Abstract:
The invention is a multi-channel loudspeaker system that provides a compact loudspeaker configuration and filter design methodology that operates in the digital signal processing domain. Further, the loudspeaker system can be designed as a multi-way loudspeaker system comprised of a symmetric arrangement of loudspeaker drivers in a two-dimensional plane and can achieve high-quality sound, constant directivity over a large area in both the vertical and horizontal planes and can be used in connection with stereo loudspeaker systems, multi-channel home entertainment systems and public address systems.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of U.S. patent application Ser. No. 10/771,190 filed on Feb. 2, 2004 titled Loudspeaker Array System, and which is incorporated into this application in its entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention generally relates to a multi-way loudspeaker system and in particular to a multi-way loudspeaker system comprised of a symmetric arrangement of loudspeaker drivers in a two-dimensional plane capable of achieving high-quality sound for use in connection with stereo loudspeaker systems, multi-channel home entertainment systems and public address systems. 
     2. Related Art 
     Loudspeaker designers are constantly striving to design controlled directivity loudspeaker systems that achieve high quality sound across a wide range of frequency bands while limiting the size and number of transducers (i.e. drivers) in the system, as well as the required number of amplifiers (i.e. ways) in the system. Achieving such a high quality sound across a wide frequency range has been challenging due to the variation in size of the transducers across the dedicated parts of the audio frequency band and the constraints in spacing between the transducers. 
     High-quality loudspeakers for the audio frequency ranges generally employ multiple, specialized drivers for dedicated parts of the audio frequency band, such as tweeters (generally 2 kHz-20 kHz), midrange drivers (generally 200 Hz-5 kHz), and woofers (generally 20 Hz-1 kHz). Typically the higher frequency drivers are smaller in size than the lower frequency drivers. 
     To achieve a high sound quality, it is desirable to position the drivers in the loudspeaker as closely as possible to one another. However, because of the physical sizes of the specialized drivers, the ability to position the drivers in close proximity to one another is limited. The farther the drivers are positioned from one another, the more acoustic problems arise. 
     Because of the spacing between drivers due to their physical size, which is comparable with the wavelength of the radiated sound, the acoustic outputs of the drivers sum up to the intended flat, frequency-independent response only on a single line perpendicular to the loudspeaker, usually at the so-called acoustic center. Outside of that axis, frequency responses are more or less distorted due to interferences caused by different path lengths of sound waves traveling from the drivers to the considered points in space. Thus, there have been many attempts in history to build loudspeakers with a controlled sound field over a larger space with smooth out-of-axis responses. 
     The current state of art for controlling sound field in large spaces, such as public spaces, is to utilize uniform coverage horns for sound reinforcement. However, the use of uniform coverage horns has its disadvantages, as the uniform coverage horns have a limited frequency range, fixed, non-steerable polar patterns, and relatively high distortion. 
     Current two-dimensional arrays for surround sound in home entertainment, so-called sound projectors, are linearly spaced arrays of identical, small wide band drivers. This type of array is capable of producing multiple sound beams, which radiate into the room, and, while bouncing back from walls to the listener, produce the desired surround effect. However, since the drivers in the two-dimensional, linearly spaced arrays are identical, the maximum sound pressure level, and sound quality of the sound projector is limited to the capabilities of the transducers, which is in general rather poor, compared with drive units that are optimized for a dedicated frequency band. Further, the sound projector employs a very high number of drivers that all need to be driven individually, which leads to high implementation complexity and high cost. 
     Thus, a need still exists for a high-quality, low-distortion, two-dimensional loudspeaker configuration that employs a minimum number of transducers, as well as amplifiers, where the transducers are optimized for high performance by utilizing specialized drivers, such as tweeters, midrange drivers or woofers, across the audio frequency band. A further need still exists for a two-dimensional loudspeaker configuration to electronically alter beam widths and steering angles on site, as opposed to fixed installations using horn arrays. 
     SUMMARY 
     The invention is a multi-way array loudspeaker that can produce high-quality sound in high fidelity stereo systems, multi-channel home entertainment systems or public address systems. 
     In one embodiment, the array includes a plurality of tweeters, mid-range drivers and woofers that are arranged in a single housing or assembled as a single unit, having sealed compartments that separate certain drivers from one another to prevent coupling of the drivers. The array may be single channel having various signal paths from the input to individual loudspeaker drivers or to a plurality of drivers. Each signal path comprises digital input and contains a digital FIR filter, a D/A converter and a power amplifier, or a so-called power D/A converter, connected to either a single driver or to multiple drivers. 
     The performance, positioning and arrangement of the loudspeaker drivers in the array may be determined by a filter design algorithm that establishes the coefficients for each FIR filter in each signal flow path of the loudspeaker. A cost minimization function is applied to prescribed frequency points, using initial driver positions and initial directivity target functions, which are defined at frequency points on a logarithmic scale within the frequency range of interest. If the obtained results from the application of the cost minimization function do not meet the performance requirements of the system, the position of the drivers may then be modified and the cost minimization function may be reapplied until the obtained results meet the system requirements. Once the obtained results meet the system requirements, the filter coefficients for each linear phase FIR filter in a signal path are computed using the Fourier approximation method or other frequency sampling method. 
     The multi-way loudspeakers of the invention may include built-in DSP processing, D/A converters and amplifiers and may be connected to a digital network (e.g. IEEE 1394 standard). Further, the multi-way loudspeaker system of the invention, due to its compact dimensions, may be designed as a wall-mountable surround system. 
     The multi-way loudspeaker system may employ drivers of different sizes, producing low distortion, high-power handling because specialized drivers can operate optimal in their dedicated frequency band, as opposed to arrays of identical wide-band drivers. The multi-way speaker design of the invention can also provide better control of in-room responses due to smooth out-of-axis responses. The system is further able to control the frequency response of reflected sound, as well as the total sound power, and to suppress floor and ceiling reflections. 
     Other systems, methods, features and advantages of the invention will be or will become apparent to one with skill in the art upon examination of the following figures and detailed description. It is intended that all such additional systems, methods, features and advantages be included within this description, be within the scope of the invention, and be protected by the accompanying claims. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       The invention can be better understood with reference to the following figures. The components in the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention. Moreover, in the figures, like reference numerals designate corresponding parts throughout the different views. 
         FIG. 1  illustrates an example of a one-dimensional four-way loudspeaker system mounted along the y-axis symmetrically to origin and a block diagram of signal flow to each of the loudspeaker drivers in the system. 
         FIG. 2  illustrates an example of a two-dimensional four-way loudspeaker system mounted along the x-axis and y-axis symmetrically to origin and a block diagram of signal flow to each of the loudspeaker drivers in the system. 
         FIG. 3  is a flow chart of a filter design algorithm used to design the loudspeaker system. 
         FIG. 4  is a graph illustrating the directivity target functions for angle-dependent attenuation. 
         FIG. 5  is a graph illustrating measured amplitude frequency responses of one mounted tweeter at various vertical out-of-axis displacement angles. 
         FIG. 6  illustrates another example of a two-dimensional four-way loudspeaker system mounted along the y and x-axis symmetrically to origin. 
         FIG. 7  is a block diagram of the signal flow to each of the loudspeaker drivers illustrated in  FIG. 6 . 
         FIG. 8  depicts the frequency responses of the four filters of the loudspeaker system in  FIG. 6 . 
         FIG. 9  illustrates the resulting horizontal (y-axis) frequency responses of the loudspeaker system in  FIG. 6  measured at various angles. 
         FIG. 10  illustrates the resulting vertical (x-axis) frequency responses of the loudspeaker system in  FIG. 6  that corresponds to the horizontal responses shown in  FIG. 9 . 
         FIG. 11  illustrates an example implementation of a one-dimensional (1D) seven-way loudspeaker system mounted symmetrically along the y-axis and a block diagram of signal flow to each of the loudspeaker drivers in the system. 
         FIG. 12  shows the frequency responses of the seven filters of the loudspeaker system in  FIG. 11 . 
         FIG. 13  illustrates the resulting horizontal (x-axis) frequency responses of the loudspeaker system in  FIG. 11  measured at various angles. 
         FIG. 14  illustrates an example implementation of a two-dimensional (2D), multi-channel, seven-way loudspeaker system mounted symmetrically along the x-axis and y-axis. 
         FIG. 15  is a block diagram of signal flow to each of the loudspeaker drivers in the loudspeaker system of  FIG. 14 . 
         FIG. 16  illustrates the resulting vertical (y-axis) frequency responses of the loudspeaker system in  FIG. 14  measured at various angles. 
         FIG. 17  illustrates an example implementation of a two-dimensional (2D), five-channel, multi-way loudspeaker system mounted symmetrically along the x-axis and y-axis designed for use for home theatre applications. 
         FIG. 18  is a block diagram of the signal flows for the right and left surround channels for the loudspeaker system in  FIG. 17 . 
         FIG. 19  is a block diagram of the signal flows for the right and left channels for the loudspeaker system in  FIG. 17 . 
         FIG. 20  is a block diagram of the signal flows for the center channel for the loudspeaker system in  FIG. 17 . 
         FIG. 21  the frequency responses of the four filters of the center channel of the loudspeaker system in  FIG. 17 . 
         FIG. 22  illustrates the resulting horizontal (x-axis) frequency responses of the center channel of the loudspeaker system in  FIG. 17  measured at various angles. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates an example implementation of a one-dimensional (1D) multi-way loudspeaker  100  which forms the bases of the invention and a block diagram of the signal flow to each of the loudspeaker drivers in the system  100 . As shown in  FIG. 1 , the multi-way loudspeaker  100  may be designed as a four-way loudspeaker having (i) a center tweeter  102  connected to a first power D/A converter  103 , (ii) two additional tweeters  104  and  106  connected to a second power D/A converter  105 , (iii) two midrange drivers  108  and  110  connected to a third power D/A converter  107 , and (v) two woofers  112  and  114  connected to a fourth power D/A converter  109 . The connection between the loudspeakers to each amplifier represents a different way in the multi-way loudspeaker. 
     In  FIG. 1 , the drivers, also referred to as transducers, may be mounted in a housing  116  comprised of separate sealed compartments  120 ,  122 , and  124 , as indicated by separators  132  and  134 . By mounting the drivers in separate sealed compartments, coupling of the neighboring drivers is minimized. Although the various compartments are visible in  FIG. 1 , the loudspeaker system may be designed such that the compartments are not visible to the consumer when embodied in a finished product. Compartment  124 , containing woofer  112  may be separated by separator  132  from compartment  120 , which contains midrange drivers  108  and  110  and tweeters  102 ,  104  and  106 . Similarly, compartment  122 , containing woofer  114  may be separated by separator  134 , from compartment  120 , which contains midrange drivers  108  and  110  and tweeters  102 ,  104  and  106 . All of the tweeters  102 ,  104 ,  106  may be contained in the same compartment  120  as the midrange drivers  108  and  110  without the necessity of separating the tweeters  102 ,  104  and  106  from the midrange drivers because the tweeters  102 ,  104  and  106  are typically sealed. 
       FIG. 1  illustrates the center tweeter  102 , tweeters  104  and  106 , midrange drivers  108 ,  110  and low-frequency woofers  112  and  114  mounted linearly along the y-axis and symmetrically about the center tweeter  102 . A typical arrangement may include tweeters  102 ,  104  and  106  of outer diameters of approximately 40-50 mm, midrange drivers  108  and  110  of outer diameters of approximately 80-110 mm, and woofers  112  and  114  of outer diameters of approximately 120-250 mm. Typically, transducer cone size may differ based on the desired application and desired size of the array. Further, the transducers may utilize neodymium magnets, although it is not necessary for the described application to utilize that particular type of magnet. 
     When utilizing tweeters of diameter 50 mm, midrange drivers of 110 mm and woofers of 160 mm, an example implementation of the system may include the center tweeter  102  mounted on the y-axis at the center point  0  at the intersection between the x and y axis. The tweeters  104  and  106  may be mounted at their centers approximately +/−60 mm from the center point. The midrange drivers  110  and  108  may then be mounted at their centers approximately +/−150 mm from the center point  0 . The low-frequency woofers  112  and  114  may then be mounted at their centers approximately +/−300 mm from the center point. 
       FIG. 1  also illustrates a block diagram  140  of the signal flow of the multi-way loudspeaker system. While  FIG. 1  illustrates four ways  142 ,  144 ,  146  and  148  of signal flow, a channel may be divided into two or more ways. The signal flow comprises a digital input  150  that may be implemented using standard interface formats, such as SPDIF or IEEE1394 and their derivatives, and that can be connected to the drivers through various paths or ways, such as those illustrated in  FIG. 1 . Each path or way  142 ,  144 ,  146  and  148  may contain a digital FIR filter  152  and a power D/A converter  103 ,  105 ,  107  and  109  connected to either a single or to multiple loudspeaker drivers. The power D/A converters  103 ,  105 ,  107  and  109  may be realized as cascades of conventional audio D/A converters (not shown) and power amplifiers (not shown), or as class-D power amplifiers (not shown) with direct digital inputs. The FIR filters  152  may be implemented with a digital signal processor (DSP) (not shown). The loudspeaker drivers may be tweeters, midrange drivers or woofers, such as those illustrated. 
     In operation, the outputs of each multiple FIR filter  152  are connected to multiple power D/A converters  103 ,  105 ,  107  and  109  that are then fed to multiple loudspeaker drivers  102 ,  104 ,  106 ,  108 ,  110 ,  112 , and  114  that are mounted on a baffle of the housing  116 . More than one driver, such as  104  and  106 , may be connected in parallel to a path or way  142 ,  144 ,  146  and  148  containing a power D/A converter  103 ,  105 ,  107  and  109 . 
       FIG. 2  illustrates a two-dimensional multi-way loudspeaker  200  that is derived by splitting the tweeters  104  and  106  and midrange drivers  108  and  110  of  FIG. 1  into pairs. As further discussed below, the paired drivers may be electrically connected with each other and may be fed by the same filters as the one-dimensional (1D) multi-way loudspeaker  100  of  FIG. 1 . Therefore, directivity along y-axis is not affected and stays the same as originally specified in far field. New directivity properties, may, however, be introduced along the x-axis, as desired. 
     In particular,  FIG. 2  illustrates a single channel, two-dimensional, four-way loudspeaker  200  having a center tweeter  202  encircled by four additional tweeters  204 ,  206 ,  208  and  210 . Additionally, the loudspeaker  200  contains four midrange drivers  212 ,  214 ,  216  and  218  and two woofers  220  and  222 . 
     Tweeters  204 ,  206 ,  208  and  210 , the midrange drivers  212 ,  214 ,  216  and  218  and the two woofers  220  and  222  are all aligned linearly along the y-axis symmetrically about the center tweeter  202 . The pair of tweeters  204  and  206  and the pair of tweeters  208  and  210  are each located on one side of the center tweeter  202 , above and below the center line defined by the x-axis. Similarly, one pair of midrange drivers  212  and  214  are positioned above the tweeters  202 ,  204 ,  206 ,  208  and  210  and the other pair of midrange drivers  216  and  218  are positioned below the tweeters  202 ,  204 ,  206 ,  208  and  210 , symmetrically with respect to the center line defined by the x-axis. 
     Similar to the loudspeaker system  100  in  FIG. 1 , the loudspeaker system in  FIG. 2  may include tweeters  202 ,  204 ,  206 ,  208  and  210  of outer diameters of approximately 40-50 mm, midrange drivers  212 ,  214 ,  216  and  218  of outer diameters of approximately 80-110 mm, and woofers  220  and  222  of outer diameters of approximately 120-250 mm. As stated previously, transducer cone size may differ based on the desired application and desired size of the array. 
     In general, the design of an n-way system results in optimum positional coordinates y 0 , +/−(y 1 , y 2 , y 3 , . . . y n−1 ), and filter coefficients for the filters FIR(0, 1, 2, 3, . . . n−1), for a specified directivity target function. In the given example n equals 4, when generating a two-dimensional array, the drivers with indices (1, . . . , m), m&lt;=n may be split into pairs (here m=1 and m=2). Thus, the corresponding x-coordinates are +/−(x 1 , x 2 , . . . , x m ), while the y-coordinates remain unchanged from the one-dimensional design. 
     The y-coordinates in the two-dimensional loudspeaker system  200  may be designed smaller than the physical dimensions of the drivers, as illustrated in  FIG. 2 , since space is gained by splitting and moving the drivers in x-direction. Thus, an additional degree of freedom is gained from the two-dimension design, which generally results in further improved performance. 
     Directivity along the x-axis can be tailored by optimizing the positioning parameters x 1 , . . . , x m , and the value of m itself. Drivers with indices (m+1) . . . n−1 are not split and remain at their original position. This means that the x-axis array is a truncated version of the original prototype array which was designed for the y-axis. Therefore, the directivity functions will exhibit a higher corner frequency. 
     The coefficients x 1  . . . x m  may be optimized such that smooth, frequency-independent directivity functions result along the x-axis. In case of x 1 &lt;y 1 , x 2 &lt;y 2 , . . . the array will be less directive in x-direction. In case of x 1 =y 1 , x 2 =y 2 , . . . , both will be equal at high frequencies. 
     In the example provided in  FIG. 2 , the center tweeter  202  may be mounted on the y-axis at the center point  0 , which is illustrated in  FIG. 2  at the intersection between the x and y axis. The tweeters  204 ,  206 ,  208  and  210  are mounted at their centers at approximately +/−30 mm along the x-axis and approximately +/−42 mm along the y-axis (+/−30 mm, +/−42 mm). 
     The midrange drivers  212 ,  214 ,  216  and  218  may then be mounted at their centers approximately +/−80 mm from the center point  0  along the x-axis and approximately +/−120 mm along the y-axis (+/−80 mm, +/−120 mm). The woofers  220  and  222  are then mounted at their centers approximately +/−300 mm from the center point (+/−0 mm, +/−300 mm). 
     Similar to the loudspeaker system  100  in  FIG. 1 , the transducers may be mounted in a housing  230  comprised of separate sealed compartments  232 ,  234  and  236 , as indicated by separators  242  and  244 . Compartment  232 , containing woofer  220 , may be separated by separator  242  from compartment  236 , which contains midrange drivers  212 ,  214 ,  216  and  218  and tweeters  202 ,  204 ,  206 ,  208  and  210 . Similarly, compartment  234 , containing woofer  222  may be separated by separator  244 , from compartment  236 , which contains midrange drivers  216 ,  214 ,  216  and  218  and tweeters  202 ,  204 ,  206 ,  208  and  210 . 
       FIG. 2  also illustrates a block diagram  250  of the signal flow of the multi-way loudspeaker system  200 .  FIG. 2  illustrates four ways  252 ,  254 ,  256  and  258  of signal flow. The signal flow comprises a digital input  264  that may be implemented using standard interface formats connected to the drivers through various paths or ways, such as the four ways illustrated in  FIG. 2 . Each path or way  252 ,  254 ,  256  and  258  may contain a digital FIR filter  260  and a power D/A converter  262  connected to either a single or to multiple loudspeaker drivers. 
       FIG. 3  is a flow chart of a filter design algorithm  300  used to design the loudspeaker system of the invention. The purpose of the filter design algorithm  300  is to determine the coefficients for each FIR filter for each signal flow path of the loudspeaker. As illustrated in further detail below, the initial driver positions and initial directivity target functions are first determined  310 . The initial positions or design configuration of the speaker and drivers may be designed in accordance with a number of different variables, depending upon the application, such as the desired size of the speaker, intended application or use, manufacturing constraints, aesthetics or other product design aspects. Driver coordinates are then prescribed for each driver along the main axis. Initial guesses for directivity target functions are then set, which includes establishing frequency points on a logarithmic scale within an interval of interest. The cost function is then minimized at the prescribed frequency points  312 . If the results do not meet the performance requirements of the system, step  314 , the position of the drivers are then modified and the cost minimization function is applied again  316 . This cycle may be repeated until the results meet the requirements. Once the results meet the requirements, the linear phase filter coefficients are computed  318 . Additionally computations  320  may also be made to equalize the drivers and to compensate for phase shifts and to allow beam steering. 
     In the first step  310 , the initial driver positions and initial directivity target functions are established. As previously mentioned, the number, position, size and orientation of the drivers are primarily determined by product design aspects. Once orientated, initial coordinate values may then be prescribed for initial driver coordinates p(n), n=1 . . . N for N drivers on the main axis. For example, in a one-dimensional (1D) array as illustrated in  FIG. 1 , N=7: p(n)=[−0.30, −0.15, −0.06, 0, 0.06, 0.15, 0.30] m (meters). In a two-dimensional (2D) array as illustrated in  FIG. 2 , N=7 p(n)=[−0.30, −0.12, −0.042, 0, 0.042, 0.12, 0.30]m. 
     If the geometry of the two-dimensional layout, as depicted in  FIG. 2 , is symmetrical along both the x and y axis, the design process for the two-dimensional layouts can be carried out in one dimension, i.e., along the main, as described above. Due to the symmetry, the same directivity characteristics will result along the opposing, except of a higher corner frequency. 
     To determine the initial directivity target functions, one must define initial guesses for directivity target functions T(f,q), which are determined based upon the desired performance of the drivers at specific angles q.  FIG. 4  is a graph illustrating an example set of target functions for angle-dependent attenuation at five specific angles q. The directivity target functions specify the intended sound level attenuation in dB (y-axis) that can be measured at various frequencies at sufficiently large distance from the speaker (larger than the dimensions of the speaker) in an anechoic environment, at an angle q degrees apart from a line perpendicular to the origin (center tweeter). Frequency vector f specifies a set of frequency points, e.g. 100, on a logarithmic scale within the interval of interest, e.g. 100 Hz . . . 20 kHz. 
     Angle vector q(i), i=1, . . . , Nq specifies a set of angles for which the optimization will be performed. While  FIG. 4 , illustrates the initial guess for directivity at five angles:
 
(Nq=5): q=[0, 10, 20, 30, 40]°,
 
in most cases it may be sufficient to prescribe directivity at only two angles, i.e., Nq=2. In this instance, targeted directivity may be specified at an outer angle, for example 40 degrees, and at 0 degrees, the prescribed zero directivity on axis, i.e., q=[0, 40]°.
 
     Except for the on-axis target function, the target functions at each angle, are linearly descending on a double logarithmic scale from T=0 dB at f=0 until a value T&lt;0 dB at a specified frequency fc (e.g. fc=350 Hz), then remain constant. The on-axis target function  402  remains constant at 0 db across the entire frequency range. The target directivity functions at ten (10) degrees  404 , twenty (20) degrees  410 , thirty (30) degrees  412  and forty (40) degrees  414 , all begin at T=0 dB and descend on a double logarithmic scale until the functions reach fc, which is represented by 350 Hz in  FIG. 4 , and then remain constant across the remaining frequency range of interest. 
     After the initial driver positions and initial directivity target functions are determined, the next step  312  is to minimize the cost function F(f) at the prescribed frequency vector points f, starting with the lowest frequency increment stepwise, e.g. 100 Hz, using the obtained solution as the initial solution for the next step, respectively, by using the following equations: 
                       F   ⁡     (   f   )       =       ∑     q   ⁡     (   i   )         ⁢       [            V   ⁡     (     f   ,   q     )            -     T   ⁡     (     f   ,   q     )         ]     2         ,   with                   V   ⁡     (     f   ,   q     )       =       ∑     n   =   1     N     ⁢           H   m     ⁡     (     n   ,   f   ,   q     )       ·       C   opt     ⁡     (     n   ,   f     )       ·   exp     ⁢     {       -   j     ·       2   ⁢   π       l   ⁡     (   f   )         ·     sin   ⁡     (       q   /   180     ·   π     )       ·     p   ⁡     (   n   )         }           ,                 l   =     c   f       ,     c   =     345   ⁢           ⁢   m   ⁢     /     ⁢   sec       ,     j   =       -   1                     
where H m (n, f, q) is a set of measured amplitude frequency responses for the considered driver n, frequency f, and angle q, normalized to the response obtained on axis (angle zero), an example of which is illustrated in  FIG. 5 .  FIG. 5  illustrates the measured frequency responses  500  of one mounted tweeter at various vertical displacement angles normalized to on axis. In  FIG. 5 , line  502  represents the on-axis response, line  504  is the measured frequency response at ten degrees, line  506  is the response at twenty degrees, line  508  is the response at thirty degrees and line  510  is the measured frequency response at forty degrees, all measured at frequencies ranging between 1 kHz and 20 kHz.
 
     Further, the minimization is performed by varying real-valued frequency points of the channel filters C opt (n,f), where n is the driver index and f is frequency, within the interval [0, 1]. In addition, the constraint
 
 C   opt ( n, f )=0,  f&gt;f   o   , f&lt;f   u  
 
must be fulfilled, depending on properties of particular driver n. For example, in case of a woofer, the upper operating limit is fo=1 kHz, for a tweeter, the lower limit is fu=2 kHz, for a midrange driver it could be fu=300 Hz, fo=3 kHz .
 
     The above described procedure for minimizing the cost function may be performed by a function “fminsearch,” that is part of the Matlab® software package, owned and distributed by The Math Works, Inc. The “fminsearch” function in the Matlab software packages uses the Nelder-Mead simplex algorithm or their derivatives. Alternatively, an exhaustive search over a predefined grid on the constrained parameter range may be applied. Other methodologies may also be used to minimize the cost function. 
     If the deviation between the obtained result and the target is sufficiently small, or acceptable as determined by one skilled in the art for the particular design application, the FIR filter coefficients for each signal path in the line array are then obtained. 
     If the deviation between the obtained results and the target are not acceptable for the particular design application, i.e. or are too large, the driver positions or geometry, and/or parameters q(i) and fc of the target function T(f,g) (see  FIG. 4 ) should then be modified. Once modified, the cost minimization function should again be applied and the process should be repeated until obtained results and the target are sufficiently small or with an acceptable range for the application. 
     Once the driver positions and driver geometry are positioned such that the algorithm as shown in  FIG. 3  yields results within an acceptable range of the target function, the FIR filter coefficients for each signal path n=1 . . . N must then be determined, depicted as step  318  in  FIG. 3 . One method for determining the FIR coefficients is to use a Fourier approximation (frequency sampling method), to obtain linear phase filters of given degree. When applying the Fourier approximation, or other frequency sampling method, a degree should be chosen such that the approximation becomes sufficiently accurate. 
     The Fourier approximation method may be performed by a function “firls,” that is part of the Matlab® software package, owned and distributed by The Math Works, Inc. Similar methodologies may be used to minimize the cost function by implementing in other software systems. 
     Additionally, modifications can be made to the FIR filters to equalize the measured frequency response of one or more drivers (in particular tweeters, midranges). The impulse response of such a filter can be obtained by well-known methods, and must be convolved with the impulse response of the linear phase channel filter when determining the FIR filter coefficients, as described above. Further, the voice coils (acoustic centers of the drivers) may not be aligned. To compensate for this, appropriate delays can be incorporated into the filters by adding leading zeros to the FIR impulse response. 
     The two-dimensional, multi-way loudspeaker system may be arranged for use in connection with a variety of applications, such as stereo loudspeaker systems, multi-channel home entertainment systems and public address systems. One skilled in the art may vary the number, type and position of the drivers, the number of channels, the number of signal flow paths or ways, as well as modify the positioning parameters along one axis to tailor directivity for a specified application. 
       FIG. 6  is yet another two-dimensional multi-way loudspeaker, similar to the loudspeaker in  FIG. 2 , except that the loudspeaker system contains four woofers  620 ,  622 ,  624  and  626 , rather than two woofers. The arrangement depicted in  FIG. 6  is a design that one skilled in the art may find desirable for use in sound reeinforcement applications. 
     In the example provided in  FIG. 6 , the center tweeter  602  may be mounted on the x-axis at the center point  0 , which is illustrated in  FIG. 6  at the intersection between the x and y axis. The tweeters  604 ,  606 ,  608  and  610  are mounted at their centers at approximately +/−42 mm along the y-axis and approximately +/−30 mm along the x-axis (+/−30 mm, +/−42 mm). 
     The midrange drivers  612 ,  614 ,  616  and  618  may then be mounted at their centers approximately +/−110 mm from the center point  0  along the y-axis and approximately +/−80 mm along the x-axis (+/−80 mm, +/−110 mm). The woofers  620 ,  622 ,  624 , and  626  are then mounted at their centers at approximately +/−300 mm along the y-axis and approximately +/−180 mm along the x-axis (+/−180 mm, +/ 300 mm). 
     Similar to the loudspeaker systems  100  and  200  in  FIGS. 1 and 2 , respectively, the transducers may be mounted in a housing  630  comprised of separate sealed compartments  630 ,  632  and  634 , as indicated by separators  636  and  642 . 
       FIG. 7  illustrates a block diagram  700  of the signal flow of the multi-way loudspeaker system  600  of  FIG. 6 .  FIG. 7  illustrates four ways  702 ,  704 ,  706  and  708  of signal flow. The signal flow comprises a digital input  710  that may be implemented using standard interface formats connected to the drivers through various paths or ways, such as the four ways illustrated in  FIG. 7 . Each path or way  702 ,  704 ,  706  and  708  may contain a digital FIR filter  712 ,  714 ,  716 ,  718  and a power D/A converter  720 ,  722 ,  724 ,  726  connected to either a single or to multiple loudspeaker drivers. 
     As illustrated in  FIG. 7 , signal flow way  702  feeds woofers  620 ,  622 ,  624  and  626  of the loudspeaker system  600  of  FIG. 6 . Signal flow way  704  feeds midrange drivers  612 ,  614 ,  616  and  618  of the loudspeaker system  600  of  FIG. 6 . Signal flow way  706  feeds tweeters  604 ,  606 ,  608  and  610  of the loudspeaker system  600  in  FIG. 6  and signal flow way  708  feeds the center tweeter  602  of the loudspeaker system  600  in  FIG. 6 . 
       FIG. 8  is a graph  800  of acceptable obtained results for the frequency responses of the four filters, illustrated in  FIG. 7 , as applied to a loudspeaker system similar to the one illustrated in  FIG. 6 . In particular, line  802  represents the results for the frequency response of FIR filter  712 . Line  804  represents the results for the frequency response of the FIR filter  714 ; line  806  represents the results for the frequency response of the FIR filter  716  and line  718  represents the results for the frequency response of the FIR filter  718 . 
       FIG. 9  is a graph  900  illustrating the resulting horizontal (y-axis) frequency response at various angles. The graph shows the obtained filter frequency responses V(f,q) after passing step  314  in  FIG. 3 . Passing means that the result met the requirements. In particular, line  902  represents the resulting horizontal on-axis response V(f,q( 1 )), line  904  is the frequency response at five degrees V(f,q( 2 )), line  906  is the response at ten degrees V(f,q( 3 )), line  908  is the response at fifteen degrees V(f,q( 4 )), line  910  is the response at twenty degrees V(f,q( 5 )), line  912  is the response at twenty-five degrees V(f,q( 6 )), line  914  is the response at thirty degrees V(f,q( 7 )), and line  916  is the response at thirty-five degrees V(f,q( 8 )), all shown at frequencies ranging between 100 Hz and 20 kHz. 
       FIG. 10  is a graph  1000  illustrating the resulting vertical (x-axis) frequency response at various angles. In particular, line  1002  represents the resulting vertical on-axis response V(f,q( 1 )), line  1004  is the frequency response at five degrees V(f,q( 2 )), line  1006  is the response at ten degrees V(f,q( 3 )), line  1008  is the response at fifteen degrees V(f,q( 4 )), line  1010  is the response at twenty degrees V(f,q( 5 )), line  1012  is the response at twenty-five degrees V(f,q( 6 )), line  1014  is the response at thirty degrees V(f,q( 7 )), and line  1016  is the response at thirty-five degrees V(f,q( 8 )), all shown at frequencies ranging between 100 Hz and 20 kHz. 
       FIGS. 11-22  represent example implementation of multi-way loudspeakers for loudspeaker systems suitable for home entertainment applications. 
       FIG. 11  illustrates an example implementation of a one-dimensional (1D), seven-way loudspeaker system  1100  mounted symmetrically along the x-axis and a block diagram  1160  of signal flow to each of the loudspeaker drivers in the system. This example implementation may serve as a basis for the two-dimensional (2D), multi-way loudspeaker system designs  1400  and  1700  illustrated in  FIGS. 14 and 17 , which may be designed for use in home entertainment applications, or other suitable applications known by those skilled in the art. 
     As illustrated in  FIG. 11 , the one-dimensional, seven-way loudspeaker system  1100  may include (i) one center tweeter  1102 , positioned at the point of origin; (ii) a first pair of tweeters  1104  and  1106 , one tweeter positioned on each side of the center tweeter  1102  at +/−0.035 m along the x-axis; (iii) a second pair of tweeters  1108  and  1110 , one positioned on each side of the first pair of tweeters at +/−0.07 m along the x-axis; (iv) a first pair of midrange drivers  1112  and  1114  positioned at +/−0.12 m along the x-axis; (v) a second pair of midrange drivers  1116  and  1118  positioned at +/−0.20 m along the x-axis; (vi) a third pair of midrange drivers  1120  and  1122  positioned at +/−0.34 m along the x-axis; and (vii) a pair of woofers  1124  and  1126  positioned at +/−0.54 m along the x-axis. 
     As in previously illustrated embodiments, the drivers may be contained with a housing having various compartments. The tweeters  1102 ,  1104 ,  1106 ,  1108  and  1110  and mid-range drivers  1112  and  1114  may be positioned within one compartment  1130 . Positioned adjacent to compartment  1130  separated by separator  1132  on one side of compartment  1136  which contains the mid-range driver  1116 . On the opposing side of compartment  1130  separated by separator  1134  is compartment  1138  which contains the mid-range driver  1118 . Compartment  1144  contains mid-range driver  1120  and is separated on one side from compartment  1136  by separator  1140  and on the other side from compartment  1152 , which contains woofer  1124 , by separator  1148 . Similarly, compartment  1146  contains mid-range driver  1122  and is separated on one side from compartment  1138  by separator  1142  and on the other side from compartment  1154 , which contains woofer  1126 , by separator  1150 . 
     The loudspeaker system  1100  may receive digital input  1180 . The signal flow diagram  1160  illustrates the center tweeter  1102  being fed by signal flow way  1174 , which includes FIR filter  1176  and a power D/A converter  1178 . The first pair of tweeters  1104  and  1106  is fed by signal flow way  1172 , which includes FIR filter  1178  and a power D/A converter  1178  and the second pair of tweeters  1108  and  1110  is fed by signal flow way  1170 , which includes FIR filter  1180  and a power D/A converter  1178 . The first pair of midrange drivers  1112  and  1114  is fed by signal flow way  1168 , which includes FIR filter  1182  and a power D/A converter  1178 , while the second pair of midrange drivers  1116  and  1118  is fed by signal flow way  1166 , which includes FIR filter  1184  and power D/A converter  1178 . The third pair of midrange drivers  1120  and  1122  is fed by signal flow way  1164 , which includes FIR filter  1186  and power D/A converter  1178 . Finally, the pair of woofers  1124  and  1126  is fed by signal flow way  1162 , which includes FIR filter  1188  and a power D/A converter  1178 . 
       FIG. 12  is a graph  1200  illustrating the frequency responses of the seven filters of the loudspeaker system in  FIG. 11  once the cost minimization function has been applied and the obtained results have been found to be sufficiently small or within the acceptable range for the desired application. The line represented by  1202  is the frequency response of FIR filter  1176 ; line  1204  is the frequency response of FIR filter  1178 ; line  1206  is the frequency response of FIR filter  1180 ; line  1208  is the frequency response of FIR filter  1182 ; line  1210  is the frequency response of FIR filter  1184 ; line  1212  is the frequency response of FIR filter  1186 ; and line  1214  is the frequency response of FIR filter  1188 . 
       FIG. 13  is a graph  1300  that illustrates the resulting horizontal (x-axis) frequency responses of the loudspeaker system in  FIG. 11  measured at various angles. The graph shows the obtained filter frequency responses V(f,q) after the requirements in step  314  in  FIG. 3  have been met. In particular, line  1302  represents the resulting horizontal on-axis response V(f,q( 1 )), line  1304  is the frequency response at ten degrees V(f,q( 2 )), line  1306  is the response at fifteen degrees V(f,q( 3 )), line  1308  is the response at twenty degrees V(f,q( 4 )), line  1310  is the response at thirty degrees V(f,q( 5 )), all shown at frequencies ranging between 100 Hz and 20 kHz. 
       FIG. 14  illustrates an example implementation of a two-dimensional (2D), multi-channel, seven-way loudspeaker system  1400  mounted symmetrically along the x-axis and y-axis. The loudspeaker system  1400  is derived by splitting the tweeters  1104 ,  1106 ,  1108  and  1110 , and the midrange drivers  1112  and  1114  of the loudspeaker system  1100  in  FIG. 11  into pairs. 
     The loudspeaker system  1400  controls directivity in two dimensions and comprises a center tweeter  1402 ; four pairs of tweeters  1404  and  1406 ,  1408  and  1410 ,  1412  and  1414 , and  1416  and  1418 ; four pairs of mid-range drivers  1420  and  1422 ,  1424  and  1426 ,  1428  and  1430  and  1432  and  1434 ; and a pair of woofers  1436  and  1438 . The first two pairs of tweeters  1404  and  1406  and  1408  and  1410  are arranged in quadratic configurations respectively about the center tweeter  1402 . A third and forth pair of tweeters  1412 ,  1414 ,  1416  and  1418  are positioned on a further distant quadrant, symmetrically along the x and y axis. The first and second pairs of mid-range drivers  1420 ,  1422 ,  1424  and  1428  are positioned on yet a further distant quadrant, symmetrically along the x and y axis. As will be explained further below, the inner quadrants are defined by a forty-five (45) degree angle relative to the x-axis. 
     Additionally, the midrange drivers  1428 ,  1430 ,  1432  and  1434  and the woofers  1436  and  1438  are linearly spaced across the x-axis. The (x, y) coordinates of the drivers of the loudspeaker  1400  may be as follows: 
     Tweeter  1402 : (0, 0) 
     Tweeters  1404 ,  1406 ,  1408  and  1410 : (+/−35, +/−35) mm 
     Tweeters  1412 ,  1414 ,  1416  and  1418 : (+/−70, +/−70) mm 
     Midrange  1420 ,  1422 ,  1424  and  1426 : (+/−120, +/−120) mm 
     Midrange  1428  and  1430 : (+/−200, 0) mm 
     Midrange  1432  and  1434 : (+/−340, 0) mm 
     Woofer  1436  and  1438 : (+/−540, 0) mm 
     As with the loudspeakers illustrated in  FIG. 11 , the drivers may be mounted in a baffle  1476  comprised of separate sealed compartments  1440 ,  1442 ,  1444 ,  1446 ,  1448 ,  1450  and  1452 . The tweeters  1402 ,  1404 ,  1406 ,  1408 ,  1410 ,  1412 ,  1414 ,  1416  and  1418  and midrange drivers  1420 ,  1422 ,  1424  and  1426  may all be contained in compartment  1440 . On the right side, compartment  1440  may be separated from compartment  1444  by a separator represented by triangular line  1460 . Compartment  1444  contains midrange driver  1430  and may be separated at its right from compartment  1448 , which contains midrange driver  1434 , by a separator represented by line  1464 . To the right of compartment  1448 , is compartment  1452 , which contains woofer  1438 . Compartments  1448  and  1452  may be separated from one another by a separator represented by line  1468 . 
     Similarly, compartment  1440  may be separated from compartment  1442  on its left by a separator represented by the triangular line  1462 . Compartment  1442  contains midrange driver  1428  and may be separated at its left from compartment  1446 , which contains midrange driver  1432 , by a separator represented by line  1466 . To the left of compartment  1446 , is compartment  1450 , which contains woofer  1436 . Compartments  1446  and  1450  may be separated from one another by a separator represented by line  1470 . 
     As with the drivers of  FIGS. 1 and 2 , the tweeters  1402 ,  1404 ,  1406 ,  1408 ,  1410 ,  1412 ,  1414 ,  1416  and  1418  may be of an outer diameter of approximately 40-50 mm, the midrange drivers  1420 ,  1422 ,  1424 ,  1426 ,  1428 ,  1430 ,  1432  and  1434  may be of an outer diameter of approximately 80-110 mm, and the woofers  1436  and  1438  may be of an outer diameter of approximately 120-160 mm. 
       FIG. 15  is a block diagram  1500  of signal flow to each of the loudspeaker drivers in the loudspeaker system  1400  of  FIG. 14 . As illustrated in  FIG. 15 , each one of the drivers having similar coordinate sets, as set forth above, is fed by different path or way, making this a seven-way loudspeaker. The loudspeaker system  1400  receives digital input  1502 . The center tweeter  1402  being fed by signal flow way  1504 . Tweeters  1404 ,  1406 ,  1408 , and  1410  are fed by signal flow way  1506 . Tweeters  1412 ,  1414 ,  1416  and  1418  are fed by signal flow way  1508 . Mid-range drivers  1420 ,  1422 ,  1424  and  1426  are fed by signal flow way  1510 , while mid-range drivers  1428  and  1430  are fed by signal flow way  1512  and mid-range drivers  1432  and  1434  are fed by signal flow way  1514 . The pair of woofers  1436  and  1438  is fed by signal flow way  1516 . Each signal flow way includes a FIR filter  1518  and power D/A converter  1520 . 
       FIG. 16  is a graph  1600  illustrates the resulting vertical (y-axis) frequency responses of the loudspeaker system  1400  in  FIG. 14  measured at various angles. The graph shows the obtained filter frequency responses V(f,q) after the requirements in step  314  in  FIG. 3  have been met. In particular, line  1602  represents the resulting horizontal on-axis response V(f,q( 1 )), line  1604  is the frequency response at ten degrees V(f,q( 2 )), line  1406  is the response at fifteen degrees V(f,q( 3 )), line  1608  is the response at twenty degrees V(f,q( 4 )), line  1610  is the response at thirty degrees V(f,q( 5 )), all shown at frequencies ranging between 100 Hz and 20 kHz. As seen by  FIG. 16 , the vertical frequency responses for the two-dimensional loudspeaker system  1400  of  FIG. 14  resembles the horizontal frequency responses, as illustrated by  FIG. 13 , for the one-dimensional loudspeaker system  1100  in  FIG. 11 , but having a considerably higher lower corner frequency above which the system becomes directive. 
       FIG. 17  illustrates an example implementation of a two-dimensional (2D), five-channel, multi-way loudspeaker system  1700  mounted symmetrically along the x-axis. The loudspeaker system  1700  is designed with a pair of integrated two-way stereo speakers mounted symmetrically along the x-axis and specifically designed for use for home theatre applications. As will be further explained below ( FIGS. 18-20 ), the loudspeaker system  1700  may have five input channels L (left), R (right), C (center), LS (left surround), and RS (right surround). 
     The loudspeaker system  1700  is similar to that in  FIG. 14  except that it provides two additional tweeters  1744  and  1746  and two additional woofers, such that the outer woofers are split into pairs  1736  and  1738  and  1740  and  1742  having the additional pair of tweeters  1744  and  1746  positioned between each pair of woofers  1736  and  1738  and  1740  and  1742 , respectively, about the y-axis. By having tweeters  1744  and  1746  assigned to the pairs  1736  and  1738  and  1740  and  1742  of woofers, respectively, the loudspeaker system  1700  may provide array independent stereo speaker channels (i.e. the tweeter may be fed a signal supplied by a separate channel). The purpose of the independent stereo speaker channels is to provide an integrated surround sound system with conventional stereo speakers and directed sound beams generated by the array to reproduce ambient rear channels indirectly using wall reflections in the listening room. 
     Like the loudspeaker system  1400  illustrated in  FIG. 14 , the loudspeaker system  1700  of  FIG. 17  has (i) a center tweeter  1702 ; (ii) two pairs of tweeters  1704  and  1706  and  1708  and  1710  arranged in a quadratic configuration about the center tweeter  1702 ; (iii) two additional pairs of tweeters  1712  and  1714 , and  1716  and  1718  positioned on a further distant quadrant, symmetrically along the x and y axis and (iv) two pairs of mid-range drivers  1720  and  1722  and  1724  and  1726  positioned on an even further distant quadrant, symmetrically along the x and y axis. The quadrants are defined by forty-five (45) degree angles relative to the x-axis. 
     Additionally, the loudspeaker system  1700  includes midrange drivers  1728 ,  1730 ,  1732  and  1743  linearly spaced across the x-axis. The (x, y) coordinates of the drivers of the loudspeaker system  1700  may be as follows: 
     Tweeter  1702 : (0, 0) 
     Tweeters  1704 ,  1706 ,  1708  and  1710 : (+/−35, +/−35) mm 
     Tweeters  1712 ,  1714 ,  1716  and  1718 : (+/−70, +/−70) mm 
     Midrange  1720 ,  1722 ,  1724  and  1726 : (+/−120, +/−120) mm 
     Midrange  1728  and  1730 : (+/−200, 0) mm 
     Midrange  1732  and  1734 : (+/−340, 0) mm 
     Tweeters  1744  and  1746 : (+/−540, 0) mm 
     Woofer  1736 ,  1738 ,  1740  and  1742 : (+/−540, +/−90) mm 
     As with the loudspeakers systems illustrated in  FIGS. 1 ,  2 ,  6 ,  11  and  14 , the drivers of the loudspeaker system  1700  may be mounted in a baffle or housing  1750  comprised of separate sealed compartments  1752 ,  1754 ,  1756 ,  1758 ,  1760 ,  1762  and  1764 , which are divided from one other by separators represented by lines  1766 ,  1768 ,  1770 ,  1772 ,  1774  and  1176 , respectively. 
       FIGS. 18-20  illustrate the block diagrams of the signal flows for the five-input signals of the loudspeaker system  1700  of  FIG. 17 .  FIG. 18  is a block diagram  1800  of the signal flows for a surround channels for the loudspeaker system  1700  in  FIG. 17 . Since the signal flows for the right and left surround channels in the system  1700  are identical except for different delay values, as further described below, the diagram  1800  in  FIG. 18  is representative of the signal flow paths for both the left and right surrounds. Thus, both the left and right surround input signals pass through a signal path system similar to that shown in  FIG. 18 . The sum of the respective output signals, as depicted in  FIG. 18 , is then computed and connected to the transducers. The outputs of the FIR filters, the frequency responses of which are shown in  FIG. 12 , are connected to delay lines D 0 , and pairs of delay lines D +/−(   1 . . . 6 ), respectively. 
     The signal flow diagram  1800  in  FIG. 18  illustrates how delays may be added to each path in accordance with the following equation:
 
Δ t=p/c· sin α, ( p =driver coordinates in  m, c= 345 m/sec speed of sound)
 
where the main sound beam, which is otherwise perpendicular to the main axis, can be steered to a desired direction with angle α. Typical values for α are −(40 . . . 60)degrees for the left surround, and +(40 . . . 60)degrees for the right surround, which means that sound beams are formed and steered towards side walls in the direction of angles α and −α bouncing against the walls and arriving at the listener as surround signals.
 
     As illustrated in  FIG. 18 , signal flow path diagram  1800  illustrates the flow paths for the digital inputs for the right and left surround sound channels  1802  and  1804 , respectively. The FIR filter  1822  output for path  1806  is connected to delay line (D 0 )  1840  which is connected to the center tweeter  1702 . The FIR filter  1824  output for path  1808  is connected in parallel to delay line (D −1 )  1842  and (D +1 ))  1844 . Delay line  1842  is connected to the right pair of tweeters  1708  and  1710  and delay line  1844  is connected to the left pair of tweeters  1704  and  1706 . Similarly, the FIR filter  1826  output for path  1810  is connected in parallel to delay line (D −2 )  1846  and (D +2 )  1848 . Delay line  1846  is connected to the right pair of tweeters  1716  and  1718  and delay line  1848  is connected to the left pair of tweeters  1712  and  1714 . Delay lines (D −3 )  1850  and (D +3 )  1852  are connected to the midrange drivers  1720  and  1722  and  1724  and  1726 , respectively, which are connected in parallel to path  1812 , which is the output path for FIR filter  1828 . 
     Midrange drivers  1728  and  1730  are connected to delay lines (D +4 )  1856  and (D −4 )  1854 , respective, which are the output path  1814  for FIR filter  1830 . Midrange drivers  1732  and  1734  are connected to delay lines (D +5 )  1862  and (D −5 )  1860 , respective, which are the output path  1816  for FIR filter  1832 . 
     The right pair of woofers  1740  and  1742  is connected to delay line (D −6 )  1864  and the left pair of woofers  1736  and  1738  is connected to the delay line (D +6 )  1866 . Delay lines (D +6 )  1866  and (D −6 )  1864  are connected in parallel to the output path  1820  for the FIR filter  1834 . 
       FIG. 19  is a block diagram of the signal flows for the right and left channels for the loudspeaker system in  FIG. 17 . The left and right channels are integrated as conventional two-way speakers. The left channel is comprised of tweeter  1744 , which is not part of the beam forming array, and woofers  1736  and  1738 . The right channel is comprised of the tweeter  1746  and woofers  1740  and  1742 . 
     As illustrated by  FIG. 19 , the signal processing  1900  for the left and right channels uses a stereo widening circuit comprised of HD filters  1910  and HI filters  1912  to widen the stereo basis and a crossover circuit with low pass filters  1914  and HP high pass filters  1916 . 
       FIG. 20  is a block diagram of the signal flows for the center channel for the loudspeaker system  1700  in  FIG. 17 . The center channel is reproduced by the inner array of tweeters  1702 ,  1704 ,  1706 ,  1708 ,  1710 ,  1712 ,  1714 ,  1716  and  1718  and mid-range drivers  1720 ,  1722 ,  1724  and  1726  with FIR filters having coefficients determined as set forth in  FIG. 3 . 
     The output of the digital signal for the center channel  2010  is divided into four signal paths  2002 ,  2004 ,  2006  and  2008 , each having a FIR filter  2012 ,  2014 ,  2016  and  2018 , respectively, and a Power D/A converter  2020 ,  2022 ,  2024  and  2026 , respectively. Path  2002  feeds the center tweeter  1702 . Path  2004  feeds the innermost quadrant of tweeters  1704 ,  1706 ,  1708  and  1710 . Path  2006  feeds the outermost quadrant of tweeters  1712 ,  1714 ,  1716  and  1718  and path  2008  feeds the center quadrant of mid-range drivers  1720 ,  1722 ,  1724  and  1726 . 
       FIG. 21  is a graph  2100  illustrating the frequency responses of the four FIR filters used in the center channel ( FIG. 20 ) of the loudspeaker system of  FIG. 17 . Line  2102  represents the frequency response of FIR filter  2012 , line  2104  represents the frequency response of FIR filter  2014 , line  2106  represents the frequency response of FIR filter  2016  and line  2108  represents the frequency response of FIR filter  2018 . 
       FIG. 22  is a graph  2200  illustrating the resulting horizontal (x-axis) and identical vertical (y-axis) frequency responses of the center channel output ( FIG. 20 ) of the loudspeaker system  1700  of  FIG. 17  measured at various angles. The graph shows the obtained filter frequency responses V(f,q) after meeting the requirement of step  314  in  FIG. 3 . In particular, line  2202  represents the resulting horizontal on-axis response V(f,q( 1 )), line  2204  is the frequency response at five degrees V(f,q( 2 )), line  2206  is the response at ten degrees V(f,q( 3 )), line  2208  is the response at fifteen degrees V(f,q( 4 )), line  2210  is the response at twenty degrees V(f,q( 5 )), line  2212  is the response at twenty-five degrees V(f,q( 6 )), line  2214  is the response at thirty degrees V(f,q( 7 )), and line  2216  is the response at thirty-five degrees V(f,q( 8 )), all shown at frequencies ranging between 100 Hz and 20 kHz. 
     While various embodiments of the invention have been described, it will be apparent to those of ordinary skill in the art that many more embodiments and implementations are possible within the scope of this invention. Accordingly, the invention is not to be restricted except in light of the attached claims and their equivalents.