Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 585962, 29 pages
http://dx.doi.org/10.1155/2015/585962
Research Article

Conflict Set and Waveform Modelling for Power Amplifier Design

Department of Power, Electronics and Communication Engineering, Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia

Received 19 July 2014; Accepted 7 February 2015

Academic Editor: Ben T. Nohara

Copyright © 2015 Anamarija Juhas and Ladislav A. Novak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Various classes of nonnegative waveforms containing dc component, fundamental and harmonic , 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 harmonic, nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of 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.

1. 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].

Although without explicit reference, many max-min/min-max engineering design problems related to nonsmooth optimizations in parameter spaces (e.g., see [2]), including problems related to the optimization of efficiency of power amplifiers (PAs) (e.g., see [312]), are connected to the concept of conflict set. The concept of conflict set has been also used in mathematics (e.g., see [1316]) and physics (e.g., see [17, 18]), including subjects like black holes [19].

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 zeromay 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 [312, 20, 21]).

Fejér in his seminal paper [22] provided general description of all nonnegative trigonometric polynomials with consecutive harmonics in terms of 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 harmonic, for arbitrary 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 so-called 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, , 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 25), 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 36 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 harmonic, is presented in Section 3.3. Description of nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of harmonic is provided in Section 4. Nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of harmonic are considered in Section 5.1. An illustration of results of Section 5.1 for particular case is given in Section 5.2. Section 6.1 is devoted to nonnegative cosine waveforms with at least one zero and arbitrary , whereas Section 6.2 considers cosine waveforms with at least one zero for . In Section 7 four case studies of application of descriptions of nonnegative waveforms with fundamental and 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.

2. 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 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 . 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 , the minimum function near the origin can be locally reduced to one of the following three normal forms [25]: , or . In this example, the conflict set is the set of all points for which minimum function is not differentiable because function possesses at least two global minima [25].

2.1. Minimum Function and Gain Function

In what follows we consider family of waveforms of typewhere stands for , ,  ,  , and . 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 satisfiesClearly, 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.”

Letbe the minimum function associated with . According to (3), and satisfy the following relation: . Since is obviously nonzero it follows immediately thatA relation analogue to (5), for (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 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 and corresponding pair that satisfiesThus the optimal waveform is determined by parameters , , 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 . Notice that it has sharp ridge and that maximum of gain function (point ) lies on the ridge. This maximum corresponds to the optimal waveform (solution of the considered extremal problem). The beginning of the ridge (point ) 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.

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

Positions of global minima of for are presented in Figure 2. According to Proposition 1, conflict set is the ray defined by and . Waveforms with parameters that belong to the conflict set have two global minima. The waveform corresponding to the end point of the ray ( and ) has global minimum at degenerate critical point (so-called maximally flat waveform [21]).

Figure 2: Positions of global minima of for .

Nonnegative waveforms of type (1) with have at least one zero. To show that, it is sufficient to see that for satisfying .

The problem of finding maximum value of fundamental harmonic cosine part of nonnegative waveform of the formwhere , is also related to the problem of finding maximum of the minimum function. Optimal waveform of family (9) has two global minima (this claim will be justified in Section 5, Remark 25), and therefore corresponding 3-tuple of parameters belongs to the conflict set in parameter space of family (9).

Let us introduce an auxiliary waveformand corresponding minimum function . Inequality can be rewritten as and therefore the highest value of is attained for . It immediately follows that nonnegative waveform of type (9) with has zero for satisfying .

2.2. 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 . In this context, for prescribed integer , conflict set is said to be a set of all pairs for which possesses multiple global minima.

Suppose that and are the positions of global minima of . Then, the conflict set is specified by the following set of relations:Relations (12) and (13) say that has minima at and , while relations (11) and (14) imply that these minima are equal and global.

The following proposition describes the conflict set of family of waveforms of type (2).

Proposition 1. Conflict set of family of waveforms of type (2) is the set of all pairs such that and .

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

Corollary 2. The conflict set has end point at . This end point corresponds to the maximally flat waveform [21].

Corollary 3. Waveforms of type (2) with parameters that belong to conflict set have two global minima at , where .

Corollary 4. Every waveform with fundamental and harmonic has either one or two global minima.

Corollary 5. Conflict set can be parameterised in terms of as follows:Notice that is monotonically increasing function on interval .

Proof of Proposition 1. Without loss of generality, we can restrict our consideration to the interval . This is an immediate consequence of the fact that is a periodic function.
Suppose that and , where , are points at which has two equal global minima. Then conflict set is specified by relations (11)–(14). From (11)–(13) it follows that relationsalso hold. Let be a pair of points associated with . ClearlyThe first and second derivatives of are equal toBy using (20)-(21), system (16) can be rewritten asFrom (19) it follows that . Multiplying (24) and (25) with and , respectively, and summing the resulting relations, we obtain . The latest relation immediately implies thatEquations (22) and (23) can be considered as a system of two linear equations in terms of and . According to (26), the determinant of this system is nonzero and therefore it has only trivial solution: According to (18), implies According to (20), impliesFurthermore, and imply that . From (29) it follows that is position of global minimum of . Clearly , which together with leads toFrom (see (19)), and (30) it follows that , which together with yields
Since is position of global minimum, it follows that . Accordingly , which together with implies that . This relation along with (19) yields
Substitution of (31) and (28) in (24) leads toNotice that is monotonically decreasing function on interval (32). Therefore parameter is monotonically increasing function on the same interval with . Consequently , which completes the proof.

2.3. 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 and . It is intimately connected to catastrophe set.

In what follows in this subsection we use polar coordinate system instead of Cartesian coordinate system . Examples of catastrophe set and conflict set for plotted in parameter space 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 and and one in the catastrophe curves for and . Notice that the end point of conflict set is the cusp point.

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

Catastrophe set divides the parameter space into disjoint subsets. In the cases and catastrophe curve defines inner and outer part. For 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).

3. Nonnegative Waveforms with at Least One Zero

In what follows let us consider a waveform containing dc component, fundamental and th () 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.

3.1. General Description of Nonnegative Waveforms with at Least One Zero

The main result of this section is presented in the following proposition.

Proposition 6. Every nonnegative waveform of type (35) with at least one zero can be expressed in the following form:whereproviding that

Remark 7. Function on the right hand side of (40) is monotonically increasing function of on interval (for more details about this function see Remark 15). From (57) and (65) it follows that relation holds for every nonnegative waveform of type (35). Notice that, according to (40), implies . Substitution of and into (55) yields . Consequently, implies that amplitude of fundamental harmonic is equal to zero.

Remark 8. Conversion of (38) into additive form leads to the following expressions for coefficients of nonnegative waveforms of type (35) with at least one zero:providing that satisfy (40) and .

Three examples of nonnegative waveforms with at least one zero for are presented in Figure 4 (examples of nonnegative waveforms with at least one zero for can be found in [12]). For all three waveforms presented in Figure 4, we assume that and . From (40) it follows that . Coefficients of waveform with (dotted line) are , , , and . Coefficients of waveform with (dashed line) are , , , and . Coefficients of waveform with (solid line) are , , , and . First two waveforms have one zero, while third waveform (presented with solid line) has two zeros.

Figure 4: Nonnegative waveforms with at least one zero for , , and .

Proof of Proposition 6. Waveform of type (35), containing dc component, fundamental and th harmonic, can be also expressed in the formwhere , , , and . It is easy to see that relations between coefficient of (35) and parameters of (47) read as follows:
Let us introduce such thatUsing (50), coefficients (49) can be expressed as (45)-(46).
Let us assume that is nonnegative waveform of type (35) with at least one zero; that is, and for some . Notice that conditions and imply that . From and , by using (50), it follows thatrespectively. On the other hand, can be rewritten asSubstitution of (51) into (52) yieldsAccording to (50), it follows that ; that is,Furthermore, substitution of (54) and (53) into (47) leads toAccording to (A.2) and (A.4) (see Appendices), there is common factor for all terms in (55). Consequently, (55) can be written in the form (38), whereFrom (56), by using (A.2), (A.4), and , we obtain (39).
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 . Relation (56) can be rewritten aswhere For , relation obviously holds. From for it follows that position of global maximum of the function of type for belongs to interval . Therefore position of global maximum of the expression in the square brackets in (60) for belongs to interval . This inequality together with leads to . Since decreases with increasing , it follows that for has global maximum on interval . For , it is easy to show that . Since is constant (see (58)), it follows from previous considerations that has global maximum on interval .
To find , let us consider first derivative of with respect to . Starting from (56), first derivative of can be expressed in the following form: where Using (A.6) (see Appendices), (62) can be rewritten asFrom and , , it follows that all summands in (63) decrease with increasing providing that . Therefore for . Consequently, and imply that .
From ,  , and it follows that or , and therefore . Since , it follows that is attained for . Furthermore, from (60) it follows that , which together with (58)-(59) leads to Both terms on the right hand side of (64) are even functions of and decrease with increase of , . Therefore, 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 From (65), it follows that (57) can be rewritten as . Finally, substitution of (64) into leads to (40), which completes the proof.

3.2. 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 and exactly two zeros. According to Remark 7, implies and . Number of zeros of on fundamental period equals , which is greater than two for and equal to two for . In the following proposition we exclude all waveforms with (the case when and 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 waveforms with also have exactly two zeros. These waveforms can be included in above proposition by substituting (69) with .

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 ).
Notice that nonnegative waveforms of type (35) with can be obtained from (66)–(68) by setting . Substitution of and into (66), along with execution of all multiplications and usage of (A.2) (see Appendices), leads to .
Also, maximally flat nonnegative waveforms (they have only one zero [21]) can be obtained from (66)–(68) by setting . Thus, substitution of into (66)–(68) leads to the following form of maximally flat nonnegative waveform of type (35):Maximally flat nonnegative waveforms of type (35) for can be expressed as

Remark 12. Every nonnegative waveform of type (35) with exactly one zero at nondegenerate critical point can be described as in Proposition 6 providing that symbol “” in relation (40) is replaced with “”. This is an immediate consequence of Propositions 6 and 9 and Remark 11.

Remark 13. Identity implies that (66) can be also rewritten asFurthermore, substitution of (67) into (72) leads to

Remark 14. According to (A.6) (see Appendices), it follows that coefficients (67) can be expressed asFurthermore, from (74) it follows that coefficients , , and are equal to
For example, for , (75) and (68) lead to and , respectively, which from (72) further imply thatAlso for , (75), (76), and (68) lead to , , and , respectively, which from (72) further imply that

Remark 15. According to (A.5) (see Appendices), relation (68) can be rewritten asClearly, amplitude of th harmonic of nonnegative waveform of type (35) with exactly two zeros is even function of . Since ,  , decreases with increase of on interval , it follows that monotonically increases with increase of . Right hand side of (68) is equal to for and to one for . Therefore, for nonnegative waveforms of type (35) with exactly two zeros, the following relation holds:The left boundary in (82) corresponds to maximally flat nonnegative waveforms (see Remark 11). The right boundary in (82) corresponds to nonnegative waveforms with zeros (also see Remark 11).
Amplitude of th harmonic of nonnegative waveform of type (35) with two zeros, as a function of parameter for , is presented in Figure 5.

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

Remark 16. Nonnegative waveform of type (35) with two zeros can be also expressed in the following form:where is given by (68) and . From (83) it follows that coefficients of fundamental harmonic of nonnegative waveform of type (35) with two zeros arewhere is amplitude of fundamental harmonic: Coefficients of th harmonic are given by (45)-(46).
Notice that (68) can be rewritten asBy introducing new variable, and using the Chebyshev polynomials (e.g., see Appendices), relations (85) and (86) can be rewritten aswhere and denote the Chebyshev polynomials of the first and second kind, respectively. From (89) it follows that which is polynomial equation of th degree in terms of variable . From and (87) it follows that Since is monotonically increasing function of , , it follows that is monotonically decreasing function of . This further implies that (90) has only one solution that satisfies (91). (For expression (91) reads .) This solution for (which can be obtained at least numerically), according to (88), leads to amplitude of fundamental harmonic.
For , solutions of (90) and (91) areInsertion of (92) into (88) leads to the following relations between amplitude of fundamental and amplitude of th harmonic, :

Proof of Proposition 9. As it has been shown earlier (see Proposition 6), nonnegative waveform of type (35) with at least one zero can be represented in form (38). Since we exclude nonnegative waveforms with , according to Remark 7, it follows that we exclude case . Therefore in the quest for nonnegative waveforms of type (35) having two zeros we will start with waveforms of type (38) for . It is clear that nonnegative waveforms of type (38) have two zeros if and only ifand . According to (64), implies . Therefore, it is sufficient to consider only the interval (69).
Substituting (96) into (38) we obtain Expression , according to (64) and (39), equals Comparison of (97) with (66) yields where coefficients , , are given by (67). In what follows we are going to show that right hand sides of (98) and (99) are equal.
From (67) it follows thatAlso, from (67) for it follows that the following relations hold:From (99), by using (75), (76), (100)-(101), and trigonometric identitieswe obtain (98). Consequently (98) and (99) are equal, which completes the proof.

3.3. 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 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 of fundamental harmonic (closed form expressions for in terms of and are given by (93)–(95)).

On the other hand, from (45)-(46) it follows that where function is defined as with the codomain . Furthermore, according to (84) and (104), the coefficients of fundamental harmonic of nonnegative waveforms with two zeros and prescribed coefficients of th harmonic are equal towhere . For chosen , according to (104) and (66), positions of zeros areFrom (106) and it follows that, for prescribed coefficients and , there are nonnegative waveforms of type (35) with exactly two zeros.

We provide here an algorithm to facilitate calculation of coefficients and of nonnegative waveforms of type (35) with two zeros and prescribed coefficients and , providing that and satisfy (103).

Algorithm 17. (i) Calculate ,
(ii) identify that satisfies both relations (90) and (91),
(iii) calculate according to (88),
(iv) choose integer , such that ,
(v) calculate and according to (106).
For , by using (93) for , (94) for , and (95) for it is possible to calculate directly from and proceed to step (iv).
For and prescribed coefficients and , there are two waveforms with two zeros, one corresponding to and the other corresponding to (see also [12]).

Let us take as an input , , and . Execution of Algorithm 17 on this input yields and (according to (94)). For we calculate and (corresponding waveform is presented by solid line in Figure 6); for we calculate and (corresponding waveform is presented by dashed line); for we calculate and (corresponding waveform is presented by dotted line).

Figure 6: Nonnegative waveforms with two zeros for , , and .

As another example of the usage of Algorithm 17, let us consider case and assume that and . Consequently and (according to (95)). For we calculate the following four pairs of coefficients of fundamental harmonic: for ,   for ,   for , and for . Corresponding waveforms are presented in Figure 7.

Figure 7: Nonnegative waveforms with two zeros for , , and .

4. 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.

Proposition 18. Every nonnegative waveform of type (35) with maximal amplitude of fundamental harmonic and prescribed amplitude of th harmonic can be expressed in the following form:if or if , providing that , , and are related to via relations (67) and (68), respectively, and .

Remark 19. Expression (108) can be obtained from (38) by setting . Furthermore, insertion of into (43)–(46) leads to the following expressions for coefficients of waveform of type (108):
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 is given by (85), while expressions for coefficients of th harmonic are given by (45)-(46).
Waveforms described by (108) have exactly one zero, while waveforms described by (109) for have exactly two zeros. As we mentioned earlier, waveforms (109) for have zeros.

Remark 20. Maximal amplitude of fundamental harmonic of nonnegative waveforms of type (35) for prescribed amplitude of harmonic can be expressed asif , or if , where is related to via (68) (or (86)) and .
From (110) it follows that (111) holds. Substitution of (86) into (85) leads to (112).
Notice that is the only common point of the intervals and . According to (111), corresponds to . It can be also obtained from (112) by setting . The waveforms corresponding to this pair of amplitudes are maximally flat nonnegative waveforms.
Maximal amplitude of fundamental harmonic of nonnegative waveform of type (35) for , as a function of amplitude of th harmonic, is presented in Figure 8.

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

Remark 21. Maximum value of amplitude of fundamental harmonic of nonnegative waveform of type (35) is This maximum value is attained for (see (112)). The corresponding value of amplitude of th harmonic is . Nonnegative waveforms of type (35) with have two zeros at and for , or at and for .
To prove that (113) holds, let us first show that the following relation holds for : From , it follows that , where , and therefore . By using trigonometric identity , we immediately obtain (114).
According to (111) and (112), it is clear that attains its maximum value on the interval . Since is monotonic function of on interval (see Remark 15), it follows that for . Therefore, to find critical points of as a function of it is sufficient to find critical points of as a function of , , and consider its values at the end points and . Plot of as a function of parameter for is presented in Figure 9. According to (112), first derivative of with respect to is equal to zero if and only if . On interval , this is true if and only if . According to (112), is equal to for , equal to zero for , and equal to for . From (114) it follows that and therefore maximum value of is given by (113). Moreover, maximum value of is attained for .