Complex Signals - DTU

1 Chapter 2 Complex SignalsA number of signal processing applications make use of Complex Signals . Someexamples include the characterization of the Fourier transform, blood velocityestimations, 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 thephases of two Complex exponentials such asx(t) =ej 1e 2, than to manipulatetrigonometric formula, such as cos( 1) cos( 2).We start by introducing Complex Signals in Section , and treating theFourier relations in Sec. Among all Complex Signals , the so-calledanalyticsignals are especially useful, and these will be considered in greater detail inSection Introduction to Complex signalsA Complex analog signalx(t) is formed by the signal pair{xR(t),xI(t)}, wherebothxR(t) andxI(t) are the ordinary real Signals . The relationship betweenthese Signals is given by:x(t) =xR(t) +jxI(t),( )wherej= 1.

2 A Complex discrete (or digital) signalx(n) is defined in asimilar manner:x(n) =xR(n) +jxI(n).( )A Complex numberxcan be represented by its real and imaginary partsxRandxI, or by its magnitude and phaseaand , respectively. The relationshipbetween these values is illustrated in Fig. Signals are defined both in continuous time and discrete time:x(t) =a(t) exp(j a(t)) andx(n) =a(n) exp(j (n)),( )wherea(t) = x2R(t) +x2I(t) anda(n) = x2R(n) +x2I(n) (t) = arctanxI(t)xR(t)and (n) = arctanxI(n)xR(n).xR(t) =a(t) cos( (t)) andxR(n) =a(n) cos( (n))xI(t) =a(t) sin( (t)) andxI(n) =a(n) sin( (n))( )The magnitudesa(t) anda(n) are also known asenvelopesofx(t) andx(n), 2. Complex SIGNALSa aej ImRexIxRFigure : Illustration of the relationship between the real and imaginary partsof the Complex numberxand its magnitude and Useful rules and identitiesMany applications require to convert between a Complex number and a trigono-metric function.

3 The transition is given by Euler s formula:Euler sformulaej = cos +jsin cos =ej +e j 2sin =ej e j 2j( )Example 1 on page 59 shows how to use these identities to plot the magnitudeof the spectral density shows some useful rej 1 0ej0= 11 e j = 11 n e jn = 1nodd integer1 2 e j2 = 1n1 2n e j2n = 1ninteger1 /2e j /2= j1 n /2e jn /2= j n= 1, 5, 6, 13, ..1 n /2e jn /2= j n= 3, 7, 11, 15, ..Table : Understanding some useful PhasorsThe wordphasoris often used by mathematicians to mean any Complex engineering, it is frequently used to denote a Complex exponential function ofconstant modulus and linear phase, that is a function of pure harmonic is an example of such a phasor :x(t) =Aej2 f0t,( )which has a constant modulusAand a linearly varying phase. It is not un-common that the modulus and phase are plotted separately. Different ways todepict phasors are illustrated in Fig. INTRODUCTION TO Complex SIGNALS47Re{x(t)}A01/f0 1/f0Im{x(t)}A01/f0 1/f0(a) Real and imaginary componentsa(t)A01/f0 1/f02 1/f0 1/f0 (t)0(b) Modulus and phase1/f0tAReA1/f0Im(c) Three-dimensional viewFigure : Different depictions of the phasorAexp(j2 f0t)48 CHAPTER 2.

4 Complex SIGNALSF inally it must be noted that a Complex valued function or phasor , whosereal part is an even function and whose imaginary part is odd, is said to behermitian. A phasor whose real part is odd and the imaginary is odd, is said tobe Spectrum of a Complex signalThe spectrum of a Complex signal can be found by using the usual expressionsfor the Fourier transform. In the following we will derive the spectrumX(f) ofthe Complex signalx(t) =x1(t) +jx2(t) as a linear combination of the spectraX1(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 typicalodd/even symmetry of the spectrum are lost. It is easy to demonstrate that thefollowing expression is valid for Complex Signals :x (t) X ( f) andx (n) X ( f),( )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),( )wherex1(t) andx2(t) are real Signals .

5 Let their spectra beX1(f) andX2(f),respectively, (t) X1(f) andx2(t) X2(f). The real part ofx(t) canbe expressed as1:x1(t) =12(x(t) +x (t)).( )Using the linear property of the Fourier transform, we get:G1(f) =12(G(f) +G ( f))( )Following the same line of considerations, one gets:g2(t) = j12(g(t) g (t)) G2(f) = j12(G(f) G ( f)).( )If one uses the indexesRandIto denote the real and imaginary parts of asignal, the following simple relations are obtained:GR(f) =G1R(f) G2I(f)GI(f) =G1I(f) +G2R(f).( )Similar relations can be derived for discrete Signals Properties of the Fourier transform for Complex sig-nalsThe basic set of properties of the Fourier transform for real Signals is also validfor Complex Signals . Table gives a short overview of the properties of theFourier transform for analog Signals . Table gives the equivalent propertiesfor digital Complex that (a+jb) =a SPECTRUM OF A Complex SIGNAL49x(t) X(f);x1(t) X1(f);x2(t) X2(f)1.

6 Linearityax1(t) +bx2(t) aX1(f) +bX2(f)( )2. SymmetryX(t) x( f)( )3. Scalingx(kt) 1|k|X(fk)( )4. Time reversalx( t) X( f)( )5. Time shifting propertyx(t+t0) X(f)ej2 ft0,wheret0is a real constant( )6. Frequency shiftx(t)e 2 f0t X(f+f0),wheref0is a real constant( )7. Time and frequency differentiationdpx(t)dtp (j2 f)pX(f),( j2 t)px(t) dpX(f)dfp, pis a real number( )8. Convolutionx1(t) x2(t) X1(f)X2(f);x1(t)x2(t) =G1(f) G2(f)( )Parseval s theorem x1(t)x 2(t)dt= X1(f)X 2(f)df( )Table : Properties of the Fourier transform for Complex analog 2. Complex SIGNALSx(n) X(f);x1(n) X1(f);x2(n) X2(f)1. Linearityax1(n) +bx2(n) aX1(f) +bX2(f),aandbconstants( )2. The symmetry property is not relevant3. The scaling property is not relevant4. Time reversalx( n) X( f)( )5. Time shiftx(n+n0) X(f)ej2 fn0 T, n0is an integer number( )6. Frequency shift propertyx(n)e j2 f0n T X(f+f0)( )7. Differentiation(j2 n T)px(n) dpX(f)dfp( )8.

7 Convolutionx1(n) x2(n) X1(f)X2(f);x1(n)x2(n) X1(f) X2(f)( )Parseval x1(n)x 2(n) =1fsfs/2 fs/2X1(f)X 2(f)df( )Table : Properties of the Fourier transform for Complex digital ANALYTIC SIGNALS51 Figure : Filtration of Complex Linear processing of Complex signalsA Complex signal consists of two real Signals - one for the real and one for theimaginary part. The linear processing of a Complex signal, such as filtrationwith a time-invariant linear filter, corresponds to applying the processing bothto the real and the imaginary part of the filtration with a filter, which impulse response is real, corresponds totwo filtration operations - one for the real and one for the imaginary part of a Complex signal using a filter with a real-valued impulse responsecan be treated as two separate processes - one for the filtration of the real andone 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).

8 ( )If the filter has a Complex impulse response, then the operation corresponds to4 real filtering operations as shown in Fig. example of an often-used filter with Complex impulse response is the filtergiven by:hm(n) ={1 Nejm2 Nn0 n N 10otherwise( )The transfer function of the filterHm(f) =1 Nsin (fN T m)sin (f T m/N)e j (N 1)(f T m/N),( )is a function of the parameterm. Figure illustrates both the impulse responseand the transfer function of the Analytic signalsAn analytic signal is a signal, which spectrum is one-sided . For analog signalsthis means that their spectrum is 0 forf >0 orf <0. Analytic discrete-timesignals have a spectrum which is 0 for fs2< f <0 and in the correspondingparts of the periodic spectrum, or 0< f <fs2and the corresponding parts ofthe periodic spectrum. The so-introduced condition for an analytic signal givesthe connection between the real and the imaginary part of the Complex 2.}

9 Complex SIGNALS051015 (n)nReal part051015 (n)nImaginary part |H2(f)|, N=16f/fsN = 16 3 2 10123arg H2(f)f/fsFigure : Impulse response and transfer function of a Complex filter used tocarry out the Discrete-time Fourier ANALYTIC Analytic analog signalsIf 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:zx(t) Zx(f) = 2X(f) forf >0X(f) forf= 00forf <0.( )This relation can be expressed in a more compact form as2:Zx(f) = [1 + sgn(f)]X(f).( )Since the sgn(f) is the fourier spectrum of the functionj1 t(j1 t sgn(f)),then the above equation is equivalent to:zx(t) =( (t) +j1 t) x(t)( )Here we introduce the signalxH(t), known as the Hilbert transform ofx(t) andgiven by:xH(t) =x(t) 1 t.( )It can be seen thatzx(t) =x(t) +jxH(t).( )Notice that the Complex conjugatez x(t) is also analytic with spectrum given byZ x(f) = 0forf >0X(0) forf= 02X(f) forfz <0( )and that consequently:x(t) =12(zx(t) +z x(t)).

10 ( )Ifzx(t) is written in the formzx(t) =az(t) exp(j (t))( )thenx(t) =az(t)cos( z(t)) andxH(t) =az(t) sin( (t))( )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).( )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).( )This operation is some times useful when one wants to work with analyticsignals, but has only a real signal to start with. Notice thathH(t) is the symmetry property of the Fourier transform it can be shown thatthe real and imaginary parts of the spectrum of a real-signal form a Hilbertpair, that is each can be obtained from the other using a Hilbert , and a number of other properties of the Hilbert transform can be found inTable??. The symbolHis used in the table to denote the Hilbert transform,and the result is x(t) =H{x(t)}.2sgn(f) returns the sign of the argument.