Patent Publication Number: US-7711057-B2

Title: Apparatus and method for providing energy—bandwidth tradeoff and waveform design in interference and noise

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
   The present invention is based upon work supported and/or sponsored by the Air Force Research Laboratory (AFRL), Rome, N.Y. under contract No. FA8750-06-C-0202 

   FIELD OF INVENTION 
   The invention relates to techniques related to a tradeoff between transmit signal Energy and its Bandwidth using new transmitter-receiver waveform design methods that are applicable for radar, sonar and wireless communications. 
   BACKGROUND OF INVENTION 
   In the general problem, a desired target is buried in both interference and noise. A transmit signal excites both the desired target and the interference simultaneously. The interference and/or interferences can be foliage returns in the form of clutter for radar, scattered returns of the transmit signal from a sea-bottom and different ocean-layers in the case of sonar, or multipath returns in a communication scene. In all of these cases, like the target return, the interference returns are also transmit signal dependent, and hence it puts conflicting demands on the receiver. In general, the receiver input is comprised of target returns, interferences and the ever present noise. The goal of the receiver is to enhance the target returns and simultaneously suppress both the interference and noise signals. In a detection environment, a decision regarding the presence or absence of a target is made at some specified instant t=t 0  using output data from a receiver, and hence to maximize detection, the Signal power to average Interference plus Noise Ratio (SINR) at the receiver output can be used as an optimization goal. This scheme is illustrated in  FIG. 1 . 
   The transmitter output bandwidth can be controlled using a known transmitter output filter having a transfer function P 1 (ω) (see  FIG. 2A ). A similar filter with transform characteristics P 2 (ω) can be used at a receiver input  22   a  shown in  FIG. 1 , to control the processing bandwidth as well. 
   The transmit waveform set f(t) at an output  10   a  of  FIG. 1 , can have spatial and temporal components to it each designated for a specific goal. The simplest situation is that shown in  FIG. 2A  where a finite duration waveform f(t) of energy E is to be designed. Thus 
   
     
       
         
           
             
               
                 
                   
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   Usually, transmitter output filter  12  characteristics P 1 (ω), such as shown in  FIG. 2B , are known and for design purposes, it is best to incorporate the transmitter output filter  12  and the receiver input filter (which may be part of receiver  22 ) along with the target and clutter spectral characteristics. 
   Let q(t) Q(ω) represent the target impulse response and its transform. In general q(t) can be any arbitrary waveform. Thus the modified target that accounts for the target output filter has transform P 1 (ω)Q(ω) etc. In a linear domain setup, the transmit signal f(t) interacts with the target q(t), or target  14  shown in  FIG. 1 , to generate the output below (referred to in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci, “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, IEEE Transactions on Information Theory, Vol. 46, No. 2, pp. 577-584, March 2000 and J. R. Guerci and S. U. Pillai, “Theory and Application of Optimum Transmit-Receive Radar,” IEEE International Radar Conference, Alexandria Va., May 2000, pp. 705-710): 
                   s   ⁡     (   t   )       =         f   ⁡     (   t   )       *     q   ⁡     (   t   )         =       ∫   0     T   o       ⁢       f   ⁡     (   τ   )       ⁢     q   ⁡     (     t   -   τ     )       ⁢     ⅆ   τ                   (   2   )               
that represents the desired signal.
 
   The interference returns are usually due to the random scattered returns of the transmit signal from the environment, and hence can be modeled as a stochastic signal w c (t) that is excited by the transmit signal f(t). If the environment returns are stationary, then the interference can be represented by its power spectrum G c (ω). This gives the average interference power to be G c (ω)|F(ω)| 2 . Finally let n(t) represent the receiver  22  input noise with power spectral density G n (ω). Thus the receiver input signal at input  22   a  equals
 
 r ( t )= s ( t )+ w   c ( t )* f ( t )+ n ( t ),  (3)
 
and the input interference plus noise power spectrum equals
 
 G   I (ω)= G   c (ω)| F (ω)| 2   +G   n (ω).  (4)
 
The received signal is presented to the receiver  22  at input  22   a  with impulse response h(t). The simplest receiver is of the noncausal type.
 
   With no restrictions on the receiver  22  of  FIG. 1 , its output signal at output  22   b  in  FIG. 1 , and interference noise components are given by 
                       y   s     ⁡     (   t   )       =         s   ⁡     (   t   )       *     h   ⁡     (   t   )         =       1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢       S   ⁡     (   ω   )       ⁢     H   ⁡     (   ω   )       ⁢     ⅇ     jω   ⁢           ⁢   t       ⁢     ⅆ   ω               ⁢     
     ⁢     and             (   5   )                   y   n     ⁡     (   t   )       =       {           w   c     ⁡     (   t   )       *     f   ⁡     (   t   )         +     n   ⁡     (   t   )         }     *       h   ⁡     (   t   )       .               (   6   )               
The output y n (t) represents a second order stationary stochastic process with power spectrum below (referred to in the previous publications and in Athanasius Papoulis, S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill Higher Education, New York 2002):
   G   o (ω)=( G   c (ω)| F (ω)| 2   +G   n (ω)| H (ω)| 2   (7) 
and hence the total output interference plus noise power is given by
 
                         σ     I   +   N     2     =       1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢         G   0     ⁡     (   ω   )       ⁢     ⅆ   ω                       =       1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢       (           G   C     ⁡     (   ω   )       ⁢            F   ⁡     (   ω   )            2       +       G   n     ⁡     (   ω   )         )     ⁢            H   ⁡     (   ω   )            2     ⁢       ⅆ   ω     .                         (   8   )               
Referring back to  FIG. 1 , the signal component y s (t) in equation (5) at the receiver output  22   b  needs to be maximized at the decision instant t o  in presence of the above interference and noise. Hence the instantaneous output signal power at t=t o  is given by the formula below shown in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci, “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, IEEE Transactions on Information Theory, Vol. 46, No. 2, pp. 577-584, March 2000, which is incorporated by reference herein:
 
                   P   O     =                y   s     ⁡     (     t   o     )            2     =                1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢       S   ⁡     (   ω   )       ⁢     H   ⁡     (   ω   )       ⁢     ⅇ     jω   ⁢           ⁢     t   o         ⁢     ⅆ   ω                2     .               (   9   )               
This gives the receiver output SINR at t=t o  be the following as specified in Guerci et. al., “Theory and Application of Optimum Transmit-Receive Radar”, pp. 705-710; and Pillai et. al., “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, incorporated herein by reference:
 
                 SINR   =         P   o       σ     I   +   N     2       =                  1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢       S   ⁡     (   ω   )       ⁢     H   ⁡     (   ω   )       ⁢     ⅇ     jω   ⁢           ⁢     t   o         ⁢     ⅆ   ω                2         1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢         G   I     ⁡     (   ω   )       ⁢            H   ⁡     (   ω   )            2     ⁢     ⅆ   ω             .               (   10   )               
We can apply Cauchy-Schwarz inequality in equation (10) to eliminate H(ω). This gives
 
                       SINR   ≤       ⁢       1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢                S   ⁡     (   ω   )            2         G   I     ⁡     (   ω   )         ⁢     ⅆ   ω                       =       ⁢       1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢                  Q   ⁡     (   ω   )            2     ⁢            F   ⁡     (   ω   )            2               G   c     ⁡     (   ω   )       ⁢            F   ⁡     (   ω   )            2       +       G   n     ⁡     (   ω   )           ⁢     ⅆ   ω                       =       ⁢       SINR   max     .                   (   11   )               
Thus the maximum obtainable SINR is given by equation (11), and this is achieved if and only if the following equation referred to in previous prior art publications, is true:
 
                           H   opt     ⁡     (   ω   )       =         S   *     (   ω   )               G   c     ⁡     (   ω   )       ⁢            F   ⁡     (   ω   )            2       +       G   n     ⁡     (   ω   )           ⁢     ⅇ       -   jω     ⁢           ⁢     t   o                       =         Q   *     (   ω   )     ⁢   F   *     (   ω   )               G   c     ⁡     (   ω   )       ⁢            F   ⁡     (   ω   )            2       +       G   n     ⁡     (   ω   )           ⁢       ⅇ       -   jω     ⁢           ⁢     t   o         .                     (   12   )               
In (12), the phase shift e −jωt     o    can be retained to approximate causality for the receiver waveform. Interestingly even with a point target (Q(ω)≡1), flat noise (G n (ω)=σ n   2 ), and flat clutter (G c (ω)=σ c   2 ), the optimum receiver is not conjugate-matched to the transmit signal, since in that case from equation (12) we have the following formula given by Pillai et. al., “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, incorporated herein by reference, Papoulis, “Probability, Random Variables and Stochastic Processes”, and H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part  1 , New York: John Wiley and Sons, 1968, incorporated by reference:
 
                     H   opt     ⁡     (   ω   )       =           F   *     (   ω   )             σ   c   2     ⁢            F   ⁡     (   ω   )            2       +     σ   n   2         ⁢     ⅇ       -   jω     ⁢           ⁢     t   o           ≠     F   *     (   ω   )     ⁢       ⅇ       -   jω     ⁢           ⁢     t   o         .                 (   13   )               
Prior Art Transmitter Waveform Design
 
   When the receiver design satisfies equation (12), the output SINR is given by the right side of the equation (11), where the free parameter |F(ω)| 2  can be chosen to further maximize the output SINR, subject to the transmit energy constraint in (1). Thus the transmit signal design reduces to the following optimization problem: 
   Maximize 
                     SINR   max     =       1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢                  Q   ⁡     (   ω   )            2     ⁢            F   ⁡     (   ω   )            2               G   c     ⁡     (   ω   )       ⁢            F   ⁡     (   ω   )            2       +       G   n     ⁡     (   ω   )           ⁢     ⅆ   ω             ,           (   14   )               
subject to the energy constraint
 
                     ∫   0     T   o       ⁢              f   ⁡     (   t   )            2     ⁢     ⅆ   t         =         1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢              F   ⁡     (   ω   )            2     ⁢     ⅆ   ω           =     E   .               (   15   )               
To solve this new constrained optimization problem, combine (14)-(15) to define the modified Lagrange optimization function (referred to in T. Kooij, “Optimum Signal in Noise and Reverberation”,  Proceeding of the NATO Advanced Study Institute on Signal Processing with Emphasis on Underwater Acoustics , Vol. I, Enschede, The Netherlands, 1968.)
 
                   Λ   =       ∫     -   ∞       +   ∞       ⁢       {                  Q   ⁡     (   ω   )            2     ⁢       y   2     ⁡     (   ω   )                 G   c     ⁡     (   ω   )       ⁢       y   2     ⁡     (   ω   )         +       G   n     ⁡     (   ω   )           -       1     λ   2       ⁢       y   2     ⁡     (   ω   )           }     ⁢     ⅆ   ω           ⁢     
     ⁢     where             (   16   )                 y   ⁡     (   ω   )       =          F   ⁡     (   ω   )                    (   17   )               
is the free design parameter. From (16) (17),
 
   
     
       
         
           
             
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                     ∂     Λ   ⁡     (   ω   )           ∂   y       =       2   ⁢     y   ⁡     (   ω   )       ⁢     {             G   n     ⁡     (   ω   )       ⁢            Q   ⁡     (   ω   )            2           {           G   c     ⁡     (   ω   )       ⁢     y   2       -     (   ω   )     -       G   n     ⁡     (   ω   )         }     2       -     1     λ   2         }       =   0.             (   18   )               
where Λ(ω) represents the quantity within the integral in (16). From (18), either
 
                   y   ⁡     (   ω   )       =   0           (   19   )             or                                     G   n     ⁡     (   ω   )       ⁢            Q   ⁡     (   ω   )            2           {           G   c     ⁡     (   ω   )       ⁢       y   2     ⁡     (   ω   )         +       G   n     ⁡     (   ω   )         }     2       -     1     λ   2         =   0     ,           (   20   )               
which gives
 
                     y   2     ⁡     (   ω   )       -             G   n     ⁡     (   ω   )         ⁢     (       λ   ⁢          Q   ⁡     (   ω   )              -         G   n     ⁡     (   ω   )           )           G   c     ⁡     (   ω   )                 (   21   )               
provided y 2 (ω)&gt;0. See T. Kooij cited above incorporated by reference herein.
 
   SUMMARY OF THE INVENTION 
   One or more embodiments of the present invention provide a method and an apparatus for transmitter-receiver design that enhances the desired signal output from the receiver while minimizing the total interference and noise output at the desired decision making instant. Further the method and apparatus of an embodiment of the present invention can be used for transmit signal energy-bandwidth tradeoff. As a result, transmit signal energy can be used to tradeoff for “premium” signal bandwidth without sacrificing performance level in terms of the output Signal to Interference plus Noise power Ratio (SINR). The two designs—before and after the tradeoff—will result in two different transmitter-receiver pairs that have the same performance level. Thus a design that uses a certain energy and bandwidth can be traded off with a new design that uses more energy and lesser bandwidth compared to the old design. In many applications such as in telecommunications, since the available bandwidth is at premium, such a tradeoff will result in releasing otherwise unavailable bandwidth at the expense of additional signal energy. The bandwidth so released can be used for other applications or to add additional telecommunications capacity. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows a diagram of a system, apparatus, and/or method including a transmitter, a transmitter output filter, a receiver, a target, interference, noise, and a switch; 
       FIG. 2A  shows a prior art graph of a prior art transmitter signal versus time, wherein the transmitter signal is output from a transmitter, such as in  FIG. 1 ; 
       FIG. 2B  shows a prior art graph of a possible frequency spectrum of a known transmitter output filter, such as in  FIG. 1 ; 
       FIG. 3A  shows a graph of target transfer function magnitude response versus frequency; 
       FIG. 3B  shows a graph of target transfer function magnitude response versus frequency; 
       FIG. 3C  shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency; 
       FIG. 3D  shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency; 
       FIG. 4A  shows graphs of three different target transfer function magnitude responses versus frequency; 
       FIG. 4B  shows a graph of noise power spectrum versus frequency; 
       FIG. 4C  shows a graph of clutter power spectrum versus frequency; 
       FIG. 4D  shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency; 
       FIG. 4E  shows a graph of transmitter threshold energy versus bandwidth; 
       FIG. 4F  shows a graph of signal to inference plus noise ratio (SINR) versus bandwidth; 
       FIG. 5A  shows graphs of three different target transfer function magnitude responses versus frequency; 
       FIG. 5B  shows a graph of noise power spectrum versus frequency; 
       FIG. 5C  shows a graph of clutter power spectrum versus frequency; 
       FIG. 5D  shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency; 
       FIG. 5E  shows a graph of transmitter threshold energy versus bandwidth; 
       FIG. 5F  shows a graph of signal to inference plus noise ratio (SINR) versus bandwidth; 
       FIG. 6A  shows a graph of signal to interference plus noise ratio versus energy for a resonant target shown in  FIG. 5A  (solid line); 
       FIG. 6B  shows a graph of signal to interference plus noise ratio versus energy for a low pass target shown in  FIG. 5A  (dashed line); 
       FIG. 6C  shows a graph of signal to interference plus noise ratio versus energy for a flat target shown in  FIG. 5A  (dotted line); 
       FIG. 7  shows a graph of signal to interference plus noise ratio versus energy and the Bandwidth-Energy swapping design; 
       FIG. 8A  shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point A in  FIG. 7 ; 
       FIG. 8B  shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point B in  FIG. 7 ; 
       FIG. 8C  shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point C in  FIG. 7 ; and 
       FIG. 9  is a graph of realizable bandwidth savings versus operating bandwidth. 
   

   DETAILED DESCRIPTION OF THE DRAWINGS 
     FIG. 1  shows a diagram of a system, apparatus, and/or method  1 , including a transmitter  10 , a transmitter output filter  12 , a target  14 , interference  16 , noise  18 , a summation block  20 , receiver  22 , and a switch  24 . The present invention, in one or more embodiments, provides a new method and apparatus, by selecting a particular transmit signal f(t), to be output from transmitter  10 , and a type of receiver or receiver transfer function for receiver  22  in accordance with criteria to be discussed below. 
   The transmitter  10  transmits an output signal f(t) at its output  10   a  and supplies this signal to the transmitter output filter  12 . As remarked earlier, for design purposes, the transmitter output filter  12  can be lumped together with the target transfer function as well as the interference spectrum. The transmit signal f(t) passes through the airwaves and interacts with a target  14  and interference  16 . The target-modified as well as the clutter-modified (or interference modified) versions of the transmit signal f(t) are supplied to the summation block  18  along with receiver noise  18 . The summation block  18  may simply be used for description purposes to indicate that the target modified, clutter modified, and noise signals combine together. A combination signal is supplied to receiver  22  at its input  22   a . The receiver  22  applies a transfer function H(ω) (which will be determined and/or selected by criteria of an embodiment of the present invention, to be described below) and a modified combination signal is provided at a receiver output  22   b . The output is accessed at time t=t o  by use of switch  24 . 
     FIG. 2A  shows a prior art graph of a prior art transmitter output signal f(t) versus time. The signal used here is arbitrary. 
     FIG. 2B  shows a prior art graph of a frequency spectrum of the transmitter output filter  12  of  FIG. 1 . 
     FIG. 3A  shows a typical graph of a target transfer function magnitude response for target  14  versus frequency; target as appearing in (14)-(21). 
     FIG. 3B  shows a typical graph of target transfer function magnitude response for target  14  versus frequency; target as appearing in (14)-(21). 
     FIG. 3C  shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency; as in right side of equation (23). 
     FIG. 3D  shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency; as in right side of equation (23). 
     FIG. 4A  shows graphs of three different target transfer function magnitude responses versus frequency; target as appearing in (14)-(21). 
     FIG. 4B  shows a graph of noise power spectrum versus frequency as appearing in equations (14)-(23). 
     FIG. 4C  shows a graph of clutter power spectrum versus frequency as appearing in equations (14)-(23). 
     FIG. 4D  shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency as in right side of equation (23). 
     FIG. 4E  shows a graph of transmitter threshold energy versus bandwidth using equation (26). 
     FIG. 4F  shows a graph of signal to inference plus noise ratio versus bandwidth using equations (27)-(31). 
     FIG. 5A  shows graphs of three different target transfer function magnitude responses versus frequency; target as appearing in (14)-(21). 
     FIG. 5B  shows a graph of noise power spectrum versus frequency as appearing in equations (14)-(23). 
     FIG. 5C  shows a graph of clutter power spectrum versus frequency as appearing in equations (14)-(23). 
     FIG. 5D  shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency as in right side of equation (23). 
     FIG. 5E  shows a graph of transmitter threshold energy versus bandwidth using equation (26). 
     FIG. 5F  shows a graph of signal to inference plus noise ratio versus bandwidth using equations (27)-(31). 
     FIG. 6A  shows a graph of signal to interference plus noise ratio versus energy for a resonant target shown in  FIG. 5A  (solid line) using equations (34)-(35). 
     FIG. 6B  shows a graph of signal to interference plus noise ratio versus energy for a low pass target shown in  FIG. 5A  (dashed line) using equations (34)-(35). 
     FIG. 6C  shows a graph of signal to interference plus noise ratio versus energy for a flat target shown in  FIG. 5A  (dotted line) using equations (34)-(35). 
     FIG. 7  shows a graph of signal to interference plus noise ratio versus energy; generated using equations (39), (48), and (51). 
     FIG. 8A  shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point A in  FIG. 7  generated using (42). 
     FIG. 8B  shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point B in  FIG. 7  generated using (42). 
     FIG. 8C  shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point C in  FIG. 7  generated using (42) for a third energy condition. 
     FIG. 9  is a realizable bandwidth savings versus operating bandwidth generated using equation (60). 
   Define Ω +  as shown in  FIGS. 3C and 3D  to represent the frequencies over which y 2 (ω) in equation (21) is strictly positive, and let Ω o  represent the complement of Ω + . As shown in  FIGS. 3C and 3D , observe that the set Ω +  is a function of the noise and target spectral characteristics as well as the constraint constant λ. In terms of Ω + , we have 
                          F   ⁡     (   ω   )            2     =     {               y   2     ⁡     (   ω   )       ,           ω   ∈     Ω   +                 0   ,           ω   ∈       Ω   o     .                       (   22   )               
From (21), y 2 (ω)&gt;0 over Ω +  gives the necessary condition
 
                 λ   ≥       max     ω   ∈     Ω   +         ⁢           G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )                        (   23   )               
and the energy constraint in (15) when applied to (21) gives
 
                 E   =         1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢         y   2     ⁡     (   ω   )       ⁢     ⅆ   ω           =         λ     2   ⁢   π       ⁢       ∫     Ω   +       ⁢               G   n     ⁡     (   ω   )         ⁢          Q   ⁡     (   ω   )                  G   c     ⁡     (   ω   )         ⁢     ⅆ   ω           -       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢           G   n     ⁡     (   ω   )           G   c     ⁡     (   ω   )         ⁢     ⅆ   ω                       (   24   )               
or, for a given value of E, we have
 
                 λ   =         E   +       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢           G   n     ⁡     (   ω   )           G   c     ⁡     (   ω   )         ⁢     ⅆ   ω                 1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢               G   n     ⁡     (   ω   )         ⁢          Q   ⁡     (   ω   )                  G   c     ⁡     (   ω   )         ⁢     ⅆ   ω             ⁢       •λ   ⁡     (   E   )       .               (   25   )               
Clearly, λ(E) in (25) must satisfy the inequality in (23) as well. This gives rise to the concept of transmitter energy threshold that is characteristic to this design approach.
 
Transmitter Threshold Energy
 
   From (23)-(25), the transmit energy E must be such that λ(E) obtained from (25) should satisfy (23). If not, E must be increased to accommodate it, and hence it follows that there exists a minimum threshold value for the transmit energy below which it will not be possible to maintain |F(ω)| 2 &gt;0. This threshold value is given by 
                   E   min     =         (       max     ω   ∈     Ω   -         ⁢           G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )                )     ⁢     1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢               G   n     ⁡     (   ω   )         ⁢          Q   ⁡     (   ω   )                  G   c     ⁡     (   ω   )         ⁢     ⅆ   ω           -       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢           G   n     ⁡     (   ω   )           G   c     ⁡     (   ω   )         ⁢     ⅆ   ω                     (   26   )               
and for any operating condition, the transmit energy E must exceed E min . Clearly, the minimum threshold energy depends on the target, clutter and noise characteristics as well as the bandwidth under consideration. With E&gt;E min , substituting (20)-(21) into the SINR max  in (14) we get
 
                         SINR   max     =       ⁢       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢                  Q   ⁡     (   ω   )            2     ⁢       y   2     ⁡     (   ω   )           λ   ⁢         G   n     ⁡     (   ω   )         ⁢          Q   ⁡     (   ω   )                ⁢     ⅆ   ω                       =       ⁢       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢            Q   ⁡     (   ω   )                λ   ⁡     (   E   )       ⁢         G   n     ⁡     (   ω   )                                 ⁢               G   n     ⁡     (   ω   )         ⁢     (         λ   ⁡     (   E   )       ⁢          Q   ⁡     (   ω   )              -         G   n     ⁡     (   ω   )           )           G   c     ⁡     (   ω   )         ⁢     ⅆ   ω                   =       ⁢       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢       (            Q   ⁡     (   ω   )            -           G   n     ⁡     (   ω   )           λ   ⁡     (   E   )           )     ⁢            Q   ⁡     (   ω   )                G   c     ⁡     (   ω   )         ⁢       ⅆ   ω     .                         (   27   )               
Finally making use of (25), the output SINR max  becomes
 
                         SINR   1     =       ⁢         1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢                Q   ⁡     (   ω   )            2         G   c     ⁡     (   ω   )         ⁢     ⅆ   ω           -         (       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢                Q   ⁡     (   ω   )            ⁢         G   n     ⁡     (   ω   )               G   c     ⁡     (   ω   )         ⁢     ⅆ   ω           )     2       E   +       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢           G   n     ⁡     (   ω   )           G   c     ⁡     (   ω   )         ⁢     ⅆ   ω                             =       ⁢     a   -     c     λ   ⁡     (   E   )                       =       ⁢     a   -       c   2       E   +   b                     =       ⁢       aE   +     (     ab   -     c   2       )         E   +   b                     (   28   )             where                           a   =       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢                Q   ⁡     (   ω   )            2         G   c     ⁡     (   ω   )         ⁢     ⅆ   ω             ,           (   29   )                 b   =       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢           G   n     ⁡     (   ω   )           G   c     ⁡     (   ω   )         ⁢     ⅆ   ω             ,           (   30   )             and                         c   =       1     2   ⁢   π       ⁢       ∫     Ω   +       ⁢               G   n     ⁡     (   ω   )         ⁢          Q   ⁡     (   ω   )                  G   c     ⁡     (   ω   )         ⁢       ⅆ   ω     .                   (   31   )               
Notice that ab−c 2 ≧0. (This was published in Waveform Diversity and Design conference, Kauai, Hi., January 2006).
 
   The optimization problem in (14)-(15) can be restated in term of Ω +  as follows: Given Q(ω), G c (ω), G n (ω) and the transmit energy E, how to partition the frequency axis into an “operating band” Ω +  and a “no show” band Ω o  so that λ +  obtained from (25) satisfies (23) and SINR max  in (27)-(28) is also maximized. In general maximization of SINR max  in (27)-(28) over Ω +  is a highly nonlinear optimization problem for arbitrary Q(ω), G c (ω) and G n (ω). 
   In what follows a new approach to this problem is presented. 
   An Embodiment of the Present Invention—Desired Band Approach 
   One approach in this situation is to make use of the “desired frequency band” of interest B 0  this is usually suggested by the target response Q(ω) (and the transmitter output filter) to determine the operating band Ω + . The desired band B 0  can represent a fraction of the total available bandwidth, or the whole bandwidth itself. The procedure for determining Ω +  is illustrated in  FIGS. 3A-3C  and  FIGS. 3B-3D  for two different situations. In  FIGS. 3A-3D , the frequency band B 0  represents the desired band, and because of the nature of the noise and clutter spectra, it may be necessary to operate on a larger region Ω +  in the frequency domain. Thus the desired band B 0  is contained always within the operating band Ω + . To determine Ω + , using equation (23) we project the band B 0  onto the spectrum √{square root over (G n (ω))}/|Q(ω)| and draw a horizontal line corresponding to 
                   λ     B   o       =       max     ω   ∈     B   o         ⁢           G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )                        (   32   )               
as shown there. Define Ω + (B 0  ) to represent the frequency region where
 
                     ω   ∈       Ω   +     ⁡     (     B   o     )         :             G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )              ≤     λ     B   o           =       max     ω   ∈     B   o         ⁢             G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )              .               (   33   )               
This procedure can give rise to two situations as shown in  FIG. 3A  and  FIG. 3B . In  FIG. 3A , the operating band Ω + (B 0  ) coincides with the desired band B 0  as shown in  FIG. 3C , whereas in  FIG. 3B , the desired band B 0  is a subset of Ω + (B 0  ) as seen from  FIG. 3D .
 
   Knowing Ω + (B 0  ), one can compute λ=λ(E) with the help of equation (25) over that region, and examine whether λ so obtained satisfies (23). If not, the transmitter energy E is insufficient to maintain the operating band Ω + (B 0  ) given in (33), and either E must be increased, or Ω + (B 0  ) must be decreased (by decreasing B 0  ) so that (23) is satisfied. Thus for a given desired band B 0  (or an operating band Ω + (B 0  )), as remarked earlier, there exists a minimum transmitter threshold energy E B     o   , below which it is impossible to maintain |F(ω) 2 &gt;0 over that entire operating band. 
   Threshold Energy 
   From equations (24) and (32), we obtain the minimum transmitter threshold energy in this case to be the following 
                         E     B   o       =       ⁢           λ     B   o         2   ⁢   π       ⁢       ∫       Ω   +     ⁡     (     B   o     )         ⁢               G   n     ⁡     (   ω   )         ⁢          Q   ⁡     (   ω   )                  G   c     ⁡     (   ω   )         ⁢     ⅆ   ω           -                     ⁢       1     2   ⁢   π       ⁢       ∫       Ω   +     ⁡     (     B   o     )         ⁢           G   n     ⁡     (   ω   )           G   c     ⁡     (   ω   )         ⁢     ⅆ   ω                       =       ⁢           λ     B   o       ⁢     c   o       -     b   o       &gt;   0.                   (   34   )               
With E≧E B     o   , the SINR max  a in (28) can be readily computed. In particular with E=E B     o   , we get
 
                   SINR   1     =         SINR   1     ⁡     (     B   o     )       =       a   o     -         c   o   2         E     B   o       +     b   o         .                 (   35   )               
Here a o , b o  and c o  are as given in (29)-(31) with Ω +  replaced by Ω + (B 0  ). Eq. (35) represents the performance level for bandwidth B 0  using its minimum threshold energy. From (21), we also obtain the optimum transmit signal transform corresponding to energy E B     o    to be
 
                                F   ⁡     (   ω   )            2     =       ⁢     {                     G   n     ⁡     (   ω   )         ⁢     (         λ     B   o       ⁢          Q   ⁡     (   ω   )              -         G   n     ⁡     (   ω   )           )           G   c     ⁡     (   ω   )         ,           ω   ∈       Ω   +     ⁡     (     B   o     )                   0   ,           ω   ∈     Ω   o                           =       ⁢     {                     G   n     ⁡     (   ω   )         ⁢     (             (       max     ω   ∈     B   o         ⁢           G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )                )     -                     G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )                    )     ⁢            Q   ⁡     (   ω   )                G   c     ⁡     (   ω   )           ,           ω   ∈       Ω   +     ⁡     (     B   o     )                   0   ,           ω   ∈     Ω   o             .                     (   36   )               
To summarize, to maintain a given desired band B 0  , there exists an operating band Ω + (B 0  )≧B 0  over which |F(ω)| 2 &lt;0 and to guarantee this, the transmit energy must exceed a minimum threshold value E B     o    given by (34).
 
     FIGS. 4A-F  shows the transmitter threshold energy E in (34) and the corresponding SINR in (35) as a function of the desired bandwidth B 0  for various target, clutter, and noise spectra. Target to noise ratio (TNR) is set at 0 dB, and the clutter to noise power ratio (CNR) is set at 20 dB here. The total noise power is normalized to unity. The desired bandwidth B 0  is normalized with respect to the maximum available bandwidth (e.g., carrier frequency). 
   In  FIGS. 4A-F , the noise and clutter have flat spectra and for the highly resonant target (solid line), the required minimum energy threshold and the SINR generated using (34)-(35) reach a saturation value for small values of the bandwidth. In the case of the other two targets, additional bandwidth is required to reach the maximum attainable SINR. This is not surprising since for the resonant target, a significant portion of its energy is concentrated around the resonant frequency. Hence once the transmit signal bandwidth reaches the resonant frequency, it latches onto the target features resulting in maximum SINR at a lower bandwidth. 
     FIGS. 5A-F  show results for a new set of clutter and noise spectra as shown there; the transmitter threshold energy E in (34) and the corresponding SINR in (35) as a function of the desired bandwidth B 0  show similar performance details. 
   From  FIG. 5F , in the case of the resonant target (solid curve) the SINR reaches its peak value resulting in saturation even when B 0  is a small fraction of the available bandwidth. This is because in that case, the transmit waveform is able to latch onto the dominant resonant frequency of the target. On the other extreme, when the target has flat characteristics (dotted curve), there are no distinguishing frequencies to latch on, and the transmitter is unable to attain the above maximum SINR even when B 0  coincides with the total available bandwidth. For a low pass target (dashed curve), the transmitter is indeed able to deliver the maximum SINR by making use of all the available bandwidth. 
   As  FIG. 3B  shows, Ω + (B 0  ) can consist of multiple disjoint frequency bands whose complement Ω o  represents the “no show” region. Notice that the “no show” region Ω o  in the frequency domain in (36) for the optimum transmit signal can be controlled by the transmit energy E in (25). By increasing E, these “no show” regions can be made narrower and this defines a minimum transmitter threshold energy E ∞  that allows Ω + (B 0  ) to be the entire available frequency axis. To determine E ∞ , let λ ∞  represent the maximum in (23) over the entire frequency axis. Thus 
                     λ   ∞     =       max          ω        &lt;   ∞       ⁢           G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )                  ,           (   37   )               
and let a ∞ , b ∞ , c ∞  refer to the constants a, b, c in (29)-(31) calculated with Ω +  representing the entire frequency axis. Then from (24)
 
                   E   ∞     =           λ   ∞     ⁢     c   ∞       -     b   ∞       =             λ   ∞       2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢               G   n     ⁡     (   ω   )         ⁢          Q   ⁡     (   ω   )                  G   c     ⁡     (   ω   )         ⁢     ⅆ   ω           -       1     2   ⁢   π       ⁢       ∫     -   ∞       +   ∞       ⁢           G   n     ⁡     (   ω   )           G   c     ⁡     (   ω   )         ⁢     ⅆ   ω             &gt;   0               (   38   )               
represents the minimum transmit energy (threshold) required to avoid partitioning in the frequency domain. With E ∞  as given by (38), we obtain SINR max  to be (use (28))
 
                     SINR   1     ⁡     (   ∞   )       =         a   ∞     -       c   ∞       λ   ∞         =         a   ∞     -       c   ∞   2         E   ∞     +     b   ∞           &gt;   0               (   39   )             and                                    F   ⁡     (   ω   )            2     =             G   n     ⁡     (   ω   )         ⁢     (         λ   ∞     ⁢          Q   ⁡     (   ω   )              -         G   n     ⁡     (   ω   )           )           G   c     ⁡     (   ω   )           ,     
     ⁢          ω        &lt;     ∞   .               (   40   )               
Clearly by further increasing the transmit energy in (39) beyond that in (38) we obtain
 
                     SINR   1     -&gt;     a   ∞       =       1     2   ⁢   π       ⁢       ∫     -   ∞       -   ∞       ⁢                Q   ⁡     (   ω   )            2         G   c     ⁡     (   ω   )         ⁢       ⅆ   ω     .                   (   41   )               
It follows that to avoid any restrictions in the frequency domain for the transmit signal, the transmitter energy E must exceed a minimum threshold value E ∞  given by (38) and (39) represents the maximum realizable SNR. By increasing E beyond E ∞ , the performance can be improved up to that in (41).
 
   In general from (34) for a given desired bandwidth B 0  , the transmit energy E must exceed its threshold value E B     o   . With E&gt;E B     o    and λ(E) as in (25), the corresponding optimum transmit signal transform is given by (see (21) (22)) 
                          F   ⁡     (   ω   )            2     =     {                     G   c     ⁡     (   ω   )         ⁢     (         λ   ⁡     (   E   )       ⁢          (   ω   )            -         G   n     ⁡     (   ω   )           )           G   c     ⁡     (   ω   )         ,           ω   ∈       Ω   +     ⁡     (     B   o     )                   0   ,           ω   ∈     Ω   o                       (   42   )               
and clearly this signal is different from the minimum threshold energy one in (36). From (28), the performance level SINR 1 (E,B 0  ) corresponding to (42) is given by (35) with E B     o    replaced by E. Thus
 
                     SINR   1     ⁡     (     E   ,     B   o       )       =         a   o     -       c   o   2       E   +     b   o           &gt;         SINR   1     ⁡     (     B   o     )       .               (   43   )               
From (43), for a given bandwidth B 0  , performance can be increased beyond that in (35) by increasing the transmit energy. Hence it follows that SINR 1 (B 0  ) represents the minimum performance level for bandwidth B 0  that is obtained by using its minimum threshold energy. It is quite possible that this improved performance SINR 1 (E,B 0  ) can be equal to the minimum performance level corresponding to a higher bandwidth B 1 &gt;B 0  . This gives rise to the concept of Energy-Bandwidth tradeoff at a certain performance level. Undoubtedly this is quite useful when bandwidth is at premium.
 
     FIGS. 5E-5F  exhibit the transmit threshold energy and maximum output SINR 1 (B 0  ) as a function of the desired bandwidth B 0  . Combining these figures using (35), an SINR vs. transmit threshold energy plot can be generated as in  FIGS. 6A-C  for each target situation. 
   For example,  FIG. 6A-C  corresponds to the three different target situations considered in  FIG. 5  with clutter and noise spectra as shown there. Notice that each point on the SINR-Energy threshold curve for each target is associated with a specific desired bandwidth. Thus for bandwidth B 1 , the minimum threshold energy required is E 1 , and the corresponding SINR equals SINR 1 (B 1 ) in (35). Let A represent the associated operating point in  FIG. 6 . Note that the operating point A corresponding to a bandwidth B 1  has different threshold energies and different performance levels for different targets. From (35), each operating point generates a distinct transmit waveform. As the bandwidth increases, from (39), SINR→SINR 1 (∞). 
   Monotonic Property of SINR 
   The threshold energy and SINR associated with a higher bandwidth is higher. To prove this, consider two desired bandwidths B 1  and B 2  with B 2 &gt;B 1 . Then from (32) we have 
                     λ   2     =           max     ω   ∈     B   2         ⁢           G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )                &gt;     λ   1       =       max     ω   ∈     B   1         ⁢           G   n     ⁡     (   ω   )                Q   ⁡     (   ω   )                    ,           (   44   )               
and from  FIG. 3 , the corresponding operating bandwidths Ω + (B 1 ) and Ω + (B 2 ) satisfy
 Ω + ( B   2 )≧Ω + ( B   1 ).  (45) 
From (34) (or (24)), the minimum threshold energies are given by
 
                     E   i     =       1     2   ⁢   π       ⁢       ∫       Ω   +     ⁡     (     B   i     )         ⁢           G   n     ⁡     (   ω   )         ⁢     (       λ   i     -           G   n     ⁡     (   ω   )           Q   ⁡     (   ω   )           )     ⁢            Q   ⁡     (   ω   )                G   c     ⁡     (   ω   )         ⁢     ⅆ   ω             ,     
     ⁢     i   =   1     ,   2           (   46   )               
and substituting (44) and (45) into (46) we get
 E 2 &gt;E 1 .  (47) 
Also from (27), the performance levels at threshold energy SINR 1 (B i ) equals
 
                     SINR   1     ⁡     (     B   i     )       =       1     2   ⁢   π       ⁢       ∫       Ω   +     ⁡     (     B   i     )         ⁢       (            Q   ⁡     (   ω   )            -           G   n     ⁡     (   ω   )           λ   i         )     ⁢            Q   ⁡     (   ω   )                G   c     ⁡     (   ω   )         ⁢     ⅆ   ω                   (   48   )               
and an argument similar to (44)-(45) gives
   SINR   1 ( B   2 )≧ SINR   1 ( B   1 )  (49) 
for B 2 &gt;B 1 . Thus as FIGS.  5 A-F- FIGS. 6A-C  show, SINR 1 (B i ) is a monotonically nondecreasing function of both bandwidth and energy.  FIG. 7  illustrates this SINR-energy relation for the target with flat spectrum shown in  FIG. 5A . In  FIG. 7 , the two operating points A and B are associated with bandwidths B 1  and B 2 , threshold energies E 1  and E 2 , and performance levels SINR 1 (B 1 ) and SINR 1 (B 2 ) respectively.
 
Since
   B   2   &gt;B   1     E   2   ≧E   1  and  SINR   1 ( B   2 )≧ SINR   1 ( B   1 ).  (50) 
The distinct transmit waveforms |F 1 (ω)| 2  and |F 2 (ω)| 2  associated with these operating point A and B are given by (36) and they are shown in  FIGS. 8A and 8B .
 
   Consider the operating point A associated with the desired bandwidth B 1 . If the transmit energy E is increased beyond the corresponding threshold value E 1  with bandwidth held constant at B 1 , the performance SINR 1 (E, B 1 ) increases beyond that at A since from (43) 
                     SINR   1     ⁡     (     E   ,     B   1       )       =           a   1     -       c   1   2       E   +     b   1           ≥       a   1     -       c   1   2         E   1     +     b   1             =       SINR   1     ⁡     (     B   1     )                 (   51   )               
and it is upper bounded by a 1 . Here a 1  corresponds to the SINR performance for bandwidth B 1  as the transmit energy E→∞. Note that a 1 , B 1  and c 1  are the constants in (29)-(31) with Ω +  replaced by Ω + (B 1 ). The dashed curve Aa 1  in  FIG. 7  represents SINR 1 (E, B 1 ) for various values of E. From (42), each point on the curve Aa 1  generates a new transmit waveform as well.
 
   Interestingly the dashed curves in  FIG. 7  cannot cross over the optimum performance (solid) curve SINR(B i ). If not, assume the performance SINR 1 (E,B 1 ) associated with the operating point A crosses over SINR(B i ) at some E 1 ′&gt;E 1 . Then from (47), there exists a frequency point B 1 ′&gt;B 1  with threshold energy E 1 ′ and optimum performance SINR 1 (B 1 ′). By assumption,
 
 SINR   1 ( E   1   ′,B   1 )&gt; SINR   1 ( B   1 ′).  (52)
 
But this is impossible since SINR 1 (B 1 ′) corresponds to the maximum SINR realizable at bandwidth B 1 ′ with energy E 1 ′, and hence performance at a lower bandwidth B 1  with the same energy cannot exceed it. Hence (52) cannot be true and we must have
 
 SINR   1 ( E   1   ′,B   1 )≦ SINR   1 ( B   1 ′),  (53)
 
i.e., the curves Aa 1 , Ba 2 , etc. does not cross over the optimum performance curve ABD.
 
   In  FIG. 7 , assume that the saturation performance value
 
 a   1   ≧SINR   1 ( B   2 ),  (54)
 
i.e., the maximum performance level for bandwidth B 1  is greater than of equal to the performance level associated with the operating point B with a higher bandwidth B 2  and a higher threshold energy E 2 . Draw a horizontal line through B to intersect the curve Aa 1  at C, and drop a perpendicular at C to intersect the x-axis at E 3 . From (51) with E=E 3  we get
 
 SINR   1 ( E   3   ,B   1 )= SINR   1 ( B   2 ).  (55)
 
Thus the operating point C on the curve Aa 1  is associated with energy E 3 , bandwidth B 1  and corresponds to a performance level of SINR 1 (B 2 ) associated with a higher bandwidth. Notice that
 
E 3 &gt;E 2 &gt;E 1 , and B 1 &lt;B 2 .  (56)
 
In other words, by increasing the transmit energy from E 1  to E 3  while holding the bandwidth constant at B 1 , the performance equivalent to a higher bandwidth B 2  can be realized provided B 2  satisfies (54). As a result, energy-bandwidth tradeoff is possible within reasonable limits. The transmit waveform |F 3 (ω)| 2  associated with the operating point C is obtained using (42) by replacing E and B 0  there with E 3  and B 1  respectively. and it is illustrated in  FIG. 8C . In a similar manner, the waveforms corresponding to the operating points A and B in  FIG. 7  can be obtained using equation (42) by replacing the energy-bandwidth pair (E,B 0 ) there with (E 1 ,B 1 ) and (E 2 ,B 2 ) respectively. These waveforms are shown in  FIG. 8A  and  FIG. 8B  respectively A comparison with  FIGS. 8A  and  8 B show that the waveform at C is different from those associated with operating point A and B.
 
   It is important to note that although the transmit waveform design |F 3 (ω)| 2  and |F 1 (ω)| 2  correspond to the same bandwidth (with different energies E 3  and E 1 ), one is not a scaled version of the other. Changing transmit energy from E 1  to E 3  unleashes the whole design procedure and ends up in a new waveform |F 3 (ω)| 2  that maintains a performance level associated with a larger bandwidth B 2 . 
   The question of how much bandwidth tradeoff can be achieved at an operating point is an interesting one. From the above argument, equality condition in (54) gives the upper bound on how much effective bandwidth increment can be achieved by increasing the transmit energy. Notice that for an operating point A, the desired bandwidth B 1  gives the operating bandwidth Ω + (B 1 ) and from (29) the performance limit 
                   a   1     =       1     2   ⁢   π       ⁢       ∫       Ω   +     ⁡     (     B   1     )         ⁢                Q   ⁡     (   ω   )            2         G   c     ⁡     (   ω   )         ⁢     ⅆ   ω                   (   57   )               
for bandwidth B 1  can be computed. Assume B 2 &gt;B 1 , and from (48) SINR 1 (B 2 ) the minimum performance at B 2  also can be computed, and for maximum bandwidth swapping the nonlinear equation
   a   1   =SINR   1 ( B   2 )  (58) 
must be solved for B 2 . Then
 Δ B ( B   1 )= B   2   −B   1   (59) 
represents the maximum bandwidth enhancement that can be realized at B 1 . This is illustrated in for the target situation in  FIG. 7 . Notice that the maximum operating bandwidth if finite in any system due to sampling considerations and after normalization, it is represented by unity. Hence the upper limit in (59) must be min(1, B 2 ). This gives
 Δ B =min(1 ,B   2 )− B   1   (60) 
and this explains the linear nature of ΔB for larger value of B i . In that case, bandwidth can be enhanced by 1−B 1  only.
 
   The design approach described in this section requires the knowledge of the target characteristics in addition to the clutter and noise spectra. 
   Although the invention has been described by reference to particular illustrative embodiments thereof, many changes and modifications of the invention may become apparent to those skilled in the art without departing from the spirit and scope of the invention. It is therefore intended to include within this patent all such changes and modifications as may reasonably and properly be included within the scope of the present invention&#39;s contribution to the art.