Abstract:
Embodiments of a multi-point push-out method are disclosed for a more accurate adjustment of minimum IQ values in order to maintain better continuity in the IQ trajectory for sampled IQ values near the origin. IQ trajectories near the origin are limited to a minimum IQ value by determining a specific scaling factor according to a position of the sampled IQ value relative to the origin and/or to the minimum IQ value. The minimum IQ value is also referred to as a threshold boundary value.

Description:
FIELD OF THE INVENTION 
     The present invention relates to modulating signals. More particularly, the present invention relates to adjusting sampled IQ values near the origin. 
     BACKGROUND OF THE INVENTION 
     An IQ modulator shapes and samples an input signal to generate a series of discrete sampled values. Each sampled value includes an in-phase (I) component and a quadrature (Q) component, collectively referred to as a sampled IQ value. An IQ trajectory is a path derived from discrete sampled IQ values sampled from the input signal. The sampled IQ values are plotted on an IQ graph, with the origin (0,0) defined as the zero value for both the in-phase (I) component and the quadrature (Q) component. The sampling rate determines the number of discrete sampled IQ values. The higher the sampling rate, the more points that define the IQ trajectory. The IQ trajectory is ideally a smooth curve representing an infinite number of discrete sampled IQ values. In practice, there is always some finite number of sampled values, as defined by the sampling rate. 
     In hardware limitations, an IQ trajectory near the origin generates distortion in a corresponding output signal. Corrective signal processing of the sampled IQ values near the origin can improve the performance of the signal processing circuit. Conventional methodologies re-assign sampled IQ values that are near the origin to either a fixed distance away from the origin or to a default IQ value. In particular, sampled IQ values are compared to a minimum threshold value, and those sampled IQ values that are lower than the minimum threshold value are assigned to a minimum circular boundary about the origin or to a new default IQ value. In terms of the IQ trajectory and the corresponding IQ plot, there is one default IQ value associated with each quadrant of the IQ plot. The minimum threshold value defines a threshold region, which is typically the form of a box centered about the origin in the IQ plot. In this case, the threshold region is defined by the coordinates (+/−x, +/−x) on the IQ plot. The default IQ value for each quadrant is defined along the minimum threshold value at 45, 135, 225, and 315 degrees measured from the positive I-axis. These locations correspond to the corners of the threshold region. 
       FIGS. 1A and 1B  illustrate an exemplary IQ trajectory and conventional approaches for revaluing IQ values near the origin. An IQ trajectory  10  is comprised of multiple sampled IQ values including a sampled IQ value  12 , a sampled IQ value  14 , a sampled IQ value  16 , a sampled IQ value  18 , and a sampled IQ value  20 . In  FIG. 1A , a minimum IQ value is defined by the threshold boundary value  24 . The threshold boundary value  24  forms an outer perimeter of a threshold region centered about the origin  22 . In simplified applications, any sampled IQ value that falls within the threshold region, such as the sampled IQ value  16 , is to be revalued to a default IQ value. This default IQ value is conventionally assigned to be at the maximum threshold distance from the origin. A default IQ value is assigned for each quadrant within the IQ plot. In the configuration shown in  FIG. 1A , the maximum threshold distance for the first quadrant is located at the point  26 . Therefore, the default IQ value for the sampled IQ value  16  is the IQ value at the point  26 . The maximum threshold distance for each quadrant is defined along the threshold boundary value  24  at 45, 135, 225, and 315 degrees measured from the positive I-axis. These locations correspond to the corners of the threshold region. Conventionally, only one default IQ value is selected per quadrant, which in this example is the IQ value  26 . As such, any sampled IQ value that is less than the threshold boundary value  24 , and therefore resides within the threshold region, is reassigned the default IQ value for the corresponding quadrant. For each quadrant in the threshold region, the default IQ value is the same regardless of where within the threshold region quadrant the sample IQ value is located. In general, this approach is referred to as a single-point push-out method, since for each quadrant, all sampled IQ values within the threshold region are “pushed-out” to the default IQ point. 
     In  FIG. 1B , a minimum IQ value is defined by the minimum circular boundary  23 . The minimum circular boundary  23  forms a perimeter of a threshold region centered about the origin  22 . In simplified applications, any sampled IQ value that falls within the threshold region, such as the sampled IQ value  16 , is to be revalued to a default IQ value. In this configuration, the default IQ value is a fixed distance away from the origin on the minimum circular boundary  23 . Therefore, the default IQ value for the sampled IQ value  16  is the IQ value at the point  25 . 
     Although the simplified approach of reassigning sampled IQ values near the origin aids in reducing distortion, using a fixed default IQ value leads to discontinuities in the IQ trajectory. Compare the IQ trajectory  10  in  FIG. 1A  with the default IQ value  26  and in  FIG. 1B  with the default IQ value  25 . Such discontinuities result in transients in the frequency response. 
     SUMMARY OF THE INVENTION 
     Embodiments of a multi-point push-out method are disclosed for a more accurate adjustment of minimum IQ values in order to maintain better continuity in the IQ trajectory for sampled IQ values near the origin. IQ trajectories near the origin are limited to a minimum IQ value by determining a specific scaling factor according to a position of the sampled IQ value relative to the origin and/or to the minimum IQ value. The minimum IQ value is also referred to as a threshold boundary value. 
     In an aspect, a method comprises generating a plurality of discrete sampled IQ values based on an input signal, wherein each sampled IQ value includes an in-phase component and a quadrature component, comparing each sampled IQ value to a defined minimum IQ value, for each specific sampled IQ value that is less than the minimum IQ value, determining a specific scaling factor to be applied to the specific sampled IQ value, wherein the specific scaling factor is determined according to a relative difference between the specific sampled IQ value and the minimum IQ value and applying the specific IQ scaling factor to the specific sampled IQ value, thereby generating a scaled IQ value, wherein the scaled IQ value exceeds the minimum IQ value, further wherein the specific IQ scaling factor is applied to the in-phase component and the quadrature component of the specific sampled IQ value. In some embodiments, the determined scaling factor is selected from a set of scaling factors, wherein each scaling factor in the set of scaling factors is associated with a range of distances from the minimum IQ value. Each scaling factor is related to another scaling factor in the set by a power of two. The method further comprises generating an IQ trajectory according to each scaled IQ value and each sampled IQ value that is equal to or greater than the minimum IQ value. In some embodiments, the input signal is sampled at an over-sampling rate. In some embodiments, the input signal is sampled at data rate. 
     In another aspect, a method comprises defining a threshold boundary value centered about a zero IQ value, thereby forming a threshold region, mapping one or more sub-threshold regions within the threshold region, each sub-threshold region is centered about the zero IQ value thereby forming a plurality of adjacent banded regions extending from the zero IQ value to the threshold boundary value, wherein each banded region is associated with a specific scaling factor, generating a plurality of discrete sampled IQ values based on an input signal, comparing each sampled IQ value to the threshold boundary value, for each sampled value that is less than the threshold boundary value, mapping the sampled IQ value to a corresponding banded region and multiplying the sampled IQ value by the specific scaling factor associated with the corresponding banded region, thereby forming a scaled IQ value. In some embodiments, each scaled IQ value is equal to or greater than the threshold boundary value and less than a maximum scaled value. Each sub-threshold region is defined by a sub-threshold value, and each sub-threshold value and the threshold boundary value are related by a power of two. In some embodiments, a first scaling factor associated with a first banded region is less than a second scaling factor associated with a second banded region if the first banded region is further away from the zero IQ value than the second banded region. In some embodiments, the threshold region forms a square region centered about the zero IQ value. Each sub-threshold region forms a square region centered about the zero IQ value. The method further comprises generating an IQ trajectory according to each scaled IQ value and each sampled IQ value that is equal to or greater than the threshold boundary value. In some embodiments, the input signal is sampled at an over-sampling rate. In some embodiments, the input signal is sampled at data rate. 
     In still another aspect, a machine comprises an IQ limiting module configured to compare a sampled IQ value based on an input signal, to a defined minimum IQ value, to determine a specific scaling factor to be applied to the specific sampled IQ value for each specific sampled IQ value that is less than the minimum IQ value, wherein the specific scaling factor is determined according to a relative difference between the specific sampled IQ value and the minimum IQ value, and to apply the specific IQ scaling factor to the specific sampled IQ value, thereby generating a scaled IQ value, wherein the scaled IQ value exceeds the minimum IQ value, further wherein the specific IQ scaling factor is applied to the in-phase component and the quadrature component of the specific sampled IQ value. In some embodiments, the IQ limiting module is configured to select the scaling factor is from a set of scaling factors, wherein each scaling factor in the set of scaling factors is associated with a range of distances from the minimum IQ value. Each scaling factor is related to another scaling factor in the set by a power of two. In some embodiments, the IQ limiting module is further configured to generate an IQ trajectory according to each scaled IQ value and each sampled IQ value that is equal to or greater than the minimum IQ value. In some embodiments, the IQ modulator is configured to sample the input signal at an over-sampling rate. 
     In another aspect, a machine comprises an IQ limiting module coupled to the IQ modulator and configured to compare a sampled IQ value based on an input signal, to a defined minimum IQ value, to determine a specific scaling factor to be applied to the specific sampled IQ value for each specific sampled IQ value that is less than the minimum IQ value, wherein the specific scaling factor is determined according to a relative difference between the specific sampled IQ value and the minimum IQ value, and to apply the specific IQ scaling factor to the specific sampled IQ value, thereby generating a scaled IQ value, wherein the scaled IQ value exceeds the minimum IQ value, further wherein the specific IQ scaling factor is applied to the in-phase component and the quadrature component of the specific sampled IQ value, a polar converter to convert each scaled IQ value and each to a corresponding amplitude value and a corresponding phase value, a voltage-controlled oscillator coupled to the polar converter and configured to output a frequency response in response to the phase value, a power amplifier coupled to the voltage controlled oscillator and the polar converter, wherein the power amplifier is configured to output an amplified signal in response to the frequency response and the amplitude response and an antenna coupled to the power amplifier and configured to transmit the amplified signal. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         FIGS. 1A and 1B  illustrate an exemplary IQ trajectory and conventional approaches for revaluing IQ values near the origin. 
         FIG. 2  illustrates an exemplary configuration of a threshold region including three banded regions. 
         FIG. 3  illustrates an exemplary implementation of the threshold region including three banded regions from  FIG. 2 . 
         FIG. 4  illustrates the implementation of  FIG. 3  including a maximum scaled value. 
         FIG. 5  further illustrates the exemplary implementation of  FIG. 4 . 
         FIG. 6  illustrates the multi-point push-out method for generating an IQ trajectory. 
         FIGS. 7A ,  7 B and  7 C illustrate block diagrams of exemplary modulation circuits for implementing the multi-point push-out method. 
     
    
    
     The present invention is described relative to the several views of the drawings. Where appropriate and only where identical elements are disclosed and shown in more than one drawing, the same reference numeral will be used to represent such identical elements. 
     DETAILED DESCRIPTION 
     Embodiments of the present invention are directed to a multi-point push-out method to re-value sampled IQ points near the origin. Those of ordinary skill in the art will realize that the following detailed description of the present invention is illustrative only and is not intended to be in any way limiting. Other embodiments of the present invention will readily suggest themselves to such skilled persons having the benefit of this disclosure. 
     Reference will now be made in detail to implementations of the present invention as illustrated in the accompanying drawings. The same reference indicators will be used throughout the drawings and the following detailed description to refer to the same or like parts. In the interest of clarity, not all of the routine features of the implementations described herein are shown and described. It will, of course, be appreciated that in the development of any such actual implementation, numerous implementation-specific decisions must be made in order to achieve the developer&#39;s specific goals, such as compliance with application and business related constraints, and that these specific goals will vary from one implementation to another and from one developer to another. Moreover, it will be appreciated that such a development effort might be complex and time-consuming, but would nevertheless be a routine undertaking of engineering for those of ordinary skill in the art having the benefit of this disclosure. 
     The multi-point push-out method determines a specific scaling factor to be applied. The specific scaling factor is determined according to a relative difference between the origin and the sampled IQ value. A threshold boundary value is defined, thereby defining a threshold region centered about the origin within which any sampled IQ value is deemed to be “too near” to the origin. The threshold region is partitioned into adjacent banded regions that expand from the origin to the threshold boundary value. Each banded region is bounded by a sub-threshold boundary value. The outermost banded region is bound on an outer edge by the threshold boundary value. A specific scaling factor is associated with each banded region. Any sampled IQ value that falls within a given banded region is scaled by the specific scaling factor associated with that given banded region. The specific scaling factor is applied to both the I-component and the Q-component of the sampled IQ value, thereby generating a scaled IQ value that is greater than the threshold boundary value. Since both the I-component and the Q-component are scaled, a vector from the origin to the scaled IQ value has the same direction and as a vector from the origin to the sampled IQ value to which the scaling factor is applied. The vector magnitude of the scaled IQ value is a multiple of the vector magnitude of the sampled IQ value. This provides the “multi-point” aspect of the method since a sampled IQ value that is scaled results in a scaled IQ value that can be located in any one of multiple points depending on the original vector of the sampled IQ value. 
     A first banded region that is closer to the origin than a second banded region has a first scaling factor that is larger than a second scaling factor of the second banded region. The specific scaling factors are determined such that the scaled IQ value is greater than or equal to the threshold boundary value and less than a maximum scaled value. The region bounded by the maximum scaled value and the threshold boundary value is referred to as a scaled region. All scaled IQ values fall within the scaled region. 
     In some embodiments, the threshold boundary value is defined by the boundary coordinates (+/−x, +/−x) on the corresponding IQ plot. In this case, the threshold boundary value forms a square threshold region centered around the origin, or zero IQ value. The sub-threshold boundary values are defined by continually dividing the threshold boundary value by powers of two. For example, a first sub-threshold boundary value is defined by the boundary coordinates (+/−x/2, +/−x/2), a second sub-threshold boundary value is defined by the boundary coordinates (+/−x/4, +/−x/4), and so on. In this embodiment, the maximum scaled value forms a square centered about the origin, and the maximum scaled value is defined by multiplying the threshold boundary value by two. In the embodiments described below, two sub-threshold boundary values are defined. It is understood that the multi-point push-out method can be implemented using more or less than two sub-threshold boundary values. It is also understood that the sub-threshold boundary values and the maximum scaled value can be defined according to other relationships than the power of two method described above. The sub-threshold boundary values can also be independently assigned, with no specific relationship to the threshold boundary value or any of the other sub-threshold boundary values other than to set boundaries for banded regions as described herein. The maximum scaled value can also be independently assigned. 
       FIG. 2  illustrates an exemplary configuration of a threshold region including three banded regions. A threshold boundary value  30  forms a square threshold region centered around the origin. The threshold region is partitioned into banded regions by two sub-threshold boundary values. A first sub-threshold boundary value  34  forms a banded region  44 . A second sub-threshold boundary value  32  forms a banded region  42 . The banded region  42  is bound by the first sub-threshold boundary value  34  and the second sub-threshold boundary value  32 . A banded region  40  is bound by the second sub-threshold boundary value  32  and the threshold boundary value  30 . In this configuration, the banded region  40  includes all IQ values less than the threshold boundary value  30  and greater than or equal to the second sub-threshold boundary value  32 . The banded region  42  includes all IQ values less than the second sub-threshold boundary value  32  and greater than or equal to the first sub-threshold boundary value  34 . The banded region  44  includes all IQ values less than the first sub-threshold boundary value  34 . Each banded region  40 ,  42 ,  44  is associated with a specific scaling factor. For example, the scaling factor associated with the banded region  40  is a scaling factor g, the scaling factor associated with the banded region  42  is a scaling factor g′, and the scaling factor associated with the banded region  44  is a scaling factor g″. 
     In some embodiments, the sub-threshold boundary values and the threshold boundary value are related by powers of two. For example, if the sub-threshold boundary value  34  is represented by a single unit, then the sub-threshold boundary value  32  is represented by two units, and the threshold boundary value  30  is represented by four units. In this configuration, the scaling factors associated with each banded region are also related by powers of two. Specifically, the scaling factor g associated with the banded region  40  is two, the scaling factor g′ associated with the banded region  42  is four, and the scaling factor g″ associated with the banded region  44  is eight. The powers of two design is complimentary with binary nature of digital design. As such, a simple bit-shifting operation is used to perform the scaling function. 
     There are two considerations for each sampled IQ value that is less than the threshold boundary value, which direction to “push” the sampled IQ value and by how much. Since both the I-component and the Q-component of the sampled IQ value are multiplied by the scaling factor, the direction that the sampled IQ value is moved is along the same vector direction as the sampled IQ value. The amount by which the sampled IQ value is moved along this vector direction is determined by the corresponding banded region within the threshold region. 
       FIG. 3  illustrates an exemplary implementation of the threshold region including three banded regions from  FIG. 2 . The point  50  represents a sampled IQ value that is less than the threshold boundary value  30 . As such, the sampled IQ value  50  is to revalued to a scaled IQ value. The sampled IQ value  50  includes an I-component and a Q-component, represented as point (a,b) and vector  58 . The sampled IQ value  50  is positioned within the banded region  40 . The scaling factor g is associated with the banded region  40 . Therefore, the scaling factor g is applied to the sampled IQ value  50 , thereby forming the scaled IQ value  54 . The scaled IQ value  54  is represented as point (a×g, b×g), since the I-component and the Q-component of the sampled IQ value  50  are each multiplied by the scaling factor g. The scaled IQ value  54  is also represented by the vector  56 . Note that the direction of the vector  56  is the same as the direction of the vector  58 . 
     Each of the boundary values  30 ,  32 ,  34  and the scaling factors g, g′, g″ are configured such that all scaled IQ values are less than a maximum scaled value.  FIG. 4  illustrates the implementation of  FIG. 3  including a maximum scaled value. A scaled region  46  is bounded by the maximum scaled value  36  and the threshold boundary value  30 . All scaled IQ values are greater than or equal to the threshold boundary value  30  and less than the maximum scaled value  36 . In the configuration where the sub-threshold boundary values and the threshold boundary value are related by powers of two, the maximum scaled value is also related by a power of two. If the threshold boundary value  30  is represented by four units, then the maximum scaled value  36  is represented by eight units. 
     The scaling methodology described above in relation to the sampled IQ value  50  and the corresponding scaled IQ value  54  is similarly implemented in each of the other banded regions  42 ,  44  ( FIG. 2 ).  FIG. 5  furthers the exemplary implementation of  FIG. 4 . The point  60  represents a sampled IQ value with vector  68 . The sampled IQ value  60  has a value that is less than the sub-threshold boundary value  32  but greater than the sub-threshold boundary value  34 . As such the sampled IQ value  60  is positioned within the banded region  42 . The scaling factor g′ is associated with the banded region  42 . Therefore, the scaling factor g′ is applied to the sampled IQ value  60 , thereby forming the scaled IQ value  64 . The I-component and the Q-component of the sampled IQ value  60  are each multiplied by the scaling factor g′. The scaled IQ value  64  is also represented by the vector  66 . Note that the direction of the vector  66  is the same as the direction of the vector  68 . 
     Similarly, the point  70  represents a sampled IQ value with vector  78 . The sampled IQ value  70  has a value that is less than the sub-threshold boundary value  34 . As such the sampled IQ value  70  is positioned within the banded region  44 . The scaling factor g″ is associated with the banded region  44 . Therefore, the scaling factor g″ is applied to the sampled IQ value  70 , thereby forming the scaled IQ value  74 . The I-component and the Q-component of the sampled IQ value  70  are each multiplied by the scaling factor g″. The scaled IQ value  74  is also represented by the vector  76 . Note that the direction of the vector  76  is the same as the direction of the vector  78 . Also note, the both the scaled IQ value  64  and the scaled IQ value  74  are positioned within the scaled region  46 . 
       FIG. 6  illustrates the multi-point push-out method for generating an IQ trajectory. At the step  200 , a threshold boundary value is defined. The threshold boundary value defines the in-line (I) and quadrature (Q) threshold values. In some embodiments, the I threshold value and the Q threshold value are the same. In other embodiments, the I threshold value and the Q threshold value are different. The threshold boundary value, as defined by the I threshold value and the Q threshold value, defines a threshold region centered about the origin of an IQ value plot. The origin is also referred to as a zero IQ value. At the step  210 , one or more sub-threshold regions are mapped within the threshold region. Each sub-threshold region is centered about the origin. The threshold region and the sub-threshold regions form a series of adjacently banded regions extending from the origin to the threshold boundary value. At the step  220 , a specific scaling factor is associated with each banded region. At the step  230 , an input signal is sampled to generate a series of discrete sampled IQ values. At the step  240 , each sampled IQ value is compared to the threshold boundary value. If at the step  240  it is determined that the sampled IQ value is less than the threshold boundary value, which is equivalent to the sampled IQ value positioned within the threshold region, then at the step  250 , the sampled IQ value is mapped to a corresponding banded region. At the step  260 , the scaling factor associated with the corresponding banded region is applied to the sampled IQ value, thereby forming a scaled IQ value. The scaled IQ value is greater than or equal to the threshold boundary value, and the scaled IQ value is less than a maximum scaled value. 
     If at the step  240  it is determined that the sampled IQ value is equal to or greater than the threshold boundary value, then the sampled IQ value is not scaled. At the step  270 , the sampled IQ values that are not scaled at the step  240  and the scaled IQ values from the step  260  are used to generate an IQ trajectory. 
       FIG. 7A  illustrates a block diagram of an exemplary modulation circuit for implementing the multi-point push-out method. The modulation circuit  100  includes an IQ modulator  102 , an IQ limiting module  104 , a polar converter  106 , a digital-to-analog converter (DAC)  108 , a DAC  110 , a voltage controlled oscillator (VCO)  112 , a power amplifier  114 , and an antenna  116 . An IQ input signal, including an in-phase (I) component and a quadrature (Q) component, is provided to the IQ modulator  102 . The IQ modulator  102  samples the IQ input signal to generate discrete sampled IQ values, which are output to the IQ limiting module  104 . The IQ limiting module  104  compares each sampled IQ value to the threshold boundary value. For each sampled IQ value that is less than the threshold boundary value, a specific scaling factor is determined and applied to form a scaled IQ value, as described above. For each sample IQ value that is greater than or equal to the threshold boundary value, a scaling factor is not determined and the sampled IQ value is essentially passed through and output from the IQ limiting module  104 . Each passed through sampled IQ value and each scaled IQ value are input to the polar converter  106 . The polar converter  106  converts each received IQ value to a corresponding digital amplitude signal and digital phase signal. 
     In the exemplary modulation circuit  100 ′ of  FIG. 7B , the IQ input signal, including an in-phase (I) component and a quadrature (Q) component, is provided to an IQ limiting module  202 . The IQ limiting module  202  first receives the IQ input signal, determines the scaling factor and applies the scaling factor to form a scaled IQ value, if appropriate. The scaled IQ values and passed through values from the IQ limiting module  202  are then output to the IQ modulator  204 , which samples these values to generate discrete values, which are then provided to the polar converter  106 . 
     In the exemplary modulation circuit  100 ″ of  FIG. 7C , the IQ input signal, including an in-phase (I) component and a quadrature (Q) component, is provided to an IQ limiting module  302 . The IQ limiting module  302  first receives the IQ input signal, determines the scaling factor and applies the scaling factor to form a scaled IQ value, if appropriate. The scaled IQ values and passed through values from the IQ limiting module  302  are then output to an IQ modulator  304  which samples these values to generate discrete values. In this embodiment, the output of the IQ modulator  304  is then provided to a second IQ limiting module  305 . The output of the second IQ limiting module  305  is then provided to the polar converter  106 . 
     In each of the exemplary modulation circuits of  FIGS. 7A ,  7 B and  7 C, the digital amplitude signal received from the polar converter  106  is converted to a corresponding analog amplitude signal by the DAC  108 . The digital phase signal received from the polar converter  106  is converted to a corresponding analog phase signal by the DAC  110 . The VCO  112  generates a frequency response according to the input analog phase signal. The frequency response output from the VCO  112  and the analog amplitude signal output from the DAC  108  are input to the power amplifier  114 . The power amplifier  114  outputs an amplified signal, which is transmitted by the antenna  116 . In some embodiments, the amplified signal is a radio frequency (RF) signal. 
     The IQ trajectory translates in part to the phase response represented by the phase signal output from the polar converter  106 . Ideally, the phase response includes smooth transitions. However, when distortion is introduced, such as when the IQ trajectory is near the origin, transients in the phase response are generated, which are manifested as irregular transitions in the phase signal input to the VCO  112 . Such irregular transitions negatively impact the performance of the VCO  112 , and therefore negatively impact the analog signal output from the modulation circuit  100 . The scaled IQ values resulting from revaluing the sampled IQ values near the origin server to reduce such irregular transitions in the phase response, thereby improving the performance of the VCO and the output analog signal. 
     In addition to providing an improved phase response, generating the scaled IQ value as described above results in reducing the dynamic range of the power amplifier. Reducing the dynamic range is often a desired design consideration. 
     Each of the boundary values is described above as a square. It is understood that the boundary values can take other shapes, such as a circle. It is also understood that it is not required that all of the boundary values take the same shape. For example, the threshold boundary value can be defined by a square and one or more of the sub-threshold boundary values can be defined by a circle. 
     The present application has been described in terms of specific embodiments incorporating details to facilitate the understanding of the principles of construction and operation of the power amplification circuit. Many of the components shown and described in the various figures can be interchanged to achieve the results necessary, and this description should be read to encompass such interchange as well. As such, references herein to specific embodiments and details thereof are not intended to limit the scope of the claims appended hereto. It will be apparent to those skilled in the art that modifications can be made to the embodiments chosen for illustration without departing from the spirit and scope of the application.