Appendix A Definitions, Conventions and Overview of Used Theory

This appendix summarizes important theoretical concepts used throughout this work (especially in Chap. 3). Since most of this theory is generally known, it will not be fully explained or proven here; for this, the interested reader is referred to specialized works. However, since many of the concepts are defined in slightly different ways by different authors, it is useful to clearly define the concepts used here. Furthermore, some important remarks are given about how to use certain theoretical concepts and notations.

A.1 The Fourier Transform

A.1.1 Definition

The Fourier transform is an indispensable tool for all signal processing and com- munication applications. This section only states some definitions and conventions for clarity’s sake. For more detailed information on Fourier analysis, the reader is referred to reference works in this field, such as [1, 4]. In this work, the continuous Fourier transform (CFT) is used. It is denoted by the operator F {.}. The Fourier transform X( f ) of a signal x(t) is defined as

 ∞ − π X( f ) = F {x(t)} = x(t) e j2 ft dt, (A.1) −∞ where j is the imaginary unit and j2 =−1. X( f ) is called the spectrum of the signal x(t). In this work, the spectrum of a time-domain signal will be denoted with the same letter as the time-domain signal, but capitalized. The independent variable f is called frequency. It should not be confused with the angular frequency ω = 2π f which is used in some works, but will not be used here. The value of X( f ) expresses the contribution of that frequency to the signal x(t).

P. A. J. Nuyts et al., Continuous-Time Digital Front-Ends for Multistandard 277 Wireless Transmission, Analog Circuits and Signal Processing, DOI: 10.1007/978-3-319-03925-1, © Springer International Publishing Switzerland 2014 278 Appendix A: Definitions, Conventions and Overview of Used Theory

The inverse continuous Fourier transform (ICFT) of a spectrum X( f ) is denoted by the operator F −1 {.} and defined as

 ∞ − π x(t) = F 1 {X( f )} = X( f ) e j2 ft d f. (A.2) −∞

The resulting signal is the same as the original signal x(t). While the CFT is not defined for all signals [1, 4], it is generally defined for signals appearing in communication applications. Note that the CFT spectrum of a signal is generally complex, even if the signal itself is real.

A.1.2 Properties

This section lists some important Fourier transform properties. A more extensive list of properties, as well as proofs for many of them, can be found in [1, 4]. Let x(t) and y(t) be signals with spectra X( f ) and Y ( f ), and let a and b be real numbers. Then the following properties hold: • The CFT and the ICFT are linear operations, i.e.

F {ax(t) + by(t)} = aX( f ) + bY( f ). (A.3)

• The spectrum of the complex conjugate of x(t) is   ∗ ∗ F x (t) = X (− f ). (A.4)

• If x(t) is real, then x(t) = x∗(t) and it follows from the above property that

∗ X(− f ) = X ( f ), (A.5)

where X ∗( f ) indicates the complex conjugate of X( f ). In other words, the spec- trum of a real signal has a real part that is even and an imaginary part that is odd. • Similarly, if x(t) is purely imaginary, then x(t) =−x∗(t) and

∗ X(− f ) =−X ( f ). (A.6)

In other words, the spectrum of an imaginary signal has a real part that is odd and an imaginary part that is even. • If x(t) is zero outside a certain time interval [t0, t1], then X( f ) is unlimited in frequency, i.e. there is no frequency f0 so that X( f ) = 0 ∀ f where | f | > f0. However |X( f )| might decrease asymptotically with increasing | f |. Appendix A: Definitions, Conventions and Overview of Used Theory 279

• If X( f ) is zero outside a certain frequency band [ f0, f1], i.e. x(t) is a band- limited signal, then x(t) is unlimited in time, i.e. there is no time t0 so that x(t) = 0 ∀ t where |t| > t0. However |x(t)| might decrease asymptotically with increasing |t|. • If x(t) is periodical with a certain period T, i.e. if x(t) = x(t + T ) ∀ t, then X( f ) is nonzero only at integer multiples of f = 1/T . In other words, a periodical signal has a discrete spectrum. • If x(t) is nonzero only at integer multiples of a period Ts, called the sampling period, then X( f ) is periodical with period fs = 1/Ts, called the sampling fre- quency. In other words, a discrete-time signal has a periodical spectrum. − j2π ft • The Fourier transform of x(t − t0) is X( f ) · e 0 , i.e. a time shift in the time domain corresponds to a linear phase shift in the frequency domain. j2π f t • The inverse Fourier transform of X( f − f0) is x(t) · e 0 , i.e. a frequency shift in the frequency domain corresponds to a linear phase shift in the time domain. • The CFT and ICFT are bijections: If x(t) = y(t) then X( f ) = Y ( f ) and vice versa. • According to Parseval’s theorem [1, Eq. (2.40)],

 ∞  ∞ ∗ ∗ x(t)y (t)dt = X( f )Y ( f )d f, (A.7) −∞ −∞

where y∗(t) denotes the complex conjugate of y(t).Ifx(t) = y(t), this theorem reduces to  ∞  ∞ |x(t)|2dt = |X( f )|2d f, (A.8) −∞ −∞

which is known as Rayleigh’s energy theorem [1, Eq. (2.41)] [4, Sect. 2.4].

A.1.3 Important Fourier Transform Pairs

Table A.1 lists some important signals and their Fourier transforms. While these results are generally known, they are not identical in all literature, as some writers use slightly different definitions of the CFT. Therefore, they are listed according to the definitions used in this work for easy reference. More Fourier transform pairs can be found in [1]. The Dirac delta function δ(t), which appears in many of the results, will be defined in Sect. A.4.1.

A.2 Convolution

A.2.1 Definition and Notation

The continuous convolution, or shorter, convolution, is an operation that operates on two functions. The convolution z(u) of two functions x(u) and y(u) is defined as 280 Appendix A: Definitions, Conventions and Overview of Used Theory

Table A.1 Some important Time domain Frequency domain fourier transform pairs 00 δ(t) 1 − j2π ft δ(t − t0) e 0 1 δ( f ) j2π f t e 0 δ( f − f0) ( π ) 1 (δ( − ) + δ( + )) cos 2 f0t 2 f f0 f f0 ( π ) 1 (δ( − ) − δ( + )) sin 2 f0t 2 j f f0 f f0

 ∞  ∞ z(u) = x(v) y(u − v) dv = y(v) x(u − v) dv (A.9) −∞ −∞ and is commonly written as

z(u) = x(u) ∗ y(u). (A.10)

While this notation is practical in many cases, and clearly indicates that convolution shares a large number of properties with multiplication (such as commutativity, associativity, etc.), it is not really unambiguous, as it does not distinguish between the variables u and v used in (A.9). For example, evaluating the function z in 2u gives  ∞ z(2u) = x(v) y(2u − v) dv (A.11) −∞ whereas the convolution of functions x(2u) and y(2u) is

 ∞ x(2u) ∗ y(2u) = x(2v) y(2(u − v)) dv −∞ 1 ∞ = x(w) y(2u − w) dw 2 −∞ 1 = z(2u), (A.12) 2 and thus z(2u) = x(2u) ∗ y(2u) even though the function z(u) was “defined” as z(u) = x(u) ∗ y(u). The reason for this is that the letter u in x(u) ∗ y(u) actually refers to the integration variable v, while in z(u) it is an independent variable that can be changed. Furthermore, in an expression like x(au) ∗ y(2a + u), it is not clear whether one should integrate over a or u. These problems can be solved by introducing a more exact notation, which uses different letters for the independent variable and the integration variable and further- more clearly indicates which variable is the integration variable:  ∞ z(u) = [x(v) ∗v y(v)] (u) = x(v) y(u − v) dv. (A.13) −∞ Appendix A: Definitions, Conventions and Overview of Used Theory 281

This notation clearly shows that the convolution of the functions x(v) and y(v) with integration variable v is a function of u. With this notation, z(2u) can be written as

z(2u) = [x(v) ∗v y(v)] (2u) , (A.14) while the convolution of the functions x(2u) and y(2u) can be written as

 ∞ z2(u) = [x(2v) ∗v y(2v)] (u) = x(2v) y(2(u − v)) dv. (A.15) −∞

The index v indicates that the integration variable is still v. If the index would be 2v, one would obtain  ∞ [x(2v) ∗2v y(2v)] (u) = x(2v) y(2(u − v)) d(2v) (A.16) −∞ = z(2u).

While this notation is unambiguous, it has the disadvantage of being very tedious. Therefore, in this work, the following conventions will be used: • Where possible, the shorter notation will be used. • In time-domain equations using the short notation, the integration variable is always t. Thus, the notation z(t) = x(t) ∗ y(t) should be interpreted as follows: Ð Replace all occurrences of t on the right-hand side by some unused symbol, e.g. u. In this case, this yields x(u) ∗ y(u). Ð Then calculate the convolution using u as the integration variable and t as the independent time variable, i.e. calculate

 ∞ [x(u) ∗u y(u)] (t) = x(u) v(t − u) du. (A.17) −∞

Ð This result, which is a function of t, and not of u, is now equal to z(t). • In frequency-domain equations using the short notation, the integration variable is always f . The procedure is similar to the one described above. • In equations where other integration variables are used, or where confusion is possible in some other way, the long notation will be used.

A.2.2 Properties

A.2.2.1 Correspondence with Multiplication

Let W( f ), X( f ), Y ( f ), and Z( f ) be the CFT spectra of signals w(t), x(t), y(t), and z(t), respectively. It can easily be shown that if 282 Appendix A: Definitions, Conventions and Overview of Used Theory

w(t) = x(t) · y(t), (A.18) then W( f ) = X( f ) ∗ Y ( f ). (A.19)

Also, if z(t) = x(t) ∗ y(t), (A.20) then Z( f ) = X( f ) · Y ( f ). (A.21)

This property shows the importance of the convolution operation in communica- tion theory.

A.2.2.2 Properties Shared with Multiplication

Since the CFT and ICFT are bijections, it follows from (A.18)Ð(A.21) that convolu- tion has a lot of properties in common with multiplication: • It is commutative: x(u) ∗ y(u) = y(u) ∗ x(u). (A.22)

• It is associative:

(x(u) ∗ y(u)) ∗ z(u) = x(u) ∗ (y(u) ∗ z(u)) = x(u) ∗ y(u) ∗ z(u). (A.23)

In the long notation, this becomes

[[x(v) ∗v y(v)] (w) ∗w z(w)] (u) = [x(v) ∗v [y(w) ∗w z(w)] (v)] (u) (A.24) = [x(v) ∗v y(v) ∗v z(v)] (u).

• It is distributive with respect to addition:

x(u) ∗ (y(u) + z(u)) = x(u) ∗ y(u) + x(u) ∗ z(u). (A.25)

• If a is a real number, then

x(u) ∗ (ay(u)) = (ax(u)) ∗ y(u) = a(x(u) ∗ y(u)). (A.26)

A.2.2.3 No Mutual Associativity

Since the convolution shares a lot of properties with multiplication, the commonly used notation x(t) ∗ y(t) was chosen very similar to multiplication. However, in Appendix A: Definitions, Conventions and Overview of Used Theory 283 calculations where both multiplication and convolution occur, one should note that while both operations are associative, they are not mutually associative, i.e.

(x(t) · y(t)) ∗ z(t) = x(t) · (y(t) ∗ z(t)), (A.27) as can easily be verified from (A.9). Therefore, the notation x(t)· y(t)∗ z(t) (without parentheses) is ambiguous and should be avoided. Of course, in the special case where x(t) = a is a constant, there is no problem as (ay(t)) ∗ z(t) = a(y(t) ∗ z(t)) = ay(t) ∗ z(t).

A.2.2.4 Convolution and Time Shift

Assume z(t) = [x(u) ∗u y(u)] (t).Using(A.9), it is easy to show that

z(t − t0) = [x(u) ∗u y(u)] (t − t0) = [x(u − t0) ∗u y(u)] (t) = [x(u) ∗u y(u − t0)] (t) , (A.28) i.e. shifting the convolution of two signals in time by t0 is equivalent to shifting either one (but only one) of the signals by t0 and then convolving them. Because of the above property, the short notation can be used as well without causing any ambiguity:

z(t − t0) = x(t − t0) ∗ y(t) = x(t) ∗ y(t − t0). (A.29)

However, note that z(t − t0) = x(t − t0) ∗ y(t − t0), even though the definition z(t) = x(t)∗ y(t) may seem to suggest this. The time shift must be applied to exactly one of the operands of the convolution operator. A consequence of this time shift property is that

x(t − t0) ∗ y(t + t0) = x(t) ∗ y(t). (A.30)

Of course similar properties hold for a frequency shift in the frequency domain. The relation between convolution and time shift will be revisited more extensively in Sect. A.4.2.4 after the introduction of the Dirac delta function.

A.3 Some Basic Functions

A.3.1 The Sinc Function

The normalized sinc function sinc(x) is defined as 284 Appendix A: Definitions, Conventions and Overview of Used Theory

Fig. A.1 The sinc function 1

0.5 sinc(x) 0

−5 −4 −3 −2 −1 0 1 2 3 4 5 x

sin(πx) sinc(x) = (A.31) πx andisshowninFig.A.1. It is equal to 0 at all nonzero integer values of x, and only there, and it is equal to 1 for x = 0. The function is called normalized because it integrates to one:  ∞ sinc(x) dx = 1. (A.32) −∞

The unnormalized sinc function is defined as   x sin(x) sinc = . (A.33) π x

In this work, only the normalized sinc function, denoted with sinc(x), will be used. Care should be taken when comparing results with other works, as some authors use the notation sinc(x) for the unnormalized sinc function.

A.3.2 The Rectangular Function

The rectangular function (x) is defined as [1]  1, |x|≤ 1 (x) = 2 (A.34) , | | > 1 0 x 2 and is plotted in Fig. A.2 Thus,  (t/T ) is a rectangular pulse with width T and height 1. It can be shown [1, 4] that  t F  = T sinc( fT). (A.35) T Appendix A: Definitions, Conventions and Overview of Used Theory 285

Fig. A.2 The rectangular 1 function 0.8 0.6 (x)

Π 0.4 0.2 0 −2 −1 0 1 2 x

Fig. A.3 The signum function 1

0.5

0 sgn(x) −0.5

−1 −2 −1 0 1 2 x

A.3.3 The Signum Function

The signum function sgn(x) is defined as ⎧ ⎨ 1, x > 0, ( ) = , = , sgn x ⎩ 0 x 0 (A.36) −1, x < 0, andshowninFig.A.3.Thevalueatx = 0 is somewhat arbitrary and is not important in most applications.

A.3.4 The Four-Quadrant Arctangent Function

The four-quadrant arctangent function atan2(y, x) is defined as [9] ⎧ ⎪ ( / ) , > , ⎪ arctan y x x 0 ⎪ ( / ) + π, ≥ , < , ⎨⎪ arctan y x y 0 x 0 ( / ) − π, < , < , ( , ) = arctan y x y 0 x 0 atan2 y x ⎪ π/ , > , = , (A.37) ⎪ 2 y 0 x 0 ⎪ −π/ , < , = , ⎩⎪ 2 y 0 x 0 undefined, y = 0, x = 0, 286 Appendix A: Definitions, Conventions and Overview of Used Theory

Fig. A.4 The four-quadrant 4 x > 0 arctangent function x < 0 2

0

atan2(y,x) −2

−4 −10 −5 0 5 10 y/x and plotted as a function of y/x in Fig. A.4. It differs from the arctangent function arctan(y/x) in that its result range is (−π, π], while arctan has range (−π/2,π/2). It is always true that    y y tan(atan2(y, x)) = tan arctan = . (A.38) x x

The atan2 function is useful to determine the angle of a complex number: any nonzero complex number z = x + jy can be written as z = aejθ , where  a = x2 + y2, (A.39)

θ = atan2(y, x). (A.40)

If z = 0, then a = 0 and θ can have any value, so if desired, any value can be assigned to atan2(0, 0).

A.4 The Dirac Delta Function

A.4.1 Definition and Notation

The Dirac delta function δ(x) (also known as Dirac impulse or delta function)is generally defined by the following two equations:

δ(x) = 0 ∀x = 0 (A.41)

 ∞ δ(x) dx = 1. (A.42) −∞

Thus, δ(x) is infinite for x = 0. The delta function is not a function in the usual sense of the word and has some anomalies that are worth noting. Appendix A: Definitions, Conventions and Overview of Used Theory 287

First of all, the actual value of the delta function in the origin depends on the dimension of the horizontal axis. For example, in the time-domain delta function δ(t), the independent variable t has units of seconds. In order for (A.42)tobe satisfied, δ(t) must have units of 1/s = Hz. In the frequency-domain delta function δ( f ), on the other hand, the independent variable f has units of Hz, and thus δ( f ) must have units of 1/Hz = s. Thus, expressions like δ(0) or δ(a) where a is a constant, do not have a very clear meaning, since their value depends on the dimension of the horizontal axis. Also, care should be taken when scaling the independent variable. That is, if one evaluates the delta function in the point 2x, either x = 2x = 0, or x = 0 and δ(x) = δ(2x) = 0. Thus one can conclude that

δ(2x) = δ(x) ∀ x. (A.43)

However, this is only correct if one also changes the integration variable in (A.42) x to 2 , i.e.  ∞  ∞ δ(2x) d(2x) = δ(u) du = 1. (A.44) −∞ −∞

If one integrates over x instead, one finds that   ∞ 1 ∞ 1 δ(2x) dx = δ(u) du = . (A.45) −∞ 2 −∞ 2 and thus 1 δ(2x) = δ(x). (A.46) 2 In order to avoid confusion, it is best to use only arguments of the form u+c, where u is the independent variable and c is a constant. The constant does not influence the integral in (A.42) and thus does not cause confusion. This is also the way the delta function normally occurs in typical signal-processing-related expressions, and the way it will be used in this work. Also, the Dirac delta will only be used with independent variables t and f .

A.4.2 Properties

A.4.2.1 Fourier Transform

It can be shown that  ∞ π e j2 uv du = δ(v). (A.47) −∞

Replacing u by t and v by − f , this means that 288 Appendix A: Definitions, Conventions and Overview of Used Theory

F {1} = δ( f ). (A.48)

Similarly, replacing u by f and v by t, one finds that

F {δ(t)} = 1. (A.49)

Furthermore, it can be seen from (A.47) and (A.41)Ð(A.42) that

− j2π ft0 F {δ(t − t0)} = e (A.50) and   j2π f0t F e = δ( f − f0). (A.51)

Using (A.51) and the fact that   1 − cos(x) = e jx + e jx , (A.52) 2   1 − sin(x) = e jx − e jx , (A.53) 2 it is easy to derive the Fourier transforms of the sine and cosine functions, which are given in Table A.1.

A.4.2.2 The Sampling Property

From the definition of the delta function, it follows [1, 4] that

 ∞ x(u)δ(u − u0) du = x(u0). (A.54) −∞

This is known as the sampling property or sifting property of the Dirac delta function.

A.4.2.3 Convolution

Since δ(u) = δ(−u), it can be seen that the integral in (A.54) is equal to the convo- lution of x(u) and δ(u), evaluated in u0. Replacing u0 by t, this gives

 ∞ [x(u) ∗u δ(u)] (t) = x(u)δ(t − u) du (A.55) −∞  ∞ = x(u)δ(u − t) du −∞ = x(t), Appendix A: Definitions, Conventions and Overview of Used Theory 289 or using the short notation:

x(t) ∗ δ(t) = x(t). (A.56)

This means that the Dirac delta function is the unity element for convolution. This also follows from the fact that its Fourier transform is 1 and that convolution corresponds to multiplication after applying the Fourier transform. Replacing δ(u) by δ(u − t0) in (A.55)gives

 ∞ [x(u) ∗u δ(u − t0)] (t) = x(u)δ(t − u − t0) du −∞  ∞ = x(u)δ(u − (t − t0)) du −∞ (A.54) = x(t − t0), or with the short notation

x(t) ∗ δ(t − t0) = x(t − t0), (A.57) i.e. convolving a signal x(t) with a Dirac impulse shifted in time by t0 shifts the signal by the same amount t0. Similarly,

X( f ) ∗ δ( f − f0) = X( f − f0), (A.58) i.e. convolving a spectrum X( f ) with a Dirac impulse shifted in frequency by f0 shifts the spectrum by the same amount f0.

A.4.2.4 Delta Function and Time Shift

After having introduced the Dirac delta function, it is useful to revisit the relation between time shift and convolution. In Sect. A.2.2.4, it was shown that if z(t) = x(t) ∗ y(t) and z0(t) = z(t − t0), then

z0(t) = x(t − t0) ∗ y(t) = x(t) ∗ y(t − t0). (A.59)

Applying (A.57) shows that

z0(t) = x(t) ∗ y(t) ∗ δ(t − t0). (A.60)

All of this can also be proven using the correspondence between multiplication and time shift and the time shift property of the Fourier transform (see Sect. A.1.2). 290 Appendix A: Definitions, Conventions and Overview of Used Theory

This property says that

j2π ft0 F {x(t − t0)} = X( f ) · e , (A.61)

j2π ft0 F {y(t − t0)} = Y ( f ) · e , (A.62)

j2π ft0 Z0( f ) = F {z(t − t0)} = Z( f ) · e . (A.63)

Since Z( f ) = X( f ) · Y ( f ) , it follows that

j2π ft0 Z0( f ) = X( f ) · Y ( f ) · e . (A.64)

j2π ft which proves (A.59) since F {δ(t − t0)} = e 0 .

A.5 Convolution Powers

For the derivations made in Chap. 3, it is useful to define the concept of convolution power. The convolution powers x∗n(u) of a signal or spectrum x(u) are defined as follows

∗ x 0(u) = δ(u) (A.65)   ∗n ∗n−1 x (u) = x (v) ∗v x(v) (u) , for all integer n ≥ 1. (A.66)

That is,

x∗0(u) = δ(u) x∗1(u) = x(u) x∗2(u) = x(u) ∗ x(u) x∗3(u) = x(u) ∗ x(u) ∗ x(u) etc.

Convolution powers correspond to multiplicative powers like convolution corre- sponds to multiplication: if X( f ) = F {x(t)}, then   ∗ X n( f ) = F xn(t) (A.67) and   ∗ X n( f ) = F x n(t) , (A.68) where xn(u) = (x(u))n is the nth power of x(u). Appendix A: Definitions, Conventions and Overview of Used Theory 291

Since any continuous function g(x) can be written as a, possibly infinite, weighted sum of powers of x using Taylor series expansion, it follows that for a time-domain signal x(t) with spectrum X( f ), the spectrum of signal g(x(t)), where g(x) is any function, can be written in terms of the convolution powers X ∗n( f ). This principle was used by Song and Sarwate [7] to calculate the spectrum of a PWM signal. While it is generally not trivial to calculate X ∗n( f ) for high values of n, this property can still be practical to get an idea of what a certain spectrum will look like. It is used extensively in Chap. 3. For readers familiar with the theory of Volterra series (see Appendix B in [8]), it can be interesting to note the relation with convolution powers1:Thenth convolution power x∗n(u) of a function x(u) can be viewed as the nth-order term in a Volterra series where all Volterra kernels are constant and equal to 1.

A.6 Power

A.6.1 Average Power and Power Spectral Density

The average power or power in a signal x(t) is defined as [1, Eq. (2.13)]  1 T/2 P {x(t)} = lim |x(t)|2dt. (A.69) T →∞ T −T/2

If one defines

t x (t) = x(t) ·  , (A.70) T T then (A.69) can be written as

 ∞ 1 2 P {x(t)} = lim |xT (t)| dt. (A.71) T →∞ T −∞

Using Parseval’s theorem (see (A.7)),itfollowsthat

 ∞ 1 2 P {x(t)} = lim |XT ( f )| d f T →∞ T −∞  ∞ = Px ( f )d f, (A.72) −∞ where

1 2 Px ( f ) = lim |XT ( f )| (A.73) T →∞ T

1 Thanks to Dimitri De Jonghe for pointing this out. 292 Appendix A: Definitions, Conventions and Overview of Used Theory is called the power spectral density (PSD) of x(t) [1, Sect. 2Ð3] [4, Sect. 4.11]. Whenever signals are treated in the frequency domain, P {X( f )} will be written instead of P {x(t)}, where X( f ) = F {x(t)}. However, there is no simple expression to calculate P {X( f )} from X( f ) without going back to the time domain.

A.6.1.1 Properties of the PSD

This section lists some properties of the PSD that are relevant in this work. Proofs for them, as well as more properties, can be found in [4, Sect. 4.11]. Assuming a signal x(t) with spectrum X( f ) and PSD Px ( f ), the following properties hold

• If x(t) is real, then Px ( f ) is even, i.e. Px ( f ) = Px (− f ). • Px ( f ) ≥ 0 ∀ f . • If a linear time-invariant filter with impulse response h(t) is applied to x(t), then the filter output y(t) = h(t) ∗ x(t) (with spectrum Y ( f ) = H( f )X( f ) where H( f ) = F {h(t)} is called the transfer function of the filter) has an average power given by

 ∞ 2 P {Y ( f )} = P {H( f ) · X( f )} = |H( f )| · Px ( f )d f. (A.74) −∞

A.6.2 In-band Power

Given a certain frequency band defined by the frequencies fa and fb with fa < fb, the in-band power of a signal x(t), denoted Pˆ {x(t)} or Pˆ {X( f )}, is defined as

  − f f ˆ ˆ a b P {x(t)} = P {X( f )} = Px ( f )d f + Px ( f )d f (A.75) − fb fa  ∞ = W( f ) · Px ( f )d f (A.76) −∞ = P {W( f ) · X( f )}, (A.77)

where

f − ( f + f )/2 f + ( f + f )/2 W( f ) =  a b +  a b . (A.78) fb − fa fb − fa and the rectangle function (x) is defined in Sect. A.3.2. This definition of in-band power assumes an RF signal band for a real signal, so that the power in both the bands [− fb, − fa] and [ fa, fb] should be taken into Appendix A: Definitions, Conventions and Overview of Used Theory 293 account. However, the formula can also be used for a baseband where the signal band ˆ is [− fb, fb] by setting fa = 0. In both cases P {x(t)} gives the total power present in the relevant signal bands only and disregards the out-of-band power. EquationA.77 shows that Pˆ {X( f )} corresponds to the total power in X( f ) after applying the ideal lowpass filter W( f ) to it, as can be seen from (A.74) (note that |W( f )|2 = W( f )). It follows that the in-band power of a signal Y ( f ) = H( f )X( f ), where H( f ) is the transfer function of a linear time-invariant filter, is given by

Pˆ {H( f )X( f )} = P {W( f )H( f )X( f )}  ∞ 2 = |W( f )H( f )| · Px ( f )d f −∞  ∞ 2 = W( f )|H( f )| · Px ( f )d f −∞   − fa fb 2 2 = |H( f )| · Px ( f )d f + |H( f )| · Px ( f )d f. − fb fa (A.79)

In addition, if x(t) and h(t) are real, it follows that |H( f )| and Px ( f ) are even functions of f and  fb ˆ 2 P {H( f )X( f )} = 2 |H( f )| · Px ( f )d f. (A.80) fa

A.6.3 Power of a Product

This section shows that the power in the product of two uncorrelated ergodic signals is equal to the product of the powers of these signals. This result is used in Chap. 3. For a real, ergodic signal x(t) with mean μx and standard deviation σx , it can be shown that the average power P {x(t)} in x(t) is equal to [1, Sect. 6.1]

P { ( )} = σ 2 + μ2. x t x x (A.81)

In this work, all signals are assumed to be ergodic. More information about ergodicity can be found in [4, Sect. 4.9] [3, Sect. 3.3.6] [1, Sect. 6.1]. It is shown in [2, Eq. (2)] that the variance of the product of two uncorrelated signals x(t) and y(t) is

σ 2 = μ2σ 2 + μ2σ 2 + σ 2σ 2. xy x y y x x y (A.82)

Using (A.81), it follows that the power in x(t)y(t) is 294 Appendix A: Definitions, Conventions and Overview of Used Theory

P { ( ) · ( )} = μ2σ 2 + μ2σ 2 + σ 2σ 2 + μ2 . x t y t x y y x x y xy (A.83)

Since x(t) and y(t) are uncorrelated, μxy = μx μy and

P { ( ) · ( )} = (μ2 + σ 2) · (μ2 + σ 2) = P { ( )} · P { ( )} . x t y t x x y y x t y t (A.84)

This proves the stated theorem.

A.7 Signal to Noise and Distortion Ratio

The signal to noise and distortion ratio (SNDR) is defined as the ratio of the in-band desired signal power to the in-band noise and distortion power. Thus, considering a signal

M X( f ) = S( f ) + i S( f ), (A.85) i=1 where S( f ) is the desired signal and i S( f ) are uncorrelated noise and distortion terms, the SNDR of X( f ) is equal to

Pˆ {S( f )} SNDR =  , (A.86) M Pˆ { ( )} i=1 i S f where the in-band power Pˆ {.} is defined in Sect. A.6.2.

A.8 Error Vector Magnitude

Most communication standards define signal quality in terms of the error vector magnitude (EVM), which is defined in this section. The definition of EVM is always based on a constellation plot of a signal, which is introduced in Sect. 2.1.3. In case of a single-carrier modulation scheme (e.g. N-QAM), one can denote the different constellation points with ci , where i = 0, 1, 2,...,N − 1 and N is the number of constellation points. Ideally, the complex envelope (before filtering) is a discrete-time signal gk where k denotes the kth symbol period, and for each k, gk is equal to some ci . However, due to nonidealities in the transmitter, the real complex envelope will deviate from this. In order to calculate EVM, a receiver is added (in order to evaluate a transmitter only, an ideal receiver is used) which tries to demodulate the signal and produces a constellation plot of the real signal. On this plot, one point rk is plotted for Appendix A: Definitions, Conventions and Overview of Used Theory 295 each received symbol. Since the points will slightly deviate from the ideal points gk, such a constellation points will show clouds of points rather than individual points. The absolute EVM is defined as the root-mean-square (RMS) deviation of each point rk from the ideal point gk:ifM symbols were demodulated, the absolute EVM is given by    M−1  1 EVM = |r − g |2. (A.87) abs M k k k=0

Note that rk and gk are complex numbers. Usually, the EVM is normalized with respect to the RMS constellation power:   / M−1 | − |2. 1 M k=0 rk gk EVM =   . (A.88) / N−1 | |2 1 N i=0 ci

This gives the relative EVM, which is usually simply referred to as EVM (including in this work). It is expressed in % or in dB. Evaluating (A.88) requires knowledge of the ideal signal gk , which is often unprac- tical in a real-world measurement setup. For this reason, it is often assumed that for every constellation point rk, the ideal point gk is the point that is closest to rk.Thisis normally correct unless the EVM is very bad and the clouds overlap. The definition of EVM then becomes   / · M−1 | − |2. 1 M k=0 mini rk ci EVM =   . (A.89) / · N−1 | |2 1 N i=0 ci

This definition is equivalent to (A.88) as long as the EVM is sufficiently good, but saturates around a certain value when the EVM becomes so bad that the clouds merge to one cloud. This is not a problem as such a transmitter has no practical use. The definition in (A.89) is used throughout this work. Care should be taken when comparing with other work, since other definitions are sometimes also used. For example, in some papers and standards (e.g. high- rate WPAN [11, Sect. 11.5.1]) the EVM is normalized with respect to the peak constellation power rather than the RMS value:   / · M−1 | − |2. 1 M k=0 mini rk ci EVM = . (A.90) maxi |ci |

The definition of EVM for OFDM signals (see Sect. 2.1.4) is similar, but the constellation points for all subcarriers are considered together. EquationA.89 then becomes 296 Appendix A: Definitions, Conventions and Overview of Used Theory      /( ) · P−1 M−1  − 2. 1 MP p=0 k=0 mini rk,p ci EVM =   , (A.91) / · N−1 | |2 1 N i=0 ci where P is the number of modulated subcarriers and rk,p is the complex envelope of the pth subcarrier during the kth symbol period. Similar definitions are also used by e.g. the WLAN [10, Sect. 17.3.9.7] and LTE [5, Sect. E.2] standards.

A.9 Notation of Noise and Distortion Terms

In Chap. 3 and the following chapters, noise and distortion signals will are using a capital letter  with an index, followed by the signal or spectrum that is influenced by the noise or distortion. The index indicates the type or cause of the nonideality. For example, if an ideal phase signal ϕ(t) is quantized to yield ϕq(t), the quantization noise on this signal will be denoted Δqϕ(t) (i.e. Δqϕ(t) = ϕq(t) − ϕ(t)), where the index “q” stands for quantization. Thus, Δqϕ should be read as “the deviation (Δqϕ) from the ideal ϕ due to quantization (q).” The spectrum of Δqϕ(t) is denoted Δq ( f ). When dealing with the nth power of a signal, square brackets will be used to distinguish two different cases: n n •[Δqϕ] (t) = (Δqϕ(t)) is the nth power of Δqϕ(t), and its spectrum is denoted ∗n [ΔqΦ] ( f ). n n n • q[ϕ ](t) indicates the deviation on ϕ (t) due to quantization noise, i.e. q[ϕ ](t) = ϕn( ) − ϕn( ) [Δ ϕ]n( ) Δ [Φ∗n]( ) q t t . The spectrum of q t will be denoted q f . n n It is important to clearly distinguish [Δqϕ] (t) and [Δqϕ] (t) as they are not equal. The signals and spectra mentioned here were just taken as examples. They are defined more clearly in Sect. 3.1.3. Appendix B Derivations and Considerations Regarding PWM

B.1 Derivation of PWM Formulas

In this section, the time- and frequency-domain PWM formulas listed in Sect. 3.2.3 are derived from equations given in [7]. Parts of the derivations made in [7] are summarized more concisely in [6], which is co-authored by the authors of [7]. Where applicable, the corresponding equations in [6] will also be noted. This section will partly use the symbols used in [7] (not [6]), while replacing certain other symbols by the symbols used in the rest of this work. For example, [7] uses the notations fc and T for the PWM frequency and period, respectively. To avoid confusion, these symbols will be replaced by fpwm and Tpwm here.

B.1.1 Generic and Signal-Dependent TEPWM Signals

As explained in [7, Sect. 2.2] [6, Sect. II-A], TEPWM signals are written as

pTE(t) = pc(t) + ps,TE(t), (B.1) where pc(t) is called the generic TEPWM signal and ps,TE(t) is called the signal- dependent part of the TEPWM signal. pc(t) represents a PWM signal with a constant duty cycle of 50 %. It is given by [7, Eq. (2)] to be equal to

∞  4 pc(t) = sin(2π(2k + 1) f t), (B.2) (2k + 1)π pwm k=0 which in this work will be written as

P. A. J. Nuyts et al., Continuous-Time Digital Front-Ends for Multistandard 297 Wireless Transmission, Analog Circuits and Signal Processing, DOI: 10.1007/978-3-319-03925-1, © Springer International Publishing Switzerland 2014 298 Appendix B: Derivations and Considerations Regarding PWM

∞  4 p (t) = sin(2πlf t). (B.3) c lπ pwm l=1,3,...

The spectrum of pTE(t) is given by

PTE( f ) = Pc( f ) + Ps,TE( f ), (B.4) where Pc( f ) is given by [7,Eq.(3)]:

∞  2   P ( f ) = δ( f − lf ) − δ( f + lf ) , (B.5) c jlπ pwm pwm l=1,3,... ±∞  2 = δ( f − lf ), (B.6) jlπ pwm l=±1,±3,... where the Dirac delta function δ( f ) is defined in Sect. A.4.1. The signal-dependent part ps,TE(t) with spectrum Ps,TE( f ) is different for UTEPWM and NTEPWM and will be calculated in the following sections.

B.1.2 Uniform-Sampling Trailing-Edge PWM

UTEPWM signals are analyzed in [7, Sect. 2.3] [6, Sect. II-B]. The signal-dependent part Ps,TE( f ; b) of their spectrum is given by [7,Eq.(6)][6, Eq. (4)]:

∞ ∞ (− π )n−1 − jπ fT j fTpwm ∗n P , ( f ; b) = e pwm B ( f − kf ), (B.7) s TE n! pwm k=−∞ n=1 where the b between the brackets (x is used in [7]) indicates the spectrum is calculated for an input b(t). Combining this with (B.4) and (B.6) readily proves (3.56). According to [7, Eq. (12)] [6, Eq. (10)],

∞    2 p (t; b) = y(t; b) + sin(2πkf t) − (−1)k sin(2πkf t − kπy(t; b)) . TE kπ pwm pwm k=1 (B.8) Here, y(t; x) is given by [7, Eq. (9)] [6, Eq. (7)] to be

∞ −   T  1 T n 1 dn−1 T n y(t; b) = b x − pwm + − pwm b t − pwm . 2 n! 2 dtn−1 2 n=2 (B.9) These equations are equal to the time-domain formulas (3.54) and (3.55). Appendix B: Derivations and Considerations Regarding PWM 299

B.1.3 Natural-Sampling Trailing-Edge PWM

NTEPWM signals are analyzed in [7, Sect. 2.5] [6, Sect. II-D]. Both papers denote NPWM signals with a hat (ˆ), so that ˆ ˆ PTE( f ) = Pc( f ) + Ps,TE( f ). (B.10)

ˆ 2 [7, Eq. (36)] gives the signal-dependent part Ps,TE( f ; b) of the spectrum as

∞ ∞ ( π)n−1  ˆ k jk ∗n P , ( f ; b) = B( f ) + (−1) B ( f + kf ) s TE n! pwm k=1 n=1  n−1 ∗n + (−1) B ( f − kfpwm) (B.11)

This can be rewritten as

∞ ∞ (− π)n−1 ˆ k jk ∗n P , ( f ; b) = B( f ) + (−1) B ( f − kf ) (B.12) s TE n! pwm k=1 n=1 −1 ∞ n−1 (− jkπ) ∗ + (−1)k B n( f − kf ) n! pwm k=−∞ n=1  ∞ n−1 (− jkπ) ∗ = B( f ) + (−1)k B n( f − kf ), n! pwm k=0 n=1  −∞ ∞ where k=0 means the sum over all integers (i.e. from to ) except 0. Combining this result with (B.10) and (B.6) leads directly to (3.58). The time domain formula (3.57) is an exact copy of (37) (replacing x(t) by b(t)).

B.1.4 Uniform-Sampling Double-Edge PWM

Symmetric UDEPWM signals are analyzed in [7, Sect. 3.1]. According to [7, Eq. (48)],

− jπ fTpwm/2 PDE,S( f ; b) = e Pc( f ) ∞ ∞ (− π )n−1 − jπ fT − jπ fT /2 j fTpwm ∗n + e pwm e pwm B ( f − kfpwm) 2n · n! k=−∞ n=1

2 The right-hand side of this equation is equal to the right-hand side in [6, Eq.(22)]. However, it seems the left-hand side in [6] was erroneously written as the complete PWM signal PWM( f ), while it should have been the signal-dependent part Pˆ( f ). 300 Appendix B: Derivations and Considerations Regarding PWM

∞ ∞ ( π )n−1 − jπ fT jπ fT /2 j fTpwm ∗n + e pwm e pwm B ( f − kfpwm), 2n · n! k=−∞ n=1 (B.13) where the capital S in the subscript stands for “symmetrical”. Using [7, Eq. (3)], this can be rewritten as ±∞ − π / 2 P ( f ; b) = e j fTpwm 2 δ( f − lf ) (B.14) DE,S jlπ pwm l=±1,±3,... ∞ ∞ − π 1 ∗ + e j fTpwm B n( f − kf ) n · ! pwm =−∞ = 2 n  k n 1

− jπ fTpwm/2 n−1 jπ fTpwm/2 n−1 e (− jπ fTpwm) + e ( jπ fTpwm) , which is equal to (3.62). The time-domain formulas are derived as follows. As explained at the end of [7, Sect. 3.1], the symmetric UDEPWM signal pDE,S(t; b) can be written as

1 + b 1 + b p (t; b) = p t; + p t; − 1, (B.15) DE,S TE 2 LE 2 where pTE(t; b) is given by (B.8) and pLE(t; b) is a uniform-sampling leading-edge PWM (ULEPWM) signal. As explained in the beginning of [7, Sect. 2.9],

pLE(t; b) =−pTE(t;−b). (B.16)

Thus, (B.15) can be written as

1 + b 1 + b p (t; b) = p t; − p t;− − 1, (B.17) DE,S TE 2 TE 2

Substituting (B.8)gives

1 + b 1 + b p (t; b) = y t; − y t;− − 1 DE,S 2 2 ∞   2 1 + b − (− )k πkf t − kπy t; π 1 sin 2 pwm = k 2 k 1 1 + b − sin 2πkf t − kπy t;− , (B.18) pwm 2 where y(t; b) is given by (B.9). Defining Appendix B: Derivations and Considerations Regarding PWM 301

1 + b y+(t) = y t; (B.19) 2 1 1 T = + b x − pwm 2 2 2   ∞ n−1 n−1 n 1 1 Tpwm d Tpwm + − x t − (B.20) 2n n! 2 dtn−1 2 n=2 1 + b y−(t) = y t;− (B.21) 2 1 1 T =− − b x − pwm 2 2 2   ∞ n−1 n−1 n 1 1 Tpwm d Tpwm + − x t − (B.22) (−2)n n! 2 dtn−1 2 n=2

leads to (3.59)Ð(3.61).

B.1.5 Natural-Sampling Double-Edge PWM

Asymmetric NDEPWM signals are analyzed in [7, Sect. 3.3]. According to [7, ˆ Eq. (60)], the signal-dependent part Ps,DE,A( f ; b) of the spectrum is equal to

∞ ∞ ( π)n−1   ˆ k jk n+k−1 Ps, , ( f ; b) = B( f ) + (− j) 1 + (−1) DE A 2n · n!  k=1 n=1  ∗n ∗n B ( f + kfpwm) + B ( f − kfpwm) , (B.23) where the A in the index stands for “asymmetrical”. Using the fact that the complete spectrum is given by ˆ ˆ PDE,A( f ; b) = Pc( f ) + Ps,DE,A( f ; b) (B.24) and substituting (B.6) and (B.23) proves (3.65). The time-domain formula (3.63) is identical to [7, Eq. (63)].

B.2 Is RF PWM a Special Case of Baseband PWM?

One may be tempted to think that an RF PWM transmitter is simply a baseband PWM transmitter where the PWM frequency fpwm has been set equal to the carrier frequency fc. This assumption will be investigated in this section, and it will be shown 302 Appendix B: Derivations and Considerations Regarding PWM

Fig. B.1 Illustration of how (a) a differential RF PWM signal 1 without PM can be generated using baseband PWM with -1 fpwm = 2 fc a Square-wave carrier psq(t) = csq(t). b . (b) Baseband PWM at 2 fc c 1 Differential RF PWM at fc = product of the above -1

(c) 1

-1

that it is not correct. However, it will also be shown that with some modifications, RF PWM can actually be considered a special case of baseband PWM. If the above assumption is true, it should be possible to produce the RF PWM signal given by (3.150) on p. 102 by applying a certain input abb(t) to a baseband PWM modulator with fpwm = fc and multiplying the result with a square-wave PMC like the one given by (3.16) on p. 55. In order to find this abb(t), assume first that there is no phase modulation, i.e. ϕ(t) =0. Since the RF PWM signal in (3.150) is a natural-sampling double-edge PWM (NDEPWM) signal, it seems logical that the baseband PWM signal should also be an NDEPWM signal. However, the RF PWM signal is also differential and thus it actually has two pulses per period Tc. For this reason, the baseband PWM frequency fpwm should not be fc but 2 fc, as can be seen from Fig. B.1. In this case, indeed the multiplication of the baseband PWM signal with the PMC does result in the desired RF PWM signal. However, if the PMC was phase- modulated, this would not be the case as can be seen in Fig. B.2. This is because in standard baseband PWM, the phase and amplitude paths are independent of each other so that the PWM pulses do not align with the phase-modulated carrier pulses. In RF PWM, on the other hand, PWM and PM happens on the same pulses. Thus, in order to generate a phase-modulated RF PWM signal using baseband PWM, not only the carrier should be phase-modulated but also the baseband PWM pulses. Clearly, implementing RF PWM this way is not very useful since it is much more complicated than the implementation used in Chap. 6. This section only serves to illustrate that RF PWM cannot generally be considered to be a special case of baseband PWM, unless the baseband PWM modulator is created in a very specific way. Going back to the case without phase-modulation, one may note what appears to be an inconsistency in the PWM models: According to (3.75) (p. 71), when a baseband PWM modulator is fed an input signal abb(t), the resulting duty cycle is given by dbb(t) = abb(t). (B.25) Appendix B: Derivations and Considerations Regarding PWM 303

Fig. B.2 Illustration of how (a) a differential RF PWM signal 1 with PM cannot be generated using simple baseband PWM. a Square-wave PMC psq(t). -1 b Baseband PWM at 2 fc. (b) c Product of the above 1

-1 (c) 1

-1

In the absence of nonidealities such as quantization, abb(t) will also be the ampli- tude of the RF carrier at fc. When considering the RF PWM signal, the RF amplitude arf (t) is given by (3.128) (p. 97)to be

arf (t) = sin(πdrf (t)). (B.26)

Clearly, since both types of PWM result in the same signal, arf (t) should be equal to abb(t). However, from Fig. B.1 it can also be seen that

dbb(t) = 2drf (t), (B.27) since the pulse widths are the same but the pulse period Tc for RF PWM is twice as long as the period Tpwm for baseband PWM. Together with (B.25) and (B.26), this contradicts the requirement that abb(t) = arf (t). This is explained by the p effect, which is discussed in Sect. 3.4.4.2. This effect can be summarized by stating that in baseband PWM, the carrier harmonics and the PWM harmonics cause intermodulation products which may fall into the signal band depending on the ratio fc/fpwm. The case under consideration here is the worst possible case of the p effect: since fc/fpwm = 1/2, the p value is 2, which is the worst possible value. Furthermore, it was explained in Sect. 3.4.4.2 that for a given p value, the p effect becomes less significant when the ratio fc/fpwm increases. However, 1/2 is the lowest possible value of this ratio for which p = 2, which makes the p effect very large in this particular case. Thus, in addition to the in-band term which has an amplitude abb(t) = dbb(t) = 2drf (t), there are a lot of intermodulation terms centered at fc, which are so large that they can no longer be considered as small noise or distortion terms, but should be considered as part of the signal. The RF PWM model shows no terms at fc apart from the desired RF signal, which has an amplitude arf (t) = sin(πdrf (t)). Since both models are exact and consider the same signal shown in Fig. B.1c, it follows that the sum of abb(t) and the amplitudes of all the p effect terms is equal to arf (t). 304 Appendix B: Derivations and Considerations Regarding PWM

In ordinary baseband PWM, the PWM input abb(t) is taken to be equal to the desired RF amplitude. In this case, this would result in very bad performance due to the extremely high p effect. In RF PWM, this effect is taken into account by recog- nizing that the RF amplitude is not equal to the duty cycle drf (t) but to sin(πdrf (t)). For this reason, RF PWM can still achieve good signal quality. To conclude, one can say that it is possible but not very useful to produce a differential RF PWM signal using baseband PWM with a PWM frequency of 2 fc— not fc as one might expect intuitively. However, in order to do so, the baseband PWM input should be taken to be

2 a (t) = d (t) = a (t), bb 2 rf π arcsin rf (B.28) and furthermore the baseband PWM output must be phase-modulated with the same phase signal ϕ(t) as the square-wave RF carrier. For these reasons, one cannot simply say that RF PWM is a special case of ordinary baseband PWM.

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Symbols C  , see delta-sigma modulation Carrier, 15 , see delta-sigma modulation Carrier frequency, 16 Carrier PWM, see pulse width modulation, baseband A CFT, see continuous Fourier transform see AM, amplitude modulation Channel spacing, 90, 203 AM signal, 16 Class-D power amplifier, 24 Amplitude, 16 Class-E power amplifier, 26 Amplitude modulation, 7, 16 Amplitude shift keying, 19 Coding efficiency, see efficiency, coding Analog-to-digital converter, 11, 131 Complex conjugate, 279 AND gate Complex envelope, 18 multiplexer-based, 162 Constellation plot, 18, 20, 204, 214, 240, 241, static CMOS, 161 294 symmetrical, 161 Continuous convolution, see convolution Angle Continuous Fourier transform, 51, 277 of complex number, 286 properties, 278 outphasing, see outphasing angle Conversion efficiency, see efficiency, Arctangent conversion four-quadrant, 285 Convolution, 34, 279 Atan2, see arctangent, four-quadrant correspondence with multiplication, 281 Average power, 291 properties, 281 properties shared with multiplication, 282 relationtotimeshift,283, 289 B unity element, 289 Baseband PWM, see pulse width modulation, baseband Convolution power, 290 Binary phase shift keying, 18 correspondence with multiplicative power, Binomial coefficient, 57 290 Binomial theorem, 57 relation to Volterra series, 291 Bit rate, 18 CooleyÐTukey algorithm, 179 Burst width modulation, see pulse width Cross point estimation, 63, 66, 69, 106, 189, modulation, baseband 202, 223 Burst-mode amplification, 33, 68, see pulse Current factor, 133, 151 width modulation, baseband Current starving, 145

P. A. J. Nuyts et al., Continuous-Time Digital Front-Ends for Multistandard 305 Wireless Transmission, Analog Circuits and Signal Processing, DOI: 10.1007/978-3-319-03925-1, © Springer International Publishing Switzerland 2014 306 Index

D power added, 208, 210 Decoder, 167, 196, 230 Envelope elimination and restoration, 32 Delay Ergodic signal, 293 propagation, see propagation delay Error vector magnitude, 20, 176, 203, 294 unit, see unit delay Delay element, 132Ð137, 190, 226 differential, 135, 190, 191, 227 F inverter, 132 Fast Fourier transform, 178 noninverting, 134 FFT, see fast Fourier transform passive, 137 FM, see frequency modulation resistive interpolation, 138, 191, 227 FM signal, 17 Delay line, see delay element, 132Ð137 Fourier transform, 277 recycling, 128 continuous, see continuous Fourier Delta function, 286, see Dirac delta function transform Delta-sigma modulation, 12, 131 inverse continuous, see inverse continuous bandpass, 40Ð41 Fourier transform baseband, 30, 32, 37Ð39 Frequency, 17, 277 complex, 35 sampling, see sampling frequency multibit, 45 Frequency modulation, 7, 17 polar, 35 Frequency shift, 279, 283 Differential nonlinearity, 156 Frequency shift keying, 18 Differential pair, 142 Digital signal processing, 3 Digital-to-analog converter, 3, 38, 131 G Digital-to-RF converter, 31, 45, 261, 268 Gated ring oscillator, 128 Digital-to-time converter, 11, 129 Dirac delta function, 286 H convolution, 288 Hardware description language, 125, 182 definition, 286 Fourier transform, 287 properties, 287 I relation to time shift, 289 I signal, see in-phase signal sampling property, 288 ICFT, see inverse continuous Fourier transform Dirac impulse, 286, see Dirac delta function Image problem, 83 Distributed active transformer, 208 In-band power, 292 DNL, see differential nonlinearity In-phase signal, 17 Double-edge PWM, see pulse width INL, see integral nonlinearity modulation, double-edge Integral noise shaping, 46 see Drain efficiency, efficiency, drain Integral nonlinearity, 156 DTC, see digital-to-time converter Intermodulation, 75, 80 Dummy circuit, 147 Interpolation factor, 138 Duty cycle, 62, 102 Inverse continuous Fourier transform, 51, 278 Dynamic range properties, 278 amplitude modulation, 213 Inverter chain, 132 power control, 213

K E Kahn transmitter, 32 Efficiency coding, 34 conversion, 24, 211 L drain, 24 Leading-edge PWM, see pulse width overall, 27 modulation, leading-edge Index 307

Load capacitance, 132 On-off keying, 18 Load pull, 28 OR gate Locking, 148Ð150, 194, 228 multiplexer-based, 162 analog, 148 static CMOS, 161 using digital components, 149 symmetrical, 161 Orthogonal frequency-division multiplexing, see OFDM M Outphasing, 35, 220, 269 Mismatch, 151 asymmetric multilevel, 269 Modulated carrier Outphasing angle, 35, 102, 103 complex representation, 18 Outphasing modulator, 35 outphasing representation, 35 Oversampling ratio, 69, 78 polar representation, 17 quadrature representation, 18 Modulation, 3, 15 P see amplitude, amplitude modulation p effect, 84Ð86, 201, 205 complex representation, 18 PAE, see efficiency, power added frequency, see frequency modulation PAPR, see peak-to-average power ratio OFDM, see OFDM Parseval’s theorem, 279, 291 outphasing modulator, 35 Peak-to-average power ratio, 23, 77, 203 outphasing representation, 35 Pelgrom constant, 152 phase, see phase modulation for delay, 154 polar modulator, 32 Pelgrom’s law, 151 polar representation, 17 for propagation delay, 152 quadrature modulator, 29 Period, 279 quadrature representation, 18 sampling, see sampling period single-carrier, 18Ð21 Phase, 16 Monte Carlo simulation, 178 Phase detector, 148 Multiplexer, 167, 196, 230 combinational, 171 Phase modulated carrier, 53 multilayer, 168 with nonidealities, 60 transmission-gate, 171 Phase modulation, 7, 16, 52 tristate, 172, 196 ideal, 53 on square wave, 54 with quantized phase, 56 N with sampled phase, 58 NAND gate Phase modulator, see variable delay block, 220, multiplexer-based, 163 223 static CMOS, 159 dummy, 222, 223 symmetrical, 159, 198 Phase shift, 279 Natural-sampling PWM, see pulse width PM, see phase modulation modulation, natural-sampling PM signal, 16 Noise shaping, 37 PMC, see phase modulated carrier Noise transfer function, 38 Polar modulator, 32 Nominal value, 152 Power, 291 NOR gate average, 291 multiplexer-based, 163 in-band, see in-band power static CMOS, 161 of a product, 293 symmetrical, 161, 198 Power amplifier, 24 Normalized sinc function, see sinc function class D, 24 class E, 26 differential, 28 O digital, 32, 268 OFDM, 21 switched-mode, 2, 7, 24Ð28, 126, 132 308 Index

Power combiner, 28, 187 Quadrature amplitude modulation, 20 Power control, 212 Quadrature modulator, 29 Power spectral density, 79, 292 Quadrature phase shift keying, 20 Process corners, 147 Quadrature signal, 18 Process variations, see variability Quantization, 10 Propagation delay, 132, 152 Quantization level, 10 PSD, see power spectral density Quantization noise, 57, 105 Pseudo-natural-sampling PWM, see pulse Quantization step, 10 width modulation, pseudo-natural- sampling Pulse shrinking, 122, 157Ð159 R Pulse swallowing, 121, 157Ð159 Ranging, 131 Pulse width, 62, 157 Rayleigh’s energy theorem, 279 Pulse width modulation, 12, 62, 131 Rectangular function, 284 analytical expressions, 64 Reference recycling, 128 baseband, 30, 32, 41Ð43, 68 Resistive interpolation, 138 ideal, 69 seedelay element, resistive interpolation, image problem, 83 138 multilevel, 118 RF PWM, see pulse width modulation, RF p effect, 84Ð86 Ring oscillator, 128 with quantized input, 71 gated, 128 with sampled input, 72 carrier, see pulse width modulation, baseband S definition, 62 Sample-and-hold, 10, 58 double-edge, 63, 66, 67, 95 Sampling, 10 asymmetric, 66 Sampling frequency, 10, 126, 279 leading-edge, 63 Sampling period, 10, 279 multilevel, 117, 214 Sampling property natural-sampling, 63, 65, 67 of Dirac delta function, see Dirac delta pseudo-natural-sampling, 63, 68, 189 function, sampling property RF, 35, 43Ð44, 95 Sampling rate, see sampling frequency differential, 37, 101 Self load, 132 ideal, 103 Series expansion, see Taylor series expansion multilevel, 35, 121 Sifting property outphasing implementation, 103, 220 of Dirac delta function, see Dirac delta required transformations on AM signal, function, sampling property 99 Sigma-delta, see delta-sigma modulation with phase modulation, 100 Signal with quantized inputs, 104 analog, 10 with sampled inputs, 106 continuous-time, 10 spectrum, 64 discrete-time, 10 trailing-edge, 63, 65, 68 band-limited, 279 types, 63 digital, 11 uniform-sampling, 63, 65, 66 continuous-time, 10 asymmetric, 66 discrete-time, 11 Pulse width modulator, 62, see pulse width discrete-time, 279 modulation periodical, 279 PushÐpull, 187, 188 Signal to noise and distortion ratio, 294 PWM, see pulse width modulation Signal transfer function, 38 Signum function, 285 Sinc function, 283 Q normalized, 283 Q signal, see quadrature signal unnormalized, 284 Index 309

Single-carrier signal, 18Ð21 Tristate inverter, 172 Single-ended-to-differential converter, 231 SNDR, see signal to noise and distortion ratio Software radio, 2, 8 U Software-defined radio, 2, 5, 8 Uniform-sampling PWM, see pulse width Spectral mask, 51 modulation, uniform-sampling Spectrum, 51, 277 Unit delay, 133, 155, 161 discrete, 279 Unit standard deviation, 155 periodical, 279 Square wave, 54 Standard cells, 126 V Surface acoustic wave filter, 9 Variability Switched-mode power amplifier, see power global, 143, 147 amplifier, switched-mode local, 151 Symbol rate, 18 Variable delay block, 187, 220, 223 Vernier delay line, 143 Volterra series, 291 T Taylor series expansion using convolution powers, 291 X TDC, see time-to-digital converter XNOR gate Threshold voltage, 133, 151 all-NOR, 164 Time shift, 53, 279, 283, 289 XOR gate, 163, 198, 231 Time-of-arrival, 131 all-NAND, 164 Time-of-flight, 131 multiplexer-based, 166 Time-to-digital converter, 11, 127 static CMOS, 165 Trailing-edge PWM, see pulse width transmission-gate-based, 166 modulation, trailing-edge two-layer gate-based, 164, 198