Abstract
General description of nonnegative waveforms up to second harmonic in terms of independent (unconstrained) parameters is provided. Three important subclasses of the class of nonnegative waveforms are also fully characterised: nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed amplitude of second harmonic, nonnegative waveforms with maximal coefficient of cosine part of fundamental harmonic for prescribed coefficients of second harmonic, and nonnegative waveforms with at least one zero. We prove that members of the first two subclasses have at least one zero; that is, they also belong to the third subclass. Nonnegative cosine waveforms up to second harmonic are also considered and characterised. A number of case studies of practical interest for power amplifier (PA) design, involving nonnegative waveforms up to second harmonic, are also considered.
1. Introduction
In electrical engineering, the problem of shaping/modelling drain (collector, plate) waveforms in PA design, in order to improve efficiency, is of ultimate interest. It is largely related to the problem of finding nonnegative waveforms and as such attracted both engineers (e.g., see [1–7]) and mathematicians (e.g., see [8, 9]).
The family of nonnegative waveforms up to second harmonic has proved to be of particular interest for range of modes named class- or class- [6, 10, 11], inverse class [12], external second harmonic injection [13], and PA with arbitrary harmonic terminations [4]. In spite of frequent usage of nonnegative waveforms up to second harmonic in PA design, according to our best knowledge, this is the first result providing general description of nonnegative waveforms up to second harmonic in terms of independent (unconstrained) parameters. This problem of finding general description which naturally arises from engineering practice has proved to be a nontrivial mathematical problem.
In this paper, we consider normalized waveform with first two harmonics (trigonometric polynomial of order two): aiming to provide closed form expressions for coefficients , , , and in terms of independent parameters such that the following condition holds: Relations (1) and (2) refer to the class of nonnegative waveforms (Section 2).
We also consider a subclass of nonnegative waveforms that satisfy additional condition: for at least one (Section 5). We prove that this subclass includes all nonnegative waveforms with maximal amplitude of fundamental harmonic, when second harmonic amplitude is prescribed (Section 3), and all nonnegative waveforms with maximal absolute value of coefficient , when coefficients and are prescribed (Section 4). General description of nonnegative cosine waveforms is provided in Section 6. In Section 7, a number of case studies related to the efficiency of PA with nonnegative waveforms up to second harmonic are considered. Figure 1 describes relationship between considered classes of nonnegative waveforms.
In the quest for shape zoology of waveforms of type (1), it is enough to consider the following generic two-parameter family of waveforms: This is because all waveforms of type (1) can be obtained from (4) by appropriate usage of the following three operations: shifting along axis and introducing additive and/or multiplicative constants. Notice that these operations do not inherently change waveform shape.
To each waveform of type (4) corresponds a pair or equivalently a pair and vice versa. The zoo of shapes is presented in Figure 2. Bolded curve in Figure 2 divides the parameter space into three disjoint subsets (inner part, outer part, and solid line itself) and helps in making classification of the shapes of waveforms. The points of this curve constitute the so-called catastrophe set and correspond to those pairs for which there exist such that both first and second derivatives of the waveform at are equal to zero: An analogous consideration of waveform shapes with first and third harmonic is presented in [5].
Waveforms that correspond to the inner points have one minimum and one maximum, whereas the waveforms that correspond to the outer points have two minima and two maxima. Points on the solid line correspond to the waveforms with one minimum, one maximum, and one inflection point. Two cusp points and correspond to the maximally flat waveforms (e.g., see [5]), with maximally flat minimum and maximally flat maximum, respectively.
2. General Description of Nonnegative Waveforms up to Second Harmonic
In this section, we provide general description of nonnegative waveforms up to second harmonic in terms of four independent (unconstrained) parameters, along with a number of special cases.
The main result of this section is stated in the following proposition.
Proposition 1. For every waveform of type (1) satisfying condition (2), coefficients , , , and can be expressed in terms of four independent (unconstrained) parameters , , , and as
Proof. Let
be a complex polynomial on the unit circle () such that parameters , , , , , and satisfy the following constraint:
According to the classical result related to trigonometric polynomials ([8]; see also [14]), all trigonometric polynomials of type (1) satisfying condition (2) can be expressed as
As shown in [8], coefficients , , , and can be expressed in terms of six parameters , , , , , and in the following form:
providing that relation (11) holds.
In what follows, we will show that coefficients , , , and can be expressed in terms of four arbitrary parameters, instead of six parameters , , , , , and constrained by (11).
We start with the observation that hypersphere (11) can be written in terms of new parameter as
Furthermore, the sphere (17) can be also parameterised by introducing new parameter as follows:
In a similar way, hypersphere (18) can be parameterised by introducing further three parameters , , and as follows:
Inserting (21)–(24) into (13)–(16) yields
By introducing parameters and defined as
coefficients , , , and (see (25)–(28)) can be expressed in four parameters only , , and , taking into consideration that
which completes the proof.
Notice that substitution of (6)–(9) into (1) lead to the following forms of nonnegative waveforms up to second harmonic:
Remark 2. Maximum value of amplitude of fundamental harmonic of nonnegative waveform of type (1) is and corresponding amplitude of second harmonic is .
In order to show that, let From (8)-(9) and (6)-(7), it follows that respectively. According to (32) and (34), amplitude of second harmonic of nonnegative waveform of type (1) is According to (33) and (35), amplitude of fundamental harmonic of nonnegative waveform of type (1) is It is easy to see that, according to (33), the maximum value can be attained if and only if and . Therefore, maximum value of amplitude of fundamental harmonic of nonnegative waveform of type (1) is . From and , it also follows that and . Consequently, (34) and (36) imply that amplitude of second harmonic of nonnegative waveform of type (1) with is .
Remark 3. A number of special cases of nonnegative waveforms (1) with or , each corresponding to a specific parameter choice, are listed in Tables 1, 2, 3, 4, 5, 6, and 7.
3. Nonnegative Waveforms with Maximal Fundamental Harmonic Amplitude
In this section, we provide general description of nonnegative waveforms of type (1) with maximal amplitude of fundamental harmonic for prescribed second harmonic amplitude .
According to (34) and (36), for nonnegative waveforms of type (1), the following relation holds: This explains why amplitude of second harmonic in Proposition 4 goes through interval only.
Proposition 4. Every nonnegative waveform of type (1) with maximal amplitude of fundamental harmonic for prescribed amplitude of second harmonic can be represented as if , or if , providing that
Remark 5. As an immediate consequence of (39) and (40), it follows that every nonnegative waveform of type (1) with maximal amplitude of fundamental harmonic for prescribed amplitude of second harmonic has at least one zero.
Remark 6. Conversion of (39) and (40) into an additive form and comparison with (1) immediately lead to the explicit form of coefficients of nonnegative waveforms of type (1) with maximal amplitude of fundamental harmonic for prescribed second harmonic amplitude.
Conversion of (39) into an additive form lead to the following explicit form of coefficients: From (39), it follows immediately that is nonnegative if and only if .
Conversion of (40) into an additive form lead to the following explicit form of coefficients: It is easy to see that implies and if and only if .
Three examples of nonnegative waveforms with maximal amplitude of fundamental harmonic for prescribed second harmonic amplitude and prescribed position of zero , are presented in Figure 3 for and . The waveform represented by solid line corresponds to and the other two waveforms correspond to (notice that , according to (41), implies ).
Remark 7. Nonnegative waveforms with maximal amplitude of fundamental harmonic when can be expressed in the form Waveforms (44) are so-called maximally flat waveform of type (1) (e.g., see [5]).
Remark 8. According to (42), (43), and (41), maximal amplitude of fundamental harmonic of nonnegative waveforms of type (1) can be expressed as a function of second harmonic amplitude (see Figure 4):
Proof of Proposition 4. Let us consider nonnegative waveform of type (1) and let be a position of its global minimum; that is, .
If , then, according to (32), and/or . For from (33) it follows that . On the other hand, , according to (33), implies . Therefore, maximal amplitude of fundamental harmonic is equal to 1 and hence , which coincides with (39) for .
In what follows, we assume that , which, from (32), implies . Since does not depend on and , according to (33), the maximal amplitude of the fundamental harmonic can be attained if and only if
Relation (46) and imply . Therefore, and hold, which further, from (6) and (7), imply
providing that
Similarly, (46) implies and , which further, according to (8), (9), and (32), imply
Insertion of (47) and (49) into (1) lead to , which can be rewritten as
Let be a position of global minimum of ; that is, . It is easy to see that is minimal when is minimal. In what follows, we will consider two cases: Case (i) and Case (ii) .
In Case (i), there are two options for expression to be minimal: either for and or for and . Furthermore, implies , which is equivalent to . Therefore, maximal amplitude of fundamental harmonic is
From and (51), it follows that and therefore
The first option of the Case (i) reduces to and , while the second option of the Case (i) reduces to and . Substitution of any of these options into (50) results in
Factorisation of (53) immediately lead to (39), which proves that (39) holds for .
From the previous consideration, it follows that, in Case (ii), it is enough to consider
only. In this case, according to (50), position of global minimum corresponds to the situation when holds. Consequently,
Also, implies and therefore maximal amplitude of fundamental harmonic is
Substitution of (55) and (56) into (50) lead to , which can be rewritten as
Also, from (55) and (56), it follows that
Substituting in (57)-(58) by lead to (40)-(41), which proves that (40)-(41) hold for .
Finally, for , from (41), it follows that , which further implies and therefore (40) and (39) coincide for . This completes the proof.
4. Nonnegative Waveforms with Maximal Coefficient of Cosine Part of Fundamental Harmonic
Problem of finding nonnegative waveforms with maximal absolute value of coefficient is of particular interest in PA efficiency analysis. In a number of cases of interest, one waveform of the voltage-current pair (e.g., current) is already known and usually has only cosine part of fundamental harmonic (e.g., see [7]). In such cases, the problem of finding maximal efficiency for prescribed second harmonic impedance can be reduced to the problem of finding nonnegative voltage waveform with maximal absolute value of coefficient providing that coefficients and are prescribed (see also Section 7.3).
The following proposition provides closed form expressions for nonnegative waveforms with maximal absolute value of coefficient for prescribed second harmonic coefficients and .
Proposition 9. Every nonnegative waveform of type (1) with maximal absolute value of coefficient for prescribed second harmonic coefficients and can be represented as if and , if and , or if , providing that
Remark 10. As an immediate consequence of (59)–(61), it follows that every nonnegative waveform of type (1) with maximal absolute value of coefficient , for prescribed second harmonic coefficients and , has at least one zero.
Remark 11. Conversion of (59)–(61) into an additive form and comparison with (1) immediately lead to the explicit form of coefficients and . For , conversion of (59) and (60) into an additive form lead to respectively. Notice that implies On the other hand, implies and therefore According to (63) and (64), fundamental harmonic amplitude of waveforms (59)-(60) is . Its value, according to (65)-(66), is less than maximal value of fundamental harmonic amplitude for prescribed second harmonic amplitude (see (45)), except for or when they are equal.
For , conversion of (61) into an additive form together with (62) lead to if , or if . According to (67) and (68), amplitude of fundamental harmonic of waveform (61) is and according to (45) (notice that implies ) it coincides with maximal amplitude of fundamental harmonic for prescribed amplitude of second harmonic.
The contours of maximal absolute value of coefficient as a function of coefficients and are plotted in Figure 5.
Two examples of nonnegative waveforms with the same maximal absolute value of coefficient (one with and the other with ) for prescribed coefficients and (corresponding to the case ) are presented in Figure 6. Another two examples of nonnegative waveforms with the same maximal absolute value of coefficient (one with and the other with ) for prescribed coefficients and (corresponding to the case ) are presented in Figure 7.
Proof of Proposition 9. Let us consider nonnegative waveform of type (1) with coefficients , , , and expressed by (6)–(9) in terms of parameters , , , and .
According to (8) and (9), (i.e., ) implies and/or . For from (6) it follows that . On the other hand, implies either or . For , according to (6)-(7), it follows that and . For , according (6)-(7), it follows that and . In both cases ( and ), maximal absolute value of is equal to 1 and corresponding coefficient is equal to zero. Consequently, if , nonnegative waveform of type (1) with maximal absolute value of coefficient is either if or if . These two waveforms coincide with (59) and (60) for , respectively.
Suppose now that . According to (8)-(9) and (32), and . Since does not depend on and , it follows that and depend on only. Therefore, for prescribed and , which further implies that . Consequently, in the course of finding maximal absolute value of coefficient , we can set first derivative of in respect to to zero. By using and , from (6), we obtain
Second derivative of with respect to is equal to . It is positive (negative) if (). Therefore, (69) lead to maximal absolute value of coefficient . Substitution of into (7) and (9) multiplied with yields
respectively. From (6), (70), and (71), it follows that
The sum of squared (6) and squared (69) equals
From (8), it follows that (73) can be expressed as . It can be rewritten in the following form:
where, according to (32), . It is easy to see that, for prescribed , is maximal when is minimal. Accordingly, in what follows, we will consider the following two cases: Case (i) and Case (ii) .
Insertion of (Case (i)) into (32) lead to . Furthermore, implies
From , (74), (70), and (72), we obtain
where denotes sign function. Substitution of and into (1) lead to (59). Furthermore, substitution of and into (1) lead to (60).
In Case (ii) from and (74) it follows that ; that is,
Clearly, , that is,
implies , which, together with , further implies . Therefore, if and only if . Since if and only if and , from (70), (72), and (77), it follows that
Notice that can be written as , which, substituted in (79), yields
Insertion of (77) and (80) into (1) lead to
Since if and only if , it follows that , which further implies that (81) can be expressed as
From and , it follows that (82) can be rewritten as
where
Relation (78) implies and therefore for some . From (83),
Substitution of (85) into (83) yields , which can be rewritten as
Replacing by in (84)–(86), where is new variable, immediately lead to (61) and (62).
Finally, for and , from (62), it follows that or . For , according to (62), it follows that and . Consequently, (61) coincides with (59) for and . Similarly, for and , from (62), it follows that ,, and . Consequently, (61) coincides with (60) for and . This completes the proof.
5. Nonnegative Waveforms with at Least One Zero
In this section, we provide general description of nonnegative waveforms of type (1) with at least one zero, that is, waveforms for which and for some . Notice that conditions and imply that . This type of nonnegative waveforms have proved to be of particular interest in PA design (e.g., see [1, 3–7]). We also provide an algorithm for calculation of coefficients and of nonnegative waveforms with at least one zero and prescribed coefficients and (Section 5.1).
Proposition 12. Every nonnegative waveform of type (1) with at least one zero can be expressed in the following form: where , , and are arbitrary real numbers.
Proof. Notice that (12) can be rewritten as
which, after inserting (10), yields
Suppose that for some . Then, for , both squared terms in (89) are equal to zero; that is,
Substituting (90) into (89) and taking into account that , , and are given by (18), (14), and (16), respectively, we obtain
It is easy to show that all terms in (91) have common factor and therefore can be written as
Substituting (21)–(24) into (90) and then inserting resulting relations into (17), we obtain
Let us denote
Inserting (27)-(28) into (92), by using (93)-(94), we finally obtain (87).
Remark 13. Conversion of (87) into an additive form and comparison with (1) immediately lead to the explicit form of coefficients of nonnegative waveforms with at least one zero: According to (95) and (96), amplitude of fundamental harmonic is Also, according to (97) and (98), amplitude of second harmonic is
Remark 14. Coefficients (95)–(98) of nonnegative waveforms with at least one zero can be obtained from coefficients (6)–(9) of general case of nonnegative waveforms by substituting (see (94) and (30)), (see (93)-(94)), and
Remark 15. In Section 3 (see Proposition 4 and Remark 5), we show that every nonnegative waveform of type (1) with maximal amplitude of fundamental harmonic for prescribed second harmonic amplitude has at least one zero. Coefficients (42) of waveforms with maximal amplitude of fundamental harmonic for prescribed , when , can be obtained from (95)–(98) by replacing with and with . Similarly, coefficients (43) of waveforms with maximal amplitude of fundamental harmonic for prescribed , when , can be obtained from (95)–(98) by replacing with 1 and with .
From the above consideration, it is easy to see that among nonnegative waveforms with at least one zero and prescribed second harmonic amplitude, there are waveforms for which the amplitude of fundamental harmonic is not maximal (e.g., see Figures 9 and 10).
Remark 16. In Section 4 (see Proposition 9 and Remark 10), we show that every nonnegative waveform of type (1) with maximal absolute value of coefficient for prescribed coefficients and has at least one zero. Coefficients (63) of waveforms with maximal coefficient , when , can be obtained from (95)–(98) by setting . Similarly, coefficients (64) of waveforms with minimal coefficient (also maximal absolute value), when , can be obtained from (95)–(98) by setting . Also, coefficients (67)-(68) of waveforms with maximal absolute value of coefficient , when , can be obtained from (95)–(98) by replacing with 1 and with .
From the above consideration, it is easy to see that among nonnegative waveforms with at least one zero and prescribed coefficients and , there are waveforms for which absolute value of coefficient is not maximal (e.g., see Figures 9 and 10).
5.1. Nonnegative Waveforms with Prescribed Second Harmonic Coefficients
In this subsection, an algorithm for calculation of fundamental harmonic coefficients of nonnegative waveforms with at least one zero for prescribed second harmonic coefficients is provided. The range of instance at which waveform takes zero value depends on the chosen pair and .
Let us show first that lead to . According to (100), implies and . These relations from (99) further imply ; that is, . Therefore, in what follows we will consider only those pairs for which .
Solving (97)-(98) in terms of and , we obtain where range of depends on the chosen pair .
A good part of this Subsection is devoted to finding this range. In this context, let us begin with substitution of (102) into . This lead to or equivalently to The expressions in the square brackets can be considered as second degree polynomials in terms of with the following discriminants: respectively. Since and for all , it follows that first factor is always positive. Therefore, inequality (103) is equivalent to Consequently, the problem of finding range of , for which (102) holds, can be reformulated as the problem of finding those for which inequality (106) holds. The discussion involves discriminant (see (105)) and leading term coefficient
In what follows, we first consider all cases with . Let and be the roots of second degree polynomial on the left side of (106); that is, According to (105) and (107), implies and (see Figure 8). Also and imply and (). Similarly, and imply and . Notice that () implies , which further implies , Thus, for : (1)if , then (106) holds for every ,(2)if and , then (106) holds for every ,(3)if and , then (106) holds for (4) if , then (106) holds for where and are given by (108). For , (106) reduces to . Thus, for , we have another two cases: