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

Authors: Anamarija Juhas; Ladislav A. Novak
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 Section2 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:
(1)f(τ;A,α)=cosτ+Acos(3τ+α),
and corresponding set of 3-tiples (τd,A,α) for which both first and second derivatives of the waveform f(τ;A,α) are equal to zero:
(2)f(1)(τd;A,α)=0,f(2)(τd;A,α)=0.
In this case, we say that τd 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 integerr≥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(3)fN(τ)=F0+∑i=1NFicos(niτ+φi),N≥1,
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 Fi>0 and nj>ni. Notice that set of positive integers ni defines the family.It is obvious that the existence of the maximally degenerate critical point is invariant of the value ofF0 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 waveformfN(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 fN(τ) of the family (3) such that
(4)fN(0)=0,(5)fN(r)(0)=0,r=1,…,rmax,
where fN(r) denotes rth derivative of the waveform fN, and rmax is maximum number of relations (5) which do not contradict (4).Lemma 1.
For nonzero waveformfN(τ) of the family (3) which satisfies (4) and (5), the following relations hold:
(6)F0≠0,(7)rmax=2N-1.Remark 2.
From (7) it is obvious that rmax is odd number which further implies that at maximally degenerate critical point fN(τ) has either minimum or maximum.Proof.
In terms of new variables(8)ai=Ficosφi,i=1,…,N,(9)bi=Fisinφi,i=1,…,N,
system (4)-(5) for rmax=2N-1 decomposes in two independent subsystems:
(10)∑i=1Nai=-F0,(11)∑i=1Nni2qai=0,q=1,…,N-1,
(written in terms of variables ai only) and
(12)∑i=1Nni2q-1bi=0,q=1,…,N,
(written in terms of variables bi only).
The matrices of subsystems (10)-(11) and (12) are
(13)Ma=[1⋯111⋯1n12⋯ni-12ni2ni+12⋯nN2⋮⋱⋮⋮⋮⋱⋮n12N-2⋯ni-12N-2ni2N-2ni+12N-2⋯nN2N-2],(14)Mb=[n1⋯ni-1nini+1⋯nNn13⋯ni-13ni3ni+13⋯nN3⋮⋱⋮⋮⋮⋱⋮n12N-1⋯ni-12N-1ni2N-1ni+12N-1⋯nN2N-1],
respectively. Matrices Ma and Mb are both Vandermonde matrices and their determinants are
(15)det(Ma)=∏1≤i<j≤N(nj2-ni2),det(Mb)=det(Ma)∏1≤j≤Nnj,
which are clearly nonzero. Therefore, each subsystem (10)-(11)and (12) has unique solution which implies that rmax≥2N-1.
It is obvious that homogeneous subsystem (12) in terms of bi has trivial solution as follows:
(16)bi=0,i=1,…,N,
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 rmax≥2N-1 and we are going to show that assumption rmax=2N leads to contradiction. For rmax=2N, the corresponding subsystem can be obtained from (10)-(11) by including additional equation fN(2N)(0)=0, which in terms of variables (8), can be written in the form
(17)∑i=1Nni2Nai=0.
Determinant of subsystem consisting of equations (11) and (17) equals det(Ma)∏1≤j≤Nnj2 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 Lemma1, the problem of finding nonzero waveform fN(τ) of the family (3) satisfying (4) and (5) can be reformulated as follows: find a nonzero waveform from the family (3) such that
(18)fN(0)=0,fN(r)(0)=0,r=1,…,2N-1.
In terms of variables ai and bi (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:
(19)Fi=|F0|[∏1≤j≤i-1nj2(ni2-nj2)]·[∏i+1≤j≤Nnj2(nj2-ni2)],i=1,…,N,φi={0iiseven,πiisodd,i=1,…,N,ifF0>0,φi={πiiseven,0iisodd,i=1,…,N,ifF0<0.Proof.
Let us find the solution of the system (10)–(12). Using Cramer’s rule, from subsystem (10)-(11) it follows that
(20)ai=-F0(-1)1+iM1idet(Ma)=(-1)iF0[∏1≤j≤i-1nj2(ni2-nj2)]·[∏i+1≤j≤Nnj2(nj2-ni2)],i=1,…,N,
where M1i 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 Fi>0  immediately follows (19), which completes the proof.Example 4.
The following two waveforms presented in Figure2:
(21)f(1,3,4)(τ)=1-65cosτ+27cos3τ-335cos4τ,f(2,4,5)(τ)=1-10063cos2τ+2527cos4τ-64189cos5τ,
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:(22)weven(τ)=1-A1cosτ-∑m=1MA2mcos(2mτ),(23)wodd(τ)=1-B1cosτ-∑k=1KB(2k+1)cos(2k+1)τ,
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 thatweven(τ) and wodd(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, nm+1=2m, and A2m=-am+1 for m=1,…,M:(24)A1=((2M)!!)2(2M-1)!!(2M+1)!!,(25)A2m=(-1)m(4m2-1)2(M!)2(M-m)!(M+m)!,m=1,…,M.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, nk+1=2k+1, F0=1, and B(2k+1)=-ak+1 for k=0,…,K:(26)B(2k+1)=(-1)k(2k+1)·((2K+1)!!)24K(K-k)!(K+k+1)!,k=0,…,K.Let us introduce “duals” of the waveforms (22) and (23):
(27)weven*(τ)=1+A1*cosτ+∑m=1MA2m*cos(2mτ),wodd*(τ)=1+B1*cosτ+∑k=1KB(2k+1)*cos(2k+1)τ,
which is defined as weven*(τ)=weven(τ+π) and wodd*(τ)=wodd(τ+π), respectively. It is easy to see that the following relations between the coefficients of waveforms (22)-(23) and their duals hold:
(28)A1*=A1,A2m*=-A2m,B1*=B1,B(2k+1)*=B(2k+1).
Notice that weven(τ) and wodd(τ) have maximally degenerate critical point at τ0=0, which implies that their duals weven*(τ) and wodd*(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 (Proposition6). We also provide closed form expression for basic waveform parameters γeven and δeven in terms of number of even harmonics. The parameter γeven is defined as the amplitude of fundamental harmonic relative to dc component of the even harmonic waveform, whereas δeven is defined as maximum value of the waveform relative to dc component (e.g., see [2]):
(29)γeven=A1,δeven=maxτweven(τ).According to (24), parameter γeven is
(30)γeven=A1=((2M)!!)2(2M-1)!!(2M+1)!!=16M2M+1((M!)2(2M)!)2.
Notice that by definition 0!!=1 and (-1)!!=1. Therefore, (30) yields γeven=1 for M=0. The relation (30) shows that γeven increases by increasing M (see Figure 3). When M tends to infinity, parameter γeven equals to Wallis product; that is,
(31)limM→∞γeven=∏n=1∞4n2(2n-1)(2n+1)=π2.Figure 3
Waveform parameterγeven 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 atτ0=0 and zero value at this point can be expressed in the following form:
(32)weven(τ)=γeven(1-cosτ-∑m=1M(2m-3)!!(2m)!!(sinτ)2m),M≥0,
where γeven 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
(33)weven(τ)=γeven(-cosτ+SM(τ)),
where S0(τ)=1 and
(34)SM(τ)=(2M+1)!!4M(2M)!!×[(2MM)-2∑m=1M(2MM-m)(-1)mcos(2mτ)(4m2-1)],M≥1.
The expression4M(sinτ)2M=(-1)M(e-jτ-ejτ)2M, where j=-1, in an expanded form reads
(35)4M(sinτ)2M=∑n=02M(2Mn)(-1)M-nej2(M-n)τ.
Substitution of n=M-m into (35) yields
(36)4M(sinτ)2M=∑m=-MM(2MM-m)(-1)mej2mτ,
that is,
(37)4M(sinτ)2M=(2MM)+2∑m=1M(2MM-m)(-1)mcos(2mτ).
Since
(38)SM(τ)-SM-1(τ)=-(2M-3)!!4M(2M)!!×[(2MM)+2∑m=1M(2MM-m)(-1)mcos(2mτ)],
using (37), we obtain
(39)SM(τ)-SM-1(τ)=-(2M-3)!!(2M)!!(sinτ)2M,S0(τ)=1.
The solution of recursive relation (39) reads
(40)SM(τ)=1-∑m=1M(2m-3)!!(2m)!!(sinτ)2m,M≥1.
Substituting (40) into (33), we finally obtain (32), which completes the proof.Using Lemma5, we prove the following proposition.Proposition 6.
Maximally flat even harmonic waveform with maximally degenerate critical point atτ0=0 and zero value at this point is nonnegative and has unique global minimum at τ0=0.Proof.
The binomial series(41)(1-x2)1/2=1-∑m=1∞(2m-3)!!(2m)!!x2m,-1≤x≤1,
for x=sinτ reads
(42)|cosτ|=(1-sin2τ)1/2=1-∑m=1∞(2m-3)!!(2m)!!(sinτ)2m.
By comparison of (32) with (42), we conclude that
(43)weven(τ)≥γeven(-cosτ+|cosτ|)≥0,
and therefore the maximally flat even harmonic waveform is nonnegative.
From (43) and (32), it is obvious that weven(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.
Sinceγeven>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,
(44)δeven=maxτweven(τ)=weven(π)=2γeven.
Raab [2] also pointed out the relation (44).Remark 8.
WhenM tends to infinity, according to (32), (42), and (31) it follows that
(45)limM→∞weven(τ)=π2(-cosτ+|cosτ|).
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 (Proposition10). We also provide closed form expression for basic waveform parameters γodd and δodd in terms of number of odd harmonics. The parameter γodd is defined as the amplitude of fundamental harmonic relative to dc component of the odd harmonic waveform, whereas δodd is defined as maximum value of the waveform relative to dc component (e.g., see [2]):
(46)γodd=B1,δodd=maxτwodd(τ).From (26) for k=0, we obtain
(47)γodd=B1=2((2K+1)!!)2(2K)!!(2K+2)!!=K+116K(2K+1K)2.
Notice that (47) yields γodd=1 for K=0. The relation (47) shows that γodd increases by increasing K (see Figure 5). When K tends to infinity, from (47), according to (31), γodd tends to
(48)limK→∞γodd=limK→∞(2K+1K+1∏n=1K(2n-1)(2n+1)4n2)=4π.Figure 5
Waveform parameterγodd 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 atτ0=0 and zero value at this point can be expressed as
(49)wodd(τ)=1-cosτ(1+∑k=1K(2k-1)!!(2k)!!(sinτ)2k),K≥0.Proof.
Taking into account (26), maximally flat odd harmonic waveform (23) with maximally degenerate critical point at τ0=0 can be expressed as
(50)wodd(τ)=1-SK(τ),
where S0(τ)=cosτ and
(51)SK(τ)=(2K+1)!!4K(2K)!!∑k=0K(2K+1K-k)(-1)k(2k+1)cos(2k+1)τ.
Hence, difference SK(τ)-SK-1(τ) is equal to
(52)SK(τ)-SK-1(τ)=1(2K+1)(2K-1)!!4K(2K)!!×∑k=0K(2K+1K-k)(-1)k(2k+1)cos(2k+1)τ.
The expression(53)4K(sinτ)2K+1=12j(-1)Kej(2K+1)τ(e-j2τ-1)2K+1,
in an expanded form reads
(54)4K(sinτ)2K+1=-12j∑n=02K+1(2K+1n)(-1)K-nej(2K+1-2n)τ.
Substitution of k=K-n into (54) yields
(55)4K(sinτ)2K+1=-12j∑k=-(K+1)K(2K+1K-k)(-1)kej(2k+1)τ,
that is,
(56)4K(sinτ)2K+1=∑k=0K(2K+1K-k)(-1)ksin(2k+1)τ.
By differentiating (56) and comparing the resulting expression with (52), we obtain
(57)SK(τ)=SK-1(τ)+(2K-1)!!(2K)!!(sinτ)2Kcosτ,S0(τ)=cosτ.
Solving recurrent relation (57), we finally obtain (49), which completes the proof.Using Lemma9, we prove the following proposition.Proposition 10.
Maximally flat odd harmonic waveform with maximally degenerate critical point atτ0=0 and zero value at this point is nonnegative and has unique global minimum at τ0=0.Proof.
First derivative of (49) reads
(58)wodd(1)(τ)=(2K+1)!!(2K)!!(sinτ)2K+1.
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 wodd(0)=0 and wodd(π)=2 it follows that wodd(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
(59)δodd=maxτwodd(τ)=wodd(π)=2.
Raab [2] also pointed out the existence of relation (59).Remark 12.
Although the binomial series(60)(1-x2)-1/2=1+∑k=1∞(2k-1)!!(2k)!!x2k,-1<x<1,
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
(61)limK→∞wodd(τ)=1-sgn(cosτ).
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.


## 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 (Proposition6). We also provide closed form expression for basic waveform parameters γeven and δeven in terms of number of even harmonics. The parameter γeven is defined as the amplitude of fundamental harmonic relative to dc component of the even harmonic waveform, whereas δeven is defined as maximum value of the waveform relative to dc component (e.g., see [2]):
(29)γeven=A1,δeven=maxτweven(τ).According to (24), parameter γeven is
(30)γeven=A1=((2M)!!)2(2M-1)!!(2M+1)!!=16M2M+1((M!)2(2M)!)2.
Notice that by definition 0!!=1 and (-1)!!=1. Therefore, (30) yields γeven=1 for M=0. The relation (30) shows that γeven increases by increasing M (see Figure 3). When M tends to infinity, parameter γeven equals to Wallis product; that is,
(31)limM→∞γeven=∏n=1∞4n2(2n-1)(2n+1)=π2.Figure 3
Waveform parameterγeven 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 atτ0=0 and zero value at this point can be expressed in the following form:
(32)weven(τ)=γeven(1-cosτ-∑m=1M(2m-3)!!(2m)!!(sinτ)2m),M≥0,
where γeven 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
(33)weven(τ)=γeven(-cosτ+SM(τ)),
where S0(τ)=1 and
(34)SM(τ)=(2M+1)!!4M(2M)!!×[(2MM)-2∑m=1M(2MM-m)(-1)mcos(2mτ)(4m2-1)],M≥1.
The expression4M(sinτ)2M=(-1)M(e-jτ-ejτ)2M, where j=-1, in an expanded form reads
(35)4M(sinτ)2M=∑n=02M(2Mn)(-1)M-nej2(M-n)τ.
Substitution of n=M-m into (35) yields
(36)4M(sinτ)2M=∑m=-MM(2MM-m)(-1)mej2mτ,
that is,
(37)4M(sinτ)2M=(2MM)+2∑m=1M(2MM-m)(-1)mcos(2mτ).
Since
(38)SM(τ)-SM-1(τ)=-(2M-3)!!4M(2M)!!×[(2MM)+2∑m=1M(2MM-m)(-1)mcos(2mτ)],
using (37), we obtain
(39)SM(τ)-SM-1(τ)=-(2M-3)!!(2M)!!(sinτ)2M,S0(τ)=1.
The solution of recursive relation (39) reads
(40)SM(τ)=1-∑m=1M(2m-3)!!(2m)!!(sinτ)2m,M≥1.
Substituting (40) into (33), we finally obtain (32), which completes the proof.Using Lemma5, we prove the following proposition.Proposition 6.
Maximally flat even harmonic waveform with maximally degenerate critical point atτ0=0 and zero value at this point is nonnegative and has unique global minimum at τ0=0.Proof.
The binomial series(41)(1-x2)1/2=1-∑m=1∞(2m-3)!!(2m)!!x2m,-1≤x≤1,
for x=sinτ reads
(42)|cosτ|=(1-sin2τ)1/2=1-∑m=1∞(2m-3)!!(2m)!!(sinτ)2m.
By comparison of (32) with (42), we conclude that
(43)weven(τ)≥γeven(-cosτ+|cosτ|)≥0,
and therefore the maximally flat even harmonic waveform is nonnegative.
From (43) and (32), it is obvious that weven(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.
Sinceγeven>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,
(44)δeven=maxτweven(τ)=weven(π)=2γeven.
Raab [2] also pointed out the relation (44).Remark 8.
WhenM tends to infinity, according to (32), (42), and (31) it follows that
(45)limM→∞weven(τ)=π2(-cosτ+|cosτ|).
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 (Proposition10). We also provide closed form expression for basic waveform parameters γodd and δodd in terms of number of odd harmonics. The parameter γodd is defined as the amplitude of fundamental harmonic relative to dc component of the odd harmonic waveform, whereas δodd is defined as maximum value of the waveform relative to dc component (e.g., see [2]):
(46)γodd=B1,δodd=maxτwodd(τ).From (26) for k=0, we obtain
(47)γodd=B1=2((2K+1)!!)2(2K)!!(2K+2)!!=K+116K(2K+1K)2.
Notice that (47) yields γodd=1 for K=0. The relation (47) shows that γodd increases by increasing K (see Figure 5). When K tends to infinity, from (47), according to (31), γodd tends to
(48)limK→∞γodd=limK→∞(2K+1K+1∏n=1K(2n-1)(2n+1)4n2)=4π.Figure 5
Waveform parameterγodd 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 atτ0=0 and zero value at this point can be expressed as
(49)wodd(τ)=1-cosτ(1+∑k=1K(2k-1)!!(2k)!!(sinτ)2k),K≥0.Proof.
Taking into account (26), maximally flat odd harmonic waveform (23) with maximally degenerate critical point at τ0=0 can be expressed as
(50)wodd(τ)=1-SK(τ),
where S0(τ)=cosτ and
(51)SK(τ)=(2K+1)!!4K(2K)!!∑k=0K(2K+1K-k)(-1)k(2k+1)cos(2k+1)τ.
Hence, difference SK(τ)-SK-1(τ) is equal to
(52)SK(τ)-SK-1(τ)=1(2K+1)(2K-1)!!4K(2K)!!×∑k=0K(2K+1K-k)(-1)k(2k+1)cos(2k+1)τ.
The expression(53)4K(sinτ)2K+1=12j(-1)Kej(2K+1)τ(e-j2τ-1)2K+1,
in an expanded form reads
(54)4K(sinτ)2K+1=-12j∑n=02K+1(2K+1n)(-1)K-nej(2K+1-2n)τ.
Substitution of k=K-n into (54) yields
(55)4K(sinτ)2K+1=-12j∑k=-(K+1)K(2K+1K-k)(-1)kej(2k+1)τ,
that is,
(56)4K(sinτ)2K+1=∑k=0K(2K+1K-k)(-1)ksin(2k+1)τ.
By differentiating (56) and comparing the resulting expression with (52), we obtain
(57)SK(τ)=SK-1(τ)+(2K-1)!!(2K)!!(sinτ)2Kcosτ,S0(τ)=cosτ.
Solving recurrent relation (57), we finally obtain (49), which completes the proof.Using Lemma9, we prove the following proposition.Proposition 10.
Maximally flat odd harmonic waveform with maximally degenerate critical point atτ0=0 and zero value at this point is nonnegative and has unique global minimum at τ0=0.Proof.
First derivative of (49) reads
(58)wodd(1)(τ)=(2K+1)!!(2K)!!(sinτ)2K+1.
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 wodd(0)=0 and wodd(π)=2 it follows that wodd(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
(59)δodd=maxτwodd(τ)=wodd(π)=2.
Raab [2] also pointed out the existence of relation (59).Remark 12.
Although the binomial series(60)(1-x2)-1/2=1+∑k=1∞(2k-1)!!(2k)!!x2k,-1<x<1,
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
(61)limK→∞wodd(τ)=1-sgn(cosτ).
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 ηN,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 (weven,wodd*) or (weven*,wodd) of waveforms, where weven,wodd, and their duals weven*,wodd* 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:
(62)i(τ)=Idcweven(τ),v(τ)=Vdcwodd*(τ),
or
(63)i(τ)=Idcweven*(τ),v(τ)=Vdcwodd(τ),
where τ stands for ωt,Idc>0 and Vdc>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,
(64)i(τ)=Idcwodd(τ),v(τ)=Vdcweven*(τ),
or
(65)i(τ)=Idcwodd*(τ),v(τ)=Vdcweven(τ),
where Idc>0 and Vdc>0.As an example, a pair(weven*,wodd) for the case N=9, is presented in Figure 7.Figure 7
Pair of maximally flat waveforms(weven*, 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 γI and γV can be expressed as η=γIγV/2 (see e.g., [2]), which according to our odd-even waveform notation leads to
(66)η=γevenγodd2.
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:
(67)ηflat=((2M)!!)2(2M-1)!!(2M+1)!!·((2K+1)!!)2(2K)!!(2K+2)!!.
As we pointed out earlier, the waveform parameters γeven and γodd increase with number of harmonics and, therefore, this is the case with ηflat. The efficiency ηflat, 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 chooseM 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 sumM+K, efficiency of class-F and inverse class-F with maximally flat waveforms (67) has maximum value if and only if
(68)0≤M-K≤1.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
(69)ηflat(M,K)ηflat(M+1,K-1)=(2M+1)(2M+3)(2M+2)2·(2K+1)22K(2K+2).
Expression (69) can be rewritten in the form
(70)ηflat(M,K)ηflat(M+1,K-1)=1+(2M+2)2-(2K+1)2(2M+2)2(4K2+4K).
It is clear that efficiency ηflat increases if and only if (2M+2)2-(2K+1)2>0; that is, if and only if
(71)M-K>-12.
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
(72)ηflat(M,K)ηflat(M-1,K+1)=(2M)2(2M-1)(2M+1)·(2K+2)(2K+4)(2K+3)2.
Expression (72) can be rewritten in the form
(73)ηflat(M,K)ηflat(M-1,K+1)=1-(2M)2-(2K+3)2(4M2-1)(2K+3)2.
It is clear that efficiency ηflat increases if and only if (2M)2-(2K+3)2<0; that is, if and only if
(74)M-K<32.
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 ηflat is achieved for M=K. If M and K are of different parity, then M=K+1 leads to the maximum value of ηflat. 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 whenM=K or M=K+1 efficiency (67) reduces to a simple form expressed through single integer N=1+M+K:(75)ηN,flat=NN+1.
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 Pmax=γIγV/(2δIδV) (e.g., see [2]). In our odd-even notation this expression can be rewritten as
(76)Pmax=γevenγodd2δevenδodd.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 Pmax=γodd/8. This result is in accordance with that obtained in [2]. Since γodd can be expressed via K (see (47)), it follows that Pmax 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 Pmax=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 Section2 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 Section3 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 ηN,flat=N/(N+1).



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