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Appendix ADefinitions, Conventions and Overviewof 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 befully explained or proven here; for this, the interested reader is referred to specializedworks. However, since many of the concepts are defined in slightly different ways bydifferent authors, it is useful to clearly define the concepts used here. Furthermore,some important remarks are given about how to use certain theoretical concepts andnotations.
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 conventionsfor clarity’s sake. For more detailed information on Fourier analysis, the reader isreferred to reference works in this field, such as [1, 4].
In this work, the continuous Fourier transform (CFT) is used. It is denoted by theoperator F {.}. The Fourier transform X ( f ) of a signal x(t) is defined as
X ( f ) = F {x(t)} =∫ ∞
−∞x(t) e− j2π f t dt, (A.1)
where j is the imaginary unit and j2 = −1. X ( f ) is called the spectrum of the signalx(t). In this work, the spectrum of a time-domain signal will be denoted with thesame letter as the time-domain signal, but capitalized.
The independent variable f is called frequency. It should not be confused withthe angular frequency ω = 2π f which is used in some works, but will not be usedhere. The value of X ( f ) expresses the contribution of that frequency to the signalx(t).
P. A. J. Nuyts et al., Continuous-Time Digital Front-Ends for Multistandard 277Wireless 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 denotedby the operator F−1 {.} and defined as
x(t) = F−1 {X ( f )} =∫ ∞
−∞X ( f ) e j2π f t 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 listof 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 realnumbers. 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 isodd.
• 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 andan imaginary part that is even.
• If x(t) is zero outside a certain time interval [t0, t1], then X ( f ) is unlimited infrequency, 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 thatx(t) = 0 ∀ t where |t | > t0. However |x(t)| might decrease asymptotically withincreasing |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 periodicalsignal has a discrete spectrum.
• If x(t) is nonzero only at integer multiples of a period Ts , called the samplingperiod, 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.
• The Fourier transform of x(t − t0) is X ( f ) · e− j2π f t0 , i.e. a time shift in the timedomain corresponds to a linear phase shift in the frequency domain.
• The inverse Fourier transform of X ( f − f0) is x(t) · e j2π f0t , i.e. a frequency shiftin 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 viceversa.
• 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). If x(t) = y(t), this theoremreduces 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 theseresults are generally known, they are not identical in all literature, as some writersuse slightly different definitions of the CFT. Therefore, they are listed according tothe definitions used in this work for easy reference. More Fourier transform pairscan be found in [1].
TheDirac delta function δ(t), which appears inmany of the results, will be definedin Sect. A.4.1.
A.2 Convolution
A.2.1 Definition and Notation
The continuous convolution, or shorter, convolution, is an operation that operates ontwo 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 importantfourier transform pairs
Time domain Frequency domain
0 0δ(t) 1δ(t − t0) e− j2π f t0
1 δ( f )
e j2π f0t δ( f − f0)cos(2π f0t) 1
2 (δ( f − f0) + δ( f + f0))
sin(2π f0t) 12 j (δ( f − f0) − δ( 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 convolutionshares a large number of properties with multiplication (such as commutativity,associativity, etc.), it is not really unambiguous, as it does not distinguish betweenthe variables u and v used in (A.9). For example, evaluating the function z in 2ugives
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
2
∫ ∞
−∞x(w) y(2u − w) dw
= 1
2z(2u), (A.12)
and thus z(2u) �= x(2u) ∗ y(2u) even though the function z(u) was “defined” asz(u) = x(u) ∗ y(u). The reason for this is that the letter u in x(u) ∗ y(u) actuallyrefers to the integration variable v, while in z(u) it is an independent variable thatcan be changed. Furthermore, in an expression like x(au) ∗ y(2a + u), it is not clearwhether one should integrate over a or u.
These problems can be solved by introducing a more exact notation, which usesdifferent 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) withintegration 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 isalways t . Thus, the notation z(t) = x(t) ∗ y(t) should be interpreted as follows:
– 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).
– Then calculate the convolution using u as the integration variable and t as theindependent time variable, i.e. calculate
[x(u) ∗u y(u)] (t) =∫ ∞
−∞x(u) v(t − u) du. (A.17)
– 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 variableis always f . The procedure is similar to the one described above.
• In equations where other integration variables are used, or where confusion ispossible 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)
thenW ( f ) = X ( f ) ∗ Y ( f ). (A.19)
Also, ifz(t) = x(t) ∗ y(t), (A.20)
thenZ( 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)–(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 commonlyused 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 thatwhile 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) (withoutparentheses) is ambiguous and should be avoided.Of course, in the special casewherex(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 eitherone (but only one) of the signals by t0 and then convolving them. Because of theabove 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 definitionz(t) = x(t)∗ y(t)may seem to suggest this. The time shift must be applied to exactlyone 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 revisitedmore 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
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.5
1
xsinc
(x)
sinc(x) = sin(πx)
πx(A.31)
and is shown in Fig. A.1. It is equal to 0 at all nonzero integer values of x , and onlythere, and it is equal to 1 for x = 0. The function is called normalized because itintegrates to one: ∫ ∞
−∞sinc(x) dx = 1. (A.32)
The unnormalized sinc function is defined as
sinc( x
π
)= sin(x)
x. (A.33)
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 usethe notation sinc(x) for the unnormalized sinc function.
A.3.2 The Rectangular Function
The rectangular function �(x) is defined as [1]
�(x) ={1, |x | ≤ 1
2
0, |x | > 12
(A.34)
and is plotted in Fig. A.2 Thus, �(t/T ) is a rectangular pulse with width T andheight 1.
It can be shown [1, 4] that
F
{�
(t
T
)}= T sinc( f T ). (A.35)
Appendix A: Definitions, Conventions and Overview of Used Theory 285
Fig. A.2 The rectangularfunction
−2 −1 0 1 2
0
0.2
0.4
0.6
0.8
1
x
Π(x)
Fig. A.3 The signumfunction
−2 −1 0 1 2
−1
−0.5
0
0.5
1
x
sgn(x)
A.3.3 The Signum Function
The signum function sgn(x) is defined as
sgn(x) =⎧⎨⎩
1, x > 0,0, x = 0,
−1, x < 0,(A.36)
and shown in Fig. A.3. The value at x = 0 is somewhat arbitrary and is not importantin most applications.
A.3.4 The Four-Quadrant Arctangent Function
The four-quadrant arctangent function atan2(y, x) is defined as [9]
atan2(y, x) =
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
arctan (y/x) , x > 0,arctan (y/x) + π, y ≥ 0, x < 0,arctan (y/x) − π, y < 0, x < 0,π/2, y > 0, x = 0,−π/2, y < 0, x = 0,undefined, y = 0, x = 0,
(A.37)
286 Appendix A: Definitions, Conventions and Overview of Used Theory
Fig. A.4 The four-quadrantarctangent function
−10 −5 0 5 10−4
−2
0
2
4
y/xat
an2(
y,x)
x > 0x < 0
and plotted as a function of y/x in Fig. A.4. It differs from the arctangent functionarctan(y/x) in that its result range is (−π, π ], while arctan has range (−π/2, π/2).It is always true that
tan(atan2(y, x)) = tan(arctan
( y
x
))= y
x. (A.38)
The atan2 function is useful to determine the angle of a complex number: any nonzerocomplex number z = x + j y can be written as z = ae jθ , 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 beassigned 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) isgenerally 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 thedimension of the horizontal axis. For example, in the time-domain delta functionδ(t), the independent variable t has units of seconds. In order for (A.42) to besatisfied, δ(t) must have units of 1/s = Hz. In the frequency-domain delta functionδ( 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 andδ(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)to 2x , i.e. ∫ ∞
−∞δ(2x) d(2x) =
∫ ∞
−∞δ(u) du = 1. (A.44)
If one integrates over x instead, one finds that
∫ ∞
−∞δ(2x) dx = 1
2
∫ ∞
−∞δ(u) du = 1
2. (A.45)
and thus
δ(2x) = 1
2δ(x). (A.46)
In order to avoid confusion, it is best to use only arguments of the form u+c, whereu is the independent variable and c is a constant. The constant does not influencethe integral in (A.42) and thus does not cause confusion. This is also the way thedelta function normally occurs in typical signal-processing-related expressions, andthe way it will be used in this work. Also, the Dirac delta will only be used withindependent variables t and f .
A.4.2 Properties
A.4.2.1 Fourier Transform
It can be shown that ∫ ∞
−∞e j2πuv 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)–(A.42) that
F {δ(t − t0)} = e− j2π f t0 (A.50)
and
F{
e j2π f0t}
= δ( f − f0). (A.51)
Using (A.51) and the fact that
cos(x) = 1
2
(e j x + e− j x
), (A.52)
sin(x) = 1
2
(e j x − e− j x
), (A.53)
it is easy to derive the Fourier transforms of the sine and cosine functions, which aregiven 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)
Thismeans that theDirac delta function is the unity element for convolution. This alsofollows from the fact that its Fourier transform is 1 and that convolution correspondsto 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 thesignal 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 f0shifts 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 relationbetween 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 multiplicationand 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
F {x(t − t0)} = X ( f ) · e j2π f t0 , (A.61)
F {y(t − t0)} = Y ( f ) · e j2π f t0 , (A.62)
Z0( f ) = F {z(t − t0)} = Z( f ) · e j2π f t0 . (A.63)
Since Z( f ) = X ( f ) · Y ( f ) , it follows that
Z0( f ) = X ( f ) · Y ( f ) · e j2π f t0 . (A.64)
which proves (A.59) since F {δ(t − t0)} = e j2π f t0 .
A.5 Convolution Powers
For the derivations made in Chap. 3, it is useful to define the concept of convolutionpower. The convolution powers x∗n(u) of a signal or spectrum x(u) are defined asfollows
x∗0(u) = δ(u) (A.65)
x∗n(u) =[x∗n−1(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
Xn( 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 bewritten as a, possibly infinite, weightedsum of powers of x using Taylor series expansion, it follows that for a time-domainsignal x(t) with spectrum X ( f ), the spectrum of signal g(x(t)), where g(x) is anyfunction, can be written in terms of the convolution powers X∗n( f ). This principlewas used by Song and Sarwate [7] to calculate the spectrum of a PWM signal. Whileit is generally not trivial to calculate X∗n( f ) for high values of n, this property canstill be practical to get an idea of what a certain spectrum will look like. It is usedextensively in Chap. 3.
For readers familiar with the theory of Volterra series (see Appendix B in [8]), itcan be interesting to note the relation with convolution powers1: The nth convolutionpower x∗n(u) of a function x(u) can be viewed as the nth-order term in a Volterraseries 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)]
P {x(t)} = limT →∞
1
T
∫ T/2
−T/2|x(t)|2dt. (A.69)
If one defines
xT (t) = x(t) · �
(t
T
), (A.70)
then (A.69) can be written as
P {x(t)} = limT →∞
1
T
∫ ∞
−∞|xT (t)|2dt. (A.71)
Using Parseval’s theorem (see (A.7)), it follows that
P {x(t)} = limT →∞
1
T
∫ ∞
−∞|XT ( f )|2d f
=∫ ∞
−∞Px ( f )d f, (A.72)
where
Px ( f ) = limT →∞
1
T|XT ( f )|2 (A.73)
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. 2–3] [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 expressionto 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. Proofsfor 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 followingproperties 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), thenthe filter output y(t) = h(t) ∗ x(t) (with spectrum Y ( f ) = H( f )X ( f ) whereH( f ) = F {h(t)} is called the transfer function of the filter) has an averagepower given by
P {Y ( f )} = P {H( f ) · X ( f )} =∫ ∞
−∞|H( f )|2 · 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 P̂ {x(t)} or P̂ {X ( f )}, is defined as
P̂ {x(t)} = P̂ {X ( f )} =∫ − fa
− fb
Px ( f )d f +∫ fb
fa
Px ( f )d f (A.75)
=∫ ∞
−∞W ( f ) · Px ( f )d f (A.76)
= P {W ( f ) · X ( f )}, (A.77)
where
W ( f ) = �
(f − ( fa + fb)/2
fb − fa
)+ �
(f + ( fa + fb)/2
fb − fa
). (A.78)
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 bandis [− fb, fb] by setting fa = 0. In both cases P̂ {x(t)} gives the total power presentin the relevant signal bands only and disregards the out-of-band power.
EquationA.77 shows that P̂ {X ( f )} corresponds to the total power in X ( f ) afterapplying 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 signalY ( f ) = H( f )X ( f ),where H( f ) is the transfer function of a linear time-invariant filter, is given by
P̂ {H( f )X ( f )} = P {W ( f )H( f )X ( f )}=
∫ ∞
−∞|W ( f )H( f )|2 · Px ( f )d f
=∫ ∞
−∞W ( f )|H( f )|2 · Px ( f )d f
=∫ − fa
− fb
|H( f )|2 · Px ( f )d f +∫ fb
fa
|H( f )|2 · Px ( f )d f.
(A.79)
In addition, if x(t) and h(t) are real, it follows that |H( f )| and Px ( f ) are evenfunctions of f and
P̂ {H( f )X ( f )} = 2∫ fb
fa
|H( f )|2 · Px ( f )d f. (A.80)
A.6.3 Power of a Product
This section shows that the power in the product of two uncorrelated ergodic signalsis 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 beshown that the average power P {x(t)} in x(t) is equal to [1, Sect. 6.1]
P {x(t)} = σ 2x + μ2
x . (A.81)
In this work, all signals are assumed to be ergodic.More information about ergodicitycan 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 uncorrelatedsignals x(t) and y(t) is
σ 2xy = μ2
xσ2y + μ2
yσ2x + σ 2
x σ 2y . (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 {x(t) · y(t)} = μ2xσ
2y + μ2
yσ2x + σ 2
x σ 2y + μ2
xy . (A.83)
Since x(t) and y(t) are uncorrelated, μxy = μxμy and
P {x(t) · y(t)} = (μ2x + σ 2
x ) · (μ2y + σ 2
y ) = P {x(t)} · P {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-banddesired signal power to the in-band noise and distortion power. Thus, considering asignal
X ( f ) = S( f ) +M∑
i=1
�i S( f ), (A.85)
where S( f ) is the desired signal and �i S( f ) are uncorrelated noise and distortionterms, the SNDR of X ( f ) is equal to
SNDR = P̂ {S( f )}∑Mi=1 P̂ {�i S( f )} , (A.86)
where the in-band power P̂ {.} is defined in Sect. A.6.2.
A.8 Error Vector Magnitude
Most communication standards define signal quality in terms of the error vectormagnitude (EVM), which is defined in this section. The definition of EVM is alwaysbased 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 thedifferent constellation points with ci , where i = 0, 1, 2, . . . , N − 1 and N is thenumber of constellation points.
Ideally, the complex envelope (before filtering) is a discrete-time signal gk wherek 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 willdeviate from this. In order to calculate EVM, a receiver is added (in order to evaluatea transmitter only, an ideal receiver is used) which tries to demodulate the signal andproduces 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 eachpoint rk from the ideal point gk : if M symbols were demodulated, the absolute EVMis given by
EVMabs =√√√√ 1
M
M−1∑k=0
|rk − gk |2. (A.87)
Note that rk and gk are complex numbers.Usually, the EVM is normalized with respect to the RMS constellation power:
EVM =√1/M
∑M−1k=0 |rk − gk |2.√
1/N∑N−1
i=0 |ci |2. (A.88)
This gives the relative EVM, which is usually simply referred to as EVM (includingin 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 forevery constellation point rk , the ideal point gk is the point that is closest to rk . This isnormally correct unless the EVM is very bad and the clouds overlap. The definitionof EVM then becomes
EVM =√1/M · ∑M−1
k=0 mini |rk − ci |2.√1/N · ∑N−1
i=0 |ci |2. (A.89)
This definition is equivalent to (A.88) as long as the EVM is sufficiently good, butsaturates around a certain value when the EVMbecomes so bad that the cloudsmergeto one cloud. This is not a problem as such a transmitter has no practical use. Thedefinition in (A.89) is used throughout this work.
Care should be taken when comparing with other work, since other definitionsare 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 peakconstellation power rather than the RMS value:
EVM =√1/M · ∑M−1
k=0 mini |rk − ci |2.maxi |ci | . (A.90)
The definition of EVM for OFDM signals (see Sect. 2.1.4) is similar, but theconstellation points for all subcarriers are considered together. EquationA.89 thenbecomes
296 Appendix A: Definitions, Conventions and Overview of Used Theory
EVM =√1/(M P) · ∑P−1
p=0∑M−1
k=0 mini∣∣rk,p − ci
∣∣2.√1/N · ∑N−1
i=0 |ci |2, (A.91)
where P is the number of modulated subcarriers and rk,p is the complex envelopeof the pth subcarrier during the kth symbol period. Similar definitions are also usedby 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 acapital letter�with an index, followed by the signal or spectrum that is influenced bythe noise or distortion. The index indicates the type or cause of the nonideality. Forexample, if an ideal phase signal ϕ(t) is quantized to yield ϕq(t), the quantizationnoise on this signal will be denoted Δqϕ(t) (i.e. Δqϕ(t) = ϕq(t) − ϕ(t)), wherethe 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 denotedΔq�( f ).
When dealing with the nth power of a signal, square brackets will be used todistinguish two different cases:
• [Δqϕ]n(t) = (Δqϕ(t))n is the nth power of Δqϕ(t), and its spectrum is denoted[ΔqΦ]∗n( f ).
• �q[ϕn](t) indicates the deviation onϕn(t) due to quantization noise, i.e.�q[ϕn](t)= ϕn
q (t) − ϕn(t). The spectrum of [Δqϕ]n(t) will be denoted Δq[Φ∗n]( f ).
It is important to clearly distinguish [Δqϕ]n(t) and [Δqϕ]n(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 BDerivations and Considerations RegardingPWM
B.1 Derivation of PWM Formulas
In this section, the time- and frequency-domain PWM formulas listed in Sect. 3.2.3are derived from equations given in [7].
Parts of the derivations made in [7] are summarized more concisely in [6], whichis co-authored by the authors of [7]. Where applicable, the corresponding equationsin [6] will also be noted.
This section will partly use the symbols used in [7] (not [6]), while replacingcertain 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. Toavoid 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
pT E (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 TEPWMsignal. pc(t) represents a PWMsignal with a constantduty cycle of 50%. It is given by [7, Eq. (2)] to be equal to
pc(t) =∞∑
k=0
4
(2k + 1)πsin(2π(2k + 1) fpwmt), (B.2)
which in this work will be written as
P. A. J. Nuyts et al., Continuous-Time Digital Front-Ends for Multistandard 297Wireless 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
pc(t) =∞∑
l=1,3,...
4
lπsin(2πl fpwmt). (B.3)
The spectrum of pT E (t) is given by
PTE( f ) = Pc( f ) + Ps,TE( f ), (B.4)
where Pc( f ) is given by [7, Eq. (3)]:
Pc( f ) =∞∑
l=1,3,...
2
jlπ
[δ( f − l fpwm) − δ( f + l fpwm)
], (B.5)
=±∞∑
l=±1,±3,...
2
jlπδ( f − l fpwm), (B.6)
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-dependentpart Ps,TE( f ; b) of their spectrum is given by [7, Eq. (6)] [6, Eq. (4)]:
Ps,TE( f ; b) = e− jπ f Tpwm∞∑
k=−∞
∞∑n=1
(− jπ f Tpwm)n−1
n! B∗n( f − k fpwm), (B.7)
where the b between the brackets (x is used in [7]) indicates the spectrum is calculatedfor an input b(t).
Combining this with (B.4) and (B.6) readily proves (3.56).According to [7, Eq. (12)] [6, Eq. (10)],
pTE(t; b) = y(t; b) +∞∑
k=1
2
kπ
[sin(2πk fpwmt) − (−1)k sin(2πk fpwmt − kπy(t; b))
].
(B.8)Here, y(t; x) is given by [7, Eq. (9)] [6, Eq. (7)] to be
y(t; b) = b
(x − Tpwm
2
)+
∞∑n=2
1
n!(
−Tpwm2
)n−1 dn−1
dtn−1
[b
(t − Tpwm
2
)]n
.
(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 denoteNPWM signals with a hat (ˆ), so that
P̂TE( f ) = Pc( f ) + P̂s,TE( f ). (B.10)
[7, Eq. (36)] gives the signal-dependent part P̂s,TE( f ; b) of the spectrum as2
P̂s,TE( f ; b) = B( f ) +∞∑
k=1
(−1)k∞∑
n=1
( jkπ)n−1
n![
B∗n( f + k fpwm)
+ (−1)n−1B∗n( f − k fpwm)]
(B.11)
This can be rewritten as
P̂s,TE( f ; b) = B( f ) +∞∑
k=1
(−1)k∞∑
n=1
(− jkπ)n−1
n! B∗n( f − k fpwm) (B.12)
+−1∑
k=−∞(−1)k
∞∑n=1
(− jkπ)n−1
n! B∗n( f − k fpwm)
= B( f ) +∑k �=0
(−1)k∞∑
n=1
(− jkπ)n−1
n! B∗n( f − k fpwm),
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)],
PDE,S( f ; b) = e− jπ f Tpwm/2Pc( f )
+ e− jπ f Tpwme− jπ f Tpwm/2∞∑
k=−∞
∞∑n=1
(− jπ f Tpwm)n−1
2n · n! B∗n( f − k fpwm)
2 The right-hand side of this equation is equal to the right-hand side in [6, Eq. (22)]. However, itseems the left-hand side in [6] was erroneously written as the complete PWM signal ̂PW M( f ),while it should have been the signal-dependent part P̂( f ).
300 Appendix B: Derivations and Considerations Regarding PWM
+ e− jπ f Tpwme jπ f Tpwm/2∞∑
k=−∞
∞∑n=1
( jπ f Tpwm)n−1
2n · n! B∗n( f − k fpwm),
(B.13)
where the capital S in the subscript stands for “symmetrical”. Using [7, Eq. (3)], thiscan be rewritten as
PDE,S( f ; b) = e− jπ f Tpwm/2±∞∑
l=±1,±3,...
2
jlπδ( f − l fpwm) (B.14)
+ e− jπ f Tpwm∞∑
k=−∞
∞∑n=1
1
2n · n! B∗n( f − k fpwm)
{e− jπ f Tpwm/2(− jπ f Tpwm)n−1 + e jπ f Tpwm/2( jπ f Tpwm)n−1
},
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
pDE,S(t; b) = pTE
(t; 1 + b
2
)+ pLE
(t; 1 + b
2
)− 1, (B.15)
where pTE(t; b) is given by (B.8) and pLE(t; b) is a uniform-sampling leading-edgePWM (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
pDE,S(t; b) = pTE
(t; 1 + b
2
)− pTE
(t;−1 + b
2
)− 1, (B.17)
Substituting (B.8) gives
pDE,S(t; b) = y
(t; 1 + b
2
)− y
(t;−1 + b
2
)− 1
−∞∑
k=1
2
kπ(−1)k
{sin
(2πk fpwmt − kπy
(t; 1 + b
2
))
− sin
(2πk fpwmt − kπy
(t;−1 + b
2
)) }, (B.18)
where y(t; b) is given by (B.9). Defining
Appendix B: Derivations and Considerations Regarding PWM 301
y+(t) = y
(t; 1 + b
2
)(B.19)
= 1
2+ 1
2b
(x − Tpwm
2
)
+ 1
2n
∞∑n=2
1
n!(
−Tpwm2
)n−1 dn−1
dtn−1
[x
(t − Tpwm
2
)]n
(B.20)
y−(t) = y
(t;−1 + b
2
)(B.21)
= −1
2− 1
2b
(x − Tpwm
2
)
+ 1
(−2)n
∞∑n=2
1
n!(
−Tpwm2
)n−1 dn−1
dtn−1
[x
(t − Tpwm
2
)]n
(B.22)
leads to (3.59)–(3.61).
B.1.5 Natural-Sampling Double-Edge PWM
Asymmetric NDEPWM signals are analyzed in [7, Sect. 3.3]. According to [7,Eq. (60)], the signal-dependent part P̂s,DE,A( f ; b) of the spectrum is equal to
P̂s,DE,A( f ; b) = B( f ) +∞∑
k=1
(− j)k∞∑
n=1
( jkπ)n−1
2n · n!(1 + (−1)n+k−1
)
{B∗n( f + k fpwm) + B∗n( f − k fpwm)
}, (B.23)
where the A in the index stands for “asymmetrical”. Using the fact that the completespectrum is given by
P̂DE,A( f ; b) = Pc( f ) + P̂s,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 basebandPWM transmitter where the PWM frequency fpwm has been set equal to the carrierfrequency fc. This assumptionwill be investigated in this section, and itwill be shown
302 Appendix B: Derivations and Considerations Regarding PWM
Fig. B.1 Illustration of howa differential RF PWM signalwithout PM can be generatedusing baseband PWM withfpwm = 2 fc a Square-wavecarrier psq(t) = csq(t). bBaseband PWM at 2 fc. cDifferential RF PWM at fc =product of the above
1(a)
(b)
(c)
-1
1
-1
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 PWMsignal given by (3.150) on p. 102 by applying a certain input abb(t) to a basebandPWMmodulator with fpwm = fc and multiplying the result with a square-wave PMClike the one given by (3.16) on p. 55. In order to find this abb(t), assume first thatthere 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 bean NDEPWM signal. However, the RF PWM signal is also differential and thus itactually has two pulses per period Tc. For this reason, the baseband PWM frequencyfpwm 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 becausein standard baseband PWM, the phase and amplitude paths are independent of eachother 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, notonly 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 muchmore complicated than the implementation used in Chap. 6. This section only servesto illustrate that RF PWM cannot generally be considered to be a special case ofbaseband PWM, unless the baseband PWM modulator is created in a very specificway.
Going back to the case without phase-modulation, one may note what appearsto be an inconsistency in the PWM models: According to (3.75) (p. 71), when abaseband PWM modulator is fed an input signal abb(t), the resulting duty cycle isgiven by
dbb(t) = abb(t). (B.25)
Appendix B: Derivations and Considerations Regarding PWM 303
Fig. B.2 Illustration of howa differential RF PWM signalwith PM cannot be generatedusing simple baseband PWM.a Square-wave PMC psq(t).b Baseband PWM at 2 fc.c Product of the above
1
(a)
(b)
(c)
-1
1
-1
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 amplitudearf(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 toabb(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 aslong as the period Tpwm for baseband PWM. Together with (B.25) and (B.26), thiscontradicts the requirement that abb(t) = arf(t).
This is explained by the p effect, which is discussed in Sect. 3.4.4.2. This effectcan be summarized by stating that in baseband PWM, the carrier harmonics andthe PWM harmonics cause intermodulation products which may fall into the signalband depending on the ratio fc/ fpwm. The case under consideration here is the worstpossible case of the p effect: since fc/ fpwm = 1/2, the p value is 2,which is theworstpossible 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 peffect very large in this particular case. Thus, in addition to the in-band term whichhas an amplitude abb(t) = dbb(t) = 2drf(t), there are a lot of intermodulation termscentered at fc, which are so large that they can no longer be considered as smallnoise or distortion terms, but should be considered as part of the signal. The RFPWM model shows no terms at fc apart from the desired RF signal, which has anamplitude arf (t) = sin(πdrf(t)). Since both models are exact and consider the samesignal shown in Fig. B.1c, it follows that the sum of abb(t) and the amplitudes of allthe 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 thedesired RF amplitude. In this case, this would result in very bad performance due tothe 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 adifferential RF PWM signal using baseband PWMwith a PWM frequency of 2 fc—not fc as onemight expect intuitively. However, in order to do so, the baseband PWMinput should be taken to be
abb(t) = 2drf(t) = 2
πarcsin arf(t), (B.28)
and furthermore the baseband PWM output must be phase-modulated with the samephase signal ϕ(t) as the square-wave RF carrier. For these reasons, one cannot simplysay that RF PWM is a special case of ordinary baseband PWM.
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Index
Symbols� , see delta-sigma modulation �, see delta-sigma modulation
AAM, see amplitude modulationAM signal, 16Amplitude, 16Amplitude modulation, 7, 16Amplitude shift keying, 19Analog-to-digital converter, 11, 131AND gate
multiplexer-based, 162static CMOS, 161symmetrical, 161
Angleof complex number, 286outphasing, see outphasing angle
Arctangentfour-quadrant, 285
Atan2, see arctangent, four-quadrantAverage power, 291
BBaseband PWM, see pulse width modulation,
basebandBinary phase shift keying, 18Binomial coefficient, 57Binomial theorem, 57Bit rate, 18Burst width modulation, see pulse width
modulation, basebandBurst-mode amplification, 33, 68, see pulse
width modulation, baseband
CCarrier, 15Carrier frequency, 16Carrier PWM, see pulse width modulation,
basebandCFT, see continuous Fourier transformChannel spacing, 90, 203Class-D power amplifier, 24Class-E power amplifier, 26Coding efficiency, see efficiency, codingComplex conjugate, 279Complex envelope, 18Constellation plot, 18, 20, 204, 214, 240, 241,
294Continuous convolution, see convolutionContinuous Fourier transform, 51, 277
properties, 278Conversion efficiency, see efficiency,
conversionConvolution, 34, 279
correspondence with multiplication, 281properties, 281properties shared with multiplication, 282relation to time shift, 283, 289unity element, 289
Convolution power, 290correspondence with multiplicative power,290
relation to Volterra series, 291Cooley–Tukey algorithm, 179Cross point estimation, 63, 66, 69, 106, 189,
202, 223Current factor, 133, 151Current starving, 145
P. A. J. Nuyts et al., Continuous-Time Digital Front-Ends for Multistandard 305Wireless Transmission, Analog Circuits and Signal Processing,DOI: 10.1007/978-3-319-03925-1, © Springer International Publishing Switzerland 2014
306 Index
DDecoder, 167, 196, 230Delay
propagation, see propagation delayunit, see unit delay
Delay element, 132–137, 190, 226differential, 135, 190, 191, 227inverter, 132noninverting, 134passive, 137resistive interpolation, 138, 191, 227
Delay line, see delay element, 132–137recycling, 128
Delta function, 286, see Dirac delta functionDelta-sigma modulation, 12, 131
bandpass, 40–41baseband, 30, 32, 37–39complex, 35multibit, 45polar, 35
Differential nonlinearity, 156Differential pair, 142Digital signal processing, 3Digital-to-analog converter, 3, 38, 131Digital-to-RF converter, 31, 45, 261, 268Digital-to-time converter, 11, 129Dirac delta function, 286
convolution, 288definition, 286Fourier transform, 287properties, 287relation to time shift, 289sampling property, 288
Dirac impulse, 286, see Dirac delta functionDistributed active transformer, 208DNL, see differential nonlinearityDouble-edge PWM, see pulse width
modulation, double-edgeDrain efficiency, see efficiency, drainDTC, see digital-to-time converterDummy circuit, 147Duty cycle, 62, 102Dynamic range
amplitude modulation, 213power control, 213
EEfficiency
coding, 34conversion, 24, 211drain, 24overall, 27
power added, 208, 210Envelope elimination and restoration, 32Ergodic signal, 293Error vector magnitude, 20, 176, 203, 294
FFast Fourier transform, 178FFT, see fast Fourier transformFM, see frequency modulationFM signal, 17Fourier transform, 277
continuous, see continuous Fouriertransform
inverse continuous, see inverse continuousFourier transform
Frequency, 17, 277sampling, see sampling frequency
Frequency modulation, 7, 17Frequency shift, 279, 283Frequency shift keying, 18
GGated ring oscillator, 128
HHardware description language, 125, 182
II signal, see in-phase signalICFT, see inverse continuous Fourier transformImage problem, 83In-band power, 292In-phase signal, 17INL, see integral nonlinearityIntegral noise shaping, 46Integral nonlinearity, 156Intermodulation, 75, 80Interpolation factor, 138Inverse continuous Fourier transform, 51, 278
properties, 278Inverter chain, 132
KKahn transmitter, 32
LLeading-edge PWM, see pulse width
modulation, leading-edge
Index 307
Load capacitance, 132Load pull, 28Locking, 148–150, 194, 228
analog, 148using digital components, 149
MMismatch, 151Modulated carrier
complex representation, 18outphasing representation, 35polar representation, 17quadrature representation, 18
Modulation, 3, 15amplitude, see amplitude modulationcomplex representation, 18frequency, see frequency modulationOFDM, see OFDMoutphasing modulator, 35outphasing representation, 35phase, see phase modulationpolar modulator, 32polar representation, 17quadrature modulator, 29quadrature representation, 18single-carrier, 18–21
Monte Carlo simulation, 178Multiplexer, 167, 196, 230
combinational, 171multilayer, 168transmission-gate, 171tristate, 172, 196
NNAND gate
multiplexer-based, 163static CMOS, 159symmetrical, 159, 198
Natural-sampling PWM, see pulse widthmodulation, natural-sampling
Noise shaping, 37Noise transfer function, 38Nominal value, 152NOR gate
multiplexer-based, 163static CMOS, 161symmetrical, 161, 198
Normalized sinc function, see sinc function
OOFDM, 21
On-off keying, 18OR gate
multiplexer-based, 162static CMOS, 161symmetrical, 161
Orthogonal frequency-division multiplexing,see OFDM
Outphasing, 35, 220, 269asymmetric multilevel, 269
Outphasing angle, 35, 102, 103Outphasing modulator, 35Oversampling ratio, 69, 78
Pp effect, 84–86, 201, 205PAE, see efficiency, power addedPAPR, see peak-to-average power ratioParseval’s theorem, 279, 291Peak-to-average power ratio, 23, 77, 203Pelgrom constant, 152
for delay, 154Pelgrom’s law, 151
for propagation delay, 152Period, 279
sampling, see sampling periodPhase, 16Phase detector, 148Phase modulated carrier, 53
with nonidealities, 60Phase modulation, 7, 16, 52
ideal, 53on square wave, 54with quantized phase, 56with sampled phase, 58
Phase modulator, see variable delay block, 220,223
dummy, 222, 223Phase shift, 279PM, see phase modulationPM signal, 16PMC, see phase modulated carrierPolar modulator, 32Power, 291
average, 291in-band, see in-band powerof a product, 293
Power amplifier, 24class D, 24class E, 26differential, 28digital, 32, 268switched-mode, 2, 7, 24–28, 126, 132
308 Index
Power combiner, 28, 187Power control, 212Power spectral density, 79, 292Process corners, 147Process variations, see variabilityPropagation delay, 132, 152PSD, see power spectral densityPseudo-natural-sampling PWM, see pulse
width modulation, pseudo-natural-sampling
Pulse shrinking, 122, 157–159Pulse swallowing, 121, 157–159Pulse width, 62, 157Pulse width modulation, 12, 62, 131
analytical expressions, 64baseband, 30, 32, 41–43, 68
ideal, 69image problem, 83multilevel, 118p effect, 84–86with quantized input, 71with sampled input, 72
carrier, see pulse width modulation,baseband
definition, 62double-edge, 63, 66, 67, 95
asymmetric, 66leading-edge, 63multilevel, 117, 214natural-sampling, 63, 65, 67pseudo-natural-sampling, 63, 68, 189RF, 35, 43–44, 95
differential, 37, 101ideal, 103multilevel, 35, 121outphasing implementation, 103, 220required transformations on AM signal,
99with phase modulation, 100with quantized inputs, 104with sampled inputs, 106
spectrum, 64trailing-edge, 63, 65, 68types, 63uniform-sampling, 63, 65, 66
asymmetric, 66Pulse width modulator, 62, see pulse width
modulationPush–pull, 187, 188PWM, see pulse width modulation
QQ signal, see quadrature signal
Quadrature amplitude modulation, 20Quadrature modulator, 29Quadrature phase shift keying, 20Quadrature signal, 18Quantization, 10Quantization level, 10Quantization noise, 57, 105Quantization step, 10
RRanging, 131Rayleigh’s energy theorem, 279Rectangular function, 284Reference recycling, 128Resistive interpolation, 138
seedelay element, resistive interpolation,138
RF PWM, see pulse width modulation, RFRing oscillator, 128
gated, 128
SSample-and-hold, 10, 58Sampling, 10Sampling frequency, 10, 126, 279Sampling period, 10, 279Sampling property
of Dirac delta function, see Dirac deltafunction, sampling property
Sampling rate, see sampling frequencySelf load, 132Series expansion, see Taylor series expansionSifting property
of Dirac delta function, see Dirac deltafunction, sampling property
Sigma-delta, see delta-sigma modulationSignal
analog, 10continuous-time, 10discrete-time, 10
band-limited, 279digital, 11
continuous-time, 10discrete-time, 11
discrete-time, 279periodical, 279
Signal to noise and distortion ratio, 294Signal transfer function, 38Signum function, 285Sinc function, 283
normalized, 283unnormalized, 284
Index 309
Single-carrier signal, 18–21Single-ended-to-differential converter, 231SNDR, see signal to noise and distortion ratioSoftware radio, 2, 8Software-defined radio, 2, 5, 8Spectral mask, 51Spectrum, 51, 277
discrete, 279periodical, 279
Square wave, 54Standard cells, 126Surface acoustic wave filter, 9Switched-mode power amplifier, see power
amplifier, switched-modeSymbol rate, 18
TTaylor series expansion
using convolution powers, 291TDC, see time-to-digital converterThreshold voltage, 133, 151Time shift, 53, 279, 283, 289Time-of-arrival, 131Time-of-flight, 131Time-to-digital converter, 11, 127Trailing-edge PWM, see pulse width
modulation, trailing-edge
Tristate inverter, 172
UUniform-sampling PWM, see pulse width
modulation, uniform-samplingUnit delay, 133, 155, 161Unit standard deviation, 155
VVariability
global, 143, 147local, 151
Variable delay block, 187, 220, 223Vernier delay line, 143Volterra series, 291
XXNOR gate
all-NOR, 164XOR gate, 163, 198, 231
all-NAND, 164multiplexer-based, 166static CMOS, 165transmission-gate-based, 166two-layer gate-based, 164, 198
