Abstract:
A digital adaptive predistorter look up table (DAPD-LUT) technique dynamically adapts a look up table (LUT) an LUT spacing for linearizing a power amplifier (PA). It optimizes the LUT spacing for the PA without prior knowledge of system state information. A size-N LUT divides a whole unsaturated PA input amplitude range into N bins, each predistorted by an entry of the LUT. The LUT is indexed by an input amplitude of a modulated signal via an index mapper to implement an unconditionally non-uniform LUT spacing. A spacing adaptor online interactively adapts the LUT spacing. The adapted LUT spacing balances the inter-modulation distortion (IMD) power at the PA output corresponding to each bin, so that the total IMD power at the PA output is minimized. This dynamically-optimum technique is practical, robust, and with low complexity.

Description:
FIELD OF THE INVENTION 
   The present invention generally relates to amplifier linearization, and more specifically to an apparatus and a method of dynamically adapting a look up table (LUT) spacing for linearizing a power amplifier (PA). 
   BACKGROUND OF THE INVENTION 
   Power efficiency of a power amplifier is a crucial issue in wireless communication systems. A stand-alone class-A PA suffers the problem of low power efficiency. On the other hand, a stand-alone power efficient PA, like class-AB or class-B amplifier, is usually highly nonlinear. When a non-constant-envelope modulated signal goes through a nonlinear PA, inter-modulation distortion (IMD) will emerges. This not only distorts the modulated signal but also causes the power spectrum of the modulated signal to overflow to the adjacent channels. As a result, both self-interference and mutual-interference among neighboring channels seriously degrade the communication quality. In order to maintain power efficiency and suppress IMD, it is a common practice to adopt a nonlinear PA with high power efficiency. 
   There exist a few schemes for PA linearization, such as the feed-forward scheme, the feedback scheme, and the predistortion scheme. Each is with either analog approaches or digital approaches. Generally speaking, the feed-forward schemes are costly and the feedback schemes are limited to only narrow band applications. All the analog approaches are inflexible. Therefore, in terms of cost effectiveness, the digital predistortion schemes are superior to the others. 
   Shown in  FIG. 1  is a block diagram illustrating the linearization of a digital predistorter (PD). The digital PD  101  predistorts a modulated input signal v m  to invert the nonlinear distortion introduced by a PA  107 . In particular, a digital adaptive PD (DAPD) employing a gain-based look up table  101   a  is very attractive for its flexibility in algorithm adaptation and its high accuracy in nonlinear compensation. As shown in  FIG. 1 , the complex baseband modulated input signal v m  carrying the payload data is fed to the cascade of the PD  101  and a radio frequency (RF) link. The PD  101  distorts the modulated input signal v m  to produce a predistorted signal v d . The RF link takes over the predistorted signal v d , to generate the transmission signal v a , through a digital-to-analog (D/A) converter  103  for transformation, a quadature modulator  105  for frequency up-conversion, and the PA  107  for power amplification. 
   Because the characteristics of a PA may vary with temperature and may be affected by aging, an adaptive algorithm is required in a DAPD-LUT scheme to update the LUT entry values. In addition, the linearization accuracy of a DAPD-LUT scheme in terms of IMD will improve 6 dB if one doubles the number of LUT entries. However, the more LUT entries one adopts, the lower LUT convergence speed it will suffer. 
   Several gain-based LUT techniques are either analyzed or implemented.  FIG. 2  is a block diagram illustrating a conventional gain-based DAPD-LUT technique that the indexing of the N-size LUT entries is uniformly spaced, wherein the normalized unsaturated input amplitude range of a PA is [0, 1] and an LUT entry&#39;s spacing d i  equals to 1/N. However, in the uplink or downlink of a wireless network, most transmitted signals do not occupy the input amplitude range of the entire PA. Some LUT entries will never be selected. Therefore, a non-uniform LUT spacing technique is highly desired to avoid wasting LUT entries. 
     FIG. 3  is a block diagram illustrating a conventional gain-based DAPD-LUT technique with an optimum non-uniform LUT spacing, wherein the LUT is indexed by the input amplitude r m  of input modulated signal via a mapper S(r m ) to implement a non-uniform LUT spacing d i , which is referred to as the conditionally-optimum spacing technique. The technique assumes knowledge of the conditions on the input signal backoff (IBO), the PA characteristics, and the probability density function (PDF) of the modulated input signal. When any of the assumed knowledge varies with time, the optimum LUT spacing needs to be recalculated. Unfortunately, the computational complexity of recalculating the LUT spacing in such a conditionally optimum technique is pretty high. 
   Since the conditionally optimum technique is optimum only under a specific set of conditions, any condition mismatch could cause significant performance degradation. However, some of conditions are difficult to accurately obtain, e.g. the PA characteristics, and some of conditions can be fast time-varying, e.g. the IBO. In addition, the computational complexity of the conditionally optimum technique thwarts any attempt to online optimize the LUT spacing for a different set of conditions. Therefore, an unconditionally optimized technique is practically useful. 
     FIG. 4  is another conventional gain-based DAPD-LUT technique with a non-uniform LUT spacing, which is referred to as the piecewise-uniform spacing technique. In the piecewise uniform spacing technique, the whole unsaturated PA input amplitude range is first artificially divided into several segments, such as 4 segments S1-S4, according to the nonlinearity of the PA characteristic curve. Each of those nonlinear segments will be assigned more LUT entries than each of those linear segments to combat the PA nonlinear distortion. Although it is still uniform spacing within each segment, this technique as a whole enjoys the advantage of non-uniform LUT spacing. The piecewise-uniform spacing technique also requires prior knowledge of the PA characteristic so as to divide the PA input amplitude range into segments of different linearities. The piecewise-uniform spacing technique focuses on the subject of PA characteristics and ignores how input signal statistics may influence the IMD performance of a PA linearization technique. 
   Because of the aforementioned problems, it is imperative to provide a technique to dynamically calculate an unconditionally-optimum LUT spacing which minimizes the overall average IMD power. 
   SUMMARY OF THE INVENTION 
   The present invention has been made to overcome the aforementioned drawback of conventional gain-based DAPD-LUT techniques for PA linearization. The primary object of the present invention is to provide an apparatus and a method of dynamically adapting the LUT spacing for linearizing a PA. Wherein the LUT spacing is decreased for the amplitude ranges with higher signal probability densities so that the overall average of IMD power is minimized. 
   The present invention is to provide an apparatus and a method to optimize the LUT spacing for PAs without prior knowledge of system state information (SSI), i.e. an SSI-learning low-complexity technique to optimize the LUT spacing for a DAPD-LUT technique. 
   The present invention is to provide an apparatus and a method capable of online adapting the LUT spacing for PAs with various nonlinear characteristics, for input signals with various statistics, and for wireless environments with various time-varying properties. 
   The present invention of an apparatus of dynamically adapting the LUT spacing for linearizing a PA includes an index mapper, a spacing adaptor, and a size-N LUT dividing a whole unsaturated PA input amplitude range into N bins. The apparatus linearizes the PA to produce an amplified output signal in response to a predistorted input derived from an input modulated signal. 
   According to the present invention, the IMD power associated with each LUT entry is in terms of variables other than the IBO, the PA characteristics and the PDF of the modulated input signal. The concerned LUT spacing problem becomes an optimization problem to minimize the total IMD power at the PA output. The existence of an optimum solution to the optimization problem is also guaranteed. The new LUT spacing balances the IMD power at the PA output corresponding to each bin, so that the total IMD power at the PA output is minimized. 
   The present invention describes an iterative procedure to approach a stationary solution which is likely to be the optimum solution. After that, it adaptively updates the index mapper through the iterative procedure. 
   Experimental results demonstrate the feasibility and robustness of the present invention with its performance close to that of the unconditionally-optimum technique and with its computational complexity much lower than that of the conditionally-optimum spacing technique. 
   The foregoing and other objects, features, aspects and advantages of the present invention will become better understood from a careful reading of a detailed description provided herein below with appropriate reference to the accompanying drawings. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a block diagram illustrating the linearization of a digital predistorter. 
       FIG. 2  is a block diagram illustrating a conventional DAPD-LUT technique with a uniform LUT spacing. 
       FIG. 3  is a block diagram illustrating a conventional gain-based DAPD-LUT technique with an optimum non-uniform LUT spacing referred to as the conditionally-optimum spacing. 
       FIG. 4  is another conventional gain-based DAPD-LUT technique with a non-uniform LUT spacing referred to as the piecewise-uniform spacing. 
       FIG. 5  shows a baseband-equivalent schematic view of a first embodiment of the apparatus according to the present invention. 
       FIG. 6  is a block diagram of the index mapper shown in  FIG. 5 . 
       FIG. 7  is a flowchart illustrating the iterative procedure. 
       FIG. 8  shows the power spectral density performance comparison among several DAPD-LUT techniques with various LUT spacings in the system scenario with IBO=−10 dB. 
       FIG. 9  shows the normalized IMD power of several DAPD-LUT techniques with various LUT spacings in system scenarios with varying IBOs and different PAs. 
       FIG. 10  shows the normalized IMD power at the PA output of several DAPD-LUT techniques with various LUT spacings. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Without prior knowledge of the system state information (SSI), the present invention provides an apparatus of SSI-learning and low complexity to optimize the LUT spacing for PAs.  FIG. 5  shows a baseband-equivalent schematic view of the apparatus according to the present invention. Referring to  FIG. 5 , the apparatus comprises an index mapper  501 , a spacing adaptor  503 , and a size-N LUT  505  containing N entries. The apparatus linearizes a PA  521  to produce an amplified output signal in response to a predistorted input derived from an input modulated signal v m . 
   An amplitude unit  520  derives the absolute amplitude of the input modulated signal v m . The size-N LUT  505  divides the unsaturated PA input amplitude range into N bins, each predistorted by an entry of the LUT  505 . The LUT  505  is indexed by an input amplitude r m  of modulated signal via the index mapper  501  to implement an unconditionally non-uniform LUT spacing. Because the characteristics of a PA may vary with temperature and may be affected by aging, an adaptive algorithm online updates the LUT value. The spacing adaptor  503  online adapts the LUT spacing. Each LUT entry corresponds to an input amplitude r m  of modulated signal. The adapted LUT spacing balances the IMD power at the PA output corresponding to each bin, so that the total IMD power at the PA output is minimized. 
     FIG. 6  is a block diagram of the index mapper shown in  FIG. 5 . As shown in  FIG. 6 , if the update indicator ω set to be 1, the spacing adaptor  503  is running. When the spacing adaptor  503  provides the index mapper  501  with a new set of bin boundary {b i }, the index mapper  501  generates an index of the LUT  505  to indicate a selected entry of LUT  505 . While the LUT  505  is indexed by an input amplitude r m  via the index mapper  501  to implement the unconditionally non-uniform LUT spacing. Thereby, each LUT entry corresponds to an input amplitude r m . 
   In order to make the LUT spacing of the present invention unconditionally-optimized, the present invention expresses the IMD power associated with each LUT entry in terms of variables other than the IBO, the PA characteristics and the PDF of the modulated input signal. In other words, the concerned LUT spacing problem becomes an optimization problem to minimize the total IMD power at the PA output. The followings describe the IMD power derivation according to the present invention to guarantee the existence of the optimum solution under some practical scenarios. 
   Without loss of generality, the present invention assume that (1) the modulated input signal v m  is real and (2) the PA has only amplitude-modulated amplitude-distortion (AM/AM) nonlinear distortion to proceed with the IMD power derivation. After that, the derivation is extended to a general scenario. 
   Since the number of the LUT entries is finite, N≠∞, so the transfer function of the PD is only piecewise continuous. The PD transfer function of the i th  bin is defined as F i (r m,i ), where r m,i =r mo,i +δr m,i  is an input signal amplitude near the amplitude midpoint r mo,i  of the i th  bin. With r m,i  as the input amplitude of the i th  bin, the PA output amplitude error associated with the i th  LUT entry is derived as
 
 e   i   =G ( F   o ( r   m,i ))− G ( F   i (r m,i ))≈ F   i   ·G′ ( F   i ( r   mo,i )),
 
   where F o (r mo,i ) is the ideal PD transfer function of the i th  bin, δF i =r m,i ·δ|f i |≈r mo,i ·|f o (r mo,i )|′·δr m,i  is the LUT approximation error of the PD output amplitude, δ|f i | is the PD gain error of the i th  LUT entry value, f o (r mo,i ) is defined as the LUT value of r mo,i  in the i th  bin, |f o (r mo,i )|′ is the derivative of |f o (r mo,i )| with respect to r m,i , and G′(F i (r mo,i )) is the slope of the tangent to the G curve, where the G curve is the transfer function of a PA. Note that we have 
                     G   ′     ⁡     (       F   i     ⁡     (     r     mo   ,   i       )       )       =         ⅆ     ⅆ       F   i     ⁡     (     r     m   ,   i       )           ⁢     G   ⁡     (       F   i     ⁡     (     r     m   ,   i       )       )         ⁢     ❘         F   i     ⁡     (     r     m   ,   i       )       =       F   i     ⁡     (     r     mo   ,   i       )                           =       (         ⅆ     ⅆ     r     m   ,   i           ⁢       F   i     ⁡     (     r     m   ,   i       )         ⁢     ❘       r     m   ,   i       =     r     mo   ,   i             )       -   1         ,                     where                         F   i   ′     ⁡     (     r     mo   ,   i       )       =         ⅆ     ⅆ     r     m   ,   i           ⁢       F   i     ⁡     (     r     m   ,   i       )         ⁢     ❘       r     m   ,   i       =     r     mo   ,   i                           =              f   o     ⁡     (     r     mo   ,   i       )            +       r     mo   ,   i       ·       Re   ⁡     (         f   o   *     ⁡     (     r     mo   ,   i       )       ·       f   o   ′     ⁡     (     r     mo   ,   i       )         )                f   o     ⁡     (     r     mo   ,   i       )                    ,               
f′ o (r mo,i ) is the derivative of f o (r mo,i ) with respect to r m,i , (·)* is the complex conjugate operation, and Re(·) denotes the real part of the enclosed argument.
 
   For a small bin, it is reasonable to expect that δr m,i  is uniformly distributed over the entire bin width. The IMD power associated with the i th  LUT entry governing a bin of width d i  can further be expressed as 
   
     
       
         
           
             
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   To generalize the derivation above, the present invention now considers the amplitude-modulated phase-distortion (AM/PM) effect of the PA having a complex modulated input signal v m . With a similar derivation, the phase error associated with the i th  LUT entry at the PA output is expressed as 
                   e     ϕ   ,   i       ≈       ⁢           (         ⅆ     ⅆ     r     m   ,   i           ⁢     arg   ⁡     (       f   o     ⁡     (     r     m   ,   i       )       )         ⁢     ❘       r     m   ,   i       =     r     mo   ,   i             )     ·   δ     ⁢           ⁢     r     m   ,   i         -       r     mo   ,   i       ·              f   o     ⁡     (     r     mo   ,   i       )            ′     ·                       ⁢     δ   ⁢           ⁢       r     m   ,   i       ·     (         ⅆ     ⅆ     r     m   ,   i           ⁢     arg   ⁡     (       f   o     ⁡     (     r     m   ,   i       )       )         ⁢     ❘       r     m   ,   i       =     r     mo   ,   i             )                       =       ⁢           [     arg   ⁡     (       f   0     ⁡     (     r     mo   ,   i       )       )       ]     ′     ·     (     1   -       r     mo   ,   i       ·              f   o     ⁡     (     r     mo   ,   i       )            ′         )     ·   δ     ⁢           ⁢     r     m   ,   i           ,               
where |f o (r mo,i )| and arg(f o (r mo,i )) respectively denote the amplitude and the phase of f o (r mo,i ), and |f o (r mo,i )|′ and [arg(f o (r mo,i ))]′ respectively denote the derivative of |f o (r mo,i )| and arg(f o (r mo,i )) with respect to r m,i .
 
   Since the amplitude error e i  and the phase error e φ,i  are orthogonal in the polar coordinate, the IMD power associated with the i th  LUT entry can thus be extended 
   
     
       
         
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   The concerned LUT spacing problem becomes an optimization problem to minimize the total IMD power at the PA output as 
               {     β   i     }     =       arg   ⁢       min     {     d   i     }       ⁢     P   ae         =     arg   ⁢       min     {     d   i     }       ⁢       ∑     i   =   1     N     ⁢       η   i     ·     d   i   2                 ,     
     ⁢     where                     η   i     =       [         (                f   o     ⁡     (     r     mo   ,   i       )            ′         F   i   ′     ⁡     (     r     mo   ,   i       )         )     2     +       (                    [     arg   ⁡     (       f   o     ⁡     (     r     mo   ,   i       )       )       ]     ′     ·               (     1   -       r     mo   ,   i       ·              f   o     ⁡     (     r     mo   ,   i       )            ′         )                )     2       ]     ·     r     mo   ,   i     2     ·       p   i     12         ,         
and p i  is the probability mass function (PMF) of r m  in the i th  bin.
 
   Next, the present invention describes an iterative procedure to approach a stationary solution which is likely to be the optimum solution. After that, the present invention further adaptively updates the index mapper through the iterative procedure. 
     FIG. 7  is a flowchart illustrating the iterative procedure. Referring to  FIG. 7 , the iterative procedure starts with the initialization of the bin boundaries and the iteration index, it then assigns the midpoint of each bin, as illustrated in step  701 . In step  702 , a long-term histogram for a plurality of modulated input signals is estimated. First, the modulated input signals are processed for a current iteration k, and a short-term histogram {ĥ i   (k) } is summarized. A long-term histogram {h i   (k) } is then approximated through the mean of the short-term histogram. However the present invention further replaces the PMF {p i } by a long-term histogram {h i }, the optimization problem becomes truly unconditional. According to the stationary solution, the bin width {d i }, for all i, are updated, as shown in step  703 . After the LUT spacing is updated, the current iteration waits for a time period until all the LUT entry values have been renewed, as shown in step  704 . If all the LUT entry values have been renewed, the update indicator is set to be 1. Otherwise the update indicator is set to be 0. The update indicator ω points out the LUT entry values are or are not updated. The renewed values are used for the next iteration. Finally, a step of check convergence with a convergence indicator ρ is taken, as in step  705 . If the LUT spacing difference 
             ∑     i   =   1     N     ⁢            d   i     (   k   )       -     d   i     (     k   -   1     )                    
between the current iteration and the previous iteration is smaller than a predetermined threshold ε., then the convergence indicator is set to be 1; otherwise the convergence indicator is set to be 0. The convergence indicator ρ serves as a quality indicator of the DAPD-LUT technique.
 
   Therefore, in order to prepare for the next iteration, the followings must be done, i e. updating the bin boundaries by 
               b   i     (     k   +   1     )       =       b     i   -   1       (     k   +   1     )       +     d   i     (   k   )           ,         
reassigning the bin midpoints by
 
               r     mo   ,   i       (   k   )       =       1   2     ⁢     (       b   i     (   k   )       +     b     i   -   1       (   k   )         )         ,         
increasing the iteration index by 1, and going back to step  702 . Please be noted that, even when the convergence indicator ρ equals to 1, the iteration of the procedure will continue so as to online adapt the LUT spacing to the variations of all kinds of system conditions.
 
   According to the present invention, in the step  701 , the initial values of the bin boundaries {b i   (k) } may be set as 
               b   0     (   1   )       =       0   ⁢           ⁢   and   ⁢           ⁢     b   i     (   1   )         =     i   N         ,         
where i is the bin index and the superscript (·) (k)  denotes the iteration index. After the iteration index k is set to 1, the midpoint of each bin is assigned as
 
             r     mo   ,   i       (   k   )       =       1   2     ⁢       (       b   i     (   k   )       +     b     i   -   1       (   k   )         )     .             
In the step  702 , the long-term histogram is estimated by h i   (k) =λ·h i   (k−1) +(1−λ)·ĥ i   (k) , where the short-term histogram {ĥ i   (k) } is averaged, λ is a forgetting factor, 0&lt;λ≦1, and
 
             h   i     (   0   )       =       1   N     .           
In the step  703 , the bin widths {d i } for k th  iteration is updated by
 
               d   i     (   k   )       =       ξ     (   k   )         η   i     (   k   )           ,     
     ⁢   where                   η   i     (   k   )       =       [         (                f   o     ⁡     (     r     mo   ,   i       )            ′         F   i   ′     ⁡     (     r     mo   ,   i       )         )     2     +       (                    [     arg   ⁡     (       f   o     ⁡     (     r     mo   ,   i       )       )       ]     ′     ·               (     1   -       r     mo   ,   i       ·              f   o     ⁡     (     r     mo   ,   i       )            ′         )                )     2       ]     ·     r     mo   ,   i     2     ·       h   i     (   k   )       12         ,     
     ⁢       ξ     (   k   )       =       (       ∑     i   =   1     N     ⁢       (     η   i     (   k   )       )       -   1         )       -   1               
is a normalization constant, f o (r mo,i ) denotes the LUT value of r mo,i  in the i th  bin, F i (r mo,i ) denotes the PD transfer function of the i th  bin, F′ i (r mo,i ) denotes the derivative of F i (r mo,i ) with respect to r m,i , |f o (r mo,i )| and arg(f o (r mo,i )) respectively denote the amplitude and the phase of f o (r mo,i ), and |f o (r mo,i )|′ and [arg(f o (r mo,i ))]′ respectively denote the derivative of |f o (r mo,i )| and arg(f o (r mo,i )) with respect to r m,i .
 
     FIG. 8  shows the power spectral density (PSD) performance comparison among several DAPD-LUT techniques with various LUT spacings in the system scenario with IBO=−10 dB, wherein the PSD performance is in terms of the normalized PSD of the PA output signal. As shown in  FIG. 8 , the dynamically-optimum of the present invention outperforms the other techniques with large gaps and approaches the unconditionally-optimum technique with a small gap. 
   Two simulation experiments are further conducted to evaluate the present invention. The first experiment tests its feasibility and compares the IMD performance among several conventional DAPD-LUT techniques with various LUT spacings. The second experiment tests the robustness of the present invention in a time-varying wireless system. 
   In the first experiment, two conditions of the IBO and the PA characteristics conditions in the system scenario are relaxed. The normalized IMD powers of several DAPD-LUT techniques with various LUT spacings are shown in  FIG. 9 . The two solid curves denote the IMD performance in the system scenario with PA # 1 . The three dashed curves denote the IMD performance in the system scenario with PA # 2 . Since the nonlinearity of PA # 2  is worse than that of PA # 1 , the unconditionally-optimum scheme in the system scenario with PA # 2  performs worse than that with PA # 1 . Nevertheless, the performance of the dynamically-optimum technique of the present invention still approaches that of the unconditionally-optimum technique regardless of the PA characteristics. 
   On the other hand, if the conditionally-optimum technique is optimized for IBO=−10 dB and PA # 1  in system scenarios with varying IBOs and with PA # 2 , as shown as the “(−10 dB, PA # 1 ) Optimum with PA # 2 ” curve in  FIG. 9 , the performance degradation is significant. Comparing point A and point B in  FIG. 9 , it can be observed that there is a 6-dB performance degradation of the conditionally-optimum technique with only the mismatch of the PA characteristics. 
   In the second experiment, the robustness of the dynamically-optimum technique of the present invention in a highly volatile system scenario is tested. The learning curve of the dynamically-optimum technique of the present invention is shown in  FIG. 10  in a time-varying system scenario with (1) IBO=−20 dB, PA # 1 , and the non-uniform OFDM input at the beginning, (2) the IBO jumping from −20 dB to −10 dB at the 50 th  iteration, (3) PA # 1  being replaced by PA # 2  at the 100 th  iteration, and (4) the non-uniform OFDM input being replaced by the uniform input at the 150 th  iteration. The horizontal axis represents the number of iteration of the iterative procedure as stated above. The vertical axis represents the normalized IMD power at the PA output. 
   As can be seen from  FIG. 10 , only the dynamically-optimum technique of the present invention can adapt itself to the variations of the system conditions. In other words, the performance of the dynamically-optimum technique of the present invention ties itself to the performance of the unconditionally-optimum technique with some transitional performance adaptation, while the performance of all the other DAPD-LUT techniques fluctuates dramatically. 
   In summary, the present invention provides a dynamically optimized non-uniform LUT spacing for the DAPD-LUT technique to linearize a PA, which has the advantages of being adaptive to all kinds of signal source going through all kinds of PA, being adaptive to time-variation of the wireless environments, low computational complexity, and reaching unconditionally-optimum performance. 
   Although the present invention has been described with reference to the preferred embodiments, it will be understood that the invention is not limited to the details described thereof. Various substitutions and modifications have been suggested in the foregoing description, and others will occur to those of ordinary skill in the art. Therefore, all such substitutions and modifications are intended to be embraced within the scope of the invention as defined in the appended claims.