 
       FIELD 
       [0002]    The embodiments discussed herein are related to a modulating device and a modulation method. 
       BACKGROUND 
       [0003]    Upon receiving signals to be transmitted, conventional wireless communication devices that conduct communication with wireless signals convert, to modulation signals, signals to be transmitted received as local signals and then transmit the converted signals by using a quadrature modulation circuit to perform quadrature modulation on the local signals that become carrier waves. Modulation systems for performing quadrature modulation use quadrature phase shift keying (QPSK) or quadrature amplitude modulation (QAM) and the like. However, a DC offset is produced as noise due to imperfections of the elements in the abovementioned quadrature modulation circuit and the DC offset is added to the modulation signals. 
         [0004]    Accordingly, Japanese Laid-open Patent Publication No. 10-079693 describes a technique for calculating a DC offset of a quadrature modulation circuit, for example, by providing a feedback circuit that provides feedback by using a quadrature demodulation circuit to perform quadrature demodulation on modulation signals, and by switching inputs and non-inputs of the modulation signals to the feedback circuit. Moreover, Japanese Laid-open Patent Publication No. 2002-077285 describes a technique for calculating a DC offset of a quadrature modulation circuit by using, for example, a frequency converting circuit to convert modulation signals to IF signals and then by using an ADC to perform digital conversion to provide feedback. 
       SUMMARY 
       [0005]    According to an aspect of the invention, a modulating device includes a first convertor configured to generate a converted analog signal by analog conversion on a input digital signal that is inputted to the first converter, a modulator configured to generate a modulated signal by quadrature modulation on the converted analog signal, a first phase shifter configured to generate a first phase shift signal by first phase rotation on a local signal by a phase shift quantity that changes with a period, a demodulator configured to generate a demodulated signal by quadrature demodulation on an output signal using the first phase shift signal, the output signal deriving from the modulated signal and being outputted from the modulating device, the quadrature demodulation corresponding to the quadrature modulation, a second convertor to generate a converted digital signal by digital conversion on the demodulated signal, a calculating circuit configured to estimate a the first direct current offset based on the input digital signal and the converted digital signal, the first direct current offset being a noise of digital current component generated between the input digital signal inputted to the first convertor and the output signal inputted to the demodulator, and a correcting circuit configured to correct at least one among from the input digital signal to the output signal based on the first direct current offset. 
         [0006]    The object and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the claims. 
         [0007]    It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention, as claimed. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0008]      FIG. 1  illustrates a baseband model for a modulating device according to an embodiment; 
           [0009]      FIG. 2  is a block diagram of a functional configuration of the modulating device; 
           [0010]      FIG. 3  illustrates a first embodiment of the modulating device; 
           [0011]      FIG. 4  illustrates a correction of a signal according to the first embodiment; 
           [0012]      FIG. 5  illustrates a second embodiment of the modulating device; 
           [0013]      FIG. 6  illustrates a third embodiment of the modulating device; 
           [0014]      FIG. 7  illustrates a correction of a signal according to the third embodiment; 
           [0015]      FIG. 8  illustrates a fourth embodiment of the modulating device; 
           [0016]      FIG. 9  illustrates a fifth embodiment of the modulating device; 
           [0017]      FIG. 10  illustrates a correction of a signal according to the fifth embodiment; 
           [0018]      FIG. 11  illustrates a sixth embodiment of the modulating device; 
           [0019]      FIG. 12  illustrates a seventh embodiment of the modulating device; 
           [0020]      FIG. 13  illustrates a correction of a signal according to the seventh embodiment; and 
           [0021]      FIG. 14  illustrates an eighth embodiment of the modulating device. 
       
    
    
     DESCRIPTION OF EMBODIMENTS 
       [0022]    However, in the abovementioned conventional techniques, the DC offset produced by the quadrature modulation circuit may not be detected accurately and the DC offset of the modulation signals may not be accurately compensated. 
         [0023]    An object in one aspect of the embodiments discussed herein is to accurately compensate the DC offset of the modulation signals. 
         [0024]    Detailed explanations of embodiments of the modulating unit and the modulation method described hereinbelow will be provided with reference to the accompanying drawings. 
         [0025]    (Baseband Model for Modulating Device  100  According to an Embodiment) 
         [0026]      FIG. 1  illustrates a baseband model for a modulating device  100  according to an embodiment. The modulating device  100  is a device for performing analog conversion and quadrature modulation of input digital signals and then transmitting the signals. In the example in  FIG. 1 , a digital signal is, for example, a signal represented by a complex number having a real number part and an imaginary number part. The modulating device  100  performs quadrature demodulation and digital conversion of the signals to be transmitted to provide feedback. 
         [0027]    In  FIG. 1 , the modulating device  100  has, for example, a DAC and a QMOD as transmission-side circuits. The modulating device  100  has, for example, a first complex multiplier, an ADC, a QDEM, and a second complex multiplier as feedback-side circuits. 
         [0028]    The digital to analog converter (DAC) is a circuit for performing analog conversion on signals. The quadrature modulating unit (QMOD) is a circuit for performing quadrature modulation on signals. 
         [0029]    The first complex multiplier is a circuit for performing phase rotation of signals in phase shift quantities that change periodically. The analog to digital converter (ADC) is a circuit for performing digital conversion on signals. The quadrature demodulating unit (QDAM) is a circuit for performing quadrature demodulation on signals. The second complex multiplier is a circuit for performing phase rotation on signals in the direction opposite that of the first complex multiplier and in the same phase shift quantity as the first complex multiplier. 
         [0030]    In  FIG. 1 , a digital signal x that is to be transmitted is input into the DAC and converted to analog by the DAC in the transmission-side circuits. Next, the signal x analog-converted by the DAC is input into the QMOD and quadrature modulation is performed by the QMOD. When processed by the DAC and the QMOD, a DC offset b produced in the DAC and the QMOD is added to the signal x. 
         [0031]    The DC offset is, for example, direct current component noise which is noise that is produced when actual circuit characteristics deviate from the ideal characteristics due to aging and production variances of circuit elements included in circuits such as the DAC, the QMOD, the QDEM, and the ADC and the like. In the following explanation, the DC offset may be referred to as a “modulating unit offset b” along with the DC offset produced by the DAC and the QDEM. 
         [0032]    In other words, in the transmission-side circuits, the digital signal x is analog-converted as indicated by reference numeral  101 , and the modulating unit offset b is added as indicated by reference numeral  102  so that the signal x becomes modulation signal x+b, which is then transmitted and input into the feedback-side circuits. 
         [0033]    In  FIG. 1 , the modulation signal x+b in the feedback-side circuits is input into the first complex multiplier and phase-rotated by θ(t) by the first complex multiplier. In this case, θ(t) is represented, for example, by multiplying an angular speed w by a time t to produce wt. Next, the signal (x+b)×exp(jθ(t)) phase-converted by the first complex multiplier is in input into the QDEM and quadrature demodulation is performed by the QDEM. The signal (x+b)×exp(jθ(t)) quadrature-demodulated by the QDEM is input into the ADC and digitally-converted by the ADC. 
         [0034]    When processed by the QDEM and the ADC, a DC offset c produced by the QDEM and the ADC is added to the signal (x+b)×exp(jθ(t)). In the following explanation, the DC offset may be referred to as a “demodulating unit offset c” along with the DC offset produced by the QMOD and the ADC. Next, the signal (x+b)×exp(jθ(t))+c to which was added the demodulating unit offset c, is input into the second complex multiplier and phase-rotated only θ(t)* by the second complex multiplier. Here, the * symbol indicates a conjugate. 
         [0035]    In other words, in the feedback-side circuits, the modulation signal x+b is phase-rotated by using a signal to be phase-rotated by the θ(t) of the reference numeral  104  as indicated by reference numeral  103 . Further, the demodulating unit offset c as indicated by the reference numeral  105  is added to the modulation signal x+b and digitally-converted as indicated by the reference numeral  106 . The modulation signal x+b is phase-rotated in the opposite direction from the reference numeral  103  by using a signal to be phase-rotated by the θ(t) of the reference numeral  104  as indicated by reference numeral  107 . Consequently, the modulation signal x+b becomes the feedback signal y={(x+b)×exp(jθ(t))+c}×exp(jθ(t))*=x+b+c×exp(jθ(t))* and is fed back. 
         [0036]    In this way, before and after the transmitted signal x+b is quadrature-demodulated, a feedback signal y=x+b+c×exp(jθ(t))* is obtained by providing feedback while performing phase rotation in the feedback-side circuits. The modulating unit offset b produced by the transmission-side circuits and the demodulating unit offset c produced by the feedback-side circuits in the feedback signal y are each elements with different frequencies. 
         [0037]    As a result, the modulating device  100  separates the modulating unit offset b produced by the transmission-side circuits and the demodulating unit offset c produced by the feedback-side circuits on the basis of the elements with the different frequencies and is able to specify the modulating unit offset b. The modulating device  100  then sets the specified modulating unit offset b as a compensation value b′ of the modulating unit offset b and inputs the compensation value b′ into the transmission-side circuits in order to subtract the compensation value b′ from the modulation signal x+b to remove the modulating unit offset b in the transmission-side circuits. As a result, the modulating device  100  is able to erase the modulating unit offset and transmit a more accurate modulation signal. 
         [0038]    (Example of Functional Configuration of Modulating Device  100 ) 
         [0039]    The following discusses an example of a functional configuration of the modulating device  100  with reference to  FIG. 2 . 
         [0040]      FIG. 2  is a block diagram of a functional configuration of the modulating device  100 . The modulating device  100  includes an input unit  201 , a first converting unit  202 , a modulating unit  203 , an amplifying unit  204 , an output unit  205 , a first phase shift unit  206 , a demodulating unit  207 , a second converting unit  208 , a second phase shift unit  209 , a calculating unit  210 , and a correcting unit  211 . 
         [0041]    &lt;Example of Transmission-Side Functions&gt; 
         [0042]    An example of the transmission-side functions of the modulating device  100  will be discussed first. The input unit  201  inputs digital signals. A digital signal includes, for example, two baseband signals that indicate information to be transmitted. The two baseband signals include an in-phase (I) signal for indicating a real number part of the information to be transmitted, and a quadrature-phase (Q) signal for indicating an imaginary number part of the information to be transmitted. The two baseband signals may be represented together as a “signal representing one complex number”. As a result, the first converting unit  202  is able to perform analog conversion on the signals input by the input unit  201 . 
         [0043]    The first converting unit  202  performs analog conversion on the signals input by the input unit  201 . The first converting unit  202  uses, for example, a DAC to perform the analog conversion on the I-signal and the Q-signal input by the input unit  201 . At this time, a DC offset produced in the first converting unit  202  is added to the I-signal and Q-signal input by the input unit  201 . As a result, the modulating unit  203  is able to perform quadrature modulation on the signals having undergone analog conversion by the first converting unit  202 . 
         [0044]    The modulating unit  203  performs quadrature modulation on the signals converted by the first converting unit  202 . The modulating unit  203  uses, for example, QMOD to perform quadrature modulation on two carrier waves that are orthogonal to each other, and produces one modulated wave for indicating the I-signal and the Q-signal converted by the first converting unit  202 . As a result, the modulating unit  203  is able to produce a modulated wave to be transmitted from an antenna. 
         [0045]    The amplifying unit  204  amplifies the signal having undergone quadrature modulation by the modulating unit  203 . The amplifying unit  204  uses, for example, an amplifier to amplify the modulated wave having undergone quadrature modulation by the modulating unit  203 . The amplifying unit  204  is able to amplify the modulated wave to be transmitted from the antenna. 
         [0046]    The output unit  205  outputs the signal having undergone quadrature modulation by the modulating unit  203 . The output unit  205  uses, for example, an antenna to transmit the modulated wave having undergone quadrature modulation by the modulating unit  203 . The output unit  205  may output the signal amplified by the amplifying unit  204 . The output unit  205  uses, for example, an antenna to transmit the modulated wave amplified by the amplifying unit  204 . As a result, the modulating device  100  is able to transmit the signal. 
         [0047]    &lt;Example of Feedback-Side Functions&gt; 
         [0048]    A first example of the functions of the feedback-side will be discussed next. The first phase shift unit  206  performs phase rotation on a local signal by a phase shift quantity that changes periodically. The first phase shift unit  206  uses, for example, a complex multiplier to perform the phase rotation on the local signal that becomes a carrier wave. As a result, the demodulating unit  207  is able to perform phase rotation on the modulated wave. 
         [0049]    In this case, the demodulating unit  207  uses the local signal that has undergone phase rotation by the first phase shift unit  206  to perform quadrature demodulation on the signal having undergone quadrature modulation by the modulating unit  203 . The demodulating unit  207  uses, for example, QDEM to produce two signals that are orthogonal to each other and that have been phase-rotated by demodulating the modulated wave using the phase-rotated carrier wave. The demodulating unit  207  may perform quadrature demodulation on the signal amplified by the amplifying unit  204 . As a result, the second converting unit  208  is able to perform digital conversion on the two signals produced by the demodulating unit  207 . 
         [0050]    The first phase shift unit  206  may perform phase rotation, by a phase shift quantity that changes periodically, on the signal having undergone quadrature modulation by the modulating unit  203 . The first phase shift unit  206  uses, for example, a complex multiplier to perform the phase rotation on the modulated wave. The first phase shift unit  206  may perform phase rotation, by the phase shift quantity that changes periodically, on the signal amplified by the amplifying unit  204 . As a result, the first phase shift unit  206  is able to perform phase rotation on the modulated wave. 
         [0051]    In this case, the demodulating unit  207  uses the first phase shift unit  206  to perform quadrature demodulation on the signal having undergone phase rotation by the first phase shift unit  206 . The demodulating unit  207  uses, for example, QDEM to produce two signals that are orthogonal to each other by performing quadrature demodulation on the modulated wave. As a result, the second converting unit  208  is able to perform digital conversion on the two signals produced by the demodulating unit  207 . 
         [0052]    The second converting unit  208  performs digital conversion on the signal having undergone quadrature demodulation by the demodulating unit  207 . The second converting unit  208  uses, for example, ADC to perform digital conversion on each of the two signals produced by the demodulating unit  207 . As a result, the second phase shift unit  209  is able to perform phase rotation on the signals having undergone digital conversion by the second converting unit  208 . 
         [0053]    The second phase shift unit  209  performs phase rotation, by a phase shift quantity in a direction opposite to the direction of the phase rotation performed by the first phase shift unit  206 , on the signal converted by the second converting unit  208 . The second phase shift unit  209  uses, for example, a complex multiplier to perform phase rotation, by the same phase shift quantity as the phase shift quantity of the phase rotation performed by the first phase shift unit  206 , in the direction opposite to the direction of the phase rotation by the first phase shift unit  206 . As a result, the second phase shift unit  209  is able to change the modulating unit offset b and the demodulating unit offset c into elements having different frequencies. 
         [0054]    The calculating unit  210  calculates an amount of direct current offset produced in a signal from when the signal is input into the input unit  201  until the signal is input into the demodulating unit  207 , by using the signal input into the input unit  201  and the signal converted by the second converting unit  208 . The calculating unit  210  may calculate the amount of direct current offset produced in a signal from when the signal is input into the input unit  201  until the signal is input into the demodulating unit  207 , by using the signal input into the input unit  201  and the signal having undergone phase rotation by the second phase shift unit  209 . 
         [0055]    The calculating unit  210  may calculate the amount of direct current offset produced in a signal from when the signal is input into the input unit  201  until the signal is input into the phase shift units, by using the signal input into the input unit  201  and the signal converted by the second converting unit  208 . The calculating unit  210  may further calculate the amount of direct current offset produced in a signal from when the signal is input into the input unit  201  until the signal is input into the first phase shift unit  206 , by using the signal input into the input unit  201  and the signal having undergone phase rotation by the second phase shift unit  209 . 
         [0056]    The calculating unit  210  obtains information indicating a relationship between, for example, the signal input by the input unit  201 , a direct current offset produced in a signal from when the signal is input into the input unit  201  until the signal is input into the demodulating unit  207 , a direct current offset produced in a signal from when the signal is input into the demodulating unit  207  until the signal is input into the calculating unit  201 , and the signal converted by the second converting unit  208 . Next, the calculating unit  210  obtains values of a signal input by the input unit  201  obtained at a plurality of points in time, and values of a signal converted by the second converting unit  208  obtained at a plurality of points in time. The calculating unit  210  uses the obtained information and the obtained values of the signals to calculate the amount of direct current offset produced in a signal from when the signal is input into the input unit  201  until the signal is input into the demodulating unit  207 . 
         [0057]    The calculating unit  210  may also calculate a value obtained by dividing, by periods of multiples, the results of an integration of a differential between the signal input by the input unit  201  and signal converted by the second converting unit  208 , in periods of multiples of cycles of the changes in the phase shift quantities. As a result, the calculating unit  210  calculates the amount of direct current offset produced in the signal from when the signal is input into the input unit  201  until the signal is input into the demodulating unit  207 . 
         [0058]    The calculating unit  210  may also calculate the amount of direct current offset produced in a signal from when the signal is input into the input unit  201  until the signal is input into the demodulating unit  207 , by using the signal input into the input unit  201  and the signal converted by the second converting unit  208 . Moreover, the calculating unit  210  may calculate a correction coefficient corresponding to a distortion produced in the signal in the amplifying unit  204  by using the signal input by the input unit  201  and the signal converted by the second converting unit  208 . The calculating unit  210  uses, for example, FPGA, to realize the above functions. 
         [0059]    The correcting section  211  corrects the signal between the input unit  201  and the output unit  205  on the basis of the direct current offset amounts calculated by the calculating unit  210 . The correcting section  211  uses, for example, a subtractor to correct the signal input into the input unit  201  by subtracting the direct current offset amounts calculated by the calculating unit  210  from the signal input into the input unit  201 . 
         [0060]    The correcting section  211  may also correct the signal between the input unit  201  and the output unit  205  on the basis of a correction coefficient and the direct current offset amounts calculated by the calculating unit  210 . The correcting section  211  uses, for example, a DPD and a subtractor to correct the signal input into the input unit  201 . As a result, the correcting section  211  is able to output a signal with high accuracy to the output unit  205 . 
         [0061]    A second example of functions on the feedback-side will be discussed. The operations by the second converting unit  208 , the second phase shift unit  209 , and the correcting section  211  are the same as those of the first example and will be omitted from the discussion of the second example. 
         [0062]    The first phase shift unit  206  may perform phase rotation, by a phase shift quantity that changes periodically, on the signal having undergone quadrature modulation by the modulating unit  203 . The first phase shift unit  206  uses, for example, a complex multiplier to perform the phase rotation on the modulated wave. The first phase shift unit  206  may perform phase rotation, by the phase shift quantity that changes periodically, on the signal amplified by the amplifying unit  204 . As a result, the first phase shift unit  206  is able to perform phase rotation on the modulated wave. 
         [0063]    The demodulating unit  207  performs quadrature demodulation on the signal having undergone phase rotation by the first phase shift unit  206 . The demodulating unit  207  uses, for example, QDEM to produce two signals that are orthogonal to each other by performing quadrature demodulation on the modulated wave. As a result, the second converting unit  208  is able to perform digital conversion on the two signals produced by the demodulating unit  207 . 
         [0064]    The calculating unit  210  uses the information indicating a relationship between the signal input by the input unit  201 , the direct current offset produced in the signal from when the signal is input into the input unit  201  until the signal is input into the first phase shift unit  206 , the direct current offset produced in the signal from when the signal is input into the first phase shift unit  206  until the signal is input into the calculating unit  210 , and the signal converted by the second converting unit  208 , and the calculating unit  210  also uses the values of the signal input by the input unit  201  obtained at the plurality of points in time, and the values of the signal converted by the second converting unit  208  obtained at the plurality of points in time, to calculate the direct current offset amount produced in the signal from when the signal is input into the input unit  201  until the signal is input into the first phase shift unit  206 . 
         [0065]    The calculating unit  210  may also calculate the amount of direct current offset produced in the signal from when the signal is input into the input unit  201  until the signal is input into the first phase shift unit  206  by calculating the value obtained by dividing, by periods of multiples, the results of the integration of the differential between the signal input by the input unit  201  and signal converted by the second converting unit  208 , in periods of multiples of cycles of the changes in the phase shift quantities. 
         [0066]    The calculating unit  210  may further calculate the amount of direct current offset produced in a signal from when the signal is input into the input unit  201  until the signal is input into the first phase shift unit  206  and the correction coefficient corresponding to the distortion produced in the signals in the amplifying unit  204 , by using the signal input into the input unit  201  and the signal having undergone phase rotation by the second phase shift unit  209 . 
         [0067]    (First Embodiment of Modulating Device  100 ) 
         [0068]      FIG. 3  illustrates a first embodiment of the modulating device  100 . As illustrated in  FIG. 3 , the modulating device  100  has two subtractors  301 , a DAC  302 , a QMOD  303 , a first oscillator  304 , an amplifier  305 , an analog EPS  306 , a second oscillator  307 , a QDEM  308 , an ADC  309 , a digital EPS  310 , and a DC offset detecting unit  311 . A processing flow of the I-signal among the I-signals and the Q-signals will be discussed hereinbelow. The processing flow of the Q-signal is the same as the processing flow of the I-signal and will be omitted. 
         [0069]    The two subtractors  301  are two-input one-output circuits that output, as an output signal, a subtraction result of the subtraction of one of the input signals from the other. In the example in  FIG. 3 , a signal line for inputting an I-signal that is a digital signal to be transmitted, a signal line for inputting a signal for correcting the I-signal from the DC offset detecting unit  311 , and a signal line for outputting the signal to the DAC  302 , are connected to one of the subtractors  301 . Moreover, a signal line for inputting a Q-signal that is a digital signal to be transmitted, a signal line for inputting a signal for correcting the Q-signal from the DC offset detecting unit  311 , and a signal line for outputting the signal to the DAC  302 , are connected to the other one of the subtractors  301 . In  FIG. 3 , the subtractors  301  subtract a compensation value b′ of the modulating unit offset b input from the DC offset detecting unit  311 , from an input I-signal x to be transmitted, and output the subtraction result u=x−b′ to the DAC  302 . 
         [0070]    The DAC  302  is a two-input two-output circuit that performs analog conversion on the two digital signals that are input signals and outputs the analog-converted signals as two output signals. In the example in  FIG. 3 , signal lines for inputting the two digital signals from the subtractors  301 , and signal lines for outputting the two signals to the QMOD  303  are connected to the DAC  302 . In  FIG. 3 , the DAC  302  performs analog conversion on input digital signals u and outputs the analog-converted signals to the QMOD  303 . The analog-converted signals may be referred to by “u” in the same way as the digital signals in the following explanation. 
         [0071]    The QMOD  303  is a three-input one-output circuit that uses a local signal that is the remaining input signal to perform quadrature modulation on the analog signals that are the two input signals, and outputs the quadrature-modulated signal as an output signal. In the example in  FIG. 3 , two signal lines for inputting the two analog signals from the DAC  302 , a signal line for inputting the local signal from the first oscillator  304 , and a signal line for outputting the signal to the amplifier  305 , are connected to the QMOD  303 . In  FIG. 3 , the QMOD  303  uses the local signal from the first oscillator  304  to perform quadrature modulation on the input signal u, and outputs the quadrature-modulated signal to the amplifier  305 . The modulating unit offset b is added to the signal u by passing through the DAC  302  and the QMOD  303 , to become the signal u+b. 
         [0072]    The first oscillator  304  is a one-output circuit that outputs a local signal as an output signal. In the example in  FIG. 3 , a signal line for outputting the local signal to the QMOD  303 , and a signal line for outputting the local signal to the analog EPS  306 , are connected to the first oscillator  304 . In  FIG. 3 , the first oscillator  304  branches one output of the local signal to output the local signal to the QMOD  303  and to the analog EPS  306 . 
         [0073]    The amplifier  305  is a one-input one-output circuit that amplifies an analog signal that is an input signal and outputs the amplified signal as an output signal. In the example in  FIG. 3 , a signal line for inputting the analog signal from the first oscillator  304 , a signal line for outputting the signal to the QDEM  308 , and a signal line for outputting the signal to the antenna, are connected to the amplifier  305 . In  FIG. 3 , the amplifier  305  outputs, to the antenna and to the QDEM  308  by branching, a one-output signal that is a quadrature-modulated signal that has been amplified. For simplification at this time, the amplifier  305  amplifies the signal by one. 
         [0074]    The analog EPS  306  is a two-input one-output circuit that performs phase rotation on a local signal that is one of the input signals by using a signal that is phase-rotated by a phase shift quantity that changes periodically and that is the other input signal, and outputs the phase-rotated signal as an output signal. In the following discussion, a signal that is phase-rotated by a phase shift quantity that changes periodically may be referred to as a “phase shift signal”. In the example in  FIG. 3 , a signal line for inputting the local signal from the first oscillator  304 , a signal line for inputting the phase shift signal from the second oscillator  307 , and a signal line for outputting the signal to the QDEM  308  are connected to the analog EPS  306 . In  FIG. 3 , the analog EPS  306  uses the phase shift signal to perform phase rotation on the local signal from the first oscillator  304 , and outputs the phase-rotated signal to the QDEM  308 . 
         [0075]    The second oscillator  307  is a one-output circuit that outputs the phase shift signal as an output signal. In the example in  FIG. 3 , a signal line for outputting the phase shift signal to the analog EPS  306  and a signal line for outputting the phase shift signal to the digital EPS  310  are connected to the second oscillator  307 . In  FIG. 3 , the second oscillator  307  outputs by branching one output phase shift signal to the analog EPS  306  and one output phase shift to the digital EPS  310 . 
         [0076]    The QDEM  308  is a two-input two-output circuit that uses the local signal that is phase-rotated by the phase shift signal that is one of the input signals to perform quadrature demodulation on an analog signal that is the other of the input signals, and outputs the quadrature-demodulated signal as an output signal. In the example in  FIG. 3 , a signal line for inputting the signal from the amplifier  305 , a signal line for inputting the signal from the analog EPS  306 , and a signal line for outputting the signal to the ADC  309  are connected to the QDEM  308 . In  FIG. 3 , the QDEM  308  uses the signals from the analog EPS  306  to perform quadrature demodulation on the analog signal from the amplifier  305 , and outputs the signal (u+b)*EXP(jwt) to the ADC  309 . 
         [0077]    The ADC  309  is a two-input two-output circuit that performs digital conversion on the two analog signals that are input signals and outputs the digital-converted signals as two output signals. In the example in  FIG. 3 , a signal line for inputting the two signals from the QDEM  308  and a signal line for outputting the signals to the digital EPS  310  are connected to the ADC  309 . In  FIG. 3 , the ADC  309  performs digital conversion on the signal (u+b)*EXP(jwt) input from the QDEM  308 , and outputs the digitally converted signal to the digital EPS  310 . The demodulating unit offset c is added to the signal (u+b)*EXP(jwt) that passes through the QDEM  308  and the ADC  309  to become the signal (u+b)*EXP(jwt)+c. 
         [0078]    The digital EPS  310  is a three-input two-output circuit that uses the phase shift signal that is the remaining input signal to perform phase rotation on the two signals that are the input signals, and outputs the phase-rotated signal as an output signal. In the example in  FIG. 3 , a signal line for inputting the two signals from the ADC  309 , a signal line for inputting the phase shift signal from the second oscillator  307 , and a signal line for outputting the signal to the DC offset detecting unit  311 , are connected to the digital EPS  310 . In  FIG. 3 , the digital EPS  310  uses the phase shift signal from the second oscillator  307  to perform phase rotation, by a phase shift quantity that is the same as that of the phase rotation performed by the analog EPS  306 , on the two input signals in a direction opposite the direction of the phase rotation by the analog EPS  306 . The digital EPS  310  outputs the phase-rotated signal (u+b)+c*EXP(jwt)* to the DC offset detecting unit  311  as the feedback signal y. 
         [0079]    The DC offset detecting unit  311  is a four-input two-output circuit that uses two feedback signals that are input signals and two digital signals to be transmitted that are input signals, to output the compensation value b′ signal of the modulating unit offset b. In the example in  FIG. 3 , a signal line for inputting the two feedback signals from the digital EPS  310 , a signal line for inputting the two digital signals that are to be transmitted, and a signal line for outputting the signal to the subtractor  301  are connected to the DC offset detecting unit  311 . In  FIG. 3 , the DC offset detecting unit  311  produces the compensation value b′ signal of the modulating unit offset b for correcting the two digital signals to be transmitted and outputs the compensation value b′ signal to the two subtractors  310  as described below with reference to  FIG. 4 . 
         [0080]    As a result, the signal input to the amplifier  305  becomes the signal u+b=u−b′+b≈u due to the DC offset detecting unit  311  inputting the compensation value b′ that becomes the same value as the modulating unit offset b into the subtractors  301 . Therefore, the modulating device  100  is able to remove the modulating unit offset b from the signal to be amplified and transmitted by the amplifier  305  and transmit a signal with high accuracy. 
         [0081]    (Correction of Signals in First Embodiment) 
         [0082]      FIG. 4  illustrates a correction of a signal according to the seventh embodiment.  FIG. 4  illustrates the flow of a signal xi until the signal x i  becomes a feedback signal y i  in the circuit illustrated in  FIG. 3 . 
         [0083]    In the circuit illustrated in  FIG. 3 , the compensation value b′ of the modulating unit offset b as indicated by reference numeral  401  is subtracted from the signal x i , and the modulating unit offset b as indicated by the reference numeral  402  is added to the signal x i . Further, the signal x i  is phase-rotated, as indicated by reference numeral  403 , using the phase shift signal indicated by reference numeral  404 , and the demodulating unit offset c is added to the signal x i  as indicated by reference numeral  405 . Further, as indicated by reference numeral  406 , the signal x i  is phase-rotated in the direction opposite to that of the phase rotation indicated by reference numeral  403 , using the phase shift signal indicated by reference numeral  404 . 
         [0084]    As a result, the signal x i  is fed back as the feedback signal y i . In this case, the signal becomes u i =x i −b′. Therefore, an error ε i  between the feedback signal y i  theoretically calculated from the signal u i  and the actually measured feedback signal y i  in the circuit illustrated in  FIG. 3  is expressed by the following equation (1). Here, 1 to n are natural numbers. “a” is a coefficient for indicating the level of amplification of the signal u i  performed by the amplifier  305 . “b” indicates the modulating unit offset. “c” indicates the demodulating unit offset. “e −jwt ” indicates the phase rotation. 
         [0000]        au   i   +b+c×e   −jwt   −y   i =ε i   (1)
 
         [0085]    The DC offset detecting unit  311  produces the signal for the compensation value b′ of the modulating unit offset b by using the signal x i  in a period I of one cycle of e −jwt  and the feedback signal y i  to apply the least-squares method in the above equation (1). In this case, the DC offset detecting unit  311  applies the least-squares method in the above equation (1) to calculate the coefficient “a”, the coefficient “b”, and the coefficient “c” when the error ε i  in the following equation (2) is the smallest. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               au 
                               i 
                             
                             + 
                             b 
                             + 
                             
                               c 
                               × 
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   w 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                             
                             - 
                             
                               y 
                               i 
                             
                           
                           ) 
                         
                          
                         
                           
                             ( 
                             
                               
                                 au 
                                 i 
                               
                               + 
                               b 
                               + 
                               
                                 c 
                                 × 
                                 
                                    
                                   
                                     
                                       - 
                                       j 
                                     
                                      
                                     
                                         
                                     
                                      
                                     w 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                               
                               - 
                               
                                 y 
                                 i 
                               
                             
                             ) 
                           
                           * 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       
                          
                         
                           ɛ 
                           i 
                         
                          
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0086]    The error ε i  in the above equation (2) becomes the smallest when the following equations (3) to (5) that are equations differentiated from the above equation (2) become 0. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       [ 
                       
                         
                           u 
                           i 
                           * 
                         
                          
                         
                           ( 
                           
                             
                               au 
                               i 
                             
                             + 
                             b 
                             + 
                             
                               c 
                               × 
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   w 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                             
                             - 
                             
                               y 
                               i 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       [ 
                       
                         ( 
                         
                           
                             au 
                             i 
                           
                           + 
                           b 
                           + 
                           
                             c 
                             × 
                             
                                
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 w 
                                  
                                 
                                     
                                 
                                  
                                 t 
                               
                             
                           
                           - 
                           
                             y 
                             i 
                           
                         
                         ) 
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       [ 
                       
                         
                            
                           
                             j 
                              
                             
                                 
                             
                              
                             w 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                          
                         
                           ( 
                           
                             
                               au 
                               i 
                             
                             + 
                             b 
                             + 
                             
                               c 
                               × 
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   w 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                             
                             - 
                             
                               y 
                               i 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0087]    The following equation (6) is obtained when the above equations (3) to (5) are expressed as a matrix. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 
                                    
                                   
                                     u 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 u 
                                 i 
                                 * 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 
                                   u 
                                   i 
                                   * 
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                       j 
                                     
                                      
                                     
                                         
                                     
                                      
                                     w 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 u 
                                 i 
                               
                             
                           
                           
                             n 
                           
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   w 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 
                                   u 
                                   i 
                                 
                                  
                                 
                                    
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     w 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   w 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                             
                           
                           
                             n 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             a 
                           
                         
                         
                           
                             b 
                           
                         
                         
                           
                             c 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               
                                 y 
                                 i 
                               
                                
                               
                                 u 
                                 i 
                                 * 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               y 
                               i 
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               
                                 y 
                                 i 
                               
                                
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   w 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0088]    The following equation (7) is obtained when the above equation (6) is transformed. 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           a 
                         
                       
                       
                         
                           b 
                         
                       
                       
                         
                           c 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                   
                                      
                                     
                                       u 
                                       i 
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                   u 
                                   i 
                                   * 
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                   
                                     u 
                                     i 
                                     * 
                                   
                                    
                                   
                                      
                                     
                                       
                                         - 
                                         j 
                                       
                                        
                                       
                                           
                                       
                                        
                                       w 
                                        
                                       
                                           
                                       
                                        
                                       t 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                   u 
                                   i 
                                 
                               
                             
                             
                               n 
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                       j 
                                     
                                      
                                     
                                         
                                     
                                      
                                     w 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                   
                                     u 
                                     i 
                                   
                                    
                                   
                                      
                                     
                                       j 
                                        
                                       
                                           
                                       
                                        
                                       w 
                                        
                                       
                                           
                                       
                                        
                                       t 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                    
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     w 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                               
                             
                             
                               n 
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 
                                   y 
                                   i 
                                 
                                  
                                 
                                   u 
                                   i 
                                   * 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 y 
                                 i 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 
                                   y 
                                   i 
                                 
                                  
                                 
                                    
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     w 
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0089]    In this case, the DC offset detecting unit  311  calculates the modulating unit offset b by substituting the signal u i  calculated from the digital signal x i  that is to be transmitted in a period I of one cycle, and the feedback signal y i  in a period I of one cycle, into the above equation (7). The DC offset detecting unit  311  then outputs the calculated modulating unit offset b as the compensation value b′ of the modulating unit offset b to the subtractors  301 . 
         [0090]    The DC offset detecting unit  311  may output, as the compensation value b′, a value obtained by dividing the modulating unit offset b by the level of amplification a of the amplifier  305  when the calculated modulating unit offset b is a value amplified by the amplifier  305 . 
         [0091]    The error ε i  between the feedback signal y i  theoretically calculated from the signal u i  and the actually measured feedback signal y i  in the circuit illustrated in  FIG. 3  is expressed by the following equation (8). Therefore, the DC offset detecting unit  311  may calculate the modulating unit offset b by using an integral operator using the signal u i  calculated from the signal x i  in the period I of one cycle of e −jwt , and the feedback signal y i  in the period I of one cycle. 
         [0000]        ax+b+c×e   −jwt   −y=ε   (8)
 
         [0092]    The following equation (9) is obtained when the above equation (8) is integrated. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       1 
                     
                      
                     
                       ( 
                       
                         
                           a 
                            
                           
                               
                           
                            
                           
                             x 
                             j 
                           
                         
                         + 
                         b 
                         + 
                         
                           c 
                           × 
                           
                              
                             
                               
                                 - 
                                 j 
                               
                                
                               
                                   
                               
                                
                               w 
                                
                               
                                   
                               
                                
                               t 
                             
                           
                         
                         - 
                         
                           y 
                           j 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       1 
                     
                      
                     
                       ɛ 
                       j 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0093]    The following equation (10) is obtained when the above equation (9) is expanded. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       a 
                        
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           1 
                         
                          
                         
                           ( 
                           
                             x 
                             j 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       1 
                        
                       
                           
                       
                        
                       b 
                     
                     - 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         1 
                       
                        
                       
                         ( 
                         
                           y 
                           j 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       1 
                     
                      
                     
                       ɛ 
                       j 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0094]    The following equation (14) is obtained when the following equations (11) to (13) are used to transpose the above equation (10). 
         [0000]    
       
         
           
             
               
                 
                   
                     a 
                      
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         I 
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         
                           x 
                           j 
                         
                         ) 
                       
                     
                   
                   = 
                   X 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       I 
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         y 
                         j 
                       
                       ) 
                     
                   
                   = 
                   X 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       I 
                     
                      
                     
                         
                     
                      
                     
                       ɛ 
                       j 
                     
                   
                   = 
                   E 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       aX 
                       i 
                     
                     + 
                     
                       1 
                        
                       b 
                     
                     - 
                     
                       Y 
                       i 
                     
                   
                   = 
                   
                     E 
                     i 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0095]    In this case, the DC offset detecting unit  311  applies the least-squares method in the above equation (14) to calculate the coefficient “a” and the coefficient “b” when an error E i  in the following equation (15) is the smallest. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               aX 
                               i 
                             
                             + 
                             
                               1 
                                
                               
                                   
                               
                                
                               b 
                             
                             - 
                             Y 
                           
                           ) 
                         
                          
                         
                           
                             ( 
                             
                               
                                 aX 
                                 i 
                               
                               + 
                               
                                 1 
                                  
                                 b 
                               
                               - 
                               Y 
                             
                             ) 
                           
                           * 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         
                           E 
                           i 
                         
                          
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0096]    The error in the above equation (15) becomes the smallest when the following equations (16) and (17) that are equations differentiated from the above equation (15) become 0. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         
                           X 
                           i 
                           * 
                         
                          
                         
                           ( 
                           
                             
                               aX 
                               i 
                             
                             + 
                             
                               1 
                                
                               b 
                             
                             - 
                             
                               Y 
                               i 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         1 
                          
                         
                           ( 
                           
                             
                               aX 
                               i 
                             
                             + 
                             
                               1 
                                
                               b 
                             
                             - 
                             
                               Y 
                               i 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0097]    The following equation (18) is obtained when the above equations (16) and (17) are expressed as a matrix. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                    
                                   
                                     X 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                           
                             
                               n 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         
                           
                             
                               1 
                                
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   X 
                                   i 
                                 
                               
                             
                           
                           
                             
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             a 
                           
                         
                         
                           
                             b 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 Y 
                                 i 
                               
                                
                               
                                 X 
                                 i 
                                 * 
                               
                             
                           
                         
                       
                       
                         
                           
                             1 
                              
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                 Y 
                                 i 
                                 * 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0098]    The following equation (19) is obtained when the above equation (18) is transformed. 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           a 
                         
                       
                       
                         
                           b 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                      
                                     
                                       X 
                                       i 
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                           
                             
                               
                                 1 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     n 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     X 
                                     i 
                                   
                                 
                               
                             
                             
                               
                                 
                                     
                                 
                                  
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   Y 
                                   i 
                                 
                                  
                                 
                                   X 
                                   i 
                                   * 
                                 
                               
                             
                           
                         
                         
                           
                             
                               1 
                                
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   Y 
                                   i 
                                   * 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0099]    In this case, the DC offset detecting unit  311  calculates the modulating unit offset b by substituting the digital signal u i  to be transmitted and the input feedback signal y i , into the above equation (19). The DC offset detecting unit  311  then outputs the calculated modulating unit offset b as the compensation value b′ of the modulating unit offset b to the subtractors  301 . 
         [0100]    (Second Embodiment of Modulating Device  100 ) 
         [0101]      FIG. 5  illustrates a second embodiment of the modulating device  100 . As illustrated in  FIG. 5 , the modulating device  100  has two subtractors  501 , a DAC  502 , a QMOD  503 , a first oscillator  504 , an amplifier  505 , an analog EPS  506 , a second oscillator  507 , a QDEM  508 , an ADC  509 , a digital EPS  510 , and a DC offset detecting unit  511  in the same way as illustrated in  FIG. 3 . A processing flow of the I-signal among the I-signals and the Q-signals will be discussed hereinbelow. The processing flow of the Q-signal is the same as the processing flow of the I-signal and will be omitted. 
         [0102]    The two subtractors  501  are circuits that are similar to the two subtractors  301  illustrated in  FIG. 3  and thus an explanation will be omitted. The DAC  502  is a circuit similar to the DAC  302  illustrated in  FIG. 3  and thus an explanation will be omitted. The QMOD  503  is a circuit similar to the QMOD  303  illustrated in  FIG. 3  and thus an explanation will be omitted. The modulating unit offset b is added to the signal u by passing through the DAC  502  and the QMOD  503 , to become the signal u+b. 
         [0103]    The first oscillator  504  is a one-output circuit that outputs a local signal as an output signal. In the example illustrated in  FIG. 5 , a signal line for outputting the local signal to the QMOD  503  and a signal line for outputting the local signal to the QDEM  508  are connected to the first oscillator  504 . In  FIG. 5 , the first oscillator  504  branches the one output of the local signal to the QMOD  503  and to the QDEM  508 . 
         [0104]    The amplifier  505  is a one-input one-output circuit that amplifies the analog signal that is the input signal and outputs the amplified signal as an output signal. In the example in  FIG. 5 , a signal line for inputting the analog signal from the first oscillator  504 , a signal line for outputting the signal to the analog EPS  506 , and a signal line for outputting the signal to the antenna, are connected to the amplifier  505 . In  FIG. 5 , the amplifier  505  outputs, to the antenna and to the analog EPS  506  by branching, the one-output signal that is the quadrature-modulated signal that has been amplified. 
         [0105]    The analog EPS  506  is a two-input one-output circuit that performs phase rotation on the signal from the amplifier  505  that is one of the input signals by using the phase shift signal from the second oscillator  507  that is the other input signal, and outputs the phase-rotated signal as an output signal. In the example illustrated in  FIG. 5 , a signal line for inputting the signal from the amplifier  505 , a signal line for inputting the phase shift signal from the second oscillator  507 , and a signal line for outputting the signal to the QDEM  508  are connected to the analog EPS  506 . In  FIG. 5 , the analog EPS  506  uses the phase shift signal to perform phase rotation on the signal from the amplifier  505 , and outputs the phase-rotated signal to the QDEM  508 . The second oscillator  507  is a circuit similar to the second oscillator  307  illustrated in  FIG. 3  and thus an explanation will be omitted. 
         [0106]    The QDEM  508  is a two-input two-output circuit that uses the local signal that is one of the input signals to perform quadrature demodulation on the analog signal that is the other of the input signals, and outputs the quadrature-demodulated signal as an output signal. In the example in  FIG. 5 , a signal line for inputting the signal from the analog EPS  506 , a signal line for inputting the local signal from the first oscillator  504 , and a signal line for outputting the signal to the ADC  509 , are connected to the QDEM  508 . In  FIG. 5 , the QDEM  508  uses the local signal from the first oscillator  504  to perform quadrature demodulation on the analog signal from the analog EPS  506  to output the signal (u+b)*EXP(jwt) to the ADC  509 . 
         [0107]    The ADC  509  is a circuit similar to the ADC  309  illustrated in  FIG. 3  and thus an explanation will be omitted. The demodulating unit offset c is added to the signal (u+b)*EXP(jwt) that passes through the QDEM  508  and the ADC  509  to become the signal (u+b)*EXP(jwt)+c. The digital EPS  510  is a circuit similar to the digital EPS  310  illustrated in  FIG. 3  and thus an explanation will be omitted. The DC offset detecting unit  511  is a circuit similar to the DC offset detecting unit  311  illustrated in  FIG. 3  and thus an explanation will be omitted. 
         [0108]    As a result, the signal input into the amplifier  505  due to the DC offset detecting unit  511  inputting the compensation value b′ that is the same value as the modulating unit offset b into the subtractors  501 , becomes the signal u+b=u−b′+b≈u. Therefore, the modulating device  100  is able to remove the modulating unit offset b from the signal to be amplified and transmitted by the amplifier  505  and is able to transmit a signal with high accuracy. 
         [0109]    (Correction of Signals in Second Embodiment) 
         [0110]    In the circuits illustrated in  FIG. 5 , the compensation value b′ of the modulating unit offset b is subtracted from the signal x i , the modulating unit offset b is added to the signal x i  and then the signal x i  is phase-rotated in the same way as illustrated in  FIG. 4 . The demodulating unit offset c is then added to the signal x i  and the signal x i  is phase-rotated in the opposite direction to be then fed back as the feedback signal y i . Therefore, the DC offset detecting unit  511  in the circuits illustrated in  FIG. 5  calculates the modulating unit offset b and outputs the modulating unit offset b to the subtractors  501  as the compensation value b′ of the modulating unit offset b in the same way as illustrated in  FIG. 4 . 
         [0111]    (Third Embodiment of Modulating Device  100 ) 
         [0112]      FIG. 6  illustrates a third embodiment of the modulating device  100 . The third embodiment is a modified example of the first embodiment. 
         [0113]    As illustrated in  FIG. 6 , the modulating device  100  has two subtractors  601 , a DAC  602 , a QMOD  603 , a first oscillator  604 , an amplifier  605 , an analog EPS  606 , a second oscillator  607 , a QDEM  608 , an ADC  609 , a digital EPS  610 , and a DC offset detecting unit  611 . A processing flow of the I-signal among the I-signals and the Q-signals will be discussed hereinbelow. The processing flow of the Q-signal is the same as the processing flow of the I-signal and will be omitted. 
         [0114]    The two subtractors  601  are two-input one-output circuits that output, as an output signal, a subtraction result of the subtraction of one of the input signals from the other. In the example in  FIG. 6 , a signal line for inputting the digital I-signal to be transmitted and a signal line for inputting the signal for correcting the I-signal from the DC offset detecting unit  611  are connected to one of the subtractors  601 . A signal line for outputting a signal to the DAC  602  and to the DC offset detecting unit  611  is connected to the other subtractor  601 . 
         [0115]    A signal line for inputting the Q-signal that is the digital signal to be transmitted and a signal line for inputting the signal for correcting the Q-signal from the DC offset detecting unit  611  are connected to the other subtractor  601 . A signal line for outputting the signal to the DAC  602  and to the DC offset detecting unit  611  is connected to the other subtractor  601 . In  FIG. 6 , the subtractors  601  subtract the compensation value b′ of the modulating unit offset b input from the DC offset detecting unit  611 , from the input I-signal x to be transmitted, and branch and output the subtraction result u=x−b′ to the DAC  602  and to the DC offset detecting unit  611 . 
         [0116]    The DAC  602  is a circuit similar to the DAC  302  illustrated in  FIG. 3  and thus an explanation will be omitted. The QMOD  603  is a circuit similar to the QMOD  303  illustrated in  FIG. 3  and thus an explanation will be omitted. The first oscillator  604  is a circuit similar to the first oscillator  304  illustrated in  FIG. 3  and thus an explanation will be omitted. The amplifier  605  is a circuit similar to the amplifier  305  illustrated in  FIG. 3  and thus an explanation will be omitted. The analog EPS  606  is a circuit similar to the analog EPS  306  illustrated in  FIG. 3  and thus an explanation will be omitted. 
         [0117]    The second oscillator  607  is a circuit similar to the second oscillator  307  illustrated in  FIG. 3  and thus an explanation will be omitted. The QDEM  608  is a circuit similar to the QDEM  308  illustrated in  FIG. 3  and thus an explanation will be omitted. The ADC  609  is a circuit similar to the ADC  309  illustrated in  FIG. 3 , and thus an explanation will be omitted. The digital EPS  610  is a circuit similar to the digital EPS  310  illustrated in  FIG. 3  and thus an explanation will be omitted. 
         [0118]    The DC offset detecting unit  611  is a four-input two-output circuit that uses two feedback signals that are input signals from the digital EPS  610  and two signals from the subtractors  601  that are input signals, to output the compensation value b′ signal of the modulating unit offset b. In the example in  FIG. 6 , a signal line for inputting the two feedback signals from the digital EPS  610 , a signal line for inputting the two signals from the subtractors  601 , and a signal line for outputting the signals to the subtractors  601  are connected to the DC offset detecting unit  611 . In  FIG. 6 , the DC offset detecting unit  611  produces the compensation value b′ signal of the modulating unit offset b for correcting the two signals to be transmitted and outputs the compensation value b′ signal to the two subtractors  601  as described below with reference to  FIG. 7 . 
         [0119]    As a result, the signal input to the amplifier  605  becomes the signal u+b=u−b′+b≈u due to the DC offset detecting unit  611  inputting the compensation value b′ that becomes the same value as the modulating unit offset b into the subtractors  601 . Therefore, the modulating device  100  is able to remove the modulating unit offset b from the signal to be amplified and transmitted by the amplifier  605  and is able to transmit a signal with high accuracy. 
         [0120]    (Correction of Signals in Third Embodiment) 
         [0121]      FIG. 7  illustrates the correction of a signal according to the third embodiment.  FIG. 7  follows a direction reverse to the flow of the signal and illustrates a flow of the feedback signal y i  being returned to the signal x i  in the circuits in  FIG. 6 . 
         [0122]    In the circuits in  FIG. 6 , the feedback signal y i  is phase-rotated, as indicated by the reference numeral  701 , by using the phase shift signal indicated by the reference numeral  702 , and the demodulating unit offset c is subtracted from the feedback signal y i  as indicated by the reference numeral  703 . Further, as indicated by reference numeral  704 , the feedback signal y i  is phase-rotated in the direction opposite to that of the phase rotation indicated by reference numeral  701 , using the phase shift signal indicated by reference numeral  702 , and the modulating unit offset b is subtracted as indicated by the reference numeral  705 . As a result, the feedback signal y i  is returned to the signal u i . Therefore, in the circuits illustrated in  FIG. 6 , an error ε i  between the signal u i  theoretically calculated from the feedback signal y i  and the actually measured signal u i  is expressed by the following equation (20). 
         [0000]        ay   i   +b+c×e   −jwt   −u   i =ε i   (20)
 
         [0123]    The DC offset detecting unit  611  produces the signal for the compensation value b′ of the modulating unit offset b by using the signal x i  and the feedback signal y i  in a period I of one cycle of e −jwt  to apply the least-squares method in the above equation (20). In this case, the DC offset detecting unit  611  applies the least-squares method in the above equation (20) to calculate the coefficient “a”, the coefficient “b”, and the coefficient “c” when the error ε i  in the following equation (21) is the smallest. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               ay 
                               i 
                             
                             + 
                             b 
                             + 
                             
                               c 
                               × 
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                             - 
                             
                               u 
                               i 
                             
                           
                           ) 
                         
                          
                         
                           
                             ( 
                             
                               
                                 ay 
                                 i 
                               
                               + 
                               b 
                               + 
                               
                                 c 
                                 × 
                                 
                                    
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     wt 
                                   
                                 
                               
                               - 
                               
                                 u 
                                 i 
                               
                             
                             ) 
                           
                           * 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         
                           ɛ 
                           i 
                         
                          
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0124]    The error ε i  in the above equation (21) becomes the smallest when the following equations (22) to (24) that are equations differentiated from the above equation (21) become 0. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         
                           y 
                           i 
                           * 
                         
                          
                         
                           ( 
                           
                             
                               ay 
                               i 
                             
                             + 
                             b 
                             + 
                             
                               c 
                               × 
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                             - 
                             
                               u 
                               i 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         ( 
                         
                           
                             ay 
                             i 
                           
                           + 
                           b 
                           + 
                           
                             c 
                             × 
                             
                                
                               
                                 j 
                                  
                                 
                                     
                                 
                                  
                                 wt 
                               
                             
                           
                           - 
                           
                             u 
                             i 
                           
                         
                         ) 
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         
                            
                           
                             
                               - 
                               j 
                             
                              
                             
                                 
                             
                              
                             wt 
                           
                         
                          
                         
                           ( 
                           
                             
                               ay 
                               i 
                             
                             + 
                             b 
                             + 
                             
                               c 
                               × 
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                             - 
                             
                               u 
                               i 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0125]    The following equation (25) is obtained when the above equations (22) to (24) are expressed as a matrix. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                    
                                   
                                     y 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                 y 
                                 i 
                                 * 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   y 
                                   i 
                                   * 
                                 
                                  
                                 
                                    
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     wt 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                 y 
                                 i 
                               
                             
                           
                           
                             n 
                           
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   y 
                                   i 
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                       j 
                                     
                                      
                                     
                                         
                                     
                                      
                                     wt 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                           
                           
                             n 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             a 
                           
                         
                         
                           
                             b 
                           
                         
                         
                           
                             c 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 u 
                                 i 
                               
                                
                               
                                 y 
                                 i 
                                 * 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               u 
                               i 
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 u 
                                 i 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0126]    The following equation (26) is obtained when the above equation (25) is transformed. 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           a 
                         
                       
                       
                         
                           b 
                         
                       
                       
                         
                           c 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
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                   ( 
                   26 
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         [0127]    In this case, the DC offset detecting unit  611  calculates the modulating unit offset b by substituting the digital signal u i  in a period I of one cycle, and the feedback signal y i  in a period I of one cycle, into the above equation (26). The DC offset detecting unit  611  then outputs the calculated modulating unit offset b as the compensation value b′ of the modulating unit offset b to the subtractors  601 . 
         [0128]    (Fourth Embodiment of Modulating Device  100 ) 
         [0129]      FIG. 8  illustrates a fourth embodiment of the modulating device  100 . The fourth embodiment is a modified example of the second embodiment. 
         [0130]    As illustrated in  FIG. 8 , the modulating device  100  has two subtractors  801 , a DAC  802 , a QMOD  803 , a first oscillator  804 , an amplifier  805 , an analog EPS  806 , a second oscillator  807 , a QDEM  808 , an ADC  809 , a digital EPS  810 , and a DC offset detecting unit  811 , in the same way as  FIG. 5 . A processing flow of the I-signal among the I-signals and the Q-signals will be discussed hereinbelow. The processing flow of the Q-signal is the same as the processing flow of the I-signal and thus an explanation will be omitted. 
         [0131]    The two subtractors  801  are two-input one-output circuits that output, as an output signal, a subtraction result of the subtraction of one of the input signals from the other. In the example in  FIG. 8 , a signal line for inputting a digital I-signal to be transmitted and a signal line for inputting a signal for correcting the I-signal from the DC offset detecting unit  811  are connected to one of the subtractors  801 . A signal line for outputting the signal to the DAC  802  and to the DC offset detecting unit  811  is connected to one of the subtractors  801 . 
         [0132]    A signal line for inputting a Q-signal that is a digital signal to be transmitted and a signal line for inputting a signal for correcting the Q-signal from the DC offset detecting unit  811  are connected to the other subtractor  801 . A signal line for outputting the signal to the DAC  802  and to the DC offset detecting unit  811  is connected to the other subtractor  801 . In  FIG. 8 , the subtractors  801  subtract the compensation value b′ of the modulating unit offset b input from the DC offset detecting unit  811 , from the input I-signal x that is to be transmitted, and output by branching one output of the subtraction result u=x-b′ to the DAC  802  and to the DC offset detecting unit  811 . 
         [0133]    The DAC  802  is a circuit similar to the DAC  502  illustrated in  FIG. 5  and thus an explanation will be omitted. The QMOD  803  is a circuit similar to the QMOD  503  illustrated in  FIG. 5  and thus an explanation will be omitted. The first oscillator  804  is a circuit similar to the first oscillator  504  illustrated in  FIG. 5  and thus an explanation will be omitted. The amplifier  805  is a circuit similar to the amplifier  805  illustrated in  FIG. 5  and thus an explanation will be omitted. The analog EPS  806  is a circuit similar to the analog EPS  506  illustrated in  FIG. 5 , and thus an explanation will be omitted. 
         [0134]    The second oscillator  807  is a circuit similar to the second oscillator  507  illustrated in  FIG. 5  and thus an explanation will be omitted. The QDEM  808  is a circuit similar to the QDEM  508  illustrated in  FIG. 5  and thus an explanation will be omitted. The ADC  809  is a circuit similar to the ADC  509  illustrated in  FIG. 5  and thus an explanation will be omitted. The digital EPS  810  is a circuit similar to the digital EPS  510  illustrated in  FIG. 5  and thus an explanation will be omitted. The DC offset detecting unit  811  is a circuit similar to the DC offset detecting unit  611  illustrated in  FIG. 6  and thus an explanation will be omitted. 
         [0135]    As a result, the signal input to the amplifier  805  becomes the signal u+b=u−b′+b≈u due to the DC offset detecting unit  811  inputting the compensation value b′ that becomes the same value as the modulating unit offset b into the subtractors  801 . Therefore, the modulating device  100  is able to remove the modulating unit offset b from the signal to be amplified and transmitted by the amplifier  805  and is able to transmit a signal with high accuracy. 
         [0136]    (Correction of Signals in Fourth Embodiment) 
         [0137]    In the circuits illustrated in  FIG. 8 , the feedback signal y i  is phase-rotated and the demodulating unit offset c is added to the feedback signal y i  in the same way as illustrated in  FIG. 7 . The feedback signal y i  is then phase-rotated in the opposite direction and the modulating unit offset b is added to the feedback signal y i  to be returned to the signal u i . Therefore, the DC offset detecting unit  811  in the circuits illustrated in  FIG. 8  calculates the modulating unit offset b and outputs the modulating unit offset b to the subtractors  801  as the compensation value b′ of the modulating unit offset b in the same way as illustrated in  FIG. 7 . 
         [0138]    (Fifth Embodiment of Modulating Device  100 ) 
         [0139]      FIG. 9  illustrates a fifth embodiment of the modulating device  100 . The fifth embodiment is a modified example of the first embodiment. 
         [0140]    As illustrated in  FIG. 9 , the modulating device  100  has two subtractors  901 , a DAC  902 , a QMOD  903 , a first oscillator  904 , an amplifier  905 , an analog EPS  906 , a second oscillator  907 , a QDEM  908 , an ADC  909 , a digital EPS  910 , and a DC offset detecting unit  911 . A processing flow of the I-signal among the I-signals and the Q-signals will be discussed hereinbelow. The processing flow of the Q-signal is the same as the processing flow of the I-signal and thus an explanation will be omitted. 
         [0141]    The two subtractors  901  are circuits that are similar to the two subtractors  301  illustrated in  FIG. 3  and thus an explanation will be omitted. The DAC  902  is a circuit similar to the DAC  302  illustrated in  FIG. 3  and thus an explanation will be omitted. The QMOD  903  is a circuit similar to the QMOD  303  illustrated in  FIG. 3  and thus an explanation will be omitted. The first oscillator  904  is a circuit similar to the first oscillator  304  illustrated in  FIG. 3  and thus an explanation will be omitted. The amplifier  905  is a circuit similar to the amplifier  305  illustrated in  FIG. 3  and thus an explanation will be omitted. The analog EPS  906  is a circuit similar to the analog EPS  306  illustrated in  FIG. 3  and thus an explanation will be omitted. 
         [0142]    The second oscillator  907  is a circuit similar to the second oscillator  307  illustrated in  FIG. 3  and thus an explanation will be omitted. The QDEM  908  is a circuit similar to the QDEM  308  illustrated in  FIG. 3  and thus an explanation will be omitted. The ADC  909  is a circuit similar to the ADC  309  illustrated in  FIG. 3 , and thus an explanation will be omitted. 
         [0143]    The DC offset detecting unit  911  is a four-input two-output circuit that uses two signals that are input signals from the ADC  909  and two signals that are input signals from the digital EPS  910 , to output the compensation value b′ signal of the modulating unit offset b. In the example in  FIG. 9 , a signal line for inputting the two signals from the ADC  909 , a signal line for inputting the two signals from the digital EPS  910 , and a signal line for outputting the signal to the subtractors  901  are connected to the DC offset detecting unit  911 . In  FIG. 9 , the DC offset detecting unit  911  produces the compensation value b′ signal of the modulating unit offset b for correcting the two signals to be transmitted, to output the compensation value b′ signal to the two subtractors  901  as described below with reference to  FIG. 4 . 
         [0144]    The digital EPS  910  is a three-input two-output circuit that uses the phase shift signal that is the remaining input signal to perform phase rotation on the two signals that are the input signals, and outputs the phase-rotated signal as an output signal. In the example in  FIG. 9 , a signal line for inputting the two digital signals to be transmitted, a signal line for inputting the phase shift signal from the second oscillator  907 , and a signal line for outputting the signal to the DC offset detecting unit  911  are connected to the digital EPS  910 . In  FIG. 9 , the digital EPS  910  performs phase rotation on the two input signals in the opposite direction and by a phase shift quantity that is the same as that of the phase rotation by the analog EPS  906 , on the two input signals, and outputs the phase-rotated signals to the DC offset detecting unit  911 . 
         [0145]    As a result, the signal input to the amplifier  905  becomes the signal u+b=u−b′+b≈u due to the DC offset detecting unit  911  inputting the compensation value b′ that becomes the same value as the modulating unit offset b into the subtractors  901 . Therefore, the modulating device  100  is able to remove the modulating unit offset b from the signal to be amplified and transmitted by the amplifier  905  and is able to transmit a signal with high accuracy. 
         [0146]    (Correction of Signals in Fifth Embodiment) 
         [0147]      FIG. 10  illustrates a correction of a signal according to the fifth embodiment.  FIG. 10  illustrates a flow of the signal x i  until the signal x i  becomes a feedback signal y i  in the circuits illustrated in  FIG. 9 . 
         [0148]    In the circuit illustrated in  FIG. 9 , the compensation value b′ of the modulating unit offset b as indicated by reference numeral  1001  is subtracted from the signal x i , and the modulating unit offset b as indicated by the reference numeral  1002  is added to the signal x i . Further, the signal x i  is phase-rotated, as indicated by reference numeral  1003 , using the phase shift signal indicated by reference numeral  1004 , and the demodulating unit offset c is added to the signal x i  as indicated by reference numeral  1005 . As a result, the signal x i  is fed back as the signal y i . Therefore, in the circuits illustrated in  FIG. 9 , an error ε i  between the signal y i  theoretically calculated from the feedback signal u i  and the actually measured signal y i  is expressed by the following equation (27). 
         [0000]      ( au   i   +b ) e   jwt   +c−y   i   ×e   jwt =ε i   (27)
 
         [0149]    The following equation (28) is obtained when the above equation (27) is expanded. 
         [0000]        au   i   e   jwt   +be   jwt   +c−y   i   ×e   jwt =ε i   (28)
 
         [0150]    The following equation (29) is obtained when the above equation (28) is transformed. 
         [0000]        au   i   +b+c×e   −jwt   −y   i   =ε×e   −jwt   (29)
 
         [0151]    The DC offset detecting unit  911  produces the signal for the compensation value b′ of the modulating unit offset b by using the signal x i  and the feedback signal y i  in a period I of one cycle of e −jwt  to apply the least-squares method in the above equation (29). In this case, the DC offset detecting unit  911  applies the least-squares method in the above equation (29) to calculate the coefficient “a”, the coefficient “b”, and the coefficient “c” when the error ε i  in the following equation (30) is the smallest. 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0152]    The subsequent processing is similar to the processing described with reference to  FIG. 4  and thus an explanation will be omitted. 
         [0153]    (Sixth Embodiment of Modulating Device  100 ) 
         [0154]      FIG. 11  illustrates a sixth embodiment of the modulating device  100 . The sixth embodiment is a modified example of the second embodiment. 
         [0155]    As illustrated in  FIG. 11 , the modulating device  100  has two subtractors  1101 , a DAC  1102 , a QMOD  1103 , a first oscillator  1104 , an amplifier  1105 , an analog EPS  1106 , a second oscillator  1107 , a QDEM  1108 , an ADC  1109 , a digital EPS  1110 , and a DC offset detecting unit  1111 . A processing flow of the I-signal among the I-signals and the Q-signals will be discussed hereinbelow. The processing flow of the Q-signal is the same as the processing flow of the I-signal and thus an explanation will be omitted. 
         [0156]    The two subtractors  1101  are circuits that are similar to the two subtractors  501  illustrated in  FIG. 5  and thus an explanation will be omitted. The DAC  1102  is a circuit similar to the DAC  502  illustrated in  FIG. 5  and thus an explanation will be omitted. The QMOD  1103  is a circuit similar to the QMOD  503  illustrated in  FIG. 5  and thus an explanation will be omitted. The first oscillator  1104  is a circuit similar to the first oscillator  504  illustrated in  FIG. 5  and thus an explanation will be omitted. The amplifier  1105  is a circuit similar to the amplifier  505  illustrated in  FIG. 5  and thus an explanation will be omitted. The analog EPS  1106  is a circuit similar to the analog EPS  506  illustrated in  FIG. 5 , and thus an explanation will be omitted. 
         [0157]    The second oscillator  1107  is a circuit similar to the second oscillator  507  illustrated in  FIG. 5  and thus an explanation will be omitted. The QDEM  1108  is a circuit similar to the QDEM  508  illustrated in  FIG. 5  and thus an explanation will be omitted. The ADC  1109  is a circuit similar to the ADC  509  illustrated in  FIG. 5  and thus an explanation will be omitted. The digital EPS  1110  is a circuit similar to the digital EPS  910  illustrated in  FIG. 9  and thus an explanation will be omitted. The DC offset detecting unit  1111  is a circuit similar to the DC offset detecting unit  911  illustrated in  FIG. 9  and thus an explanation will be omitted. 
         [0158]    As a result, the signal input to the amplifier  1105  becomes the signal u+b=u−b′+b≈u due to the DC offset detecting unit  1111  inputting the compensation value b′ that becomes the same value as the modulating unit offset b input into the subtractors  1101 . Therefore, the modulating device  100  is able to remove the modulating unit offset b from the signal to be amplified and transmitted by the amplifier  1105  and is able to transmit a signal with high accuracy. 
         [0159]    (Correction of Signals in Sixth Embodiment) 
         [0160]    In the circuits illustrated in  FIG. 11 , the compensation value b′ of the modulating unit offset b is subtracted from the signal x i , the modulating unit offset b is added to the signal x i , and then the signal x i  is phase-rotated in the same way as illustrated in  FIG. 10 . The demodulating unit offset c is then added to the signal x i  and the signal x i  is then fed back as the signal y i . Therefore, the DC offset detecting unit  1111  in the circuits illustrated in  FIG. 11  calculates the modulating unit offset b and outputs the modulating unit offset b to the subtractors  1101  as the compensation value b′ of the modulating unit offset b in the same way as illustrated in  FIG. 10 . 
         [0161]    (Seventh Embodiment of Modulating Device  100 ) 
         [0162]      FIG. 12  illustrates a seventh embodiment of the modulating device  100 . The seventh embodiment is one in which a further function for compensating a distortion produced in an analog signal in the amplifier is added to the modulating device  100  according to the first embodiment. 
         [0163]    As illustrated in  FIG. 12 , the modulating device  100  has a digital pre-distortion (DPD)  1201 , an adder  1202 , a DAC  1203 , a QMOD  1204 , a first oscillator  1205 , an amplifier  1206 , an analog EPS  1207 , a second oscillator  1208 , a QDEM  1209 , an ADC  1210 , a digital EPS  1211 , an identifier  1212 , and a subtractor  1213 . In the following explanation, the signal x is described as a signal representing a complex number formed by combining the I-signal and the Q-signal. 
         [0164]    The DPD  1201  is a two-input one-output circuit that compensates a signal to be transmitted that is one of the input signals by using a signal for correcting distortion that is produced in the analog signal in the amplifier  1206  and that is the other of the input signals, and outputs the compensated signal as an output signal. In the example illustrated in  FIG. 12 , a signal line for inputting a digital signal to be transmitted and a signal line for inputting, from the identifier  1212 , a signal for compensating the distortion produced in the analog signal in the amplifier  1206 , are connected to the DPD  1201 . A signal line for outputting the signal to the adder  1202  is also connected to the DPD  1201 . In  FIG. 12 , the DPD  1201  uses values a1 and a3 that have nonlinear reverse characteristics, that are input from the identifier  1212 , and that become signals for compensating the distortion produced in the analog signal in the amplifier  1206 , to compensate the input signal x that is to be transmitted. The DPD  1201  then outputs a compensation result x′ to the adder  1202 . 
         [0165]    The adder  1202  is a two-input one-output circuit that outputs, as an output signal, an addition result of the addition of one of the input signals to the other. In the example illustrated in  FIG. 12 , a signal line for inputting signals from the DPD  1201  and a signal line for inputting the compensation value −b′ signal of the modulating unit offset b for compensating the DC offset from the identifier  1212 , are connected to the adder  1202 . A signal line for outputting the signal to the DAC  1203  and to the subtractor  1213  is also connected to the adder  1202 . In  FIG. 12 , the adder  1202  adds the compensation value b′ of the modulating unit offset b input from the identifier  1212  to the signal xi′ and outputs the one output addition result u=x′+b′ by branching the result to the DAC  1203  and to the subtractor  1213 . 
         [0166]    The DAC  1203  is a one-input one-output circuit that performs analog conversion on a digital signal that is an input signal, and outputs the analog-converted signals as an output signal. In the example in  FIG. 12 , a signal line for inputting the digital signal from the adder  1202 , and a signal line for outputting the signal to the QMOD  1204  are connected to the DAC  1203 . In FIG.  12 , the DAC  1203  performs analog conversion on the input digital signal u and outputs the analog-converted signal to the QMOD  1204 . The analog-converted signal may be referred to by “u” in the same way as the digital signal in the following explanation. 
         [0167]    The QMOD  1204  is a two-input one-output circuit that uses the local signal that is the remaining input signal to perform quadrature modulation on the analog signal that is the input signal, and outputs the quadrature-modulated signal as an output signal. In the example illustrated in  FIG. 12 , a signal line for inputting the analog signal from the DAC  1203 , a signal line for inputting the local signal from the first oscillator  1205 , and a signal line for outputting the signal to the amplifier  1206  are connected to the QMOD  1204 . In  FIG. 12 , the QMOD  1204  performs quadrature modulation on the input signal u by using the local signal from the first oscillator  1205  and outputs the quadrature-modulated signal to the amplifier  1206 . The modulating unit offset b is added to the signal u by passing through the DAC  1203  and the QMOD  1204 , to become the signal u+b. 
         [0168]    The first oscillator  1205  is a circuit similar to the first oscillator  304  illustrated in  FIG. 3  and thus an explanation will be omitted. The amplifier  1206  is a circuit similar to the amplifier  305  illustrated in  FIG. 3  and thus an explanation will be omitted. The analog EPS  1207  is a circuit similar to the analog EPS  306  illustrated in  FIG. 3  and thus an explanation will be omitted. The second oscillator  1208  is a circuit similar to the second oscillator  307  illustrated in  FIG. 3  and thus an explanation will be omitted. 
         [0169]    The QDEM  1209  is a two-input one-output circuit that uses the local signal that is phase-rotated by the phase shift signal that is one of the input signals to perform quadrature demodulation on the analog signal that is the other of the input signals, and outputs the quadrature-demodulated signal as an output signal. In the example illustrated in  FIG. 12 , a signal line for inputting the signal from the amplifier  1206 , a signal line for inputting the signal from the analog EPS  1207 , and a signal line for outputting the signal to the ADC  1210  are connected to the QDEM  1209 . In  FIG. 12 , the QDEM  1209  uses the signal from the analog EPS  1207  to perform quadrature demodulation on the analog signal from the amplifier  1206  and outputs the signal (u+b)*EXP(jwt) to the ADC  1210 . 
         [0170]    The ADC  1210  is a one-input one-output circuit that performs digital conversion on the analog signal that is the input signal, and outputs the digital-converted signal as an output signal. In the example in  FIG. 12 , a signal line for inputting the signal from the QDEM  1209 , and a signal line for outputting the signal to the digital EPS  1211  are connected to the ADC  1210 . In  FIG. 12 , the ADC  1210  performs digital conversion on the signal (u+b)*EXP(jwt) input from the QDEM  1209 , and outputs the digitally converted signal to the digital EPS  1211 . The demodulating unit offset c is added to the signal (u+b)*EXP(jwt) that passes through the QDEM  1209  and the ADC  1210  to become the signal (u+b)*EXP(jwt)+c. 
         [0171]    The digital EPS  1211  is a two-input one-output circuit that uses the phase shift signal that is the remaining input signal to perform phase rotation on the signal that is one of the input signals, and outputs the phase-rotated signal as an output signal. In the example illustrated in  FIG. 12 , a signal line for inputting the signal from the ADC  1210 , a signal line for inputting the phase shift signal from the second oscillator  1208 , and a signal line for outputting the signal to the identifier  1212 , are connected to the digital EPS  1211 . In  FIG. 12 , the digital EPS  1211  uses the phase shift signal from the second oscillator  1208  to perform phase rotation on the input signal in the opposite direction and by a phase shift quantity that is the same as that of the phase rotation by the analog EPS  1207 , and outputs the phase-rotated signal to the identifier  1212 . 
         [0172]    The identifier  1212  is a two-input three-output circuit that uses the signal from the digital EPS  1211  that is an input signal, and a signal from the subtractor  1213  that is an input signal, to output the nonlinear reverse characteristic values a1 and a3 and the compensation value b′ signal of the modulating unit offset b. In the example illustrated in  FIG. 12 , a signal line for inputting the signal from the digital EPS  1211 , a signal line for inputting the signal from the subtractor  1213 , and a signal line for outputting the signal to the DPD  1201 , the adder  1202 , and to the subtractor  1213 , are connected to the identifier  1212 . In  FIG. 12 , the identifier  1212  produces signals of the nonlinear reverse characteristic values a1 and a3 for compensating the distortion produced on the analog signal in the amplifier  1206  and outputs the produced signals to the DPD  1201  as described below in reference to  FIG. 13 . The identifier  1212  produces the compensation value b′ signal of the modulating unit offset b for correcting the digital signal to be transmitted and outputs the compensation value b′ signal to the adder  1202  and to the subtractor  1213  as described below with reference to  FIG. 13 . 
         [0173]    The subtractor  1213  is a two-input one-output circuit that outputs, as an output signal, a subtraction result of the subtraction of one of the input signals from the other. In the example illustrated in  FIG. 12 , a signal line for inputting the signal from the adder  1202 , a signal line for inputting the signal from the identifier  1212 , and a signal line for outputting the signal to the identifier  1212 , are connected to the subtractor  1213 . In  FIG. 12 , the subtractor  1213  subtracts the signal input from the identifier  1212  from the signal u input from the adder  1202 , and outputs the subtraction result to the identifier  1212 . As a result, the modulating device  100  is able to compensate the distortion of the signal in the amplifier  1206  and the modulating unit offset b, and is able to transmit a signal with high accuracy. 
         [0174]    (Correction of Signal in Seventh Embodiment) 
         [0175]      FIG. 13  illustrates a correction of a signal according to the seventh embodiment. 
         [0176]    In the circuits in  FIG. 12 , the feedback signal y i  is phase-rotated, as indicated by the reference numeral  1301 , by using the phase shift signal indicated by the reference numeral  1302 , and the demodulating unit offset c is subtracted from the feedback signal y i  as indicated by the reference numeral  1303 . Further, as indicated by reference numeral  1304 , the feedback signal y i  is phase-rotated in the direction opposite to that of the phase rotation indicated by reference numeral  1301 , by using the phase shift signal indicated by reference numeral  1302 . The feedback signal y i  is then amplified in the negative direction as indicated by the reference numeral  1305  and the modulating unit offset b is subtracted as indicated by the reference numeral  1306 . As a result, the feedback signal y i  is returned to the signal u i . Therefore, in the circuits illustrated in  FIG. 12 , an error ε i  between the signal u i  theoretically calculated from the feedback signal y i  and the actually measured signal u i  is expressed by the following equation (31). 
         [0000]        a   1 ( y   i   +c×e   jwt )+ a   3   |y   i   +c×e   jwt | 2 ( y   i   +c×e   jwt )+ b−u   i =ε i   (31)
 
         [0177]    Since the offset is a small value with respect to the signal, the above equation (31) approximates the following equation (32). 
         [0000]        a   1   y   1   +a   1   c×e   jwt   +a   3   |y   i | 2   y   i   +b−u   i =ε i   (32)
 
         [0178]    The identifier  1212  produces the signal for compensating the two digital signals that are to be transmitted, by applying the least-squares method to the above equation (32). In this case, the identifier  1212  applies the least-squares method in the above equation (32) to calculate the coefficient “a 1 ”, the coefficient “a 3 ”, the coefficient “b”, and the coefficient “c” when the error ε i  in the following equation (33) is the smallest. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 a 
                                 1 
                               
                                
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               
                                 a 
                                 3 
                               
                                
                               
                                 
                                    
                                   
                                     y 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                 y 
                                 i 
                               
                             
                             + 
                             b 
                             + 
                             
                               
                                 a 
                                 1 
                               
                                
                               c 
                               × 
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   w 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                             
                             - 
                             
                               u 
                               i 
                             
                           
                           ) 
                         
                          
                         
                           
 
                         
                          
                         
                           
                             ( 
                             
                               
                                 
                                   a 
                                   1 
                                 
                                  
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   3 
                                 
                                  
                                 
                                   
                                      
                                     
                                       y 
                                       i 
                                     
                                      
                                   
                                   2 
                                 
                                  
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               b 
                               + 
                               
                                 
                                   a 
                                   1 
                                 
                                  
                                 c 
                                 × 
                                 
                                    
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     wt 
                                   
                                 
                               
                               - 
                               
                                 u 
                                 i 
                               
                             
                             ) 
                           
                           * 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         
                           ɛ 
                           i 
                         
                          
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
         [0179]    The error ε i  in the above equation (33) becomes the smallest when the following equations (34) to (37) that are equations differentiated from the above equation (33) become 0. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         
                           y 
                           i 
                           * 
                         
                          
                         
                           ( 
                           
                             
                               
                                 a 
                                 1 
                               
                                
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               
                                 a 
                                 3 
                               
                                
                               
                                 
                                    
                                   
                                     y 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                 y 
                                 i 
                               
                             
                             + 
                             b 
                             + 
                             
                               
                                 a 
                                 1 
                               
                                
                               c 
                               × 
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                             - 
                             
                               u 
                               i 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         
                           
                              
                             
                               y 
                               i 
                             
                              
                           
                           2 
                         
                          
                         
                           
                             y 
                             i 
                             * 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   a 
                                   1 
                                 
                                  
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               
                                 
                                   a 
                                   3 
                                 
                                  
                                 
                                   
                                      
                                     
                                       y 
                                       i 
                                     
                                      
                                   
                                   2 
                                 
                                  
                                 
                                   y 
                                   i 
                                 
                               
                               + 
                               b 
                               + 
                               
                                 
                                   a 
                                   1 
                                 
                                  
                                 c 
                                 × 
                                 
                                    
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     wt 
                                   
                                 
                               
                               - 
                               
                                 u 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         ( 
                         
                           
                             
                               a 
                               1 
                             
                              
                             
                               y 
                               i 
                             
                           
                           + 
                           
                             
                               a 
                               3 
                             
                              
                             
                               
                                  
                                 
                                   y 
                                   i 
                                 
                                  
                               
                               2 
                             
                              
                             
                               y 
                               i 
                             
                           
                           + 
                           b 
                           + 
                           
                             
                               a 
                               1 
                             
                              
                             c 
                             × 
                             
                                
                               
                                 j 
                                  
                                 
                                     
                                 
                                  
                                 wt 
                               
                             
                           
                           - 
                           
                             u 
                             i 
                           
                         
                         ) 
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       [ 
                       
                         
                            
                           
                             j 
                              
                             
                                 
                             
                              
                             wt 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 a 
                                 1 
                               
                                
                               
                                 y 
                                 i 
                               
                             
                             + 
                             
                               
                                 a 
                                 3 
                               
                                
                               
                                 
                                    
                                   
                                     y 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                 y 
                                 i 
                               
                             
                             + 
                             b 
                             + 
                             
                               
                                 a 
                                 1 
                               
                                
                               c 
                               × 
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                             - 
                             
                               u 
                               i 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0180]    The following equation (38) is obtained when the above equations (34) to (37) are expressed as a matrix. 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                  
                                 
                                   y 
                                   i 
                                 
                                  
                               
                               2 
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                  
                                 
                                   y 
                                   i 
                                 
                                  
                               
                               4 
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               y 
                               i 
                               * 
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 y 
                                 i 
                                 * 
                               
                                
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                  
                                 
                                   y 
                                   i 
                                 
                                  
                               
                               4 
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                  
                                 
                                   y 
                                   i 
                                 
                                  
                               
                               6 
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                    
                                   
                                     y 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                 y 
                                 i 
                                 * 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                    
                                   
                                     y 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                 y 
                                 i 
                                 * 
                               
                                
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               y 
                               i 
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                    
                                   
                                     y 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                 y 
                                 i 
                               
                             
                           
                         
                         
                           n 
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                                
                               
                                 j 
                                  
                                 
                                     
                                 
                                  
                                 wt 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 y 
                                 i 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                    
                                   
                                     y 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                 y 
                                 i 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   wt 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                                 
                             
                              
                             
                                
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 wt 
                               
                             
                           
                         
                         
                           n 
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 a 
                                 1 
                               
                             
                           
                           
                             
                               
                                 a 
                                 3 
                               
                             
                           
                           
                             
                               b 
                             
                           
                           
                             
                               
                                 
                                   a 
                                   1 
                                 
                                  
                                 c 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     u 
                                     i 
                                   
                                    
                                   
                                     y 
                                     i 
                                     * 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                        
                                       
                                         y 
                                         i 
                                       
                                        
                                     
                                     2 
                                   
                                    
                                   
                                     y 
                                     i 
                                     * 
                                   
                                    
                                   
                                     u 
                                     i 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   u 
                                   i 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     n 
                                   
                                    
                                   
                                     
                                       u 
                                       i 
                                     
                                      
                                     
                                        
                                       
                                         
                                           - 
                                           j 
                                         
                                          
                                         
                                             
                                         
                                          
                                         wt 
                                       
                                     
                                   
                                 
                                  
                                 
                                     
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
         [0181]    The following equation (39) is obtained when the above equation (38) is transformed. 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             a 
                             1 
                           
                         
                       
                       
                         
                           
                             a 
                             3 
                           
                         
                       
                       
                         
                           b 
                         
                       
                       
                         
                           
                             
                               a 
                               1 
                             
                              
                             c 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                      
                                     
                                       y 
                                       i 
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                      
                                     
                                       y 
                                       i 
                                     
                                      
                                   
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   y 
                                   i 
                                   * 
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     y 
                                     i 
                                     * 
                                   
                                    
                                   
                                      
                                     
                                       j 
                                        
                                       
                                           
                                       
                                        
                                       wt 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                      
                                     
                                       y 
                                       i 
                                     
                                      
                                   
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                      
                                     
                                       y 
                                       i 
                                     
                                      
                                   
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                        
                                       
                                         y 
                                         i 
                                       
                                        
                                     
                                     2 
                                   
                                    
                                   
                                     y 
                                     i 
                                     * 
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                        
                                       
                                         y 
                                         i 
                                       
                                        
                                     
                                     2 
                                   
                                    
                                   
                                     y 
                                     i 
                                     * 
                                   
                                    
                                   
                                      
                                     
                                       j 
                                        
                                       
                                           
                                       
                                        
                                       wt 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
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                                         y 
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                         ] 
                       
                       
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                         1 
                       
                     
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                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0182]    In this case, the coefficient c is found with the following equation (40). 
         [0000]    
       
         
           
             
               
                 
                   c 
                   = 
                   
                     
                       
                         a 
                         1 
                       
                        
                       c 
                     
                     
                       a 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0183]    In this case, the identifier  1212  calculates the coefficients a 1  and a 3  and the compensation value b′ of the modulating unit offset b by substituting the digital signal u i  in the period I of one cycle and that is to be transmitted and the feedback signal y i  in the period I of one cycle, into the above equation (39). The identifier  1212  then outputs the calculated coefficients a l  and a 3  to the DPD  1201  and outputs the calculated compensation value b′ of the modulating unit offset b to the adder  1202 . A further function for compensating the distortion produced in an analog signal in the amplifier  1206  may be further added to the modulating device  100  according to the third and fifth embodiments in the same way as illustrated in  FIG. 12 . 
         [0184]    (Eighth Embodiment of Modulating Device  100 ) 
         [0185]      FIG. 14  illustrates an eighth embodiment of the modulating device  100 . The eighth embodiment is one in which a further function for compensating the distortion produced in an analog signal in the amplifier is added to the modulating device  100  according to the second embodiment. 
         [0186]    As illustrated in  FIG. 14 , the modulating device  100  has a DPD  1401 , an adder  1402 , a DAC  1403 , a QMOD  1404 , a first oscillator  1405 , an amplifier  1406 , an analog EPS  1407 , a second oscillator  1408 , a QDEM  1409 , an ADC  1410 , a digital EPS  1411 , an identifier  1412 , and a subtractor  1413 . In the following explanation, the signal x is described as a signal representing a complex number formed by combining the I-signal and the Q-signal. 
         [0187]    The DPD  1401  is a circuit similar to the DPD  1201  illustrated in  FIG. 12  and thus an explanation will be omitted. The adder  1402  is a circuit similar to the adder  1202  illustrated in  FIG. 12  and thus an explanation will be omitted. The DAC  1403  is a circuit similar to the DAC  1203  illustrated in  FIG. 12  and thus an explanation will be omitted. The QMOD  1404  is a circuit similar to the QMOD  1204  illustrated in  FIG. 12  and thus an explanation will be omitted. 
         [0188]    The first oscillator  1405  is a circuit similar to the first oscillator  504  illustrated in  FIG. 5  and thus an explanation will be omitted. The amplifier  1406  is a circuit similar to the amplifier  505  illustrated in  FIG. 5  and thus an explanation will be omitted. The analog EPS  1407  is a circuit similar to the analog EPS  506  illustrated in  FIG. 5 , and thus an explanation will be omitted. The second oscillator  1408  is a circuit similar to the second oscillator  507  illustrated in  FIG. 5  and thus an explanation will be omitted. 
         [0189]    The QDEM  1409  is a two-input one-output circuit that uses the local signal that is one of the input signals to perform quadrature demodulation on the analog signal that is the other of the input signals, and outputs the quadrature-demodulated signal as an output signal. In the example in  FIG. 14 , a signal line for inputting the signal from the analog EPS  1407 , a signal line for inputting the local signal from the first oscillator  1405 , and a signal line for outputting the signal to the ADC  1410 , are connected to the QDEM  1409 . In  FIG. 14 , the QDEM  1409  uses the local signal from the first oscillator  1405  to perform quadrature demodulation on the analog signal from the analog EPS  1407  to output the signal (u+b)*EXP(jwt) to the ADC  1410 . 
         [0190]    The ADC  1410  is a circuit similar to the ADC  1210  illustrated in  FIG. 12  and thus an explanation will be omitted. The digital EPS  1411  is a circuit similar to the digital EPS  1211  illustrated in  FIG. 12  and thus an explanation will be omitted. The identifier  1412  is a circuit similar to the identifier  1212  illustrated in  FIG. 12  and thus an explanation will be omitted. The subtractor  1413  is a circuit similar to the subtractor  1213  illustrated in  FIG. 12  and thus an explanation will be omitted. As a result, the modulating device  100  is able to compensate the distortion of the signal in the amplifier  1406  and the modulating unit offset b, and is able to transmit a signal with high accuracy. 
         [0191]    (Correction of Signal in Eighth Embodiment) 
         [0192]    In the circuits illustrated in  FIG. 14 , the feedback signal y i  is phase-rotated, the demodulating unit offset c is subtracted from the feedback signal y i , and the feedback signal y i  is then phase-rotated in the reverse direction in the same way as illustrated in  FIG. 13 . The feedback signal y i  is then amplified in the negative direction and the modulating unit offset b is subtracted from the feedback signal y i  to obtain the signal u i . Therefore, in the circuits illustrated in  FIG. 14 , the identifier  1412  calculates the coefficients a 1  and a 3  in the same way as in  FIG. 13  and outputs the coefficients to the DPD  1401 . The identifier  1412  also calculates the compensation value b′ of the modulating unit offset b and outputs the compensation value b′ to the adder  1402  in the same way as in  FIG. 13 . 
         [0193]    As described above, based on the modulating device  100  according to the embodiments discussed herein, the modulation signal may be fed back while being phase-rotated, and the DC offset produced in the quadrature modulation circuit may be calculated on the basis of the signal to be transmitted and the fed back signal. As a result, the modulating device  100  according to the embodiments discussed herein removes the calculated DC offset from the modulation signal and is able to transmit the modulation signal with high accuracy. 
         [0194]    Similarly, based on the modulating device  100  according to the embodiments discussed herein, the modulation signal is phase-rotated, demodulated, and then fed back so that the DC offset produced in the quadrature modulation circuit may be calculated on the basis of the signal to be transmitted and the fed back signal. As a result, based on the modulating device  100  according to the embodiments discussed herein, the modulating unit offset is set as a direct current element and the demodulating unit offset is set as a periodically changing element so that the time desired for calculating the modulating unit offset may be reduced. Moreover, based on the modulating device  100  according to the embodiments discussed herein, a coefficient for compensating the distortion produced on the modulation signal in the amplifier may be calculated. As a result, the modulating device  100  according to the embodiments discussed herein uses the calculated coefficient to compensate the modulation signal and thus is able to transmit the modulation signal with high accuracy. 
         [0195]    Moreover, for example, a conventional modulating device may have a feedback circuit that provides feedback by performing quadrature demodulation on the modulation signal with a quadrature demodulation circuit. In this case, the conventional modulation device inputs the modulation signal into the feedback circuit to calculate an average value of the sum of the modulating unit offset and the demodulating unit offset within the time period of the input. Further, the conventional modulation device does not input the modulation signal into the feedback circuit to calculate an average value of the sum of the demodulating unit offset within the time period of the non-input. Thus in the conventional modulating device, the modulating unit offset is calculated by subtracting the average value of the demodulating unit offset from the average value of the sum of the modulating unit offset and the demodulating unit offset. However, at the point in time that the modulation signal is compensated, the actual modulating unit offset and the calculated modulating unit offset are different and the accuracy of the calculated modulating unit offset may be poor. Conversely, based on the modulating device  100  according to the embodiments discussed herein, the calculation of the modulating unit offset may be performed continuously and the accuracy of the modulating unit offset may be improved. 
         [0196]    Similarly, a conventional modulating device may calculate the modulating unit offset by using a frequency converting circuit to convert the modulation signal to an IF signals and then use an ADC to perform digital conversion to provide feedback. However, in this case, the conventional modulating device uses a quadrature demodulation circuit realized by a high-speed ADC and digital signal processing thus resulting in an increase in cost and an increase in power consumption. Conversely, the modulating device  100  according to the embodiments discussed herein exhibits a lower cost and reduced power consumption in comparison to the case of using the quadrature demodulation circuit realized by the high-speed ADC and the digital signal processing. 
         [0197]    All examples and conditional language recited herein are intended for pedagogical purposes to aid the reader in understanding the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although the embodiments of the present invention have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.