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Chapter 2
Complex Signals
A number of signal processing applications make use of complex signals. Some examples include the characterization of the Fourier transform, blood velocity estimations, and modulation of signals in telecommunications. Furthermore, a number of signal-processing concepts are easier to derive, explain and un- derstand using complex notation. It is much easier, for example to add the phases of two complex exponentials such asx(t) =ejφ1e?φ2, than to manipulate trigonometric formula, such as cos(φ1)cos(φ2). We start by introducing complex signals in Section 2.1, and treating the Fourier relations in Sec. 2.2. Among all complex signals, the so-calledanalytic signals are especially useful, and these will be considered in greater detail in
Section 2.3.1.
2.1 Introduction to complex signals
A complex analog signalx(t) is formed by the signal pair{x
R(t),xI(t)}, where
bothxR(t) andxI(t) are the ordinary real signals. The relationship between these signals is given by: x(t) =xR(t) +jxI(t),(2.1) wherej=⎷-1. A complex discrete (or digital) signalx(n) is defined in a similar manner: x(n) =xR(n) +jxI(n).(2.2) A complex numberxcan be represented by its real and imaginary partsxR andxI, or by its magnitude and phaseaandθ, respectively. The relationship between these values is illustrated in Fig. 2.1. Complex signals are defined both in continuous time and discrete time: x(t) =a(t)exp(jθa(t)) andx(n) =a(n)exp(jθ(n)),(2.3) where a(t) =?x
2R(t) +x2I(t) anda(n) =?x
2R(n) +x2I(n)
θ(t) = arctanxI(t)x
R(t)andθ(n) = arctanxI(n)x
R(n). x
R(t) =a(t)cos(θ(t)) andxR(n) =a(n)cos(θ(n))
x I(t) =a(t)sin(θ(t)) andxI(n) =a(n)sin(θ(n))(2.4) The magnitudesa(t) anda(n) are also known asenvelopesofx(t) andx(n), respectively.45
46CHAPTER 2. COMPLEX SIGNALSaθae
jθIm RexI x RFigure 2.1: Illustration of the relationship between the real and imaginary parts of the complex numberxand its magnitude and phase.2.1.1 Useful rules and identities Many applications require to convert between a complex number and a trigono- metric function. The transition is given by Euler"s formula:Euler"s formula e jθ= cosθ+jsinθ cosθ=ejθ+e-jθ2 sinθ=ejθ-e-jθ2j(2.5) Example 1 on page 59 shows how to use these identities to plot the magnitude of the spectral density function. Table 2.1 shows some useful identities.Useful results r θ re jθ1 0ej0= 1
1±π e±jπ=-1
1±nπ e±jnπ=-1nodd integer
1±2π e±j2π= 1n
1±2nπ e±j2nπ= 1ninteger
1±π/2e±jπ/2=±j
1±nπ/2e±jnπ/2=±j n= 1, 5, 6, 13, ...
1±nπ/2e±jnπ/2=?j n= 3, 7, 11, 15, ...Table 2.1: Understanding some useful identities.
2.1.2 Phasors
The wordphasoris often used by mathematicians to mean any complex number. In engineering, it is frequently used to denote a complex exponential function of constant modulus and linear phase, that is a function of pure harmonic behavior.
Here is an example of such a phasor:
x(t) =Aej2πf0t,(2.6) which has a constant modulusAand a linearly varying phase. It is not un- common that the modulus and phase are plotted separately. Different ways to depict phasors are illustrated in Fig. 2.2.
2.1. INTRODUCTION TO COMPLEX SIGNALS47Re{x(t)}
A 0 1/f0 -1/f0
Im{x(t)}
A 0 1/f0 -1/f0(a) Real and imaginary components a(t) A 0 1/f0 -1/f0 2π
1/f0-1/f0
θ(t)
0(b) Modulus and phase
1/f0 t A Re A 1/f 0
Im(c) Three-dimensional view
Figure 2.2: Different depictions of the phasorAexp(j2πf0t)
48CHAPTER 2. COMPLEX SIGNALSFinally it must be noted that a complex valued function or phasor, whose
real part is an even function and whose imaginary part is odd, is said to be hermitian. A phasor whose real part is odd and the imaginary is odd, is said to be antihermitian.
2.2 Spectrum of a complex signal
The spectrum of a complex signal can be found by using the usual expressions for the Fourier transform. In the following we will derive the spectrumX(f) of the complex signalx(t) =x1(t) +jx2(t) as a linear combination of the spectra X
1(f) andX2(f) of the real-valued signalsx1(t) andx2(t).
One consequence of the fact thatx(t) orx(n) is complex, is that the typical odd/even symmetry of the spectrum are lost. It is easy to demonstrate that the following expression is valid for complex signals: x ?(t)↔X?(-f) andx?(n)↔X?(-f),(2.7) wherex?(t) is the complex conjugate ofx(t), and (↔) denotes a Fourier trans- form pair. Let the complex signalx(t) be expressed in the form: x(t) =x1(t) +jx2(t),(2.8) wherex1(t) andx2(t) are real signals. Let their spectra beX1(f) andX2(f), respectively, i.e.x1(t)↔X1(f) andx2(t)↔X2(f). The real part ofx(t) can be expressed as 1: x
1(t) =12
(x(t) +x?(t)).(2.9) Using the linear property of the Fourier transform, we get: G
1(f) =12
(G(f) +G?(-f)) (2.10) Following the same line of considerations, one gets: g
2(t) =-j12
(g(t)-g?(t))↔G2(f) =-j12 (G(f)-G?(-f)).(2.11) If one uses the indexesRandIto denote the real and imaginary parts of a signal, the following simple relations are obtained: G
R(f) =G1R(f)-G2I(f)
G
I(f) =G1I(f) +G2R(f).(2.12)
Similar relations can be derived for discrete signals too.
2.2.1 Properties of the Fourier transform for complex sig-
nals The basic set of properties of the Fourier transform for real signals is also valid for complex signals. Table 2.2 gives a short overview of the properties of the Fourier transform for analog signals. Table 2.3 gives the equivalent properties for digital complex signals.1
Remember that (a+jb)?=a-jb
2.2. SPECTRUM OF A COMPLEX SIGNAL49x(t)↔X(f);x1(t)↔X1(f);x2(t)↔X2(f)1. Linearity
ax
1(t) +bx2(t)↔aX1(f) +bX2(f) (2.13)2. Symmetry
X(t)↔x(-f) (2.14)3. Scaling
x(kt)↔1|k|X?fk (2.15)4. Time reversal x(-t)↔X(-f) (2.16)5. Time shifting property x(t+t0)↔X(f)ej2πft0,wheret0is a real constant (2.17)6. Frequency shift x(t)e-2πf0t↔X(f+f0),wheref0is a real constant (2.18)7. Time and frequency differentiation d px(t)dt p, pis a real number (2.19)8. Convolution x
1(t)?x2(t)↔X1(f)X2(f);x1(t)x2(t) =G1(f)?G2(f) (2.20)Parseval"s theorem
x
1(t)x?2(t)dt=?
∞X1(f)X?2(f)df(2.21)Table 2.2: Properties of the Fourier transform for complex analog signals
50CHAPTER 2. COMPLEX SIGNALSx(n)↔X(f);x1(n)↔X1(f);x2(n)↔X2(f)1. Linearity
ax
1(n) +bx2(n)↔aX1(f) +bX2(f), aandbconstants (2.22)2. The symmetry property is not relevant
3. The scaling property is not relevant
4. Time reversal
x(-n)↔X(-f) (2.23)5. Time shift x(n+n0)↔X(f)ej2πfn0ΔT, n0is an integer number (2.24)6. Frequency shift property x(n)e-j2πf0nΔT↔X(f+f0) (2.25)7. Differentiation (j2πnΔT)px(n)↔dpX(f)df p(2.26)8. Convolution x
1(n)?x2(n)↔X1(f)X2(f);x1(n)x2(n)↔X1(f)?X2(f) (2.27)Parseval
-∞x
1(n)x?2(n) =1f
sf s/2? -fs/2X
1(f)X?2(f)df(2.28)Table 2.3: Properties of the Fourier transform for complex digital signals.
2.3. ANALYTIC SIGNALS51Figure 2.3: Filtration of complex signals.
2.2.2 Linear processing of complex signals
A complex signal consists of two real signals - one for the real and one for the imaginary part. The linear processing of a complex signal, such as filtration with a time-invariant linear filter, corresponds to applying the processing both to the real and the imaginary part of the signal. The filtration with a filter, which impulse response is real, corresponds to two filtration operations - one for the real and one for the imaginary part of the signal. Filtering a complex signal using a filter with a real-valued impulse response can be treated as two separate processes - one for the filtration of the real and one for the filtration of the imaginary component of the input signal: h(t)?(a(t) +jb(t)) =h(t)?a(t) +jh(t)?b(t).(2.29) If the filter has a complex impulse response, then the operation corresponds to
4 real filtering operations as shown in Fig. 2.3
An example of an often-used filter with complex impulse response is the filter given by: h m(n) =? 1N ejm2πN
0 otherwise(2.30)
The transfer function of the filter
H m(f) =1N is a function of the parameterm. Figure 2.4 illustrates both the impulse response and the transfer function of the filter.
2.3 Analytic signals
An analytic signal is a signal, which spectrum is "one-sided". For analog signals this means that their spectrum is≡0 forf >0 orf <0. Analytic discrete-time signals have a spectrum which is≡0 for-fs2 < f <0 and in the corresponding parts of the periodic spectrum, or 0< f
52CHAPTER 2. COMPLEX SIGNALS051015 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 h2R(n) nReal part 051015
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 h2I(n) nImaginary part -0.5-0.4-0.3-0.2-0.100.10.20.30.40.50 0.2 0.4 0.6 0.8 1 |H2(f)|, N=16 f/fsN = 16 -0.5-0.4-0.3-0.2-0.100.10.20.30.40.5-3 -2 -1 0 1 2 3 arg H2(f) f/fsFigure 2.4: Impulse response and transfer function of a complex filter used to carry out the Discrete-time Fourier transform. 2.3. ANALYTIC SIGNALS532.3.1 Analytic analog signals
If a real signalx(t) with frequency spectrum X(f) is taken as a starting point, then the following relations will be valid for the respective analytic signalzx(t) and its spectrum: z x(t)↔Zx(f) =? ?2X(f) forf >0 X(f) forf= 0
0 forf <0.(2.32)
This relation can be expressed in a more compact form as 2: Z x(f) = [1 + sgn(f)]X(f).(2.33) Since the sgn(f) is the fourier spectrum of the functionj1πt (j1πt ↔sgn(f)), then the above equation is equivalent to: z x(t) =? δ(t) +j1πt
?x(t) (2.34) Here we introduce the signalxH(t), known as the Hilbert transform ofx(t) and given by: x H(t) =x(t)?1πt
.(2.35) It can be seen that
z x(t) =x(t) +jxH(t).(2.36) Notice that the complex conjugatez?x(t) is also analytic with spectrum given by Z ?x(f) =? ?0 forf >0 X(0) forf= 0
2X(f) forfz <0(2.37)
and that consequently: x(t) =12 (zx(t) +z?x(t)).(2.38) Ifzx(t) is written in the form
z x(t) =az(t)exp(jθ(t)) (2.39) then x(t) =az(t)cos(θz(t)) andxH(t) =az(t)sin(θ(t)) (2.40) If the analytic signalzx(t) is filtered with a filter with a real impulse response, then the output signaly(t) will be: y(t) =h(t)?zx(t) =h(t)?x(t) +jh(t)?xH(t).(2.41) If the Hilbert transform ofh(t) is denoted byhH(t), then one gets: y(t) =x(t)?(h(t) +jhH(t)) =x(t)?zh(t).(2.42) This operation is some times useful when one wants to work with analytic signals, but has only a real signal to start with. Notice thathH(t) is usually noncausal. Using the symmetry property of the Fourier transform it can be shown that the real and imaginary parts of the spectrum of a real-signal form a Hilbert pair, that is each can be obtained from the other using a Hilbert transform. This, and a number of other properties of the Hilbert transform can be found in Table??. The symbolHis used in the table to denote the Hilbert transform, and the result is ˜x(t) =H{x(t)}.2 sgn(f) returns the sign of the argument. It returns +1 iff >0,-1 iff <0, and 0 if f= 0. 54CHAPTER 2. COMPLEX SIGNALS-0.500.50
0.005 0.01 0.015 0.02 fg |G(f)| f -0.500.50 0.005 0.01 0.015 0.02quotesdbs_dbs19.pdfusesText_25
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