Conflict Set and Waveform Modelling for Power Amplifier Design

Various classes of nonnegative waveforms containing dc component, fundamental and kth harmonic (k ≥ 2), which proved to be of interest in waveform modelling for power amplifier (PA) design, are considered in this paper. In optimization of PA efficiency, nonnegative waveforms with maximal amplitude of fundamental harmonic and those with maximal coefficient of cosine term of fundamental harmonic (optimal waveforms) play an important role. Optimal waveforms have multiple global minima and this fact closely relates the problem of optimization of PA efficiency to the concept of conflict set. There is also keen interest in finding descriptions for various classes of suboptimal waveforms, such as nonnegative waveforms with at least one zero, nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of kth harmonic, nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of kth harmonic, and nonnegative cosine waveforms with at least one zero. Closed form descriptions for all these suboptimal types of waveforms are provided in this paper. Suboptimal waveforms may also have multiple global minima and therefore be related to the concept of conflict set. Four case studies of usage of closed form descriptions of nonnegative waveforms in PA modelling are also provided.


Introduction
The origin of the concept of conflict set goes back to J. C. Maxwell (Maxwell 1831-1879), who informally introduced most of features of what today is called conflict set [1].From this reason Maxwell set or Maxwell stratum is also used as synonyms for conflict set.Roughly speaking, conflict set associated with a smooth function  with  parameters is the set of -tuples in parameter space for which  has multiple global minima.Conflict set is also intimately related to singularity theory and catastrophe theory [1].
Nonnegative waveforms with maximal amplitude of fundamental harmonic and those with maximal coefficient of cosine term of fundamental harmonic (optimal waveforms) have multiple global minima and therefore are closely related to the concept of conflict set.The suboptimal waveforms such as (i) nonnegative waveforms with at least one zero, (ii) nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of th harmonic, (iii) nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of th harmonic, (iv) nonnegative cosine waveforms with at least one zero may also have multiple global minima [9,11,12] and therefore be related to the concept of conflict set, as well.These suboptimal waveforms are clearly of interest in shaping/modelling drain (collector/plate) waveforms in PA design (e.g., see [3-12, 20, 21]).
Fejér in his seminal paper [22] provided general description of all nonnegative trigonometric polynomials with  consecutive harmonics in terms of 2 + 2 parameters satisfying one nonlinear constraint.He also derived closed form solution to the problem of finding maximum possible amplitude of the first harmonic of nonnegative cosine polynomials with consecutive harmonics.
Fuzik [3] (see also [10]) considered cosine polynomials with dc, fundamental and th harmonic, for arbitrary  ≥ 2 and provided closed form solution for coefficients of optimal waveform.Rhodes in [7] provided closed form expression for maximum possible amplitude of fundamental harmonic of nonnegative waveforms containing consecutive odd harmonics.A subclass of nonnegative cosine waveforms with dc, fundamental and third harmonic, having factorized form description has been considered in [23].
High efficiency PA with arbitrary output harmonic terminations has been analysed in [9], along with maximal efficiency, fundamental output power, and load impedance.
Factorized form of nonnegative waveforms up to second harmonic with at least one zero has been suggested in [11] in the context of continuous class B/J mode of PA operation.
General description of all nonnegative waveforms up to second harmonic in terms of four independent parameters has been provided in [12].This includes nonnegative waveforms with at least one zero, as a special case.
End point of conflict set normally corresponds to socalled maximally flat waveform, which also belongs to class of suboptimal waveforms.First comprehensive usage of maximally flat waveforms, in the context of analysis of PA, goes to Raab [20].General description of maximally flat waveforms with arbitrary number of harmonics has been presented in [21], along with closed form expressions for efficiency of class-F and inverse class-F PA with maximally flat waveforms.Description of maximally flat cosine waveforms with consecutive harmonics has been presented in [8] in the context of finite harmonic class-C PA.
In this paper we provide general descriptions of a number of optimal and suboptimal nonnegative waveforms containing dc component, fundamental and an arbitrary th harmonic,  ≥ 2, and show how they are related to the concept of conflict set.According to our best knowledge, this paper provides the very first usage of conflict set in the course of solving problems related to optimization of PA efficiency.Main results are stated in six propositions (Propositions 1, 6, 9, 18, 22, and 26), four corollaries (Corollaries 2-5), twenty remarks, and three algorithms.Four case studies of usage of closed form descriptions of nonnegative waveforms in PA efficiency analysis are considered in detail in Section 7.This paper is organized in the following way.In Section 2 we introduce concepts of minimum function and gain function (Section 2.1), conflict set (Section 2.2), and parameter space (Section 2.3).In Sections 3-6 we provide general descriptions of various classes of nonnegative waveforms containing dc component, fundamental and th harmonic with at least one zero.General case of nonnegative waveforms with at least one zero is presented in Section 3.1.The case with exactly two zeros is considered in Section 3.2.An algorithm for calculation of coefficients of fundamental harmonic of nonnegative waveforms with two zeros, for prescribed coefficients of th harmonic, is presented in Section 3.3.Description of nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of th harmonic is provided in Section 4. Nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of th harmonic are considered in Section 5.1.An illustration of results of Section 5.1 for particular case  = 3 is given in Section 5.2.Section 6.1 is devoted to nonnegative cosine waveforms with at least one zero and arbitrary  ≥ 2, whereas Section 6.2 considers cosine waveforms with at least one zero for  = 3.In Section 7 four case studies of application of descriptions of nonnegative waveforms with fundamental and th harmonic in PA modelling are presented.In the Appendices, list of some finite sums of trigonometric functions, widely used throughout the paper, and brief account of the Chebyshev polynomials are provided.

Minimum Function, Gain Function, and Conflict Set
In this section we consider minimum function and gain function (Section 2.1), conflict set (Section 2.2), and parameter space (Section 2.3) in the context of nonnegative waveforms with fundamental and th harmonic.
We start with provision of a brief account of the facts related to the concepts of minimum function and conflict set.For this purpose let us denote by (; ) a family of smooth functions of  variables depending on  parameters, where  ∈   is -tuple of variables and  ∈   is -tuple of parameters.The minimum function  :   → , associated with the function , is defined as () = min  (; ).Therefore, the domain of the minimum function is parameter space of the function .The minimum function () is continuous, but not necessarily smooth function of parameters [13,24].It is a smooth function if (; ) possesses unique global minimum at nondegenerate critical point [13] (critical point is degenerate if at least first two consecutive derivatives are equal to zero).In this context, the conflict set can be defined as the set of the parameters for which function  has global minimum at a degenerate critical point or/and multiple global minima [13].
For a wide class of minimum functions, when the number of parameters is not greater than four, the behaviour of minimum function in a neighbourhood of any point can be described by one of "normal forms" from a finite list as stated in [24].For example, for smooth function  :  ×  2 → , the minimum function ( 1 ,  2 ) = min  (,  1 ,  2 ) near the origin can be locally reduced to one of the following three normal forms [25]: −| 1 |, min( 1 ,  2 ,  1 +  2 ), or min  ( 4 +  1  2 +  2 ).In this example, the conflict set is the set of all points ( 1 ,  2 ) for which minimum function ( 1 ,  2 ) is not differentiable because function (,  1 ,  2 ) possesses at least two global minima [25].

Minimum Function and Gain Function.
In what follows we consider family of waveforms of type  (; , , ) = 1 −  (cos  +  cos ( + )) , (1) where  stands for ,  > 0,  ≥ 2,  > 0, and  ∈ [0, 2).Waveforms of type (1) include all possible shapes which can occur, but not all possible waveforms containing fundamental and th harmonic.However, shifting of waveforms of type (1) along -axis could recover all possible waveforms with fundamental and th harmonic.
The problem of finding nonnegative waveform of type (1) having maximum amplitude of fundamental harmonic plays an important role in optimization of PA efficiency.This extremal problem can be reformulated as problem of finding nonnegative waveform from family (1) having maximum possible value of coefficient .Nonnegative waveform of family (1) with maximum possible value of coefficient  is called "optimal" or "extremal" waveform.
Furthermore, let us introduce an auxiliary waveform which is smooth function of one variable  and two parameters  and .In terms of (; , ), the above extremal problem reduces to the problem of finding maximum possible value of coefficient  that satisfies Clearly, for any prescribed pair (, ), there is a unique maximal value of coefficient  for which inequality (3) holds for all .This maximal value of  associated with the pair (, ) we denote it by (, ) and call it "gain function." Let  min (, ) = min   (; , ) be the minimum function associated with (; , ).According to (3), (, ) and  min (, ) satisfy the following relation: 1 + (, ) min (, ) = 0. Since  min (, ) is obviously nonzero it follows immediately that A relation analogue to (5), for  = 2 (fundamental and second harmonic), has been derived in [4].According to our best knowledge, it was the first appearance of gain function expressed via associated minimum function.The consideration presented in [4] has been restricted to the particular case when  = .The same problem for  =  and arbitrary  ≥ 2 has been investigated in [3] (see also [10]).
According to above consideration, the problem of finding 3-tuple (, , ) with maximum possible value of  for which (3) holds is equivalent to the problem of finding maximum value of gain function Thus the optimal waveform  * () is determined by parameters  max ,  * , and  * ; that is, Optimal waveform has two global minima (this claim will be justified in Section 4, Remark 21).Consequently, the pair ( * ,  * ), which corresponds to maximum of gain function (, ), belongs to conflict set in (, ) parameter space.
Figure 1 shows graph of gain function (, ) for  = 2. Notice that it has sharp ridge and that maximum of gain function (point  2 ) lies on the ridge.This maximum corresponds to the optimal waveform (solution of the considered extremal problem).The beginning of the ridge (point  1 ) corresponds to the waveform which possesses global minimum at degenerate critical point, that is, corresponds to maximally flat waveform (e.g., see [21]).Gain function (, ) is not differentiable on the ridge and consequently is not differentiable at the point where it has global maximum.This explains why the approach based on critical points does not work and why conflict set is so important in the considered problem.
Positions of global minima of (; , ) for  = 2 are presented in Figure 2. According to Proposition 1, conflict set is the ray defined by  > 1/4 and  = .Waveforms (; , ) with parameters that belong to the conflict set have two global minima.The waveform corresponding to the end point of the ray ( = 1/4 and  = ) has global minimum at degenerate critical point (so-called maximally flat waveform [21]).

Conflict Set.
Historically, conflict set came into being from the problems in which families of smooth functions (such as potentials, distances, and waveforms) with two competing minima occur.The situation when competing minima become equal refers to the presence of conflict set (Maxwell set, Maxwell strata) in the associated parameter space.
There are many facets of conflict set.For example, in the problem involving distances between two sets of points, the conflict set is the intersections between iso-distance lines [14].Conflict set also arises in the situation when two wave fronts coming from different objects meet [15,25].In the study of black holes, conflict set is the line of crossover of the horizon formed by the merger of two black holes [19].In the classical Euler problem, conflict set is a set of points where distinct extremal trajectories with the same value of the cost functional meet one another [18].
Conflict set is very difficult to calculate, both analytically and numerically (e.g., see [15]), because of apparent nondifferentiability in some directions.In optimization of PA efficiency, some authors already reported difficulties in finding optimum via standard analytical tools [4,5].
In this section, we consider conflict set in the context of family of waveforms of type (2) for arbitrary  ≥ 2. In this context, for prescribed integer  ≥ 2, conflict set is said to be a set of all pairs (, ) for which (; , ) possesses multiple global minima.
The following proposition describes the conflict set of family of waveforms of type (2).(2) is the set of all pairs (, ) such that  > 1/ 2 and  = .

Proposition 1. Conflict set of family of waveforms of type
The proof of Proposition 1, which is provided at the end of this section, also implies that the following four corollaries hold.

Corollary 4. Every waveform with fundamental and 𝑘th harmonic has either one or two global minima.
Corollary 5. Conflict set can be parameterised in terms of  Δ as follows: Notice that ( Δ ) is monotonically increasing function on interval 0 <  Δ ≤ /.

Parameter Space.
In parameter space of family of waveforms (2) there are two subsets playing important role in the classification of the family instances.These are conflict set and catastrophe set.Catastrophe set is subset of parameter space of waveform (; , ).It consists of those pairs (, ) for which the corresponding waveforms (; , ) have degenerate critical points at which first and second derivatives are equal to zero.Thus, for finding catastrophe set we have to consider the following system of equations: where   is a degenerate critical point of waveform (; , ).
Conflict set in parameter space of waveform (; , ), as shown in Proposition 1, is the ray described by  > 1/ 2 and  = .It is intimately connected to catastrophe set.
In what follows in this subsection we use polar coordinate system ( cos ,  sin ) instead of Cartesian coordinate system (, ).Examples of catastrophe set and conflict set for  ≤ 5 plotted in parameter space ( cos ,  sin ) are presented in Figure 3. Solid line represents the catastrophe set while dotted line describes conflict set.The isolated pick points (usually called cusp) which appear in catastrophe curves correspond to maximally flat waveforms, with maximally flat minimum and/or maximally flat maximum.There are two such picks in the catastrophe curves for  = 2 and  = 4 and one in the catastrophe curves for  = 3 and  = 5.Notice that the end point of conflict set is the cusp point.
Catastrophe set divides the parameter space ( cos ,  sin ) into disjoint subsets.In the cases  = 2 and  = 3 catastrophe curve defines inner and outer part.For  > 3 catastrophe curve makes partition of parameter space in several inner subsets and one outer subset (see Figure 3).
Notice also that multiplying (; , ) with a positive constant and adding in turn another constant, which leads to waveform of type (; , , ) (see (1) and (2)), do not make impact on the character of catastrophe and conflict sets.This is because in the course of finding catastrophe set first and second derivatives of (; , ) are set to zero.Clearly (34) in terms of (; , ) are equivalent to the analogous equations in terms of (; , , ).Analogously, in the course of finding conflict set we consider only the positions of global minima (these positions for waveforms (; , ) and (; , , ) are the same).

Nonnegative Waveforms with at Least One Zero
In what follows let us consider a waveform containing dc component, fundamental and th ( ≥ 2) harmonic of the form The amplitudes of fundamental and th harmonic of waveform of type (35), respectively, are As it is shown in Section 2.1, nonnegative waveforms with maximal amplitude of fundamental harmonic or maximal coefficient of fundamental harmonic cosine part have at least one zero.It is also shown in Section 2.2 (Corollary 4) that waveforms of type (35) with nonzero amplitude of fundamental harmonic have either one or two global minima.Consequently, if nonnegative waveform of type (35) with nonzero amplitude of fundamental harmonic has at least one zero, then it has at most two zeros.
In Section 3.1 we provide general description of nonnegative waveforms of type (35) with at least one zero.In Sections 3.2 and 3.3 we consider nonnegative waveforms of type (35) with two zeros.
In what follows we are going to prove that (40) also holds.According to (38),   () is nonnegative if and only if Let us first show that position of global maximum of   () belongs to the interval | −  0 | ≤ 2/.Relation (56) can be rewritten as where where Using (A.6) (see Appendices), (62) can be rewritten as (64) Both terms on the right hand side of (64) are even functions of  and decrease with increase of ||, || ≤ .Therefore, max    () attains its lowest value for || = .It is easy to show that right hand side of (64) for || =  is equal to 1, which further implies that max From (65), it follows that (57) can be rewritten as   ≤ [max    ()] −1 .Finally, substitution of (64) into   ≤ [max    ()] −1 leads to (40), which completes the proof.

Nonnegative Waveforms with Two Zeros.
Nonnegative waveforms of type (35) with two zeros always possess two global minima.Such nonnegative waveforms are therefore related to the conflict set.
In this subsection we provide general description of nonnegative waveforms of type (35) for  ≥ 2 and exactly two zeros.According to Remark 7,   = 1 implies || =  and   () = 1 − cos ( −  0 ).Number of zeros of   () = 1 − cos (− 0 ) on fundamental period equals , which is greater than two for  > 2 and equal to two for  = 2.In the following proposition we exclude all waveforms with   = 1 (the case when  = 2 and  2 = 1 is going to be discussed in Remark 10).

Proposition 9. Every nonnegative waveform of type (35) with exactly two zeros can be expressed in the following form:
where Remark 10.For  = 2 waveforms with  2 = 1 also have exactly two zeros.These waveforms can be included in above proposition by substituting (69) with 0 < || ≤ .
Remark 11.Apart from nonnegative waveforms of type (35) with two zeros, there are another two types of nonnegative waveforms which can be obtained from (66)-(68).These are (i) nonnegative waveforms with  zeros (corresponding to || = ) and (ii) maximally flat nonnegative waveforms (corresponding to  = 0).
Remark 16.Nonnegative waveform of type (35) with two zeros can be also expressed in the following form: where   is given by (68) and 0 < || < .From (83) it follows that coefficients of fundamental harmonic of nonnegative waveform of type (35) with two zeros are where  1 is amplitude of fundamental harmonic: Coefficients of th harmonic are given by ( 45)-(46).Notice that (68) can be rewritten as By introducing new variable, and using the Chebyshev polynomials (e.g., see Appendices), relations (85) and ( 86) can be rewritten as where   () and   () denote the Chebyshev polynomials of the first and second kind, respectively.From (89) it follows that Insertion of (92) into (88) leads to the following relations between amplitude  1 of fundamental and amplitude   of th harmonic,  ≤ 4: and max    () ̸ =   ( 0 ).According to (64), max    () ̸ =   ( 0 ) implies || ̸ = 0. Therefore, it is sufficient to consider only the interval (69).

Nonnegative Waveforms with Two Zeros and Prescribed
Coefficients of th Harmonic.In this subsection we show that, for prescribed coefficients   and   , there are  nonnegative waveforms of type (35) with exactly two zeros.According to (37) and (82), coefficients   and   of nonnegative waveforms of type (35) with exactly two zeros satisfy the following relation: According to Remark 16, the value of  (see ( 87)) that corresponds to   = √ 2  +  2  can be determined from ( 90)-(91).As we mentioned earlier, (90) has only one solution that satisfies (91).This value of , according to (88), leads to the amplitude  1 of fundamental harmonic (closed form expressions for  1 in terms of   and  ≤ 4 are given by ( 93)-( 95)).

Mathematical Problems in Engineering
For  = 2 and prescribed coefficients  2 and  2 , there are two waveforms with two zeros, one corresponding to  1 < 0 and the other corresponding to  1 > 0 (see also [12]).

Nonnegative Waveforms with Maximal Amplitude of Fundamental Harmonic
In this section we provide general description of nonnegative waveforms containing fundamental and th harmonic with maximal amplitude of fundamental harmonic for prescribed amplitude of th harmonic.
The main result of this section is presented in the following proposition.Angle / q = 0 q = 1 q = 2 q = 3 Proposition 18.Every nonnegative waveform of type ( 35) with maximal amplitude  1 of fundamental harmonic and prescribed amplitude   of th harmonic can be expressed in the following form: On the other hand, (109) coincides with (66).Therefore, the expressions for coefficients of (109) and (66) also coincide.Thus, expressions for coefficients of fundamental harmonic of waveform (109) are given by (84), where  1 is given by (85), while expressions for coefficients of th harmonic are given by ( 45)-(46).
Option 1.According to (115),   = 0 implies  1 = 1 (notice that this implication shows that  1 does not depend on  and therefore we can set  to zero value).

Nonnegative Waveforms with Maximal Absolute Value of the Coefficient of Cosine Term of Fundamental Harmonic
In this section we consider general description of nonnegative waveforms of type (35) with maximal absolute value of coefficient  1 for prescribed coefficients of th harmonic.This type of waveform is of particular interest in PA efficiency analysis.In a number of cases of practical interest either current or voltage waveform is prescribed.In such cases, the problem of finding maximal efficiency of PA can be reduced to the problem of finding nonnegative waveform with maximal coefficient  1 for prescribed coefficients of th harmonic (see also Section 7).In Section 5.1 we provide general description of nonnegative waveforms of type (35) with maximal absolute value of coefficient  1 for prescribed coefficients of th harmonic.In Section 5.2 we illustrate results of Section 5.1 for particular case  = 3.

Nonnegative Waveforms with Maximal Absolute Value of
Coefficient  1 for  ≥ 2. Waveforms   () of type (35) with  1 ≥ 0 can be derived from those with  1 ≤ 0 by shifting by , and therefore we can assume without loss of generality that  1 ≤ 0. Notice that if  is even, then shifting   () by  produces the same result as replacement of  1 with − 1 (  remains the same).On the other hand, if  is odd, then shifting   () by  produces the same result as replacement of  1 with − 1 and   with −  .
According to (37), coefficients of th harmonic can be expressed as where Conversely, for prescribed coefficients   and   ,  can be determined as where definition of function atan 2(, ) is given by (105).
The main result of this section is stated in the following proposition.
Introducing new variable, and using the Chebyshev polynomials (e.g., see Appendices), relations   =   cos  and (125) can be rewritten as where   () and   () denote the Chebyshev polynomials of the first and second kind, respectively.Substitution of (128) into (127) leads to which is polynomial equation of th degree in terms of variable .From || ≤  and (126) it follows that cos (   ) ≤  ≤ 1.

Proof of Proposition 22.
As it was mentioned earlier in this section, we can assume without loss of generality that  1 ≤ 0. We consider waveforms   () of type (35) such that   () ≥ 0 and   () = 0 for some  0 .From assumption that nonnegative waveform   () of type (35) has at least one zero, it follows that it can be expressed in form (38).

Nonnegative Cosine Waveforms with at Least One Zero
Nonnegative cosine waveforms have proved to be of importance for waveform modelling in PA design (e.g., see [10]).
In this section we consider nonnegative cosine waveforms  containing fundamental and th harmonic with at least one zero.
Cosine waveform with dc component, fundamental and th harmonic, can be obtained from (35) by setting  1 =   = 0; that is, In Section 6.1 we provide general description of nonnegative cosine waveforms of type (153) with at least one zero.We show that nonnegative cosine waveforms with at least one zero coincide with nonnegative cosine waveforms with maximal absolute value of coefficient  1 for prescribed coefficient   .In Section 6.2 we illustrate results of Section 6.1 for particular case  = 3.

Nonnegative Cosine Waveforms with at Least One Zero
for  ≥ 2. Amplitudes of fundamental and th harmonic of cosine waveform of type (153) are  1 = | 1 | and   = |  |, respectively.According to (42), for nonnegative cosine waveforms of type (153) the following relation holds: This explains why th harmonic coefficient   in Proposition 26 goes through interval [−1, 1].Waveforms (153) with  1 ≥ 0 can be obtained from waveforms with  1 ≤ 0 by shifting by , and therefore, without loss of generality, we can assume that  1 ≤ 0. Proposition 26.Each nonnegative cosine waveform of type (153) with  1 ≤ 0 and at least one zero can be represented as where implies that (156) can be rewritten as Furthermore, substitution of (157) into (160) leads to Remark 28.All nonnegative cosine waveforms of type (153) with at least one zero and  1 ≤ 0, except one of them, can be represented either in form (155) or form (156).This exception is maximally flat cosine waveform with  1 < 0 which can be obtained from (155) for   = 1/( 2 − 1) or from (156) for  0 = 0. Maximally flat cosine waveform with  1 < 0 can also be obtained from (70) by setting  0 = 0. Furthermore, setting  0 = 0 in (71) leads to maximally flat cosine waveforms for  ≤ 4 and  1 < 0.
Remark 31.Nonnegative cosine waveforms of type ( 153) with at least one zero coincide with nonnegative cosine waveforms with maximal absolute value of coefficient  1 for prescribed coefficient   .

Nonnegative Cosine Waveforms with at Least
One Zero for  = 3.In this subsection we consider nonnegative cosine waveforms with at least one zero for  = 3 (for case  = 2 see [12]).

Four Case Studies of Usage of Nonnegative Waveforms in PA Efficiency Analysis
In this section we provide four case studies of usage of description of nonnegative waveforms with fundamental and th harmonic in PA efficiency analysis.In first two case studies, to be presented in Section 7.1, voltage is nonnegative waveform with fundamental and second harmonic with at least one zero.In remaining two case studies, to be considered in Section 7.2, voltage waveform contains fundamental and third harmonic.Let us consider generic PA circuit diagram, as shown in Figure 15.We assume here that voltage and current waveforms at the transistor output are where  stands for .Both waveforms are normalized in the sense that dc components of voltage and current are  dc = 1 and  dc = 1, respectively.Under assumption that blocking capacitor   behaves as short-circuit at the fundamental and higher harmonics, current and voltage waveforms at the load are In terms of coefficients of voltage and current waveforms, the load impedance at fundamental harmonic is  1 = −( 1V −  1V )/ 1 , whereas load impedance at th harmonic is   = −( V − V )/  .All other harmonics are short-circuited (  = 0 for  ̸ = 1 and  ̸ = ).Time average output power of PA (e.g., see [10]) with waveform pair (175) at fundamental frequency can be expressed as For normalized waveforms (175) with  dc = 1 and  dc = 1, dc power is  dc = 1.Consequently, PA efficiency  =  1 / dc (e.g., see, [10,26]) is equal to Thus, time average output power  1 of PA with pair of normalized waveform (175) is equal to efficiency (178).Power utilization factor (PUF) is defined [26] as "the ratio of power delivered in a given situation to the power delivered by the same device with the same supply voltage in Class A mode. " Since the output power in class-A mode is  1,class-A = max[V()] ⋅ max[()]/8 (e.g., see [9]), it follows that power utilization factor PUF =  1 / 1,class-A for PA with pair of normalized waveforms (175) can be expressed as 7.1.Nonnegative Waveforms for  = 2 in PA Efficiency Analysis.
In this subsection we provide two case studies of usage of description of nonnegative waveforms with fundamental and second harmonic ( = 2) in PA efficiency analysis.For more examples of usage of descriptions of nonnegative waveforms with fundamental and second harmonic in PA efficiency analysis see [12].
Case Study 7.1.In this case study we consider efficiency of PA for given second harmonic impedance, providing that voltage is nonnegative waveform with fundamental and second harmonic and current is "half-sine" waveform frequently used in efficiency analysis of classical PA operation (e.g., see [10]).Standard model of current waveform for classical PA operation has the form (e.g., see [10,26]) where  is conduction angle and   > 0. Since   () is even function, it immediately follows that its Fourier series contains only dc component and cosine terms: The dc component of the waveform (180) is where sinc  = (sin )/.The coefficient of the fundamental harmonic component reads and the coefficient of th harmonic component can be written in the form For "half-sine" current waveform, conduction angle is equal to  (class-B conduction angle).According to (182), this further implies that  dc =   /.To obtain normalized form of waveform (180), we set  dc = 1 which implies that   = .Furthermore, substitution of  =  and   =  in (180) leads to Similarly, substitution of   =  and  =  into (183) and (184) leads to the coefficients of waveform (185).Coefficients of fundamental and second harmonic, respectively, are On the other hand, voltage waveform of type (35) for  = 2 reads This waveform contains only fundamental and second harmonic, and therefore all harmonics of order higher than two are short-circuited (  = 0 for  > 2).For current voltage pair (185) and (187), load impedance at fundamental harmonic is  1 = −( 1V −  1V )/ 1 , whereas load impedance at second harmonic is  2 = −( 2V − 2V )/ 2 .According to our assumption, the load is passive and therefore Re{ 1 } > 0 and Re{ 2 } ≥ 0, which further imply  1  1V < 0 and  2  2V ≤ 0, respectively.
It is easy to see that problem of finding maximal efficiency of PA with current-voltage pair (185) and (187) for prescribed second harmonic impedance can be reduced to the problem of finding voltage waveform of type (187) with maximal coefficient | 1V |, for prescribed coefficients of second harmonic (see Section 5).
The following algorithm (analogous to Algorithm 22 presented in [12]) provides the procedure for calculation of maximal efficiency with current-voltage pair (185) and (187) for prescribed second harmonic impedance.The definition of function atan 2(, ), which appears in the step (iii) of the following algorithm, is given by (105).
In this case study, coefficients of fundamental and second harmonic of current waveform are given by (186).Maximal efficiency of PA associated with the waveform pair (185) and (187), as a function of normalized second harmonic impedance  2 =  2 / Re{ 1 }, is presented in Figure 16(a).As can be seen from Figure 16(a), efficiency of 0.78 is achieved at the edge of Smith chart, where second harmonic impedance has small resistive part.Corresponding PUF calculated according to (179) is presented in Figure 16(b).Peak efficiency  = /4 = 0.7854 and peak value of PUF = 1 are attained when second harmonic is short-circuited (which corresponds to ideal class-B operation [10,26]).

Case Study 7.2.
As another case study, let us consider the efficiency of PA, providing that current waveform is nonnegative cosine waveform up to third harmonic with maximum value of amplitude of fundamental harmonic [22] (see also [8]): and voltage waveform is nonnegative waveform of type (187).Load impedances at fundamental, second, and third harmonic are  1 = −( 1V − 1V )/ 1 ,  2 = −( 2V − 2V )/ 2 , and  3 = 0, respectively.According to our assumption, the load is passive and therefore Re{ 1 } > 0 and Re{ 2 } ≥ 0, which further imply  1  1V < 0 and  2  2V ≤ 0, respectively.Because current waveform (188) contains only cosine terms and voltage waveform is the same as in previous case Mathematical Problems in Engineering study, the procedure for calculation of maximal efficiency of PA with waveform pair (187)-( 188) is the same as presented in Algorithm 32.In this case study the coefficients of fundamental and second harmonic of current waveform are  1 = (1 + √ 5)/2 and  2 = 2 √ 5/5, respectively.Maximal efficiency of PA associated with the waveform pair (187)-(188), as a function of normalized second harmonic impedance  2 =  2 / Re{ 1 }, is presented in Figure 19(a).Efficiency of 0.8 is achieved at the edge of Smith chart, where second harmonic impedance has small resistive part.The theoretical upper bound  = (1 + √ 5)/4 ≈ 0.8090 is attained when second harmonic is short-circuited.When this upper bound is reached, both second and third harmonic are short-circuited which implies that we are dealing with finite harmonic class-C [6,8], or dually, when current and voltage interchange their roles, with finite harmonic inverse class-C [6,9].Corresponding PUF, calculated according to (179), is presented in Figure 19(b).Peak value of PUF ≈ 0.8541 is attained when second harmonic is short-circuited.

Nonnegative Waveforms for 𝑘 = 3 in PA Efficiency
Analysis.In this subsection we provide another two case studies of usage of description of nonnegative waveforms in PA efficiency analysis, this time with fundamental and third harmonic ( = 3).Case Study 7.3.Let us consider current-voltage pair such that voltage is nonnegative waveform with fundamental and third harmonic: and current is nonnegative cosine waveform given by (188).Load impedances at fundamental, second, and third harmonic are  1 = −( 1V −  1V )/ 1 ,  2 = 0, and  3 = −( 3V −  3V )/ 3 , respectively.According to our assumption, the load is passive and therefore Re{ 1 } > 0 and Re{ 3 } ≥ 0, which further imply  1  1V < 0 and  3  3V ≤ 0.
In this subsection we consider the problem of finding maximal efficiency of PA with waveform pair (188)-(189) for given third harmonic impedance.As we mentioned earlier, problem of finding maximal efficiency of PA with current-voltage pair (188)-(189) for prescribed third harmonic impedance can be reduced to the problem of finding voltage waveform of type (189) with maximal coefficient | 1V |, for prescribed coefficients of third harmonic (see Section 5.2).
The following algorithm provides the procedure for calculation of maximal efficiency with current-voltage pair (188)-(189).The definition of function atan 2(, ), which appears in step (iii) of the following algorithm, is given by (105).In this case study coefficients of fundamental and third harmonic of current waveform are  1 = (1 + √ 5)/2 and  3 = (5 − √ 5)/10, respectively.For the waveform pair (188)-(189), maximal efficiency of PA as a function of normalized third harmonic impedance  3 =  3 / Re{ 1 } is presented in Figure 22.Efficiency of 0.8 is reached when third harmonic impedance has small resistive part.Peak efficiency  = (1 + √ 5)/4 ≈ 0.8090 is achieved when third harmonic is shortcircuited.For the present case study, in what follows we show that power utilization factor is proportional to efficiency.For voltage waveform of type (189) it is easy to see that V( + ) = 2 − V() holds.This relation along with the fact that waveform V() that provides maximal efficiency has at least one zero implies that max[V()] = 2. On the other hand, current waveform (188) is cosine waveform with positive coefficients and therefore max[()] = (0) = 2 + 4/ √ 5. Consequently, according to (179), the following relation holds: PUF Case study 7.3 = 2 (5 − 2 √ 5)  = 1.0557. (190) Clearly, the ratio PUF/ is constant and therefore in this case study PUF can be easily calculated from the corresponding efficiency.Accordingly, peak efficiency and peak value of PUF Case study 7.3 = 3 √ 5/2 − 5/2 = 0.8541 are attained for the same voltage waveform (when third harmonic is shortcircuited).
In the course of finding power utilization factor, notice that current waveform of type (191) attains its maximum value for  = 0. Insertion of max[()] = (0) = 2.78 and max[V()] = 2 for voltage waveform of type (189) into (179) leads to PUF Case study 7.4 = 1.439. (193) Again, the ratio PUF/ is constant and PUF can be easily calculated from the corresponding efficiency.Accordingly, peak value of PUF Case study 7.4 ≈ 1.2118 and peak efficiency are attained for the same voltage waveform.

Conclusion
In this paper we consider a problem of finding general descriptions of various classes of nonnegative waveforms with fundamental and th harmonic.These classes include nonnegative waveforms with at least one zero, nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of th harmonic, nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of th harmonic, and nonnegative cosine waveforms with at least one zero.Main results are stated in six propositions (Propositions 1, 6, 9, 18, 22, and 26), four corollaries (Corollaries 2-5), twenty remarks, and three algorithms.Four case studies of usage of closed form descriptions of nonnegative waveforms in PA efficiency analysis are considered in detail in Section 7.

Figure 1 :
Figure 1: Graph of (, ) for  = 2. Points  1 and  2 denote beginning of the ridge and maximum of gain function, respectively.

Figure 3 :
Figure 3: Catastrophe set (solid line) and corresponding conflict set (dotted line) for  ≤ 5.In each plot, white triangle dot corresponds to optimal waveform and white circle dot corresponds to maximally flat waveform.

Figure 5 :
Figure 5: Amplitude of th harmonic of nonnegative waveform with two zeros as a function of parameter .

Figure 8 :
Figure 8: Maximal amplitude of fundamental harmonic as a function of amplitude of th harmonic.

Figure 9 :
Figure 9: Maximal amplitude of fundamental harmonic as a function of parameter .
Points on the respective curve correspond to the waveforms which can be expressed in both forms (121) and (122).