Maximally Flat Waveforms with Finite Number of Harmonics in Class-𝐹 Power Amplifiers

Abstract

In this paper general solution to the problem of finding maximally flat waveforms with finite number of harmonics (maximally flat trigonometric polynomials) is provided. Waveform coefficients are expressed in closed form as functions of harmonic orders. Two special cases of maximally flat waveforms (so-called maximally flat even harmonic and maximally flat odd harmonic waveforms), which proved to play an important role in class-𝐹 and inverse class-𝐹 power amplifier (PA) operations, are also considered. For these two special types of waveforms, coefficients are expressed as functions of two parameters only. Closed form expressions for efficiency and power output capability of class-𝐹 and inverse class-𝐹 PA operations with maximally flat waveforms are also provided as explicit functions of number of a harmonics.

1. Introduction

Roughly speaking, maximally flat waveform (maximally flat trigonometric polynomial) of a family of waveforms is a waveform which possesses degenerate critical point with highest degree (maximally degenerate critical point), among all members of the family. At the maximally degenerate critical point, maximum possible consecutive derivatives of waveform are equal to zero, starting from the first. The existence of maximally flat trigonometric polynomials, according to our best knowledge, has been reported for the first time in [1] (in Serbian), in the context of catastrophes in parameter space of trigonometric polynomials with two harmonics of arbitrary order.

In the context of the analysis of power amplifiers (PAs) in electrical engineering, the benefits of flattening of the bottom of voltage waveform were known as early as 1919 (e.g., see [2]), but the credit for the first comprehensive usage of maximally flat waveforms, in this context, goes to Raab [2]. In this pioneering paper a subclass of maximally flat waveforms with fundamental harmonic and two prescribed higher harmonics 𝑃,𝑄≤5, that can be obtained from cosine polynomials after shifting by 𝜋/2, has been considered. Grebennikov and Sokal [3] extended Raab’s result up to seventh harmonic. In [4], the coefficients of maximally flat trigonometric polynomials with fundamental and additional two harmonics with arbitrary order 𝑃 and 𝑄 have been provided in closed form as functions of 𝑃 and 𝑄. A closed form solution to the general case of maximally flat cosine polynomials with 𝑁 consecutive harmonics has been presented in [5].

In this paper, we consider the general case of maximally flat trigonometric polynomials. In Section 2 we provide general solution to the problem of finding closed form representations of maximally flat trigonometric polynomials using rigorous mathematical tools. Two special subclasses of maximally flat trigonometric polynomials with natural application in class-𝐹 and inverse class-𝐹 PA operations are considered in Section 3. In Section 4, based on results of Section 3, closed form expressions of efficiency and power-output capability of class-𝐹 and inverse class-𝐹 PA with maximally flat waveforms are derived.

Motivating Example. Let us consider the following family with two parameters, consisting of waveforms with first and third harmonics: 𝑓(𝜏;𝐴,𝛼)=cos𝜏+𝐴cos(3𝜏+𝛼), (1) and corresponding set of 3-tiples (𝜏_𝑑,𝐴,𝛼) for which both first and second derivatives of the waveform 𝑓(𝜏;𝐴,𝛼) are equal to zero: 𝑓⁽¹⁾𝜏_𝑑;𝐴,𝛼=0,𝑓⁽²⁾𝜏_𝑑;𝐴,𝛼=0. (2) In this case, we say that 𝜏_𝑑 is a degenerate critical point of the waveform 𝑓(𝜏;𝐴,𝛼). The pairs (𝐴,𝛼) that correspond to the waveforms with degenerate critical points form so-called catastrophe set, illustrated in Figure 1 by solid line. For the family (1), the corresponding catastrophe set divides the parameter space (𝐴cos𝛼,𝐴sin𝛼) into three disjoint subsets (inner and outer part of the solid line and solid line itself) and helps in making classification of the zoology of waveforms 𝑓(𝜏;𝐴,𝛼). Waveforms that correspond to the inner points have one minimum and one maximum, whereas the waveforms that correspond to the outer points have three minima and three maxima. Points on the solid line correspond to the waveforms with one minimum, one maximum and two inflection points. The cusp point (𝐴=1/9, 𝛼=𝜋) corresponds to the maximally flat waveform shown in the shadowed frame in Figure 1. The fact that maximally flat waveforms are related to the cusp points of the catastrophe set illustrates that they are rather exceptional.

Figure 1 

Parameter space (𝐴cos𝛼,𝐴sin𝛼) and waveforms of the family 𝑓(𝜏;𝐴,𝛼)=cos𝜏+𝐴cos(3𝜏+𝛼).

2. General Case of Maximally Flat Waveforms

Critical point of a waveform is a point at which first derivative is equal to zero. Critical point of waveform is degenerate if at least first two derivatives are equal to zero. An integer 𝑟≥2 is said to be the degree of degeneracy of critical point if first 𝑟 consecutive derivatives are equal to zero and (𝑟+1)th derivative is not.

In this Section we consider the following problem: find a waveform from the family 𝑓_𝑁(𝜏)=𝐹₀+^𝑁_𝑖=1𝐹_𝑖cos𝑛_𝑖𝜏+𝜑_𝑖,𝑁≥1, (3) which possess critical point of highest degree of degeneracy. Such a waveform is said to be maximally flat waveform of the family. We can assume without loss of generality that 𝐹_𝑖>0 and 𝑛_𝑗>𝑛_𝑖. Notice that set of positive integers 𝑛_𝑖 defines the family.

It is obvious that the existence of the maximally degenerate critical point is invariant of the value of 𝐹₀ and therefore, without loss of generality we can assume that the waveform is equal to zero at the maximally degenerate critical point. It is also clear that existence of the maximally degenerate critical point is invariant of the translation along the 𝜏-axis and consequently maximally degenerate critical point can be placed at any point along 𝜏-axis. In what follows it is convenient to assume that maximally degenerate critical point is 𝜏₀=0.

The above problem of finding waveform 𝑓_𝑁(𝜏) of the family (3) with maximally degenerate critical point, providing that the value of the waveform equals zero at this degenerate critical point, can be replaced by an equivalent problem of finding a nonzero waveform 𝑓_𝑁(𝜏) of the family (3) such that 𝑓_𝑁(0)=0,𝑓_𝑁^(𝑟)(0)=0,𝑟=1,…,𝑟_max, (4)(5) where 𝑓_𝑁^(𝑟) denotes 𝑟th derivative of the waveform 𝑓_𝑁, and 𝑟_max is maximum number of relations (5) which do not contradict (4).

Lemma 1. For nonzero waveform 𝑓_𝑁(𝜏) of the family (3) which satisfies (4) and (5), the following relations hold: 𝐹₀≠0,𝑟_max=2𝑁−1. (6)(7)

Remark 2. From (7) it is obvious that 𝑟_max is odd number which further implies that at maximally degenerate critical point 𝑓_𝑁(𝜏) has either minimum or maximum.

Proof. In terms of new variables 𝑎_𝑖=𝐹_𝑖cos𝜑_𝑖,𝑖=1,…,𝑁,𝑏_𝑖=𝐹_𝑖sin𝜑_𝑖,𝑖=1,…,𝑁, (8)(9) system (4)-(5) for 𝑟_max=2𝑁−1 decomposes in two independent subsystems: ^𝑁_𝑖=1𝑎_𝑖=−𝐹₀,_𝑁𝑖=1𝑛_𝑖^2𝑞𝑎_𝑖=0,𝑞=1,…,𝑁−1, (10)(11) (written in terms of variables 𝑎_𝑖 only) and ^𝑁_𝑖=1𝑛_𝑖^2𝑞−1𝑏_𝑖=0,𝑞=1,…,𝑁, (12) (written in terms of variables 𝑏_𝑖 only).
The matrices of subsystems (10)-(11) and (12) are 𝑀_𝑎=𝑛1⋯111⋯1₂₁⋯𝑛²_𝑖−1𝑛²_𝑖𝑛²_𝑖+1⋯𝑛²_𝑁𝑛⋮⋱⋮⋮⋮⋱⋮_12𝑁−2⋯𝑛^2𝑁−2_𝑖−1𝑛_𝑖^2𝑁−2𝑛^2𝑁−2_𝑖+1⋯𝑛_𝑁^2𝑁−2,𝑀_𝑏=𝑛₁⋯𝑛_𝑖−1𝑛_𝑖𝑛_𝑖+1⋯𝑛_𝑁𝑛³₁⋯𝑛³_𝑖−1𝑛³_𝑖𝑛³_𝑖+1⋯𝑛³_𝑁𝑛⋮⋱⋮⋮⋮⋱⋮_12𝑁−1⋯𝑛^2𝑁−1_𝑖−1𝑛_𝑖^2𝑁−1𝑛^2𝑁−1_𝑖+1⋯𝑛_𝑁^2𝑁−1, (13)(14) respectively. Matrices 𝑀_𝑎 and 𝑀_𝑏 are both Vandermonde matrices and their determinants are det𝑀_𝑎=_{1≤𝑖<𝑗≤𝑁}𝑛²_𝑗−𝑛²_𝑖,det𝑀_𝑏=det𝑀_{𝑎1≤𝑗≤𝑁}𝑛_𝑗, (15) which are clearly nonzero. Therefore, each subsystem (10)-(11)and (12) has unique solution which implies that 𝑟_max≥2𝑁−1.
It is obvious that homogeneous subsystem (12) in terms of 𝑏_𝑖 has trivial solution as follows: 𝑏_𝑖=0,𝑖=1,…,𝑁, (16) and it is unique.
Let us first show that (6) holds. Suppose in the contrary that 𝐹₀=0. Then subsystem (10)-(11) has also trivial solution only, which implies that waveform (3) is identically equal to zero.
Let us now show that (7) also holds. We already have proved that 𝑟_max≥2𝑁−1 and we are going to show that assumption 𝑟_max=2𝑁 leads to contradiction. For 𝑟_max=2𝑁, the corresponding subsystem can be obtained from (10)-(11) by including additional equation 𝑓_𝑁^(2𝑁)(0)=0, which in terms of variables (8), can be written in the form ^𝑁_𝑖=1𝑛_𝑖^2𝑁𝑎_𝑖=0. (17) Determinant of subsystem consisting of equations (11) and (17) equals det(𝑀_𝑎)∏_{1≤𝑗≤𝑁}𝑛²_𝑗 and it is clearly nonzero. Hence, this subsystem has only trivial solution, which contradicts (10) since 𝐹₀≠0. This completes the proof.

As a consequence of Lemma 1, the problem of finding nonzero waveform 𝑓_𝑁(𝜏) of the family (3) satisfying (4) and (5) can be reformulated as follows: find a nonzero waveform from the family (3) such that 𝑓_𝑁(0)𝑓=0,_𝑁(𝑟)(0)=0,𝑟=1,…,2𝑁−1. (18) In terms of variables 𝑎_𝑖 and 𝑏_𝑖 (see (8) and (9)) system (18) can be transformed into the system (10)–(12). In what follows we will provide closed form solution of this system, which leads to the maximally flat waveform.

Proposition 3. Maximally flat waveform of the family (3) with maximally degenerate critical point at 𝜏₀=0, having zero value at this point is fully described by the following set of parameters: 𝐹_𝑖=𝐹_{01≤𝑗≤𝑖−1}𝑛²_𝑗𝑛²_𝑖−𝑛²_𝑗⋅_{𝑖+1≤𝑗≤𝑁}𝑛²_𝑗𝑛²_𝑗−𝑛²_𝑖,𝜑𝑖=1,…,𝑁,_𝑖=0𝑖𝑖𝑠𝑒𝑣𝑒𝑛,𝜋𝑖𝑖𝑠𝑜𝑑𝑑,𝑖=1,…,𝑁,𝑖𝑓𝐹₀𝜑>0,_𝑖=𝜋𝑖𝑖𝑠𝑒𝑣𝑒𝑛,0𝑖𝑖𝑠𝑜𝑑𝑑,𝑖=1,…,𝑁,𝑖𝑓𝐹₀<0. (19)

Proof. Let us find the solution of the system (10)–(12). Using Cramer’s rule, from subsystem (10)-(11) it follows that 𝑎_𝑖=−𝐹₀(−1)^1+𝑖𝑀_1𝑖det𝑀_𝑎=(−1)^𝑖𝐹_{01≤𝑗≤𝑖−1}𝑛²_𝑗𝑛²_𝑖−𝑛²_𝑗⋅_{𝑖+1≤𝑗≤𝑁}𝑛²_𝑗𝑛²_𝑗−𝑛²_𝑖,𝑖=1,…,𝑁, (20) where 𝑀_1𝑖 is (1𝑖)th minor of the matrix (13). Notice that the first product is empty for 𝑖=1 and second for 𝑖=𝑁 (by definition an empty product is equal to 1). From (8), (9), (16), (20) and 𝐹_𝑖>0 immediately follows (19), which completes the proof.

Example 4. The following two waveforms presented in Figure 2: 𝑓_(1,3,4)(𝜏)6=1−52cos𝜏+73cos3𝜏−𝑓35cos4𝜏,_(2,4,5)(𝜏)=1−10063cos2𝜏+2527cos4𝜏−64189cos5𝜏, (21) are maximally flat waveforms of the family with 1st, 3rd and 4th harmonic and family with 2nd, 4th and 5th harmonic, respectively, with maximally degenerate critical point at 𝜏₀=0.

Figure 2 

Examples of two maximally flat waveforms.

3. Two Special Cases: Maximally Flat Even and Odd Harmonic Waveforms

The following are two special types of waveforms: 𝑤_even(𝜏)=1−𝐴₁cos𝜏−^𝑀_𝑚=1𝐴_2𝑚cos(2𝑚𝜏),𝑤_odd(𝜏)=1−𝐵₁cos𝜏−^𝐾_𝑘=1𝐵_(2𝑘+1)cos(2𝑘+1)𝜏, (22)(23) where 𝐴₁>0 and 𝐵₁>0, which are of particular interest in PA efficiency analysis. Waveform (22) which contains dc component, fundamental harmonic, and 𝑀 consecutive even harmonics is said to be an even harmonic waveform. Waveform (23) which contains dc component, fundamental, and 𝐾 consecutive odd harmonics is said to be an odd harmonic waveform.

In what follows we assume that 𝑤_even(𝜏) and 𝑤_odd(𝜏) refer to maximally flat even and odd harmonic waveforms, respectively, with maximally degenerate critical point at 𝜏₀=0 and zero value at this point.

In this Section, we show that maximally flat even and odd harmonic waveforms of the form (22) and (23) have global minimum at maximally degenerate critical point. Such a minimum we will call “maximally flat minimum.” Since the waveforms have zero values at this point, it immediately follows that they are nonnegative. We also derive basic parameters of these waveforms.

The coefficients of maximally flat even harmonic waveform (22), with maximally degenerate critical point at 𝜏₀=0 and zero value at this point, can be obtained straightforward from (20) for 𝑁=𝑀+1, 𝑛₁=1, 𝐹₀=1, 𝐴₁=−𝑎₁, 𝑛_𝑚+1=2𝑚, and 𝐴_2𝑚=−𝑎_𝑚+1 for 𝑚=1,…,𝑀:𝐴₁=((2𝑀)!!)²()(),2𝑀−1!!2𝑀+1!!𝐴_2𝑚=(−1)^𝑚4𝑚²2()−1𝑀!²()!()!,𝑀−𝑚𝑀+𝑚𝑚=1,…,𝑀. (24)(25)

The coefficients of maximally flat odd harmonic waveform (23), with maximally degenerate critical point at 𝜏₀=0 and zero value at this point, can be obtained straightforward from (20) for 𝑁=𝐾+1, 𝑛_𝑘+1=2𝑘+1, 𝐹₀=1, and 𝐵_(2𝑘+1)=−𝑎_𝑘+1 for 𝑘=0,…,𝐾:𝐵_(2𝑘+1)=(−1)^𝑘()⋅))2𝑘+1((2𝐾+1!!²4^𝐾()!()!,𝐾−𝑘𝐾+𝑘+1𝑘=0,…,𝐾. (26)

Let us introduce “duals” of the waveforms (22) and (23): 𝑤^∗_even(𝜏)=1+𝐴^∗₁cos𝜏+^𝑀_𝑚=1𝐴^∗_2𝑚(),𝑤cos2𝑚𝜏_∗odd(𝜏)=1+𝐵^∗₁cos𝜏+^𝐾_𝑘=1𝐵^∗_()2𝑘+1cos(2𝑘+1)𝜏, (27) which is defined as 𝑤^∗_even(𝜏)=𝑤_even(𝜏+𝜋) and 𝑤^∗_odd(𝜏)=𝑤_odd(𝜏+𝜋), respectively. It is easy to see that the following relations between the coefficients of waveforms (22)-(23) and their duals hold: 𝐴^∗₁=𝐴₁,𝐴^∗_2𝑚=−𝐴_2𝑚,𝐵^∗₁=𝐵₁,𝐵^∗_(2𝑘+1)=𝐵_(2𝑘+1). (28) Notice that 𝑤_even(𝜏) and 𝑤_odd(𝜏) have maximally degenerate critical point at 𝜏₀=0, which implies that their duals 𝑤^∗_even(𝜏) and 𝑤^∗_odd(𝜏) have maximally degenerate critical point at 𝜏₀=𝜋.

3.1. Maximally Flat Even Harmonic Waveform

In this subsection we show that maximally flat even harmonic waveform is nonnegative and has global minimum at maximally degenerate critical point (Proposition 6). We also provide closed form expression for basic waveform parameters 𝛾_even and 𝛿_even in terms of number of even harmonics. The parameter 𝛾_even is defined as the amplitude of fundamental harmonic relative to dc component of the even harmonic waveform, whereas 𝛿_even is defined as maximum value of the waveform relative to dc component (e.g., see [2]): 𝛾_even=𝐴₁,𝛿_even=max_𝜏𝑤_even(𝜏). (29)

According to (24), parameter 𝛾_even is 𝛾_even=𝐴₁=((2𝑀)!!)²()()=2𝑀−1!!2𝑀+1!!16^𝑀()2𝑀+1𝑀!²(2𝑀)!². (30) Notice that by definition 0!!=1 and (−1)!!=1. Therefore, (30) yields 𝛾_even=1 for 𝑀=0. The relation (30) shows that 𝛾_even increases by increasing 𝑀 (see Figure 3). When 𝑀 tends to infinity, parameter 𝛾_even equals to Wallis product; that is, lim_𝑀→∞𝛾_even=^∞_𝑛=14𝑛²()=𝜋2𝑛−1)(2𝑛+12. (31)

Figure 3 

Waveform parameter 𝛾even for maximally flat even harmonic waveform as function of number of even harmonics 𝑀.

We first prove the following statement:

Lemma 5. Maximally flat even harmonic waveform with maximally degenerate critical point at 𝜏₀=0 and zero value at this point can be expressed in the following form: 𝑤_{𝑒𝑣𝑒𝑛}(𝜏)=𝛾_{𝑒𝑣𝑒𝑛}1−cos𝜏−^𝑀_𝑚=1(2𝑚−3)!!()()2𝑚!!sin𝜏^2𝑚,𝑀≥0, (32) where 𝛾_{𝑒𝑣𝑒𝑛} is given by (30).

Proof. Taking into account (30) and (25), maximally flat even harmonic waveform (22) with maximally degenerate critical point at 𝜏₀=0 can be expressed as 𝑤_even(𝜏)=𝛾_even−cos𝜏+𝑆_𝑀(𝜏), (33) where 𝑆₀(𝜏)=1 and 𝑆_𝑀(𝜏)=(2𝑀+1)!!4^𝑀()×𝑀2𝑀!!2𝑀−2^𝑀_𝑚=12𝑀()𝑀−𝑚−1^𝑚cos(2𝑚𝜏)4𝑚²,−1𝑀≥1. (34)
The expression 4^𝑀(sin𝜏)^2𝑀=(−1)^𝑀(𝑒^−𝑗𝜏−𝑒^𝑗𝜏)^2𝑀, where 𝑗=√−1, in an expanded form reads 4^𝑀(sin𝜏)^2𝑀=^2𝑀_𝑛=0𝑛()2𝑀−1^𝑀−𝑛𝑒^{𝑗2(𝑀−𝑛)𝜏}. (35) Substitution of 𝑛=𝑀−𝑚 into (35) yields 4^𝑀(sin𝜏)^2𝑀=^𝑀_𝑚=−𝑀()2𝑀𝑀−𝑚−1^𝑚𝑒^{𝑗2𝑚𝜏}, (36) that is, 4^𝑀(sin𝜏)^2𝑀=𝑀2𝑀+2^𝑀_𝑚=1()2𝑀𝑀−𝑚−1^𝑚cos(2𝑚𝜏). (37) Since 𝑆_𝑀(𝜏)−𝑆_𝑀−1(𝜏)()=−2𝑀−3!!4^𝑀()×𝑀2𝑀!!2𝑀+2^𝑀_𝑚=1()2𝑀𝑀−𝑚−1^𝑚cos(2𝑚𝜏), (38) using (37), we obtain 𝑆_𝑀(𝜏)−𝑆_𝑀−1(𝜏)()=−2𝑀−3!!()()2𝑀!!sin𝜏^2𝑀,𝑆₀(𝜏)=1. (39) The solution of recursive relation (39) reads 𝑆_𝑀(𝜏)=1−^𝑀_𝑚=1(2𝑚−3)!!()()2𝑚!!sin𝜏^2𝑚,𝑀≥1. (40) Substituting (40) into (33), we finally obtain (32), which completes the proof.

Using Lemma 5, we prove the following proposition.

Proposition 6. Maximally flat even harmonic waveform with maximally degenerate critical point at 𝜏₀=0 and zero value at this point is nonnegative and has unique global minimum at 𝜏₀=0.

Proof. The binomial series 1−𝑥^21/2=1−^∞_𝑚=1(2𝑚−3)!!()𝑥2𝑚!!^2𝑚,−1≤𝑥≤1, (41) for 𝑥=sin𝜏 reads |cos𝜏|=1−sin²𝜏^1/2=1−^∞_𝑚=1(2𝑚−3)!!()()2𝑚!!sin𝜏^2𝑚. (42) By comparison of (32) with (42), we conclude that 𝑤_even(𝜏)≥𝛾_even(−cos𝜏+|cos𝜏|)≥0, (43) and therefore the maximally flat even harmonic waveform is nonnegative.
From (43) and (32), it is obvious that 𝑤_even(𝜏) is equal to zero if and only if 𝜏=0; that is, it has unique global minimum at maximally degenerate critical point 𝜏₀=0. This completes the proof.

Remark 7. Since 𝛾_even>0, it is obvious that the maximum value of (32) is achieved for sin𝜏=0 and cos𝜏=−1, that is, for 𝜏=𝜋, only. Thus, 𝛿_even=max_𝜏𝑤_even(𝜏)=𝑤_even(𝜋)=2𝛾_even. (44) Raab [2] also pointed out the relation (44).

Remark 8. When 𝑀 tends to infinity, according to (32), (42), and (31) it follows that lim_𝑀→∞𝑤_even(𝜏)=𝜋2(−cos𝜏+|cos𝜏|). (45) Clearly, (45) is a “half-sine” waveform.

Maximally flat even harmonic waveforms, including limit waveform (𝑀→∞), are presented in Figure 4.

Figure 4 

Maximally flat even harmonic waveforms.

3.2. Maximally Flat Odd Harmonic Waveform

In this subsection, we show that maximally flat odd harmonic waveform is nonnegative and has global minimum at maximally degenerate critical point (Proposition 10). We also provide closed form expression for basic waveform parameters 𝛾_odd and 𝛿_odd in terms of number of odd harmonics. The parameter 𝛾_odd is defined as the amplitude of fundamental harmonic relative to dc component of the odd harmonic waveform, whereas 𝛿_odd is defined as maximum value of the waveform relative to dc component (e.g., see [2]): 𝛾_odd=𝐵₁,𝛿_odd=max_𝜏𝑤_odd(𝜏). (46)

From (26) for 𝑘=0, we obtain 𝛾_odd=𝐵₁))=2((2𝐾+1!!²()()=2𝐾!!2𝐾+2!!𝐾+116^𝐾𝐾2𝐾+1². (47) Notice that (47) yields 𝛾_odd=1 for 𝐾=0. The relation (47) shows that 𝛾_odd increases by increasing 𝐾 (see Figure 5). When 𝐾 tends to infinity, from (47), according to (31), 𝛾_odd tends to lim_𝐾→∞𝛾_odd=lim_𝐾→∞2𝐾+1𝐾+1^𝐾_𝑛=1(2𝑛−1)(2𝑛+1)4𝑛²=4𝜋. (48)

Figure 5 

Waveform parameter 𝛾odd for maximally flat odd harmonic waveform as function of number of odd harmonics 𝐾.

Let us first prove the following Lemma.

Lemma 9. Maximally flat odd harmonic waveform with maximally degenerate critical point at 𝜏₀=0 and zero value at this point can be expressed as 𝑤_𝑜𝑑𝑑(𝜏)=1−cos𝜏1+^𝐾_𝑘=1(2𝑘−1)!!()()2𝑘!!sin𝜏^2𝑘,𝐾≥0. (49)

Proof. Taking into account (26), maximally flat odd harmonic waveform (23) with maximally degenerate critical point at 𝜏₀=0 can be expressed as 𝑤_odd(𝜏)=1−𝑆_𝐾(𝜏), (50) where 𝑆₀(𝜏)=cos𝜏 and 𝑆_𝐾(𝜏)=(2𝐾+1)!!4^𝐾(2𝐾)!!^𝐾_𝑘=02𝐾+1()𝐾−𝑘−1^𝑘()()2𝑘+1cos2𝑘+1𝜏. (51) Hence, difference 𝑆_𝐾(𝜏)−𝑆_𝐾−1(𝜏) is equal to 𝑆_𝐾(𝜏)−𝑆_𝐾−1(𝜏)=1()()2𝐾+12𝐾−1!!4^𝐾()×2𝐾!!_𝐾𝑘=0()2𝐾+1𝐾−𝑘−1^𝑘(2𝑘+1)cos(2𝑘+1)𝜏. (52)
The expression 4^𝐾(sin𝜏)^2𝐾+1=12𝑗(−1)^𝐾𝑒^{𝑗(2𝐾+1)𝜏}𝑒^−𝑗2𝜏−1^2𝐾+1, (53) in an expanded form reads 4^𝐾(sin𝜏)^2𝐾+11=−2𝑗^2𝐾+1_𝑛=0𝑛()2𝐾+1−1^𝐾−𝑛𝑒^{𝑗(2𝐾+1−2𝑛)𝜏}. (54) Substitution of 𝑘=𝐾−𝑛 into (54) yields 4^𝐾(sin𝜏)^2𝐾+11=−2𝑗^𝐾_{()𝑘=−𝐾+1}()2𝐾+1𝐾−𝑘−1^𝑘𝑒^{𝑗(2𝑘+1)𝜏}, (55) that is, 4^𝐾(sin𝜏)^2𝐾+1=^𝐾_𝑘=0()2𝐾+1𝐾−𝑘−1^𝑘sin(2𝑘+1)𝜏. (56) By differentiating (56) and comparing the resulting expression with (52), we obtain 𝑆_𝐾(𝜏)=𝑆_𝐾−1(𝜏)+(2𝐾−1)!!()()2𝐾!!sin𝜏^2𝐾𝑆cos𝜏,₀(𝜏)=cos𝜏. (57) Solving recurrent relation (57), we finally obtain (49), which completes the proof.

Using Lemma 9, we prove the following proposition.

Proposition 10. Maximally flat odd harmonic waveform with maximally degenerate critical point at 𝜏₀=0 and zero value at this point is nonnegative and has unique global minimum at 𝜏₀=0.

Proof. First derivative of (49) reads 𝑤⁽¹⁾_odd(𝜏)=(2𝐾+1)!!()()2𝐾!!sin𝜏^2𝐾+1. (58) It is obvious that (58) has only two zeros; namely, 𝜏=0 and 𝜏=𝜋. Therefore, (49) has only one minimum and only one maximum. Since 𝑤_odd(0)=0 and 𝑤_odd(𝜋)=2 it follows that 𝑤_odd(𝜏) is nonnegative and has unique global minimum at maximally degenerate critical point 𝜏₀=0. This completes the proof.

Remark 11. As it was pointed out earlier, the maximum value of (49) is achieved for 𝜏=𝜋, which implies 𝛿_odd=max_𝜏𝑤_odd(𝜏)=𝑤_odd(𝜋)=2. (59) Raab [2] also pointed out the existence of relation (59).

Remark 12. Although the binomial series 1−𝑥^2−1/2=1+^∞_𝑘=1(2𝑘−1)!!()𝑥2𝑘!!^2𝑘,−1<𝑥<1, (60) does not converge for 𝑥²=1, the above formula holds when 𝑥² tends to 1 from the left side (Abel’s limit theorem). Since cos𝜏=sgn(cos𝜏)⋅|cos𝜏|, according to the definition of the sign function, from (49), (60), and (48), for 𝑥=sin𝜏, it follows that lim_𝐾→∞𝑤_odd(𝜏)=1−sgn(cos𝜏). (61) Clearly, (61) is a square waveform.

Maximally flat odd harmonic waveforms, including limit waveform (𝐾→∞), are presented in Figure 6.

Figure 6 

Maximally flat odd harmonic waveforms.

4. Efficiency and Power-Output Capability of Class-𝐹 and Inverse Class-𝐹 PA with Maximally Flat Waveforms

There is a continuous interest in shaping current and voltage waveforms (e.g., see [2–7]). The case when both voltage and current waveforms contain only finite number of harmonics is of particular importance (e.g., see [2, 3, 5–7]). It has been notified [2, 3] that maximally flat waveforms could offer an approximate solution for current and voltage waveforms of finite harmonic class-𝐹 and inverse class-𝐹 PA.

In this Section, even harmonic and odd harmonic waveforms with maximally flat global minimum combined with their duals play central role in finding efficiency and power-output capability of class-𝐹 PA and inverse class-PA with maximally flat waveforms.

Raab [2] noticed that maximally flat waveforms could offer a good approximation for current and voltage waveforms for class-𝐹 PA operation. He investigated maximally flat waveforms up to fifth harmonic, and efficiencies of class-𝐹 PA operation for various combinations of harmonics up to five.

In this Section, we provide closed form expression for the efficiency of class-𝐹 PA and inverse class-𝐹 PA with maximally flat waveforms. We also provide proof of an interesting statement with practical implications saying that there is more benefit in consecutive inclusion of harmonics alternatively in current and voltage waveforms, than in inclusion of several harmonics in one waveform only. This statement was originally formulated in [2] based on the consideration of instances up to fifth harmonic, without general proof. When all even harmonics up to 𝑁 are included in current (voltage) waveform and all odd harmonics up to 𝑁 are included in voltage (current) waveform, we say that we are dealing with class-𝐹 (inverse class-𝐹) PA with 𝑁 harmonics. For this case, we show that the efficiency of the class-𝐹 PA and the inverse class-𝐹 PA with maximally flat waveforms is 𝜂_𝑁,flat=𝑁/(𝑁+1). We would like to call attention to the fact that this efficiency is identical to the efficiency of finite harmonic class-𝐶 PA with maximally flat current waveform, when it contains first 𝑁 harmonics [5].

For class-𝐹 PA and inverse class-𝐹 PA one of the waveforms possesses minimum at 𝜏₀ and the other at 𝜏₀+𝜋. Without loss of generality, we assume 𝜏₀=0.

For the purpose of the analysis of efficiency for class-𝐹 PA and inverse class-𝐹 PA with maximally flat waveforms we introduce either pair (𝑤_even,𝑤^∗_odd) or (𝑤^∗_even,𝑤_odd) of waveforms, where 𝑤_even,𝑤_odd, and their duals 𝑤^∗_even,𝑤^∗_odd are defined in Section 3.

For class-𝐹 PA with maximally flat waveforms, the pair of current and voltage waveforms can be expressed in two ways: 𝑖(𝜏)=𝐼_dc𝑤_even(𝜏),𝑣(𝜏)=𝑉_dc𝑤^∗_odd(𝜏), (62) or 𝑖(𝜏)=𝐼_dc𝑤^∗_even(𝜏),𝑣(𝜏)=𝑉_dc𝑤_odd(𝜏), (63) where 𝜏 stands for 𝜔𝑡,𝐼_dc>0 and 𝑉_dc>0.

For the inverse class-𝐹 PA with maximally flat waveforms, the pair of current and voltage waveforms can also be expressed in two ways, 𝑖(𝜏)=𝐼_dc𝑤_odd(𝜏),𝑣(𝜏)=𝑉_dc𝑤^∗_even(𝜏), (64) or 𝑖(𝜏)=𝐼_dc𝑤^∗_odd(𝜏),𝑣(𝜏)=𝑉_dc𝑤_even(𝜏), (65) where 𝐼_dc>0 and 𝑉_dc>0.

As an example, a pair (𝑤^∗_even,𝑤_odd) for the case 𝑁=9, is presented in Figure 7.

Figure 7 

Pair of maximally flat waveforms (𝑤∗even, 𝑤odd) for class-𝐹 PA and inverse class-𝐹 PA for 𝑁=9.

The efficiency of class-𝐹 and inverse class-𝐹 PA via current and voltage waveform parameters 𝛾_𝐼 and 𝛾_𝑉 can be expressed as 𝜂=𝛾_𝐼𝛾_𝑉/2 (see e.g., [2]), which according to our odd-even waveform notation leads to 𝛾𝜂=^even𝛾_odd2. (66) From (30), (47), and (66) it follows that the efficiency of class-𝐹 PA (inverse class-𝐹 PA) with maximally flat waveforms can be expressed as an explicit function of number of even harmonics 𝑀 in voltage (current) waveform and number of higher odd harmonics 𝐾 in another waveform: 𝜂_flat=((2𝑀)!!)²()()⋅))2𝑀−1!!2𝑀+1!!((2𝐾+1!!²()().2𝐾!!2𝐾+2!! (67) As we pointed out earlier, the waveform parameters 𝛾_even and 𝛾_odd increase with number of harmonics and, therefore, this is the case with 𝜂_flat. The efficiency 𝜂_flat, as a function of 𝑀 and 𝐾, is plotted in Figure 8.

Figure 8 

Efficiency of class-𝐹 and inverse class-𝐹 PA with maximally flat waveforms.

In what follows, we show how to choose 𝑀 and 𝐾, for the prescribed sum 𝑀+𝐾, in order to ensure that efficiencies of class-F and inverse class-F with maximally flat waveforms are maximal.

Proposition 13. For a prescribed sum 𝑀+𝐾, efficiency of class-𝐹 and inverse class-𝐹 with maximally flat waveforms (67) has maximum value if and only if 0≤𝑀−𝐾≤1. (68)

Proof. Let us first consider the quotient of efficiencies related to the pairs (𝑀,𝐾) and (𝑀+1,𝐾−1). According to (67), this quotient can be expressed as 𝜂_flat(𝑀,𝐾)𝜂_flat()=()𝑀+1,𝐾−12𝑀+1)(2𝑀+3(2𝑀+2)²⋅(2𝐾+1)²().2𝐾2𝐾+2 (69) Expression (69) can be rewritten in the form 𝜂_flat(𝑀,𝐾)𝜂_flat()()𝑀+1,𝐾−1=1+2𝑀+2²−(2𝐾+1)²(2𝑀+2)²4𝐾².+4𝐾 (70) It is clear that efficiency 𝜂_flat increases if and only if (2𝑀+2)²−(2𝐾+1)²>0; that is, if and only if 1𝑀−𝐾>−2. (71)
Furthermore, let us consider the quotient of efficiencies related to the pairs (𝑀,𝐾) and (𝑀−1,𝐾+1). According to (67), this quotient can be expressed as 𝜂_flat(𝑀,𝐾)𝜂_flat()=()𝑀−1,𝐾+12𝑀²()⋅()2𝑀−1)(2𝑀+12𝐾+2)(2𝐾+4(2𝐾+3)². (72) Expression (72) can be rewritten in the form 𝜂_flat(𝑀,𝐾)𝜂_flat()()𝑀−1,𝐾+1=1−2𝑀²−(2𝐾+3)²4𝑀²−1(2𝐾+3)². (73) It is clear that efficiency 𝜂_flat increases if and only if (2𝑀)²−(2𝐾+3)²<0; that is, if and only if 3𝑀−𝐾<2. (74)
By combining (71) and (74), we finely obtain −1/2<𝑀−𝐾<3/2, which clearly implies (68), since 𝑀 and 𝐾 are integers. This completes the proof.

Remark 14. From (68), if 𝑀 and 𝐾 are of the same parity (both are either even or odd), it follows that maximum value of 𝜂_flat is achieved for 𝑀=𝐾. If 𝑀 and 𝐾 are of different parity, then 𝑀=𝐾+1 leads to the maximum value of 𝜂_flat. For maximally flat waveforms, the above consideration proves the fact that there is more benefit for efficiency in consecutive inclusion of harmonics alternatively in current and voltage waveforms than in inclusion of several harmonics in one waveform only (originally stated in [2]). As an illustration see bold zigzag line in Figure 8.

Remark 15. In the cases when 𝑀=𝐾 or 𝑀=𝐾+1 efficiency (67) reduces to a simple form expressed through single integer 𝑁=1+𝑀+𝐾:𝜂_𝑁,flat=𝑁.𝑁+1 (75) The cases of class-𝐹 or inverse class-𝐹 PA with 𝑀=𝐾 or 𝑀=𝐾+1 can be called “class-𝐹 or inverse class-𝐹 PA with 𝑁 harmonics.”

The power-output capability of class-𝐹 and inverse class-𝐹 PA can be expressed via current and voltage waveform parameters as 𝑃_max=𝛾_𝐼𝛾_𝑉/(2𝛿_𝐼𝛿_𝑉) (e.g., see [2]). In our odd-even notation this expression can be rewritten as 𝑃_max=𝛾_even𝛾_odd2𝛿_even𝛿_odd. (76)

Using (44) and (59), power-output capability of finite harmonic class-𝐹 PA and inverse class-𝐹 PA with maximally flat waveforms can be expressed as 𝑃_max=𝛾_odd/8. This result is in accordance with that obtained in [2]. Since 𝛾_odd can be expressed via 𝐾 (see (47)), it follows that 𝑃_max can be also expressed as an explicit function of 𝐾. When number of harmonics tends to infinity, according to (48), power-output capability tends to 𝑃_max=1/(2𝜋).

5. Conclusion

In this paper, we consider general case of a problem of finding maximally flat waveforms with finite number of harmonics (maximally flat trigonometric polynomials). In Section 2 we prove that maximal degree of a degenerate critical point of waveforms with 𝑁 harmonics, not necessarily consecutive, is equal to 2𝑁−1 (Lemma 1) and provide a closed form expressions for coefficients of such waveforms (Proposition 3).

In Section 3 we consider the so-called maximally flat even harmonic and maximally flat odd harmonic waveforms. We prove that these waveforms are nonnegative and have global minimum at maximally degenerate critical point (Propositions 6 and 10).

We provide closed form expressions for efficiency and power output capability of class-𝐹 and inverse class-𝐹 PA operations with maximally flat waveforms as functions of number of harmonics. We prove that maximal benefit in the increasing efficiency of class-𝐹 and inverse class-𝐹 PA operations with maximally flat waveforms can be achieved when harmonics are consecutively included in current and voltage waveforms. It is also shown that the efficiency of both class-𝐹 PA and inverse class-𝐹 PA operation with maximally flat waveforms in the case of 𝑁 harmonics has a particular simple form 𝜂_𝑁,flat=𝑁/(𝑁+1).

Acknowledgment

This work is supported by the Serbian Ministry of Education, Science and Technology Development as a part of the Project TP32016.