Abstract:
An ultrasonic diagnostic apparatus capable of obtaining phase information of ultrasonic waves reflected at respective sampling points within an object to be inspected and displaying the phase information in an easy-to-understand-way. The ultrasonic diagnostic apparatus includes: a transmitting and receiving unit for converting reception signals outputted from plural ultrasonic transducers, which have transmitted ultrasonic waves and received ultrasonic echoes, into digital signals; reception focus processing means for performing reception focus processing on the digital signals to generate a sound ray signal along a reception direction of ultrasonic waves; first calculating means for performing quadrature detection processing on the sound ray signal to generate a complex baseband signal; second calculating means for obtaining phase information of the complex baseband signal; and image signal generating means for generating an image signal representing phase rotation of the complex baseband signal at plural sampling points along the reception direction of ultrasonic waves based on the phase information of the complex baseband signal.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The present invention relates to an ultrasonic diagnostic apparatus for imaging organs within a living body and so on by transmitting and receiving ultrasonic waves to generate ultrasonic images to be used for diagnoses. 
         [0003]    2. Description of a Related Art 
         [0004]    In medical fields, various imaging technologies have been developed for diagnoses by observing inside of an object to be inspected. Especially, ultrasonic imaging for acquiring interior information of the object by transmitting and receiving ultrasonic waves enables image observation in real time and provides no exposure to radiation unlike other medical image technologies such as X-ray photography or RI (radio isotope) scintillation camera. Accordingly, ultrasonic imaging is utilized as an imaging technology at a high level of safety in a wide range of departments including not only the fetal diagnosis in obstetrics, but gynecology, circulatory system, digestive system, and so on. 
         [0005]    The principle of ultrasonic imaging is as below. Ultrasonic waves are reflected at a boundary between regions with different acoustic impedances like a boundary between structures within the object. Therefore, by transmitting ultrasonic beams into the object such as a human body, receiving ultrasonic echoes generated within the object, and obtaining reflection points where the ultrasonic echoes are generated or reflection intensity, outlines of structures (e.g., internal organs, diseased tissues, and so on) existing within the object can be extracted. 
         [0006]    The acoustic impedance is a constant intrinsic to a material as expressed by equation (1) or (2), and the unit of MRayl (mega Rayl) is generally used therefor and 1 Mrayl=1×10 6  kg·m −2 ·s −1 . 
         [0000]        Z=ρ·C    (1) 
         [0000]        Z =(ρ· K ) 1/2    (2) 
         [0000]    where “ρ” represents density of an acoustic medium, “C” represents acoustic velocity within the acoustic medium, and “K” represents a bulk modulus of the acoustic medium. 
         [0007]    Further, given that the acoustic impedance of the first medium is Z 1  and the acoustic impedance of the second medium adjacent to the first medium is Z 2 , the vertical reflectance “R” of ultrasonic waves at the interface between the first medium and the second medium is given by the following equation (3). 
         [0000]        R =( Z   2   −Z   1 )/( Z   2   +Z   1 )   (3) 
         [0008]    Generally, an ultrasonic image is generated based on the intensity of ultrasonic waves reflected at the respective sampling points within the object, but an attempt to obtain information within the object based on phases of ultrasonic waves has been made. Especially, when Z 1 &gt;Z 2 , the reflectance R is negative and the phases of ultrasonic waves are inverted, and therefore, the attempt is considered to be effective for acquiring tissue properties within the object. 
         [0009]    As related technologies, Japanese Patent Application Publication JP-A-11-113893 discloses an ultrasonic diagnostic apparatus including complex signal converting means for converting a reception signal obtained by transmitting and receiving ultrasonic waves into a complex signal, and image forming means for forming an ultrasonic image based on only one of a real part and an imaginary part of the complex signal. 
         [0010]    Japanese Patent Application Publication JP-A-11-113894 discloses an ultrasonic diagnostic apparatus including complex signal converting means for converting a reception signal obtained by transmitting and receiving ultrasonic waves into a complex signal, phase difference calculating means for calculating phases of respective sampling points on an ultrasonic beam from the complex signal to obtain phase differences between the respective sampling points by phase comparison, and variance calculating means for calculating a variance of the phase differences. 
         [0011]    Japanese Patent Application Publication JP-A-11-113895 discloses an ultrasonic diagnostic apparatus including complex signal converting means for converting a reception signal obtained by transmitting and receiving ultrasonic waves into a complex signal, phase difference calculating means for calculating phases of the respective sampling points from the complex signal, and phase display means for displaying the phases. 
         [0012]    Japanese Patent Application Publication JP-A-11-113896 discloses an ultrasonic diagnostic apparatus including complex signal converting means for converting a reception signal obtained by transmitting and receiving ultrasonic waves into a complex signal, ratio calculating means for calculating a ratio between a real part and an imaginary part of the complex signal, and display means for displaying the ratio between the real part and the imaginary part. 
         [0013]    Japanese Patent Application Publication JP-A-11-137546 discloses an ultrasonic diagnostic apparatus including complex signal converting means for converting a reception signal obtained by transmitting and receiving ultrasonic waves into a complex signal, and phase difference calculating means for calculating phases of respective sampling points on an ultrasonic beam from the complex signal to obtain phase differences by phase comparison between the sampling points, wherein an ultrasonic image representing properties of living body tissues based on the phase differences is displayed. 
         [0014]    In the above-mentioned documents, the complex signal is obtained by quadrature detection processing of the reception signal, however, the documents do not disclose display of the phase information in an easy-to-understand way based on the results. Further, the amount of information of the complex signal in the low-frequency band is small because it is sampled at a lower sampling rate than that for the original reception signal and the noise contained in the complex signal affects thereon, and accordingly, there is a problem that the phase calculation accuracy can not be obtained sufficiently. 
       SUMMARY OF THE INVENTION 
       [0015]    The present invention has been achieved in view of the above-mentioned problems. A purpose of the present invention is to provide an ultrasonic diagnostic apparatus capable of obtaining phase information of ultrasonic waves reflected at the respective sampling points within the object and displaying the phase information in an easy-to-understand way. A further purpose of the present invention is to realize sufficient phase calculation accuracy for detecting inversion of phase at a boundary between different regions. 
         [0016]    In order to accomplish the above-mentioned purposes, an ultrasonic diagnostic apparatus according to one aspect of the present invention includes: a transmitting and receiving unit for supplying drive signals to plural ultrasonic transducers to transmit ultrasonic waves and converting reception signals outputted from the plural ultrasonic transducers, which have received ultrasonic echoes, into digital signals; reception focus processing means for performing reception focus processing on the digital signals to generate a sound ray signal along a reception direction of ultrasonic waves; first calculating means for performing quadrature detection processing on the sound ray signal generated by the reception focus processing means to generate a complex baseband signal; second calculating means for obtaining phase information of the complex baseband signal; and image signal generating means for generating an image signal representing phase rotation of the complex baseband signal at plural sampling points along the reception direction of ultrasonic waves based on the phase information of the complex baseband signal. 
         [0017]    According to the present invention, the phase information can be displayed in an easy-to-understand-way by obtaining the phase information of the complex baseband signal and generating the image signal representing phase rotation of the complex baseband signal at plural sampling points along the reception direction of ultrasonic waves. Furthermore, in the case where the complex baseband signal, or the phase information and/or amplitude information of the complex baseband signal is interpolated, sufficient phase calculation accuracy can be realized. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0018]      FIG. 1  is a block diagram showing a configuration of an ultrasonic diagnostic apparatus according to the first embodiment of the present invention; 
           [0019]      FIG. 2  shows a first example of a diagnostic image displayed on a display unit; 
           [0020]      FIG. 3  shows a second example of a diagnostic image displayed on the display unit; 
           [0021]      FIG. 4  is a block diagram showing a configuration of an ultrasonic diagnostic apparatus according to the second embodiment of the present invention; 
           [0022]      FIG. 5  is a diagram for explanation of the difference between the interpolation methods in the first embodiment and the second embodiment; 
           [0023]      FIG. 6  is a diagram for comparison between the amplitudes of the complex baseband signal obtained by the interpolation methods in the first embodiment and the second embodiment; 
           [0024]      FIG. 7  is a diagram for comparison between the phases of the complex baseband signal obtained by the interpolation methods in the first embodiment and the second embodiment; and 
           [0025]      FIG. 8  is a diagram for comparison between the vectors of the complex baseband signal obtained by the interpolation methods in the first embodiment and the second embodiment. 
       
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0026]    Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings. 
         [0027]      FIG. 1  is a block diagram showing a configuration of an ultrasonic diagnostic apparatus according to the first embodiment of the present invention. The ultrasonic diagnostic apparatus includes an ultrasonic probe  10 , a scan control unit  11 , a transmission delay pattern storage unit  12 , a transmission control unit  13 , a drive signal generating unit  14 , a reception signal processing unit  21 , a reception delay pattern storage unit  22 , a reception control unit  23 , a B-mode image generating unit  24 , a complex signal calculating unit  25 , a complex signal interpolating unit  26 , an amplitude and phase calculating unit  27 , an image signal generating unit  28 , a D/A converter  29 , a display unit  30 , a console  31 , a control unit.  32 , and a storage unit  33 . 
         [0028]    The ultrasonic probe  10  is used in contact with an object to be inspected, and includes plural ultrasonic transducers  10   a  forming a one-dimensional or two-dimensional transducer array. These ultrasonic transducers  10   a  transmit an ultrasonic beam based on applied drive signals, and receive propagating ultrasonic echoes to output reception signals. 
         [0029]    Each ultrasonic transducer includes a vibrator having electrodes formed on both ends of a material having a piezoelectric property (piezoelectric material) such as a piezoelectric ceramic represented by PZT (Pb(lead) zirconate titanate), a polymeric piezoelectric element represented by PVDF (polyvinylidene difluoride), or the like. When a pulse or continuous wave voltage is applied to the electrodes of the vibrator, the piezoelectric material expands and contracts. By the expansion and contraction, pulse or continuous wave ultrasonic waves are generated from the respective vibrators, and an ultrasonic beam is formed by synthesizing these ultrasonic waves. Further, the respective vibrators expand and contract by receiving the propagating ultrasonic waves to generate electric signals. These electric signals are outputted as reception signals of ultrasonic waves. 
         [0030]    The scan control unit  11  sequentially sets the transmission direction of an ultrasonic beam and the reception direction of ultrasonic echoes. The transmission delay pattern storage unit  12  has stored plural transmission delay patterns to be used when the ultrasonic beam is formed. The transmission control unit  13  selects one of the plural transmission delay patterns stored in the transmission delay pattern storage unit  12  according to the transmission direction set in the scan control unit  11 , and sets delay times to be provided to drive signals for the plural ultrasonic transducers  10   a  based on the selected transmission delay pattern. 
         [0031]    The drive signal generating unit  14  includes plural pulsers corresponding to the plural ultrasonic transducers  10   a,  for example. The drive signal generating unit  14  may adjust the delay amounts of the drive signals based on the transmission delay pattern selected by the transmission control unit  13  and supply the drive signals to the ultrasonic probe  10  such that the ultrasonic waves transmitted from the plural ultrasonic transducers  10   a  may form an ultrasonic beam, or may supply drive signals such that the ultrasonic waves transmitted at a time from the plural ultrasonic transducers  10   a  may reach the entire imaging region of the object. 
         [0032]    The reception signal processing unit  21  includes plural preamplifiers  21   a  and plural A/D converters  21   b  corresponding to the plural ultrasonic transducers  10   a.  The reception signals outputted from the respective ultrasonic transducers  10   a  are amplified in the amplifiers  21   a  and the analog reception signals outputted from the amplifiers  21   a  are converted into digital reception signals by the A/D converters  21   b.  The A/D converters  21   b  output the digital reception signals to the reception control unit  23 . 
         [0033]    The reception delay pattern storage unit  22  has stored plural reception delay patterns to be used when reception focus processing is performed on the reception signals outputted from the plural ultrasonic transducers  10   a.  The reception control unit  23  selects one of the plural reception delay patterns stored in the reception delay pattern storage unit  22  according to the reception direction set in the scan control unit  11 , and performs reception focus processing by providing delays to the reception signals based on the selected reception delay pattern and adding the reception signals to one another. By the reception focus processing, a sound ray signal, in which the focus of the ultrasonic echoes is narrowed, is formed. 
         [0034]    The B-mode image generating unit  24  generates a B-mode image signal as tomographic image information on tissues within the object based on the sound ray signal formed by the reception control unit  23 . The B-mode image generating unit  24  includes an STC (sensitivity time control) part  24   a,  an envelope detection part  24   b,  and a DSC (Digital Scan Converter)  24   c.    
         [0035]    The STC part  24   a  performs correction of attenuation due to a distance on the sound ray signal formed by the reception control unit  23  according to the depths of the reflection positions of ultrasonic waves. The envelope detection part  24   b  performs envelope detection processing on the sound ray signal corrected by the STC part  24   a  to generate an envelope signal. The DSC  24   c  converts (raster-converts) the envelope signals generated by the envelope detection part  24   b  into an image signal that follows the normal scan system of television signals and performs necessary image processing such as gradation processing to generate a B-mode image signal. 
         [0036]    The complex signal calculating unit  25  performs quadrature detection processing on the sound ray signal formed by the reception control unit  23  to generate a complex baseband signal. For explanation of the quadrature detection processing, assume that ultrasonic wave (plane wave) Φ traveling in the z-axis direction as the reception direction of ultrasonic waves is expressed by the equation (4). 
         [0000]      Φ=Φ 0 exp( jωt−kz )   (4) 
         [0000]    where “Φ 0 ” represents an initial value of amplitude of ultrasonic waves, “j” represents an imaginary unit, “ω” represents an angular frequency of ultrasonic waves, “t” represents time, and “k” represents a variable determined depending on a tissue within the object. 
         [0037]    What is actually measured as the sound ray signal is only the real component in the equation (4), but complex baseband signal “V” expressed by the equation (5) can be generated by performing quadrature detection processing on the measured sound ray signal. 
         [0000]        V=x+jy    (5) 
         [0000]    That is, by multiplying the ultrasonic wave “Φ” by I-signal and Q-signal having substantially the same angular frequency as the angular frequency “ω” of the ultrasonic wave “Φ” with a 90° phase shift relative to each other, the ultrasonic wave “Φ” is detected in I-phase (real number axis) and Q-phase (imaginary number axis) orthogonal to each other. Thereby, the complex baseband signal “V” has I-phase component (real component) “x” and Q-phase component (imaginary component) “y”. 
         [0038]    The complex baseband signal “V” is obtained at a predetermined number of sampling points along the reception direction of ultrasonic waves. However, for convenience of measurement, the number of sampling points is limited. On this account, the complex signal interpolating unit  26  interpolates the complex baseband signal “V” generated by the complex signal calculating unit  25 , and thereby, the number of sampling points for display is made larger than the number of sampling points at measurement. 
         [0039]    Then, the amplitude and phase calculating unit  27  obtains amplitude “A” and phase “θ” of the complex baseband signal interpolated by the complex signal interpolating unit  26  according to the equations (6) and (7). 
         [0000]        A =( x   2   +y   2 ) 1/2    (6) 
         [0000]      θ=tan −1 ( y/x )   (7) 
         [0000]    Here, the amplitude and phase calculating unit  27  obtains the amplitude information and phase information of the complex baseband signal, however, only the phase information of the complex baseband signal may be obtained according to need. 
         [0040]    The image signal generating unit  28  generates an image signal representing phase rotation of the complex baseband signal at the plural sampling points along the reception-direction of ultrasonic waves based on the phase information of the complex baseband signal obtained by the amplitude and phase calculating unit  27 . Further, the image signal generating unit  28  may generate an image signal representing vectors of the complex baseband signal at the plural sampling points along the reception direction of ultrasonic waves based on the phase information and the amplitude information of the complex baseband signal obtained by the amplitude and phase calculating unit  27 . 
         [0041]    For example, the image signal generating unit  28  generates an image signal for display formed by synthesizing a B-mode image based on the B-mode image signal generated by the B-mode image generating unit  24  and an image representing phase rotation of the complex baseband signal at plural sampling points along a segment of line designated in the B-mode image. Thereby, the image representing the phase rotation of the complex baseband signal is synthesized with the B-mode image of the object. 
         [0042]    The D/A converter  29  converts the digital image signal outputted from the image signal generating unit  28  into an analog image signal. The display unit  30  includes a display device such as a CRT, LCD, or the like, and displays diagnostic images based on the analog image signal. 
         [0043]    The control unit  32  controls the scan control unit  11  and the complex signal calculating unit  25  to image signal generating unit  28  according to the operation of an operator using the console  31 . The above-mentioned scan control unit  11 , transmission control unit  13 , reception control unit  23 , B-mode image generating unit  24  to image signal generating unit  28 , and control unit  32  can be realized by a CPU and software (programs). The software (programs) is stored in the storage unit  33 . As a recording medium in the storage unit  33 , not only a built-in hard disk but also a flexible disk, MO, MT, RAM, CD-ROM, DVD-ROM, or the like may be used. 
         [0044]      FIG. 2  shows a first example of a diagnostic image displayed on the display unit. This diagnostic image is obtained by imaging a carotid artery of an object to be inspected. The diagnostic image shown in  FIG. 2  includes a (a) B-mode image of the object, (b) an image representing amplitudes of a complex baseband signal, (c) an image representing phases of the complex baseband signal, and (d) an image representing phase differences (amounts of phase rotation) relative to the linear approximation of the phase of the complex baseband signal. In  FIG. 2 , the vertical axis indicates the depth within the object. 
         [0045]    First, (a) the B-mode image is displayed on the display unit  30  shown in  FIG. 1 . When an operator uses the console  31  to designate a vertical line (broken line A-A′ in the image) representing the reception direction of ultrasonic waves, under the control of the control unit  32 , the complex signal calculating unit  25  obtains a complex baseband signal at plural sampling points along the designated vertical line. Then, the complex signal interpolating unit  26  interpolates the complex baseband signal, and the amplitude and phase calculating unit  27  obtains amplitudes, phases, and amounts of phase rotation of the interpolated complex baseband signal. The image signal generating unit  28  generates an image signal for displaying the amplitudes, phases, and amounts of phase rotation of the complex baseband signal, and the amplitudes, phases, and amounts of phase rotation of the complex baseband signal as well as the B-mode image are displayed on the display unit  30 . In this manner, it becomes easier to acquire the amounts of phase rotation of the complex baseband signal by displaying the amounts of phase rotation of the complex baseband signal. 
         [0046]      FIG. 3  shows a second example of a diagnostic image displayed on the display unit. This diagnostic image is obtained by imaging a carotid artery of an object to be inspected. The diagnostic image shown in  FIG. 3  includes (a) a B-mode image of the object, (b) an image representing amplitudes of a complex baseband signal, and (c) a vector diagram of the complex baseband signal. In  FIG. 3 , the vertical axis indicates the depth within the object. 
         [0047]    First, (a) the B-mode image is displayed on the display unit  30  shown in  FIG. 1 . When an operator uses the console  31  to designate a vertical line (broken line A-A′ in the image) representing the reception direction of ultrasonic waves, under the control of the control unit  32 , the complex signal calculating unit  25  obtains a complex baseband signal at plural sampling points along the designated vertical line. Then, the complex signal interpolating unit  26  interpolates the complex baseband signal, and the amplitude and phase calculating unit  27  obtains amplitudes and phases of the interpolated complex baseband signal. The image signal generating unit  28  generates an image signal for displaying the amplitudes of the complex baseband signal, and the amplitudes of the complex baseband signal as well as the B-mode image are displayed on the display unit  30 . 
         [0048]    Furthermore, when the operator uses a mouse or the like to click a start button, the horizontal line (the broken line B-B′ in the image) moves from the upper end to the lower end in (a) the B-mode image, and the image signal generating unit  28  generates an image signal for displaying leading end positions of vectors of the complex baseband signal corresponding to intersection points of the vertical line and the horizontal line, and the leading end positions of the vectors of the complex baseband signal are cumulatively displayed in (c) the vector diagram. In this manner, it becomes easier to acquire the movement of phase rotation of the complex baseband signal by displaying the vectors of the complex baseband signal as moving images. 
         [0049]    Next, the second embodiment of the present invention will be explained. 
         [0050]      FIG. 4  is a block diagram showing a configuration of an ultrasonic diagnostic apparatus according to the second embodiment of the present invention. An ultrasonic diagnostic apparatus according to the second embodiment is provided with an amplitude and phase interpolating unit  34  in place of the complex signal interpolating unit  26  in the first embodiment shown in  FIG. 1 , and the rest of the configuration is the same as that of the first embodiment. 
         [0051]    The complex signal calculating unit  25  generates a complex baseband signal “V” expressed by the equation (5) by performing quadrature detection processing on the sound ray signal formed by the reception control unit  23 . 
         [0000]        V=x+jy    (5) 
         [0052]    Then, the amplitude and phase calculating unit  27  obtains amplitude “A” and phase “θ” of the complex baseband signal generated by the complex signal calculating unit  25  according to the equations (6) and (7). 
         [0000]        A =( x   2   +y   2 ) 1/2    (6) 
         [0000]      θ=tan −1 ( y/x )   (7) 
         [0000]    Here, the amplitude and phase calculating unit  27  obtains the amplitude information and phase information of the complex baseband signal, however, only the phase information of the complex baseband signal may be obtained according to use. 
         [0053]    The amplitudes “A” and phases “θ” of the complex baseband signal are obtained at a predetermined number of sampling points along the reception direction of ultrasonic waves. However, for convenience of measurement, the number of sampling points is limited. On this account, the amplitude and phase interpolating unit  34  interpolates the amplitudes “A” and phases “θ” obtained by the amplitude and phase calculating unit  27 , and thereby, the number of sampling points is increased. 
         [0054]    The image signal generating unit  28  generates an image signal representing phase rotation of the complex baseband signal at the plural sampling points along the reception direction of ultrasonic waves based on the phase information of the complex baseband signal interpolated by the amplitude and phase interpolating unit  34 . Furthermore, the image signal generating unit  28  may generate an image signal representing vectors of the complex baseband signal at the plural sampling points along the reception direction of ultrasonic waves based on the phase information and the amplitude information interpolated by the amplitude and phase interpolating unit  34 . 
         [0055]    Next, the difference between the interpolation methods in the above explained first embodiment and second embodiment will be explained.  FIG. 5  is a diagram for explanation of the difference between the interpolation methods in the first embodiment and the second embodiment. The complex baseband signal has an I-phase component and a Q-phase component (also referred to as “IQ data” as below). In  FIG. 5 , the horizontal axis indicates the depth within the object and the vertical axis indicates amplitude “I” of the I-phase component or amplitude “Q” of the Q-phase component. 
         [0056]    As shown in  FIG. 5 , according to the interpolation method  1  in the first embodiment, the measurement values of the complex baseband signal are interpolated in the stage of IQ data. On the other hand, according to the interpolation method  2  in the second embodiment, the measurement values of the complex baseband signal are converted into amplitudes and phases, and the amplitudes and phases are interpolated. In either case, because of the interpolation, the polygonal line showing changes in amplitude or phase depending on the depth becomes smoother and the image accuracy to be displayed is improved. 
         [0057]      FIG. 6  is a diagram for comparison between the amplitudes of the complex baseband signal obtained by the interpolation methods in the first embodiment and the second embodiment.  FIG. 6  shows the measurement values of the amplitude of the complex baseband signal, the amplitude obtained based on the IQ data interpolated according to the interpolation method  1  and the amplitude interpolated according to the interpolation method  2 . As shown in  FIG. 6 , there is not so much of difference between the amplitudes depending on the interpolation methods. 
         [0058]      FIG. 7  is a diagram for comparison between the phase of the complex baseband signal obtained by the interpolation methods in the first embodiment and the second embodiment.  FIG. 7  shows the measurement values of the phase of the complex baseband signal, the phase obtained based on the IQ data interpolated according to the interpolation method  1  and the phase interpolated according to the interpolation method  2 . As shown in  FIG. 7 , there is a difference between the phases depending on the interpolation methods. 
         [0059]      FIG. 8  is a diagram for comparison between the vectors of the complex baseband signals obtained by the interpolation methods in the first embodiment and the second embodiment. Here, the horizontal axis indicates the amplitude of I-signal and the vertical axis indicates the amplitude of Q-signal. In  FIG. 8 , the measurement values (circles) of the vectors of the complex baseband signal at nine sampling points # 1  to # 9  and the vector locus (broken line) having the amplitudes and phases obtained based on the IQ data interpolated according to the interpolation method  1  and the vector locus (solid line) having the amplitudes and phases interpolated according to the interpolation method  2 . As shown in  FIG. 8 , there is a large difference between the rotational directions of the vectors, i.e., the amounts of phase rotation from the sampling point # 8  to the sampling point # 9  depending on the interpolation methods. According to the interpolation method  2 , there is shown the condition that the vector rotation changes from the clockwise rotation to the counter-clockwise rotation, and the occurrence of phase inversion can be clearly observed.