Abstract:
To provide a quadrature detector wherein unnecessary frequency components are not produced, the following procedure is carried out: an analog input signal is converted into a digital signal f 1 ( t ), and the signal f 1 ( t ) is delayed by a sampling time τ to form a signal f 2 ( t ). Then, letting the reference frequency be ω 0 , the I- and Q-components in quadrature detection by the following expression: 
 
 I={f   2 ( t )*sin ω 0   t−f   1 ( t )*sin ω 0 ( t−τ)}/sin ω   0τ 
 
 Q={f   2 ( t )*cos ω 0   t−f   1 ( t )*cos ω 0 ( t−τ )}/sin ω 0τ 
The signal processing based on the above expression is carried out by a circuit comprising delaying means, multiplying means, and subtracting means.

Description:
BACKGROUND OF THE INVENTION  
       [0001]     The present invention relates to a method and an apparatus for quadrature detection and an MRI (Magnetic Resonance Imaging) system. More particularly, the present invention relates to a method and an apparatus for quadrature detection wherein high-frequency input signals are converted into low-frequency signals and an MRI system equipped with such a quadrature detector.  
         [0002]     An MRI system subjects a magnetic resonance signal, received as RF (Radio Frequency) signal, to quadrature detection by reference signals to obtain a baseband signal. Then, based on the baseband signal, the MRI system reconstructs an image. In quadrature detection, in addition to the baseband signal, a signal having a center frequency two times that of the reference signals. Since this signal is unnecessary, however, it is removed through a filter. (Refer to Non-patent Document 1, for example.)  
         [0003]     [Non-patent Document 1]  NMR Medicine, Basic and Clinical , ed. Society for the Study of Nuclear Magnetic Resonance Medicine (3rd ed.), Maruzen, Nov. 20, 1987, pp. 106-108.  
         [0004]     The frequency of the reference signal is brought into correspondence with the center frequency of the RF signal. Therefore, if the center frequency of the RF signal varies, the frequency of the reference signal must be accordingly varied. As a result, the filter for unnecessary component removal must be changed.  
       SUMMARY OF THE INVENTION  
       [0005]     Therefore, an object of the present invention is to implement a method and an apparatus for quadrature detection wherein unnecessary frequency components are not produced and an MRI system equipped with such a quadrature detector.  
         [0006]     (1) According to an aspect for solving the problem, the invention is a method for quadrature detection. The method for quadrature detection is characterized in that: an analog input signal is converted into a digital signal f 1 ( t ), and the signal f 1 ( t ) is delayed by a sampling time τ to form a signal f 2 ( t ). Then, letting the reference frequency be ω 0 , the I- and Q-components in quadrature detection are determined by Expression 2: 
 
 I={f   2 ( t )*sin ω 0   t−f   1 ( t )*sin ω 0 ( t−τ )}/sin ω 0 τ
 
 Q={f   2 ( t )*cos ω 0   t−f   1 ( t )*cos ω 0 ( t− τ)}/sin ω 0 τ  [Ex. 2]
 
         [0007]     (2) According to another aspect for solving the problem, the invention is a quadrature detector characterized in that the quadrature detector comprises: an analog-to-digital converting means which converts an analog input signal into a digital signal f 1 ( t ); a first delaying means which delays the signal f 1 ( t ) by a sampling time τ to form a signal f 2 ( t ); a second delaying means which delays a reference signal sin ω 0   t  by a sampling time τ to form a delay reference signal sin ω 0 ( t −τ); a third delaying means which delays a reference signal cos ω 0   t  by a sampling time τ to form a delay reference signal cos ω 0 ( t −τ); a first multiplying means which multiplies the signal f 2 ( t ) by the reference signal sin ω 0   t ; a second multiplying means which multiples the signal f 1 ( t ) by the delay reference signal sin ω 0 ( t −τ); a third multiplying means which multiples the signal f 2 ( t ) by the reference signal cos ω 0   t ; a fourth multiplying means which multiplies the signal f 1 ( t ) by the delay reference signal cos ω 0 ( t −τ); a first subtracting means which determines the difference between the output signal of the first multiplying means and the output signal of the second multiplying means; a second subtracting means which determines the difference between the output signal of the third multiplying means and the output signal of the fourth multiplying means; a fifth multiplying means which multiplies the output signal of the first subtracting means by a predetermined coefficient; and a sixth multiplying means which multiplies the output signal of the second subtracting means by a predetermined coefficient.  
         [0008]     (3) According to a further aspect for solving the problem, the invention is an MRI system. The MRI system applies a static magnetic field, gradient fields, and a high-frequency magnetic field to an object by a magnet system, subjects resulting magnetic resonance signals to quadrature detection by a quadrature detector, and reconstructs an image based on the signals which underwent quadrature detection. The MRI system is characterized in that it is equipped with the quadrature detector comprising: an analog-to-digital converting means which converts an analog input signal into a digital signal f 1 ( t ); a first delaying means which delays the signal f 1 ( t ) by a sampling time τ to form a signal f 2 ( t ); a second delaying means which delays a reference signal sin ω 0   t  by a sampling time τ to form a delay reference signal sin ω 0 ( t −τ); a third delaying means which delays a reference signal cos ω 0   t  by a sampling time τ to form a delay reference signal cos ω 0 ( t −τ); a first multiplying means which multiplies the signal f 2 ( t ) by the reference signal sin ω 0   t ; a second multiplying means which multiples the signal f 1 ( t ) by the delay reference signal sin ω 0 ( t −τ); a third multiplying means which multiples the signal f 2 ( t ) by the reference signal cos ω 0   t ; a fourth multiplying means which multiplies the signal f 1 ( t ) by the delay reference signal cos ω 0 ( t −τ); a first subtracting means which determines the difference between the output signal of the first multiplying means and the output signal of the second multiplying means; a second subtracting means which determines the difference between the output signal of the third multiplying means and the output signal of the fourth multiplying means; a fifth multiplying means which multiplies the output signal of the first subtracting means by a predetermined coefficient; and a sixth multiplying means which multiplies the output signal of the second subtracting means by a predetermined coefficient.  
         [0009]     The coefficient is preferably 1/sin ω 0 τ for the purpose of making appropriate the amplitude of an output signal according to the frequency of a reference signal. The first to third delaying means, the first to sixth multiplying means, and the first and second subtracting means are preferably constituted of DSP as a whole in terms of versatility.  
         [0010]     The first to third delaying means, the first to sixth multiplying means, and the first and second subtracting means are preferably constituted of FPGA as a whole in terms of flexibility. The first to third delaying means, the first to sixth multiplying means, and the first and second subtracting means are preferably constituted of ASIC as a whole because of the absence of redundancy.  
         [0011]     The first to third delaying means, the first to sixth multiplying means, and the first and second subtracting means preferably operates on a clock whose frequency is 1/τ in terms of synchronous operation. The first to sixth multiplying means and the first and second subtracting means preferably perform pipeline operation because asynchronous operation is feasible.  
         [0012]     According to the invention according to the aspect described in (1) above, the following is carried out: an analog input signal is converted into a digital signal f 1 ( t ), and the signal f 1 ( t ) is delayed by the sampling time τ to form a signal f 2 ( t ). Then, letting the reference frequency be ω 0 , the I- and Q-components in quadrature detection are determined by Expression 3: 
 
 I={f   2 ( t )*sin ω 0   t−f   1 ( t )*sin ω 0 ( t−τ )}/sin ω 0 τ
 
 Q={f   2 ( t )*cos ω 0   t−f   1 ( t )*cos ω 0 ( t−τ )}/sin ω 0 τ  [Ex. 3]
 
         [0013]     Therefore, unnecessary frequency components are not produced.  
         [0014]     The invention according to the another or further aspect described in (2) or (3) above comprises the first to third delaying means, first to sixth multiplying means, and first and second subtracting means. Then, signal processing is carried out based on the above expression. Therefore, unnecessary frequency components are not contained in output signals.  
         [0015]     Further objects and advantages of the present invention will be apparent from the following description of the preferred embodiments of the invention as illustrated in the accompanying drawings. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0016]      FIG. 1  is a block diagram of the MRI system.  
         [0017]      FIG. 2  is a block diagram of the quadrature detection circuit. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0018]     Referring to drawings, the best mode for carrying out the invention will be described in detail below. The present invention is not limited to the best mode for carrying out the invention.  FIG. 1  is a block diagram of an MRI system. This apparatus is an example of the best mode for carrying out the invention. The constitution of the apparatus illustrates an example of the best mode for carrying out the invention with respect to MRI system.  
         [0019]     As illustrated in the figure, the apparatus includes a magnet system  10 . The magnet system  10  include a main field magnet unit  102 , a gradient coil unit  104 , and an RF coil unit  106 . The magnet system  10  has an imaging space in it, and an object whose image is to be picked up is carried into and out of the imaging space.  
         [0020]     The main field magnet unit  102  form a static magnetic field in the imaging space. The main field magnet unit  102  is constituted using, for example, superconducting electromagnets. The constitution of the main field magnet unit  102  is not limited to this, and may be constituted using normal conduction electromagnets, permanent magnets, or the like.  
         [0021]     The gradient coil unit  104  generates three gradient fields. This is for providing the static magnetic field strength with a gradient, respectively, in the directions of three axes orthogonal to one another: slice axis, phase axis, and frequency axis. To enable the generation of such gradient fields, the gradient coil unit  104  has three systems of gradient coils.  
         [0022]     The RF coil unit  106  forms an RF magnetic field for exciting spins in the body of an object in the static magnetic field space. Formation of RF magnetic field is also designated as transmission of RF pulse. An electromagnetic wave which produced by excited spins, that is, a magnetic resonance signal is detected by the RF coil unit  106 . The magnetic resonance signal is an RF signal.  
         [0023]     The magnetic resonance signal becomes a signal in the frequency domain, that is, Fourier space. The magnetic resonance signal is encoded on two axes by gradient in the direction of phase axis and in the direction of frequency axis. Therefore, the magnetic resonance signal is obtained as a signal in the two-dimensional Fourier space. The two-dimensional Fourier space is also designated as k-space.  
         [0024]     The gradient coil unit  104  is connected with a gradient drive unit  20 . The gradient drive unit  20  supplies a driving signal to the gradient coil unit  104  to generate gradient fields. The gradient drive unit  20  has three systems of drive circuits in correspondence with the three systems of gradient coils in the gradient coil unit  104 .  
         [0025]     The RF coil unit  106  is connected with a transmitter-receiver unit  30 . The transmitter-receiver unit  30  supplies a driving signal to the RF coil unit  106  to transmit an RF pulse. Also, the transmitter-receiver unit  30  receives a detection signal from the RF coil unit  106 . The reception signal is converted into a digital signal in the transmitter-receiver unit  30 , as described later. Further, the reception signal is subjected to quadrature detection through a quadrature detection circuit, and is inputted as a digital baseband signal to a computer  40 .  
         [0026]     The computer  40  stores data, inputted from the transmitter-receiver unit  30 , in memory. A data space is formed in the memory. This data space corresponds to the k-space. The computer  40  subjects the data in the k-space to two-dimensional inverse Fourier transform, and thereby reconstructs an image. Further, the computer  40  controls the gradient drive unit  20  and the transmitter-receiver unit  30  to carry out imaging.  
         [0027]     The computer  40  is connected with a display unit  50  and an operating unit  60 . The display unit  50  comprises a graphic display or the like. The operating unit  60  comprises a keyboard or the like equipped with a pointing device.  
         [0028]     The display unit  50  displays reconstructed images and varied information outputted from the computer  40 . The operating unit  60  is operated by the user, and inputs varied instructions, information, and the like to the computer  40 . The user operates the MRI system in an interactive manner through the display unit  50  and the operating unit  60 .  
         [0029]      FIG. 2  is a block diagram illustrating an example of the quadrature detection circuit. The quadrature detection circuit forms part of the transmitter-receiver unit  30 . This circuit is an example of the best mode for carrying out the invention. The constitution of the circuit illustrates an example of the best mode for carrying out the invention with respect to quadrature detector. The operation of the circuit illustrates an example of the best mode for carrying out the invention with respect to method for quadrature detection.  
         [0030]     As illustrated in the figure, the circuit comprises an analog-to-digital converter  302 , data buffers  402 ,  404 , and  406 , multipliers  502 ,  504 ,  506 ,  508 ,  510 , and  512 , and subtracters  602  and  604 .  
         [0031]     The analog-to-digital converter  302  converts an analog input signal into a digital signal f 1 ( t ). The input signal is an RF signal received from the RF coil unit  106 . The sampling period of the analog-to-digital converter  302  is τ. The analog-to-digital converter  302  is an example of the analog-to-digital converting means in the present invention.  
         [0032]     The signal f 1 ( t ) is inputted to the buffer  402 , which then outputs a signal f 2 ( t ). The signal f 2 ( t ) is obtained by delaying the signal f 1 ( t ) by τ. The buffer  402  functions as a delay device whose delay time is τ. The buffer  402  is an example of the first delaying means in the present invention.  
         [0033]     The signal f 2 ( t ) is multiplied by a reference signal sin ω 0   t  at the multiplier  502 . The signal f 1 ( t ) is multiplied by a delay reference signal sin ω 0 ( t −τ) at the multiplier  504 . The delay reference signal sin ω 0 ( t −τ) is obtained by delaying the reference signal sin ω 0   t  by τ at the buffer  404 .  
         [0034]     The multiplier  502  is an example of the first multiplying means in the present invention. The multiplier  504  is an example of the second multiplying means in the present invention. The buffer  404  is an example of the second delaying means in the present invention.  
         [0035]     The subtracter  602  determines the difference between the output signals of the multipliers  502  and  504 . This differential signal is multiplied by a coefficient 1/sin ω 0 τ at the multiplier  510 . Setting the coefficient to 1/sin ω 0 τ is preferable for the purpose of making appropriate the amplitude of an output signal according to the frequency of a reference signal. The coefficient by which multiplication is carried out may be an appropriate constant value. A signal indicating the result of multiplication is outputted as an I-component of quadrature detection signal. The subtracter  602  is an example of the first subtracting means in the present invention. The multiplier  510  is an example of the fifth multiplying means in the present invention.  
         [0036]     The signal f 2 ( t ) is multiplied by a reference signal cos ω 0   t  at the multiplier  506 , and the signal f 1 ( t ) is multiplied by a delay reference signal cos ω 0 ( t −τ) at the multiplier  508 . The delay reference signal cos ω 0 ( t −τ) is obtained by delaying the reference signal cos ω 0   t  by τ at the buffer  406 .  
         [0037]     The multiplier  506  is an example of the third multiplying means in the present invention. The multiplier  508  is an example of the fourth multiplying means in the present invention. The buffer  406  is an example of the third delaying means in the present invention.  
         [0038]     The subtracter  604  determines the difference between the output signals of the multipliers  506  and  508 . This differential signal is multiplied by the coefficient 1/sin ω 0 T at the multiplier  512 . Setting the coefficient to 1/sin ω 0 T is preferable for the purpose of making appropriate the amplitude of an output signal according to the frequency of a reference signal. The coefficient by which multiplication is carried out may be an appropriate constant value. A signal indicating the result of multiplication is outputted as a Q-component of quadrature detection signal. The subtracter  604  is an example of the second subtracting means in the present invention. The multiplier  512  is an example of the sixth multiplying means in the present invention.  
         [0039]     The I- and Q-components obtained by the above-mentioned data processing become signals which are represented by Expression 4: 
 
 I={f   2 ( t )*sin ω 0   t−f   1 ( t )*sin ω 0 ( t−τ )}/sin ω 0 τ
 
 Q={f   2 ( t )*cos ω 0   t−f   1 ( t )*cos ω 0 ( t−τ )}/sin ω 0 τ  [Ex. 4]
 
         [0040]     When the frequency ω 0  of the reference signal is matched with the center frequency of the RF signal, both of these I- and Q-components become baseband signals for the reason described below. At the same time, the baseband signals do not contain a signal whose frequency is 2ω 0 . Therefore, a filter for removing signals whose frequency is 2ω 0  is unnecessary. For this reason, even if the frequency of a reference signal is varied in accordance with variation in the center frequency of an RF signal, that can be coped with without any modification.  
         [0041]     This quadrature detection circuit can be used to convert the band into any frequency band as well as baseband. Even in such a case, only the frequency component of the difference between the center frequency of the input signal and the frequency of the reference signal is obtained. The frequency component of the sum of them is not generated. Therefore, a filter for removing unnecessary components is not required.  
         [0042]     The circuit is constituted of, for example, DSP (Digital Signal Processor) as a whole excepting the analog-to-digital converter  302 . Thus, a quadrature detection circuit excellent in versatility is obtained. Alternatively, the circuit may be constituted of FPGA (Field Programable Gate Array) or ASIC (Application Specific Integrated Circuit). Thus, a quadrature detection circuit excellent in flexibility or free from redundancy, respectively, is obtained. The circuit may be constituted of appropriate discrete components, needless to add.  
         [0043]     This circuit operates on a clock whose frequency is 1/τ. Thus, the operations of the individual parts of the circuit can be synchronized with one another. Alternatively, the circuit may be so constituted that it performs pipeline operation. Thus, asynchronous operation is feasible.  
         [0044]     The reason why the I- and Q-components become only baseband signals will be described. The signal f 1 ( t ) is represented by Expression 5: 
 
 f   1 ( t )=∫ F (ω)*cos{ω t+θ (ω)} dω   [Ex. 5]
 
         [0045]     The signal f 2 ( t ) is represented by Expression 6:  
                     f2   ⁢     (   t   )       =       ⁢     f1   ⁡     (     t   -   τ     )                   =       ⁢     ∫       F   ⁡     (   ω   )       *   cos   ⁢     {       ω   ⁡     (     t   -   τ     )       +     θ   ⁡     (   ω   )         }     ⁢     ⅆ   ϖ                     =       ⁢     ∫       F   ⁡     (   ω   )       *   cos   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }     *   cos   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }     *                         ⁢     sin   ⁢     {   ωτ   }       ]     ⁢     ⅆ   ω                   [     Ex   .           ⁢   6     ]             
 
         [0046]     If the frequency bandwidth 2ω 1  of the signal f 1 ( t ) is sufficiently narrower than the frequency bandwidth determined by the sampling period τ, the approximation expressed by Expression 7 holds within the band of from ω−ω 1  to ω+ω 1 : 
 
cos{ωτ}≈cos{ω 0 τ}
 
sin{ωτ}≈sin{ω 0 τ}  [Ex. 7]
 
         [0047]     Therefore, the right side can be rewritten as follow:  
                     Right   ⁢           ⁢   side     ≈       ⁢       ∫       F   ⁡     (   ω   )       *   cos   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }     *   cos   ⁢     {   ω0τ   }         +                       ⁢     sin   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }     *   sin   ⁢     {   ω0τ   }       ]     ⁢     ⅆ   ω                 =       ⁢       cos   ⁢     {   ω0τ   }     *     ∫       F   ⁡     (   ω   )       *     [     cos   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }       ]     ⁢     ⅆ   ω           +                     ⁢     sin   ⁢     {   ω0τ   }     *     ∫       F   ⁡     (   ω   )       *     [     sin   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }       ]     ⁢     ⅆ   ω                         [     Ex   .           ⁢   8     ]             
 
         [0048]     The expression is rewritten as mentioned above.  
         [0049]     Consequently, consideration will be given to a signal g 1 ( t ) expressed by Expression 9, where j is an imaginary unit.  
                     g1   ⁡     (   t   )       =       ⁢       f1   ⁡     (   t   )       +     j   *       [       f2   ⁡     (   t   )       -       f1   ⁡     (   t   )       *   cos   ⁢     {   ω0τ   }         ]     /   sin     ⁢     {   ω0τ   }                     =       ⁢       ∫       F   ⁡     (   ω   )       *   cos   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }     ⁢     ⅆ   ω         +     j   *     [     cos   ⁢     {   ω0τ   }     *                           ⁢       ∫       F   ⁡     (   ω   )       *     [     cos   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }       ]     ⁢     ⅆ   ω         +     sin   ⁢     {   ω0τ   }     *                       ⁢       ∫       F   ⁡     (   ω   )       *     [     sin   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }       ]     ⁢     ⅆ   ω         -                     ⁢     ∫       F   ⁡     (   ω   )       *   cos   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }     ⁢     ⅆ   ω     *       cos   ⁡     [     {   ω0τ   }     ]       /   sin     ⁢     {   ω0τ   }                     =       ⁢       ∫       F   ⁡     (   ω   )       *   cos   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }     ⁢     ⅆ   ω         +     j   *                       ⁢     ∫       F   ⁡     (   ω   )       *   sin   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }     ⁢     ⅆ   ω                         =       ⁢       ∫       F   ⁡     (   ω   )       *   cos   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }         +     j   *   sin   ⁢     {       ω   ⁢           ⁢   t     +     θ   ⁡     (   ω   )         }           ]     ⁢     ⅆ   ω                   [     Ex   .           ⁢   9     ]             
 
         [0050]     As described above, the signal g 1 ( t ) becomes the complex representation of a signal whose frequency is (o. Such a signal becomes a signal only in baseband by subjecting the signal to quadrature detection using a signal represented by Expression 10 as reference signal. 
 
cos{−ω 0   t}+j *sin{−ω 0   t}   [Ex. 10]
 
         [0051]     This fact is publicly known.  
         [0052]     Such a reference signal is multiplied by the signal g 1 ( t ) as follows:  
                     g2   ⁡     (   t   )       =       ⁢       g1   ⁡     (   t   )       *     {       cos   ⁡     (       -   ω0     ⁢           ⁢   t     )       +     j   *     sin   ⁡     (       -   ω0     ⁢           ⁢   t     )           }                   =       ⁢       [       f1   ⁡     (   t   )       +     j   *       {       f2   ⁡     (   t   )       -       f1   ⁡     (   t   )       *     cos   ⁡     (   ω0τ   )           }     /     sin   ⁡     (   ω0τ   )             ]     *                     ⁢     {       cos   ⁡     (     ω0   ⁢           ⁢   t     )       -     j   *     sin   ⁡     (     ω0   ⁢           ⁢   t     )           }                 =       ⁢         f1   ⁡     (   t   )       *     cos   ⁡     (     ω0   ⁢           ⁢   t     )         +       {       f2   ⁡     (   t   )       -       f1   ⁡     (   t   )       *     cos   ⁡     (     ω0   ⁢           ⁢   t     )           }     /                         ⁢       sin   ⁡     (   ω0τ   )       *     sin   ⁡     (     ω0   ⁢           ⁢   t     )         }     +     j   *     [         -     f1   ⁡     (   t   )         *     sin   ⁡     (     ω0   ⁢           ⁢   t     )         +                         ⁢         {       f2   ⁡     (   t   )       -       f1   ⁡     (   t   )       *     cos   ⁡     (   ω0τ   )           }     /     sin   ⁡     (   ω0τ   )         *     cos   ⁡     (     ω0   ⁢           ⁢   t     )         ]               =       ⁢     [     {         f1   ⁡     (   t   )       *     cos   ⁡     (     ω0   ⁢           ⁢   t     )       *     sin   ⁡     (   ω0τ   )         +     {       f2   ⁡     (   t   )       -       f1   ⁡     (   t   )       *                                     ⁢     cos   ⁡     (   ω0τ   )       }     *     sin   ⁡     (     ω0   ⁢           ⁢   t     )         }     -     j   *     {       f1   ⁡     (   t   )       *     sin   ⁡     (     ω0   ⁢           ⁢   t     )       *                             ⁢       sin   ⁡     (   ω0τ   )       -       {       f2   ⁡     (   t   )       -       f1   ⁡     (   t   )       *     cos   ⁡     (   ω0τ   )           }     *     cos   ⁡     (     ω0   ⁢           ⁢   t     )           }     ]     /                   ⁢     sin   ⁡     (   ω0τ   )                   =       ⁢     [     {         f2   ⁡     (   t   )       *     sin   ⁡     (     ω0   ⁢           ⁢   t     )         +       f1   ⁡     (   t   )       *     {         cos   ⁡     (     ω0   ⁢           ⁢   t     )       *     sin   ⁡     (     ω0   ⁢           ⁢   t     )         -                                   ⁢       cos   ⁡     (     ω0   ⁢           ⁢   t     )       *     sin   ⁡     (     ω0   ⁢           ⁢   t     )         }     }     +     j   *     {         f2   ⁡     (   t   )       *     cos   ⁡     (     ω0   ⁢           ⁢   t     )         -                         ⁢       f1   ⁡     (   t   )       *     {         sin   ⁡     (     ω0   ⁢           ⁢   t     )       *     sin   ⁡     (     ω0   ⁢           ⁢   τ     )         +       cos   ⁡     (     ω0   ⁢           ⁢   τ     )       *                                 ⁢     cos   ⁢     (     ω0   ⁢           ⁢   t     )       }     }     ]     /     sin   ⁡     (   ω0τ   )                   =       ⁢     [     {         f2   ⁡     (   t   )       *     sin   ⁡     (     ω0   ⁢           ⁢   t     )         -       f1   ⁡     (   t   )       *   sin   ⁢     {     ω0   ⁡     (     t   -   τ     )       }       +                         ⁢     j   *       {         f2   ⁡     (   t   )       *     cos   ⁡     (     ω0   ⁢           ⁢   t     )         -       f1   ⁡     (   t   )       *   cos   ⁢     {     ω0   ⁡     (     t   -   τ     )       }         ]     /                       ⁢     sin   ⁡     (   ω0τ   )                     [     Ex   .           ⁢   11     ]             
 
         [0053]     As described above, the real number portion is expressed by Expression 12: 
 
 I={f   2 ( t )*sin ω t−f   1 ( t )*sin ω 0 ( t−τ)}/sin ω 0 τ   [Ex. 12]
 
         [0054]     The imaginary number portion is expressed by Expression 13: 
 
 Q={f   2 ( t )*cosω 0   t−f   1 ( t )*cos ω 0 ( t−τ )}/sin ω 0 τ  [Ex. 13]
 
         [0055]     These I and Q are none other than the output signal of the quadrature detection circuit illustrated in  FIG. 2 .  
         [0056]     Many widely different embodiments of the invention may be configured without departing from the spirit and the scope of the present invention. It should be understood that the present invention is not limited to the specific embodiments described in the specification, except as defined in the appended claims.