Patent Publication Number: US-7916049-B2

Title: Group delay characteristic correcting device and group delay characteristic correcting method

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is based upon and claims the benefit of priority of Japanese Patent Application 2008-233438, filed on Sep. 11, 2008, the entire contents of which are hereby incorporated herein by reference. 
     FIELD 
     The present disclosure is directed to a group delay characteristic correcting device and a group delay characteristic correcting method for correcting group delay characteristics of an analog low-pass filter used for removing aliasing. 
     BACKGROUND 
     In order to remove aliasing, direct conversion transceivers, for example, require an analog low-pass filter (LPF) at a stage subsequent to a digital-analog converter (DAC) or at a stage previous to an analog-digital converter (ADC). 
     Transmission systems using millimeter-wave bands or broadband signals, such as UWB (Ultra Wide Band), have become more common in recent years, and therefore, demand has been raised for a large increase in the sampling rates of digital-analog converters and analog-digital converters. 
     For example, in the case where the baseband frequency is 600 MHz, a sampling rate of at least 1.2 GHz is required. Digital-analog converters and analog-digital converters having such sampling rates have already been in the marketplace; however, even when a 1.2 GHz digital-analog converter is used, aliasing occurs in the bandwidth between 600 and 1200 MHz. Therefore, unless aliasing in this bandwidth is removed by a low-pass filter, adjacent channel interference occurs. Similarly in the case of analog-digital converters, it is necessary to cut off adjacent channels in advance by a low-pass filter in order to prevent adjacent channel interference. 
     Low-pass filters used for this purpose need to have steep cut-off characteristics. Examples of common low-pass filters having steep cut-off characteristics are Chebyshev filters, inverse Chebyshev filters and elliptic filters. Although such a filter has a steep cut-off characteristic, phase linearity is lost particularly around the cutoff frequency, and the group delay characteristics are degraded. 
     In the case where the group delay characteristics of a filter are degraded, when an input waveform passes through the filter, an output waveform becomes distorted. As a result, significant degradation is observed in the transmission characteristics. Accordingly, methods for improving the group delay characteristics of low-pass filters have conventionally been studied. 
     One such method is to insert an equalizer in a transmitter to produce a preliminarily distorted output, thereby correcting the distortion caused when the input waveform passes through the low-pass filter (see Patent Documents 1 and 2, for example). 
     Another method suggests the use of an equalizer with such a filter that causes the group delay characteristics to have a concave shape in the middle of the spectral frequencies and have a convex shape at edges of the spectral frequencies (see Patent Document 3, for example). 
     Yet another method is to improve the group delay characteristics by devising the shape of an analog filter (see Patent Document 4, for example).
     [Patent Document 1] Japanese Laid-open Patent Application Publication No. 2000-299652   [Patent Document 2] Japanese Laid-open Patent Application Publication No. 2001-257564   [Patent Document 3] Japanese Laid-open Patent Application Publication No. S62-227208   [Patent Document 4] Japanese Laid-open Patent Application Publication No. H11-261303   

     However, neither Patent Document 1 nor Patent Document 2 discloses a method for calculating the inverse characteristics of the group delay characteristics of the filter. The correction technique of Patent Document 3 is directed only to a filter whose group delay characteristics have a concave shape in the middle of the spectral frequencies and a convex shape at edges of the spectral frequencies. 
     The technique of Patent Document 4 requires a large cost since the analog filter itself is devised, and also leaves the problem of less design freedom in terms of the pass band, cutoff band and the like. 
     SUMMARY 
     According to an aspect of the invention, a group delay characteristic correcting device for correcting group delay characteristics of an analog low-pass filter used to remove aliasing of a digital-analog converter or an analog-digital converter. The group delay characteristic correcting device includes a digital signal processing unit configured to have an all-pass phase circuit at a stage previous to the digital-analog converter or at a stage subsequent to the analog-digital converter so as to correct the group delay characteristics of the analog low-pass filter. 
     The object and advantages of the disclosure will be realized and attained by means of the elements and combinations particularly pointed out in the claims. 
     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 present disclosure as claimed. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a block diagram of a transmitter according to one embodiment of the present disclosure; 
         FIG. 2  is a block diagram of a receiver according to one embodiment of the present disclosure; 
         FIG. 3  is a flowchart of a process of designing a digital signal processing unit according to one embodiment of the present disclosure; 
         FIG. 4  is a structural diagram of a Butterworth normalized low-pass filter; 
         FIG. 5  is a structural diagram showing a designed analog low-pass filter; 
         FIG. 6  is an amplitude characteristic diagram of the analog low-pass filter; 
         FIG. 7  is a phase characteristic diagram of the analog low-pass filter; 
         FIGS. 8A through 8C  show polar locations of the Butterworth low-pass filter; 
         FIG. 9  shows amplitude frequency characteristics of an analog low-pass filter and a digital low-pass filter; 
         FIG. 10  shows phase frequency characteristics of the analog low-pass filter and the digital low-pass filter; 
         FIGS. 11A and 11B  show amplitude and phase frequency characteristics of a first-order all-pass phase circuit; 
         FIGS. 12A and 12B  show amplitude and phase frequency characteristics of a second-order all-pass phase circuit; 
         FIGS. 13A and 13B  show characteristics of the analog low-pass filter and an all-pass-phase-circuit cascade-connected filter, obtained by a simulation; and 
         FIG. 14  shows group delay frequency characteristics of the analog low-pass filter and the cascade-connected filter, obtained by a simulation. 
     
    
    
     DESCRIPTION OF EMBODIMENT 
     Embodiments that describe the best mode for carrying out the present invention are explained next with reference to the drawings. 
     [Transmitter] 
       FIG. 1  is a block diagram of a direct conversion transmitter according to one embodiment. The transmitter has a structure designed to be used in the OFDM (Orthogonal Frequency Division Multiplexing) scheme. In  FIG. 1 , a data generating unit  11  generates I data and Q data to be transmitted. The I and Q data are respectively converted from frequency-domain signals into time-domain signals at an IFFT (Inverse Fast Fourier Transform) unit  12 , and then sent to a digital signal processing unit  13 . 
     The digital signal processing unit  13  performs a signal process in order to improve the group delay characteristics of low-pass filters  15 A and  15 B operating at a subsequent stage. The I and Q data output from the digital signal processing unit  13  are converted into analog signals by digital-analog converters (DAC)  14 A and  14 B, respectively. Then, the low-pass filters (LPF)  15 A and  15 B remove unnecessary high-frequency components from the respective analog signals, thereby removing aliasing created by the digital-analog converters  14 A and  14 B. 
     Subsequently, a quadrature modulator (IQMOD)  16  modulates the carrier frequencies of the respective signals with the I and Q data, and a power amplifier (PA)  17  power-amplifies the modulated signals, which is then output from an antenna  18 . 
     [Receiver] 
       FIG. 2  is a block diagram of a direct conversion receiver according to one embodiment. The receiver has a structure designed to be used in the OFDM scheme. In  FIG. 2 , a signal received by an antenna  21  is amplified by a preamplifier (LNA)  22 , and then supplied to a quadrature demodulator (IQDEMOD)  23 , in which the signal is demodulated into I and Q signals. 
     Low-pass filters  24 A and  24 B remove unnecessary high-frequency components from the respective I and Q signals, respectively, thereby preventing aliasing otherwise created by analog-digital converters (ADC)  25 A and  25 B. Subsequently, the signals are digitized by the analog-digital converters (ADC)  25 A and  25 B, and then supplied to a digital signal processing unit  26 . 
     The digital signal processing unit  26  performs a signal process in order to improve the group delay characteristics of the low-pass filters  24 A and  24 B. I and Q data output from the digital signal processing unit  26  are respectively converted from time-domain signals into frequency-domain signals at an FFT (Fast Fourier Transform) unit  27 , and then supplied to a data restoring unit  28 , in which the signals are restored to original data. 
     [Design of Digital Signal Processing Unit] 
       FIG. 3  is a flowchart showing a process of designing the digital signal processing units  13  and  26 . In Step S 1  of  FIG. 3 , the analog low-pass filters  15 A and  15 B (or  24 A and  24 B) having desired characteristics are designed. In this example, assume that each of the low-pass filters  15 A and  15 B (or  24 A and  24 B) is a second-order Butterworth analog low-pass filter having a cutoff frequency of 10 kHz. 
     An analog low-pass filter is generally designed using data normalized to a cutoff frequency of ½π and an impedance of 1Ω. Each of a band-pass filter (BPF), a high-pass filter (HPF) and a band elimination filter (BEF) can be created by changing values of a low-pass filter, and the same technique is employed to create them. 
     The second-order Butterworth normalized low-pass filter includes an inductance of 1.41421 [H] and a capacitance of 1.41421 [F], as illustrated in  FIG. 4 . Desired frequency and impedance for an analog low-pass filter can be obtained by the following equations. In Step S 2 , a frequency conversion coefficient M and an impedance conversion coefficient K are obtained. 
     
       
         
           
             
               
                 
                   M 
                   = 
                     
                   ⁢ 
                   
                     target 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       frequency 
                       / 
                       original 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     frequency 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     as 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     a 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     reference 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     10 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       kHz 
                       / 
                       
                         ( 
                         
                           
                             1 
                             / 
                             2 
                           
                           ⁢ 
                           Π 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     6.28 
                     × 
                     
                       10 
                       4 
                     
                   
                 
               
             
             
               
                 
                   K 
                   = 
                     
                   ⁢ 
                   
                     target 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       impedance 
                       / 
                       original 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     impedance 
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                     as 
                     ⁢ 
                     
                         
                     
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                     reference 
                   
                 
               
             
             
               
                 
                   = 
                     
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                     50 
                     ⁢ 
                     
                         
                     
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                     ⁢ 
                     
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                     ⁢ 
                     
                         
                     
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                     Ω 
                   
                 
               
             
             
               
                 
                   = 
                     
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                   50 
                 
               
             
           
         
       
     
     Using M and K above, constants L and C are calculated in Step S 3  by the following equations. 
     
       
         
           
             
               
                 
                   L 
                   = 
                     
                   ⁢ 
                   
                     
                       1.42 
                       ⁢ 
                       
                           
                       
                       [ 
                       H 
                       ] 
                     
                     × 
                     K 
                     ⁢ 
                     
                       / 
                     
                     ⁢ 
                     M 
                   
                 
               
             
             
               
                 
                   = 
                     
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                       1.42 
                       ⁢ 
                       
                           
                       
                       [ 
                       H 
                       ] 
                     
                     × 
                     
                       50 
                       / 
                       6.28 
                     
                     × 
                     
                       10 
                       4 
                     
                   
                 
               
             
             
               
                 
                   ≈ 
                     
                   ⁢ 
                   
                     1.13 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     kH 
                   
                 
               
             
             
               
                 
                   C 
                   = 
                     
                   ⁢ 
                   
                     
                       1.42 
                       ⁢ 
                       
                           
                       
                       [ 
                       F 
                       ] 
                     
                     / 
                     
                       ( 
                       
                         M 
                         × 
                         K 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       1.42 
                       ⁢ 
                       
                           
                       
                       [ 
                       F 
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                       ( 
                       
                         6.28 
                         × 
                         
                           10 
                           4 
                         
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                         50 
                       
                     
                   
                 
               
             
             
               
                 
                   ≈ 
                     
                   ⁢ 
                   
                     452.23 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     nF 
                   
                 
               
             
           
         
       
     
       FIG. 5  shows an analog low-pass filter designed in the above manner. Frequency characteristics (amplitude characteristic and phase characteristic) are calculated by incorporating the analog low-pass filter in a circuit simulator. The results of the calculation are illustrated in  FIGS. 6 and 7 . 
     Next, the analog low-pass filter is digitized. In Step S 11 , a formulated transfer function G(s) is obtained for the analog normalized low-pass filter. The transfer function G(s) is obtained by an expression s=jω, where ω=2πf(rad/sec), and f is the frequency [Hz]. In this step, the transfer function having amplitude and phase information is derived from the amplitude characteristic. In the case of a Butterworth analog low-pass filter, the amplitude characteristic is expressed as follows.
 
| G (ω)| 2 =1/(1+ε 2 ω 2N )
 
ε is found by ε=(10 AC −1) 1/2 , where AC is an amount of attenuation at the cutoff frequency, which is generally 3 dB. In this case, ε=1.
 
     In order to calculate G(s), ω=s/j is obtained from s=jω, and then substituted into the above equation of |G(ω)| 2 . 
                            G   ⁡     (   s   )            2     =       ⁢            G   ⁡     (   s   )       ⁢     G   ⁡     (     -   s     )                          =       ⁢     1   /     [     1   +       (     -   js     )       2   ⁢   N         ]                   =       ⁢     1   /     [     1   +       (     -     s   2       )     N       ]                   
s to make the above denominator zero, i.e. a pole, is calculated.
 
     In the case where N is an odd number, the following is obtained by solving s 2N =1. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           s 
                           k 
                         
                         = 
                           
                         ⁢ 
                         
                           ⅇ 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
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                         = 
                           
                         ⁢ 
                         
                           
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                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
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                                 ( 
                                 
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                                   ⁢ 
                                   
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                   ( 
                   
                     
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                     , 
                     1 
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                     , 
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                     ⁢ 
                     
                         
                     
                     , 
                     
                       
                         2 
                         ⁢ 
                         N 
                       
                       - 
                       1 
                     
                   
                   ) 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     In the case where N is an even number, the following is obtained by solving s 2N =−1. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                                   ( 
                                   
                                     
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                               ] 
                             
                           
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                                     + 
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                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   ( 
                   
                     
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                         ⁢ 
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                       - 
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                   ) 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     As for a Butterworth low-pass filter, all poles lie in s planes, on a unit circle at π/N intervals, as illustrated in  FIGS. 8A through 8C .  FIG. 8A  shows the case of N=1;  FIG. 8B  shows the case of N=2; and  FIG. 8C  shows the case of N=3. All poles are obtained from G(s)G(−s), and a transfer function including only poles on the left-hand half-plane is stable. 
     In the case of N=1, since a stable pole is s 1 =−1, G(s)=1/(s+1). 
     In the case of N=2, since stable poles are s 1 =e 3π/4  and s 2 =e 5π/4 , 
     
       
         
           
             
               
                 
                   
                     G 
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
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                       [ 
                       
                         
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                             s 
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                               s 
                               1 
                             
                           
                           ) 
                         
                         ⁢ 
                         
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                             s 
                             - 
                             
                               s 
                               2 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     1 
                     / 
                     
                       
                         [ 
                         
                           
                             s 
                             2 
                           
                           + 
                           
                             
                               √ 
                               2 
                             
                             ⁢ 
                             s 
                           
                           + 
                           1 
                         
                         ] 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
     In the case of N=3, since stable poles are s 2 =e 2π/3 , s 3 =−1 and s 4 =e 4π/3 , 
     
       
         
           
             
               
                 
                   
                     G 
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     1 
                     / 
                     
                       [ 
                       
                         
                           ( 
                           
                             s 
                             - 
                             
                               s 
                               3 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             s 
                             - 
                             
                               s 
                               1 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             s 
                             - 
                             
                               s 
                               2 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     1 
                     / 
                     
                       
                         [ 
                         
                           
                             ( 
                             
                               s 
                               + 
                               1 
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 s 
                                 2 
                               
                               + 
                               s 
                               + 
                               1 
                             
                             ) 
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
     Even if the order of N is increased more than 3, the transfer function of the analog normalized low-pass filter can be obtained. The frequency characteristics of the analog low-pass filter are calculated using s=jω. 
     [Frequency Pre-Warping] 
     In order to approximate the analog normalized low-pass filter with an IIR (Infinite Impulse Response) digital low-pass filter, a frequency conversion and a bilinear s-z transform are performed; however, prior to these processes, frequency pre-warping is required in Step S 12 . 
     In the bilinear s-z transform, s=(2/T s )[(1−z −1 )/(1+z −1 )] needs to be calculated; however, because of s=jω A  for the analog angular frequency and z=exp(jω D T s ) for the digital angular frequency, the relationship ω A =(2/T s )tan[(ω D T s )/2] is obtained from jω A =(2/T s )[1−(exp(jω D T s )) −1 /1+(exp(jω D T s )) −1 ]. 
     That is, by performing the frequency pre-warping, the following equation is satisfied in the case where the cutoff frequency of a digital low-pass filter is desired to be ω C =10 kHz.
 
ω C, Analog =(2/ T   s )tan[(ω C   T   s )/2]
 
In order to do the design using the analog normalized low-pass filters, ω C, Analog  is used as the cutoff frequency below.
 
[Frequency Conversion]
 
     In the frequency conversion using the analog normalized low-pass filters, s=s/ω C  is used as the conversion equation. In Step S 13 , the frequency conversion is performed by assigning the conversion equation in the equation of the normalized low-pass filter. In the case of N=2, the frequency conversion is as follows. 
                     G   ⁡     (   s   )       =       ⁢     1   /     [       s   2     +       √   2     ⁢   s     +   1     ]                   =       ⁢     1   /     [         (     s   /     ω   C       )     2     +       √   2     ⁢     (     s   /     ω   C       )       +   1     ]                   
[Bilinear s-z Transform]
 
     Next in Step S 14 , the bilinear s-z transform is performed, and the analog low-pass filter is converted into a digital low-pass filter (Step S 15 ). As described above, the equation of the bilinear s-z transform is s=(2/T s )[(1−Z −1 )/(1+Z −1 )]. This is substituted into the equation of the normalized low-pass filter as follows.
 
 H ( z )=1/[{(2/ T   s )[(1 −z   −1 )/(1+ z   −1 )]} 2 +√2 s {(2/ T   s )[(1 −z   −1 )/(1 +z   −1 )]}+1]
 
In order to obtain a frequency characteristic H(ω) of H(z), z=exp(jωT s ) is substituted into the above equation.
 
       FIG. 9  shows amplitude frequency characteristics of an analog low-pass filter and a digital low-pass filter which approximates the analog low-pass filter.  FIG. 10  shows phase frequency characteristics of the analog low-pass filter and the digital low-pass filter. The analog low-pass filter is indicated by the dashed-dotted line, and the digital low-pass filter is indicated by the solid line. The cutoff frequency is 10 kHz, and both the amplitude and phase exhibit substantially matched characteristics in the signal pass band. 
     Next is described correction of the phase characteristic performed by an all-pass phase circuit. 
     [Setting of All-Pass Phase Circuit] 
     In order to correct the phase characteristic of the analog low-pass filter, an all-pass phase circuit (also referred to as “all-pass filter”) is provided. The all-pass phase circuit is expressed by the following equation (Step S 21 ). 
                           X   ⁡     (   z   )       =       ⁢         z     -   N       +       a   1     ⁢     z     -     (     N   -   1     )           +       a   2     ⁢     z     -     (     N   -   2     )         ⁢           ⁢   …     +     a   N         1   +       a   1     ⁢     z     -   1         +       a   2     ⁢     z     -   2         +   …   +       a   N     ⁢     z     -   N                         =       ⁢       1   +       ∑     n   =   1     N     ⁢       a   n     ⁢     z     -     (     N   -   n     )                 1   +       ∑     n   =   1     N     ⁢       a   n     ⁢     z     -   n                             [     Equation   ⁢           ⁢   1     ]               
The all-pass phase circuit has an amplitude characteristic of 1 in all frequency bands, and is capable of providing a desired phase characteristic by setting coefficients a n .
 
     In order to design an all-pass phase circuit, first, the order of the circuit needs to be determined. A larger order allows finer phase adjustments. For example, the following shows a first-order, a second-order and a third-order all-pass phase circuit.
     First Order: X(z)=(z −1 +a 1 )/(1+a 1 z −1 )   Second Order: X(z)=(z −2 +a 1 z −1 +a 2 )/(1+a 1 z −1 +a 2 z −2 )   Third Order: X(z)=(z −3 +a 1 z −2 +a 2 z −1 +a 3 )/(1+a 1 z −1 +a 2 z −2 +a 2 z −2 )
 
As the order of z increases, finer phase adjustments can be made; however, with the increase in the order of z, the digital circuit increases in size, and the time delay of the entire signal also increases. Therefore, the all-pass phase circuit should be designed with an appropriate order.
   

       FIGS. 11A and 11B  show amplitude and phase frequency characteristics of a first-order all-pass phase circuit (a 1 32 −0.5), and  FIGS. 12A and 12B  show amplitude and phase frequency characteristics of a second-order all-pass phase circuit (a 1 =0.6 and a 2 =0.3). The following describes designing in which a second-order all-pass phase circuit is used. 
     [Characteristics Obtained When Digital Low-pass Filter and All-pass Phase Circuit are Cascade-connected] 
     In Step S 22 , a transfer function F(z) of a cascade-connected filter, in which the second-order all-pass phase circuit is cascade-connected to a digital low-pass filter, is obtained.
 
 F ( z )= X ( z ) H ( z )
 
F(z) is calculated as follows.
 
                           F   ⁡     (   z   )       =       ⁢       X   ⁡     (   z   )       ⁢     H   ⁡     (   z   )                     =       ⁢           z     -   2       +       a   1     ⁢     z     -   1         +     a   2         1   +       a   1     ⁢     z     -   1         +       a   2     ⁢     z     -   2             ×                     ⁢     1         (       2       ω   C     ⁢     T   S         ·       1   -     z     -   1           1   +     z     -   1             )     2     +       2     ⁢     (       2       ω   C     ⁢     T   S         ·       1   -     z     -   1           1   +     z     -   1             )       +   1                     [     Equation   ⁢           ⁢   2     ]               
The all-pass phase circuit is designed to have flat group delay characteristics of F(z). In Step S 23 , the frequency characteristic is obtained by substituting z=exp(jωT s ) into F(z), and a phase angle ∠F(ω) is obtained.
 
                       ∠   ⁢           ⁢     F   ⁡     (   ω   )         =     ∠   ⁢     {                 z     -   2       +       a   1     ⁢     z     -   1         +     a   2         1   +       a   1     ⁢     z     -   1         +       a   2     ⁢     z     -   2             ×               1               (       2       ω   C     ⁢     T   S         ·       1   -     z     -   1           1   +     z     -   1             )     2     +                   2     ⁢     (       2       ω   C     ⁢     T   S         ·       1   -     z     -   1           1   +     z     -   1             )       +   1                   }         ,     
     ⁢     (     z   =     exp   ⁡     (     jω   ⁢           ⁢     T   S       )         )             [     Equation   ⁢           ⁢   3     ]               
[Calculation of Group Delay Amount u(ω) and Setting of Evaluation Function ν]
 
     A group delay amount u(ω) can be obtained from the following equation using an increase of the phase angle d∠F(ω) and an increase in the angular frequency dω (Step S 24 ).
 
 u (ω)=− d∠F (ω)/ dω 
 
     Since the group delay characteristics in the signal pass band should eventually become flat, the difference between the maximum value and the minimum value of the group delay in the signal pass band is calculated, and the difference is used as an evaluation function ν to be minimized. For example, for a low-pass filter having a cutoff frequency of 10 kHz, the difference between the maximum value and the minimum value of the group delay amount is calculated in the range between 0 Hz and 10 kHz.
 
ν= u (ω) max   −u (ω) min  (ωis the signal pass band)
 
[Calculation of Coefficients a n ]
 
     In Step S 25 , the coefficients a n  of the all-pass phase circuit, which minimize the evaluation function ν, are calculated. If the coefficients a n  are set only at the stage of designing the analog low-pass filter and the all-pass phase circuit, the minimum value of the evaluation function ν can be obtained by calculating the evaluation function ν for all combinations of the coefficients a n . In the case of the second-order all-pass phase circuit, for example, the evaluation function ν is calculated for all combinations of the coefficients a n  when −1≦a 1 ≦1 and −1≦a 2 ≦1. 
     [Calculation Example] 
       FIGS. 13 and 14  relate to the analog low-pass filter and the cascade-connected filter, in which the all-pass phase circuit is cascade-connected to the digital low-pass filter as designed in the above embodiment, and show their characteristics obtained by a simulation. As illustrated in  FIG. 13A , in the signal pass band, the amplitude frequency characteristic of the cascade-connected filter indicated by the solid line matches the characteristic of the analog low-pass filter indicated by the dashed-dotted line. In  FIG. 13B , the phase frequency characteristic of the cascade-connected filter exhibits linearity. 
       FIG. 14  shows group delay frequency characteristics of the analog low-pass filter and the cascade-connected filter obtained by a simulation. In the signal pass band (10 kHz and lower), the cascade-connected filter indicated by the solid line exhibits a flat group delay characteristic compared to the analog low-pass filter indicated by the dashed-dotted line. Note that the values of a 1  and a 2  in this case are 0.0700 and −0.0800. 
     In Step S 26 , the all-pass phase circuit defined by the coefficients a 1 , a 2 , . . . , and a n  obtained as described above is set in the digital signal processing unit  13  (or  26 ). 
     In this manner, it is possible to readily correct the group delay characteristics of the analog low-pass filter. 
     All examples and conditional language used herein are intended for pedagogical purposes to aid the reader in understanding the present disclosure 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 or inferiority of the present disclosure. Although the embodiments of the present disclosure have been described in detail, it should be understood that various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the present disclosure.