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Appendix a Definitions, Conventions and Overview of Used Theory 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: – Replace all occurrences of t on the right-hand side by some unused symbol, e.g.
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