Abstract:
A pseudo-stereo circuit is provided which processes an input monophonic signal into stereophic audio signals. A phase-shift circuit shifts a phase of the input monophonic signal by a phase shift amount that depends upon a frequency of the monophonic signal, to produce an output signal having a gain with respect to the input monophonic signal which is equal to or larger than a predetermined level over an entire frequency band thereof, and reaches a peak at a frequency at which the phase shift amount of the output signal with respect to the input monophonic signal assumes a value equal or closer to −π. A mixing circuit produces a first mixed signal by mixing a signal obtained by inverting a phase of the output signal of the phase-shift circuit with the input monophonic signal by a first mixing ratio, and produces a second mixed signal obtained by mixing the output signal of the phase-shift circuit with the input monophonic signal by a second mixing ratio. The mixing circuit generates the first mixed signal as a first audio signal carried by one of left and right channels that provide stereophonic audio signals, and generates the second mixed signal as a second audio signal carried by the other of the left and right channels.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a pseudo-stereo circuit for converting monophonic audio signals into stereophonic audio signals. 
     2. Prior Art 
     FIG. 1 shows an example of a conventional pseudo-stereo circuit. The pseudo-stereo circuit is principally comprised of L-channel phase-shift circuit  100 L and R-channel phase-shift circuit  100 R for shifting the phase of a monophonic audio signal Min, to generate respective output signals, and a stereo coordination circuit  200  that receives the output signals of the phase-shift circuits  100 L and  100 R, and produces stereophonic audio signals carried by two channels, i.e., L and R channels. 
     The L-channel phase-shift circuit  100 L includes, for example, three all-pass filters  101 L,  102 L and  103 L that are cascade-connected in this order. Similarly, the R-channel phase-shift circuit  100 R includes three all-pass filters  101 R,  102 R, and  103 R similar in structure to the filters  101 L,  102 L and  103 L, that are cascade-connected in this order. Each of these all-pass filters will be described in detail below. 
     The all-pass filter  101 L is comprised of an operational amplifier  301 , resistors  302 - 304 , and a capacitor  305 , that are connected in the manner as shown in FIG.  1 . The resistors  303  and  304  have the same resistance value. Accordingly, the input voltage Vn of the inverting input terminal (−) of the operational amplifier  301  is given by the following expression (1): 
     
       
           Vn= (Min+ Vo )/2  (1) 
       
     
     where Vo is the output voltage of the operational amplifier  301 . 
     On the other hand, the input voltage Vp of the noninverting input terminal (+) of the operational amplifier  301  is given by the following expression (2): 
     
       
           Vp= Min/(1+ jωC   1   R   1 )  (2) 
       
     
     where R 1  represents the resistance value of the resistor  302 , C 1  the capacitance value of the capacitor  305 , and ω the angular frequency of the input monophonic signal Min. 
     In the circuit arrangement shown in FIG. 1, since the inverting input terminal (−) and noninverting input terminal (+) of the operational amplifier  301  are virtually short-circuited to each other due to negative-feedback operation of the circuit, the input voltage Vp becomes equal to the input voltage Vn, and the following expression (3) is established: 
     
       
         (Min+ Vo )/2=Min/(1+ jωC   1   R   1 )  (3) 
       
     
     By transforming the above expression (3), the transfer function of the all-pass filter  101 L is obtained as follows:                    H   =     Vo   /   Min                   =       (     1   -     j                 ω                 C1R1       )     /     (     1   +     j                 ω                 C1R1       )                                  (   4   )                                
     Thus, the gain G of the all-pass filter  101 L with respect to the input monophonic signal Min is obtained from the above expression (4), and expressed by:                    G   =        H                    =            (     1   -     j                 ω                 C1R1       )     /     (     1   +     j                 ω                 C1R1       )                          =            (     1   -     j                 ω                 C1R1       )          /          (     1   +     j                 ω                 C1R1       )                                     =   1                 (   5   )                                
     Accordingly, the input monophonic signal Min of any level of frequency passes through the all-pass filter  101 L while keeping its amplitude at the same value. 
     The phase of the input monophonic signal Min is shifted when the signal passes through the all-pass filter  101 L. The phase shift amount or phase angle θ is determined depending upon the frequency of the input signal Min, as shown in the following expression (6):                    θ   =     arg                   (   H   )                     =       -   2                       tan     -   1            (     ω                 C1R1     )                                    (   6   )                                
     The all-pass filter  101 L has the above described construction and frequency characteristics. 
     The other all-pass filters  102 L and  103 L subsequent to the all-pass filter  101 L have exactly the same structure as the all-pass filter  101 L. As is apparent from the above expression (6), the phase shift amount given by each of the all-pass filters  101 L- 103 L to the input monophonic signal Min varies from 0 to −π, as the frequency f=ω/2π changes. Accordingly, the phase shift amount given by the L-channel phase-shift circuit  100 L as a whole to the input signal Min varies from 0 to −3π as the frequency f of the input signal changes. The phase shift amount θL given by the whole L-channel phase-shift circuit  100 L is illustrated in FIG.  2 . 
     The R-channel phase-shift circuit  100  R has basically the same structure as the L-channel phase-shift circuit  100 L as explained above, but the resistance value of the resistor  302  and the capacitance value of the capacitor  305  of each of the all-pass filters  101 R- 103 R are different from the values R 1  and C 1  of the all-pass filters  101 L- 103 L, such that, as shown in FIG. 2, the curve representing the frequency characteristic of the phase shift amount θR of the R-channel phase-shift circuit  100 R as a whole is shifted with respect to the curve representing the frequency characteristic of the phase shift amount θL of the L-channel phase-shift circuit  100 L in the direction of the X-axis representing the frequency of the input signal. 
     By appropriately selecting the resistance value of the resistor  302  and the capacitance value of the capacitor  305  in each of the L-channel phase-shift circuit  100 L and R-channel phase-shift circuit  100 R, a difference (θL−θR) between the phase shift amounts of these circuits  100 L,  100 R can be controlled to approximately π/2 over almost the entire audio frequency band, as shown in FIG.  2 . In the circuit shown in FIG. 1, the resistance and capacitance values are suitably selected so that the above requirement is satisfied. 
     In the circuit arrangement shown in FIG. 1, therefore, the L-channel phase-shift circuit  100 L and the R-channel phase-shift circuit  100 R output respective audio signals whose phases are shifted with respect to the phase of the input monophonic signal Min and are different from each other by π/2. 
     The stereo coordination circuit  200  functions to produce stereophonic audio signals based on the respective output signals of the L-channel phase-shift circuit  100 L and R-channel phase-shift circuit  100 R as explained above. The stereo coordination circuit  200  is comprised of a subtracter  201 , a filter  202 , an adder  203  and a subtracter  204 . In the thus constructed stereo coordination circuit  200 , the subtracter  201  produces a signal corresponding to a difference between the output signals of the L-channel phase-shift circuit  100 L and the R-channel phase-shift circuit  100 R, and the filter  202  limits the frequency range of the output signal of the subtracter  201 . The adder  203  performs addition of the output signal of the filter  202  and the output signal of the L-channel phase-shift circuit  100 L. The subtracter  204  performs subtraction between the output signal of the filter  202  and the output signal of the R-channel phase-shift circuit  100 R. The adder  203  and the subtracter  204  then generate stereophonic audio signals carried by two channels, or L and R channels, so as to produce sound that affords the listener a sense of the spatial distribution of the sound sources. 
     When the above-described pseudo-stereo circuit is produced as an integrated circuit or IC, the resulting IC chip has a relatively large area since the circuit requires a large number of constituent components, such as operational amplifiers. Also, the known pseudo-stereo circuit requires six capacitors only in the phase-shift circuits for the L and R channels, and these capacitors are generally required to have large capacitance values. It is, therefore, difficult to form these capacitors on the IC board, in view of the limitation of the chip area, and the capacitors need to be provided outside the IC chip, resulting in an increased number of pins needed to be used in the IC. Under these circumstances, the known pseudo-stereo circuit suffers from undesirably high manufacturing cost. 
     SUMMARY OF THE INVENTION 
     It is therefore an object of the invention to provide an inexpensive pseudo-stereo circuit having a simplified structure. 
     To attain the above object, the present invention provides a pseudo-stereo circuit comprising an input terminal that receives an input monophonic signal to be processed, a phase-shift circuit that shifts a phase of the input monophonic signal by a phase shift amount that depends upon a frequency of the monophonic signal, to produce an output signal having a gain with respect to the input monophonic signal which is equal to or larger than a predetermined level over an entire frequency band thereof, and reaches a peak at a frequency at which the phase shift amount of the output signal with respect to the input monophonic signal assumes a value equal or closer to −π, and a mixing circuit that produces a first mixed signal by mixing a signal obtained by inverting a phase of the output signal of the phase-shift circuit with the input monophonic signal by a first mixing ratio, and produces a second mixed signal obtained by mixing the output signal of the phase-shift circuit with the input monophonic signal by a second mixing ratio, the mixing circuit generating the first mixed signal as a first audio signal carried by one of left and right channels that provide stereophonic audio signals, and generating the second mixed signal as a second audio signal carried by the other of the left and right channels. 
     Preferably, the phase shift amount of the output signal of the phase-shift circuit with respect to the input monophonic signal changes in a range from 0π to −2π depending upon a frequency of the input monophonic signal. 
     In a preferred form, the phase-shift circuit comprises first and second phase-shift filters that are cascade-connected, Each of the first and second phase-shift filters comprises an operational amplifier having an inverting input terminal, a noninverting input terminal, and an output terminal, a time-constant circuit formed of a resistance through which an input signal of the filter is transmitted to the noninverting input terminal of the operational amplifier, and a capacitance, an input resistance through which the input signal is transmitted to the inverting input terminal of the operational amplifier, and a feedback resistance interposed between the inverting input terminal and the output terminal of the operational amplifier. A resistance value ratio of the input resistance to the feedback resistance of the first phase-shift filter is set to be greater than 1, and a resistance value ratio of the input resistance to the feedback resistance of the second phase-shift filter is set to be smaller than 1. 
     Preferably, the first and second phase-shift filters each shift the phase of an input signal thereof by a phase shift amount which changes in e range from 0π to −2π depending upon a frequency of the input monophonic signal, to produce a output signal which is shifted in phase with respect to the input signal. 
     Also preferably, the first phase-shift filter generates an output signal which has a gain with respect to an input signal thereof, which progressively increases from 1 to a predetermined value as a frequency of the input signal increases, and the second phase-shift filter generates an output signal which has a gain with respect to an input signal thereof, which progressively decreases from 1 to a second predetermined value as a frequency of the input signal increases. 
     Preferably, the first predetermined value has a reciprocal thereof almost equal to the second predetermined value. 
     Advantageously, the phase shift amount of the first mixed signal with respect to the input monophonic signal progressively changes in a predetermined direction as a frequency of the monophonic signal changes, and the phase shift amount of the second mixed signal with respect to the input monophonic signal is maintained at an almost constant value irrespective of changes in the frequency of the monophonic signal, the first and second mixing ratios being determined so that frequency characteristics of the gains of the first and second mixed signals with respect to the input monophonic signal are substantially identical to each other over the entire frequency band. 
     Preferably, the first and second mixed signals each have a gain which reaches a peak at or about a frequency at which a phase difference between the first and second mixed signals is equal to π. 
     The above and other objects, features, and advantages of the invention will be become more apparent from the following detailed description taken in conjunction with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram showing the construction of a known pseudo-stereo circuit; 
     FIG. 2 is a graph showing frequency characteristics of phase-shift circuits corresponding to two channels, i.e., L channel and R channel, in the pseudo-stereo circuit of FIG. 1; 
     FIG. 3 is a block diagram showing the construction of a pseudo-stereo circuit according to one embodiment of the present invention; 
     FIG. 4A is a graph showing, by way of example, the frequency characteristic of the gain of a phase-shift circuit in the pseudo-stereo circuit of FIG. 3; 
     FIG. 4B is a graph showing, by way of example, the frequency characteristic of the phase shift amount of the phase-shift circuit in the pseudo-stereo circuit of FIG. 3; 
     FIG. 5A is a graph showing, by way of example, the frequency characteristic of the gain of a signal processing system that produces an L-channel audio signal, in the pseudo-stereo circuit of FIG. 3; 
     FIG. 5B is a graph showing, by way of example, the frequency characteristic of the phase shift amount of the signal processing system that produces the L-channel audio signal, in the pseudo-stereo circuit of FIG. 3; 
     FIG. 5C is a graph showing, by way of example, the frequency characteristic of the gain of a signal processing system that produces an R-channel audio signal, in the pseudo-stereo circuit of FIG. 3; 
     FIG. 5D is a graph showing, by way of example, the frequency characteristic of the phase-shift amount of the signal processing system that produces the R-channel audio signal, in the pseudo-stereo circuit of FIG. 3; 
     FIG. 6 is a circuit diagram showing the construction of a specific example of the pseudo-stereo circuit of FIG. 3; and 
     FIG. 7 is a block diagram showing the arrangement of a surround system as an example in which the pseudo-stereo circuit of FIG. 3 is used. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The invention will now be described in detail with reference to drawings showing a preferred embodiment thereof. 
     FIG. 3 shows the construction of a pseudo-stereo circuit according to one embodiment of the invention. As shown in FIG. 3, the pseudo-stereo circuit of the present embodiment is principally comprised of a phase-shift circuit  1 , multipliers  2  and  3 , and adders  4 ,  5 , and thus has a considerably simple structure. 
     The phase-shift circuit  1  serves to shift the phase of an input monophonic signal Min to be processed in the present embodiment, and includes two phase-shift filters  11 ,  12  that are cascade-connected. Each of the phase filters  11 ,  12  is adapted to shift the phase of an input signal thereto, such that the phase shift amount given by each of the filters  11 ,  12  varies in the range of 0 to −π. The gain, namely, the ratio of the output signal of each phase-shift filter  11 ,  12  to the corresponding input signal, is not constant for changes in the frequency of the input signal. Namely, the gain of one ( 11 ) of the phase-shift filters progressively increases from 1 to a certain value (&gt;1) as the frequency increases, and the gain of the other phase-shift filter ( 12 ) progressively decreases from 1 to a certain value (&lt;1) as the frequency increases. The structures of the phase-shift filters  11  and  12  will be more specifically described later. 
     FIG.  4 A and FIG. 4B show respective frequency characteristics of the gain and phase shift amount of the phase-shift circuit  1  as a whole, which is comprised of the phase-shift filters  11  and  12 . As indicated in FIG. 4B, the phase shift amount given to the input signal by means of the phase-shift circuit  1  varies in the range of 0 to −π, depending upon the frequency of the input signal. Also, as indicated in FIG. 4A, the gain given to the input signal by means of the phase-shift circuit  1  is kept being equal to or higher than a certain value throughout the entire frequency band, and it reaches a peak at a given frequency where the phase shift amount is approximately equal to −π. 
     The multiplier  2  multiplies the output signal of the phase-shift circuit  1  by a predetermined coefficient “−a”. The multiplier  3 , on the other hand, multiplies the output signal of the phase-shift circuit  1  by a predetermined coefficient “b”. Then, the adder  4  adds the output signal of the multiplier  2  and the original input monophonic signal Min, and the adder  5  adds the output signal of the multiplier  3  and the original input monophonic signal Min. The results of addition of the adders  4 ,  5  are produced as stereophonic audio signals carried by two channels, i.e., L and R channels. 
     FIGS. 5A and 5B show respective frequency characteristics of the gain and phase shift amount of a signal processing system (comprised of the phase-shift circuit  1 , multiplier  2 , and adder  4 ) associated with production of the L-channel audio signal in the pseudo-stereo circuit of the present embodiment. FIGS. 5C and 5D show respective frequency characteristics of the gain and phase shift amount of a signal processing system (comprised of the phase-shift circuit  1 , multiplier  3 , and adder  5 ) associated with production of the R-channel audio signal. 
     As shown in FIG. 5B, the phase shift amount of the signal processing system that produces the L-channel audio signal varies in the range of 0 to −2π, depending upon the frequency of the input signal. As shown in FIG. 5A by way of example, the gain of the signal processing system for producing the L-channel audio signal is kept being equal to or larger than a certain value throughout the entire frequency band, and it reaches a peak at a given frequency where the phase shift amount is approximately equal to −π. 
     On the other hand, the phase shift amount of the signal processing system that produces the R-channel audio signal is almost 0 and hardly changes throughout the entire frequency band, as shown in FIG.  5 D. However, the frequency characteristic of the gain of the signal processing system for producing the R-channel audio signal is substantially the same as that of the gain of the signal processing system for producing the L-channel audio signal, as shown in FIG.  5 C. 
     The frequency characteristics of the respective signal processing systems as described above can be obtained by suitably controlling the multiplication coefficients “−a” and “b” of the multipliers  2  and  3 . 
     In the present embodiment having the above-described frequency characteristics, the input monophonic signal Min is converted into audio signals of L and R channels whose intensity ratio and phase difference depend upon the frequency of the input signal Min, and these audio signals are generated from the respective adders  4 ,  5 . In this case, the gains of both of the signal processing systems for producing the audio signals of the L and R channels reach their peaks, at around the frequency where the phase difference of the L-channel audio signal and the R-channel audio signal is approximately equal to π. This arrangement can avoid destructive interference (in which the sounds of the L and R channels cancel each other in the air, and cannot be heard), which would otherwise occur when a sound speaker generates sound represented by audio signals of L and R channels having a phase difference of π. Thus, the pseudo-stereo circuit of the present embodiment has a remarkably simple structure as shown in FIG. 3, and still provides such a good performance as that of the known pseudo-stereo circuit. 
     Referring next to FIG. 6, a specific example of the circuitry of the pseudo-stereo circuit according to the present embodiment will be now described. In FIG. 6, the same reference numerals as used in FIG. 3 are used for identifying corresponding components or elements, so as to clarify the relationship with the known circuitry of FIG. 3 described above. 
     In FIG. 6, the phase-shift filter  11  is comprised of an operational amplifier  51 , resistors  52 - 54  and a capacitor  55 . In the phase-shift filter  11 , the input monophonic signal Min enters the noninverting input terminal (+) of the operational amplifier  51 , through a time-constant circuit (RC circuit) formed of the resistor  52  and the capacitor  55 , and also enters the inverting input terminal (−) of the operational amplifier  51  through the resistor  53 . Also, the output signal of the operational amplifier  51  is fed back to the inverting input terminal (−) through the resistor  54 . In the above-described all-pass filter  101 L shown in FIG. 1, the input resistor  303  on the side of the inverting input terminal (−) and the feedback resistor  304  have the same resistance value. In the phase-shift filter  11 , however, the resistance value of the feedback resistor  54  of the operational amplifier  51  is twice as large as that of the input resistor  53  on the side of the inverting input terminal (−). Other than this aspect, the phase-shift filter shown in FIG. 6 has the same structure or arrangement as the all-pass filter  101 L of FIG.  1 . 
     In the circuit described above, if the resistance value of the resistor  53  is designated by R, and the resistance value of the resistor  54   2 R, the following expression (7) is established: 
      (Min− Vn )/ R= ( Vn−Vo )/( 2   R )  (7) 
     where Vn represents the input voltage of the inverting input terminal (−) of the operational amplifier  51 , and Vo the output voltage of the operational amplifier  51 . 
     Accordingly, the input voltage Vn of the inverting input terminal (−) of the operational amplifier  51  is given by the following expression (8): 
     
       
           Vn= (2Min+ Vo )/3  (8) 
       
     
     On the other hand, the input voltage Vp of the noninverting input terminal (+) of the operational amplifier  51  is given by the following expression (9): 
     
       
           Vp= Min/(1+ jωC   1   R   1 )  (9) 
       
     
     where R 1  represents the resistance value of the resistor  52 , C 1  the capacitance value of the capacitor  55 , and ω the angular frequency of the input monophonic signal Min. 
     Since the input voltage Vp is equal to the input voltage Vn due to the feedback operation in the phase-shift filter  11 , the following expression (10) is established: 
     
       
         (2Min+ Vo )/3=Min/(1+ jωC   1   R   1 )  (10) 
       
     
     By transforming the above expression (10), the transfer function H 1  of the phase-shift filter  11  is obtained as follows:                      H1   =     Vo   /   Min                 =       (     1   -     2      jω                 C1R1       )     /     (     1   +     jω                 C1R1       )                     (   11   )                                             
     From the above expression (11), the phase shift amount θ 1  of the phase-shift filter  11  is obtained as follows:                    θ1   =     arg                   (   H1   )                     =         -     tan     -   1            2      ω                 C1R1     -       tan     -   1          ω                 C1R1                                  (   12   )                                
     It will be understood from the above expression (12) that the phase shift amount θ 1  of the phase-shift filter  11  changes from 0 to −π while the angular frequency ω changes from zero to infinity (∞). 
     The gain G 1  of the phase-shift filter  11  is also obtained from the above expression (10), as follows:                    G1   =        H1                      =       {       (     1   +       (     2                 ω                 C1R1     )     2       )     /     (     1   +       (     ω                 C1R1     )     2       )       }       1   /   2                                  (   13   )                                
     It will be understood from the above expression (13) that the gain G 1  of the phase-shift filter  11  changes from 1 to 2 while the angular frequency ω changes from zero to infinity (∞). 
     The phase-shift filter  12  is comprised of an operational amplifier  61 , resistors  62 - 64  and a capacitor  65 . The phase-shift filter  12  has substantially the same structure as the above-described phase-shift filter  11 , except that the input resistor  63  on the side of the inverting input terminal (−) of the operational amplifier  61  has a resistance value that is twice as large as that of the feedback resistor  64 . 
     Since the resistance value of the input resistor  63  is twice as large as that of the feedback resistor  64  in the phase-shift filter  12 , the input voltage Vn′ of the inverting input terminal (−) is given as follows: 
       Vn′= (Min′+2 Vo ′)/3  (14) 
     where Min′ represents the input signal received by the phase-shift filter  12 , and Vo′ the output signal of the phase-shift filter  12 . 
     Based on the above expression (14), the transfer function H 2  of the phase-shift filter  12  is obtained by similar calculations as performed in the case of the phase-shift filter  11 , as follows:                    H2   =       Vo   ′     /     Min   ′                     =       (     1   -     j   (                ω                   C2R2   /   2       )       )     /     (     1   +     j                 ω                 C2R2       )                                  (   15   )                                
     From the above expression (15), the phase shift amount θ 2  of the phase-shift filter  12  is given by the following expression (16):                    θ2   =     arg                   (   H2   )                     =       -       tan     -   1            (     ω                   C2R2   /   2       )         -       tan     -   1          ω                 C2R2                                  (   16   )                                
     It will be understood from the above expression (16) that the phase shift amount θ 2  of the phase-shift filter  12  also changes from 0 to −π while the angular frequency ω changes from zero to infinity (∞). 
     The gain G 2  of the phase-shift filter  12  is also derived from the above expression (15), as follows:                    G2   =        H2                    =       {       (     1   +       (     ω                   C2R2   /   2       )     2       )     /     (     1   +       (     ω                 C2R2     )     2       )       }       1   /   2                     (   17   )                                
     It will be understood from the above expression (17) that the gain G 2  of the phase-shift filter  12  changes from 1 to ½ while the angular frequency ω changes from zero to infinity (∞). 
     Next, the phase shift amount θ and gain G of the phase-shift circuit  1  as a whole that is comprised of the phase-shift filters  11  and  12  will be now explained. 
     The phase shift amount θ of the phase-shift circuit  1  as a whole is derived from the above expressions (12) and (16), as follows:                    θ   =                  θ                 1     +     θ                 2                   =                    -     tan     -   1            2      ω                 C1R1     -       tan     -   1          ω                 C1R1     -                                  tan     -   1            (     ω                   C2R2   /   2       )       -       tan     -   1          ω                 C2R2                     (   18   )                                
     Thus, the phase shift amount θ changes from 0 to −2π while the angular frequency ω changes from zero to infinity (∞). 
     The gain G of the whole phase-shift circuit  1  is derived from the above expressions (13) and (17), as follows:                    G   =              G1G2               =                {       (     1   +       (     2                 ω                 C1R1     )     2       )                     (     1   +         (     ω                 C1R1     )     2     /     (     1   +                               (                          (     ω                 C1R1     )     2     )                       (     1   +       (   ωC2R2   )     2       )       }     )       1   /   2                 =                {     (     1   +     4        ω   2          C1   2          R1   2       +       ω   2          C2   2            R2   2     /   4       +                                                   ω   4          C1   2          R1   2          C2   2          R2   2       )       /     (     1   +       ω   2          C1   2          R1     -   2         +       ω   2          C2   2          R2   2       +                     (                        ω   4          C1   2          R1   2          C2   2          R2   2       )       }     )       1   /   2                   (   19   )                                
     As already explained above, while the angular frequency ω changes from zero to infinity (∞), the gain G 1  of the phase-shift filter  11  changes from 1 to 2, and the gain G 2  of the phase-shift filter  12  changes from 1 to ½. Accordingly, the gain G of the whole phase-shift circuit  1  given by the above expression (19) increases from 1 as the angular frequency ω increases from 0, and reaches a peak value at a certain angular frequency ω 0 . The gain G of the phase-shift circuit  1  then decreases as the angular frequency ω increases, and becomes equal to 1 when the angular frequency goes to infinity (∞). 
     The angular frequency ω 0  at which the gain G reaches the peak can be obtained by calculating dG/dω based on the above expression (19), and solving the equation dG/dω=0 with respect to ω. The result of the calculation is expressed as follows: 
     
       
         ω 0 ={(12 C   1   2   R   1   2 −3 C   2   2   R   2   2 )/(12 C   1   4   R   1   4   C   2   2   R   2   2 −3 C   1   2   R   1   2   C   2   4   R   2   4 )} ¼   (20) 
       
     
     For the sake of simplicity, the condition: 
     
       
           C   1   R   1 = C   2   R   2 =τ  (21) 
       
     
     is now added. Under this condition, the following expression (22) is derived from the above expression (20): 
     
       
         ω 0 =1/τ  (22) 
       
     
     If the above-indicated ω 0  is substituted for ω in the above expression (18), under the above condition (21), the phase shift amount θ of the phase-shift circuit  1  is given by the following expression (23):                      θ   =         -     tan     -   1            2     -       tan     -   1          1     -       tan     -   1            (     1   /   2     )       -       tan     -   1          1                                =     -   π                   (   23   )                                
     In the circuit shown in FIG. 6, the phase shift amount θ of the phase-shift circuit  1  becomes equal to −π and the gain G reaches its peak when the angular frequency ω of the input monophonic signal Min is equal to ω 0 =1/τ. 
     The phase-shift circuit  1  has been described above in detail. While the frequency at which the phase shift amount θ of the phase-shift circuit  1  becomes equal to −π precisely coincides with the frequency at which the gain G reaches its peak in the example of the circuit shown in FIG. 6, these frequencies need not strictly coincide with each other, and the effect or advantages of the present embodiment can be obtained provided that the difference between these frequencies is sufficiently small. 
     A signal processing system will be now described which produces stereophonic audio signals of L and R channels, from the output signal of the phase-shift circuit  1  and the input monophonic signal. 
     A phase inverter circuit  70  is comprised of an operational amplifier  71  and resistors  72  and  73 . The phase inverter circuit  70  serves to invert the phase of the output signal of the phase-shift circuit  1 , and generates the resulting signal to a multiplier/adder  80  in the next stage. 
     The multiplier/adder  80  is comprised of an operational amplifier  81  and resistors  82 - 83 . The multiplier/adder  80  multiplies the output signal of the phase inverter circuit  70  and the input monophonic signal Min by respective coefficients, adds the results of multiplication together, and outputs the resulting signal as an L-channel audio signal. The above-described phase inverter circuit  70  and the multiplier/adder  80  correspond to the multiplier  2  and the adder  4  shown in FIG.  3 . The coefficient by which the multiplier/adder  80  multiplies the output signal of the phase inverter circuit  70  can be adjusted by suitably selecting the resistance value Ra 1  of the resistor  82 , and the coefficient by which the multiplier/adder  80  multiplies the input monophonic signal Min can be adjusted by suitably selecting the resistance value Ra 2  of the resistor  83 . 
     A multiplier/adder  90  is comprised of an operational amplifier  91  and resistors  92 - 95 . The multiplier/adder  90  multiplies the output signal of the phase-shift circuit  1  and the input monophonic signal Min by respective coefficients, adds the results of multiplication together, and outputs the resulting signal as an R-channel audio signal. Thus, the multiplier/adder  90  correspond to the multiplier  3  and the adder  5  shown in FIG.  3 . The coefficients by which the output signal of the phase-shift circuit  1  and the input monophonic signal Min are multiplied can be respectively adjusted by suitably selecting the resistance value Rb 1  of the resistor  92  and the resistance value Rb 2  of the resistor  93 . The multiplication coefficients of the multiplier/adder  90  and the multiplication coefficients of the multiplier/adder  80  are respectively set to such optimum values that the frequency characteristics as shown in FIG.  5 A through FIG. 5D can be obtained. 
     FIG. 7 schematically shows a surround system as a specific example in which the pseudo-stereo circuit of the present embodiment described above is used, wherein the pseudo-stereo circuit  21 , a surround circuit  22 , and a tone control circuit  23  are cascade-connected in this order. Since the pseudo-stereo circuit  21  of the present embodiment is relatively simple in construction and small in size, as compared with the known counterpart, the surround system as a whole is available at a reduced cost. Furthermore, the pseudo-stereo circuit of the present embodiment provides such a good performance as that of the known circuit, in spite of a reduced number of components, and therefore the surround system including the present pseudo-stereo circuit has a high performance, and is available at a relatively low cost.