A complex discrete (or digital) signal x(n) is defined in a

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1 Chaper Complex Signals A number of signal processing applicaions make use of complex signals. Some examples include he characerizaion of he Fourier ransform, blood velociy esimaions, and modulaion of signals in elecommunicaions. Furhermore, a number of signal-processing conceps are easier o derive, explain and undersand using complex noaion. I is much easier, for example o add he phases of wo complex exponenials such as x = e jφ e jφ, han o manipulae rigonomeric formula, such as cos(φ ) cos(φ ). We sar by inroducing complex signals in Secion., and reaing he Fourier relaions in Sec... Among all complex signals, he so-called analyic signals are especially useful, and hese will be considered in greaer deail in Secion.3... Inroducion o complex signals A complex analog signal x is formed by he signal pair {x R,x I }, where boh x R and x I are he ordinary real signals. The relaionship beween hese signals is given by: x = x R + jx I, (.) where j =. similar manner: A complex discree (or digial) signal x(n) is defined in a x(n) = x R (n) + jx I (n). (.) A complex number x can be represened by is real and imaginary pars x R and x I, or by is magniude and phase a and θ, respecively. The relaionship beween hese values is illusraed in Fig... Complex signals are defined boh in coninuous ime and discree ime: x = a exp(jθ a ) and x(n) = a(n) exp(jθ(n)), (.3) where a = x R + xi and a(n) = x R (n) + x I (n) θ = arcan x I x R and θ(n) = arcan x I(n) x R (n). x R = a cos(θ) and x R (n) = a(n) cos(θ(n)) x I = a sin(θ) and x I (n) = a(n) sin(θ(n)) (.4) The magniudes a and a(n) are also known as envelopes of x and x(n), respecively. 45

2 46 CHAPTER. COMPLEX SIGNALS Im ae jθ x I a θ x R Re Figure.: Illusraion of he relaionship beween he real and imaginary pars of he complex number x and is magniude and phase. Euler s formula.. Useful rules and ideniies Many applicaions require o conver beween a complex number and a rigonomeric funcion. The ransiion is given by Euler s formula: e jθ = cos θ + j sin θ cos θ = ejθ + e jθ sin θ = ejθ e jθ j (.5) Useful resuls Example on page 59 shows how o use hese ideniies o plo he magniude of he specral densiy funcion. Table. shows some useful ideniies. r θ re jθ e j = ±π e ±jπ = ±nπ e ±jnπ = n odd ineger ±π e ±jπ = n ±nπ e ±jnπ = n ineger ±π/ e ±jπ/ = ±j ±nπ/ e ±jnπ/ = ±j n =, 5, 6, 3,... ±nπ/ e ±jnπ/ = j n = 3, 7,, 5,... Table.: Undersanding some useful ideniies... Phasors The word phasor is ofen used by mahemaicians o mean any complex number. In engineering, i is frequenly used o denoe a complex exponenial funcion of consan modulus and linear phase, ha is a funcion of pure harmonic behavior. Here is an example of such a phasor: x = Ae jπf, (.6) which has a consan modulus A and a linearly varying phase. I is no uncommon ha he modulus and phase are ploed separaely. Differen ways o depic phasors are illusraed in Fig...

3 .. INTRODUCTION TO COMPLEX SIGNALS 47 Re{x} Im{x} A A /f /f /f /f (a) Real and imaginary componens A a π θ /f /f /f /f (b) Modulus and phase Im Re A /f A /f (c) Three-dimensional view Figure.: Differen depicions of he phasor A exp(jπf )

4 48 CHAPTER. COMPLEX SIGNALS Finally i mus be noed ha a complex valued funcion or phasor, whose real par is an even funcion and whose imaginary par is odd, is said o be hermiian. A phasor whose real par is odd and he imaginary is odd, is said o be anihermiian.. Specrum of a complex signal The specrum of a complex signal can be found by using he usual expressions for he Fourier ransform. In he following we will derive he specrum X(f) of he complex signal x = x + jx as a linear combinaion of he specra X (f) and X (f) of he real-valued signals x and x. One consequence of he fac ha x or x(n) is complex, is ha he ypical odd/even symmery of he specrum are los. I is easy o demonsrae ha he following expression is valid for complex signals: x X ( f) and x (n) X ( f), (.7) where x is he complex conjugae of x, and ( ) denoes a Fourier ransform pair. Le he complex signal x be expressed in he form: x = x + jx, (.8) where x and x are real signals. Le heir specra be X (f) and X (f), respecively, i.e. x X (f) and x X (f). The real par of x can be expressed as : x = (x + x ). (.9) Using he linear propery of he Fourier ransform, we ge: G (f) = (G(f) + G ( f)) (.) Following he same line of consideraions, one ges: g = j (g g ) G (f) = j (G(f) G ( f)). (.) If one uses he indexes R and I o denoe he real and imaginary pars of a signal, he following simple relaions are obained: G R (f) = G R (f) G I (f) G I (f) = G I (f) + G R (f). (.) Similar relaions can be derived for discree signals oo... Properies of he Fourier ransform for complex signals The basic se of properies of he Fourier ransform for real signals is also valid for complex signals. Table. gives a shor overview of he properies of he Fourier ransform for analog signals. Table.3 gives he equivalen properies for digial complex signals. Remember ha (a + jb) = a jb

5 .. SPECTRUM OF A COMPLEX SIGNAL 49 x X(f); x X (f); x X (f). Lineariy ax + bx ax (f) + bx (f) (.3). Symmery X x( f) (.4) 3. Scaling x(k) k X ( ) f k (.5) 4. Time reversal x( ) X( f) (.6) 5. Time shifing propery x( + ) X(f)e jπf, where is a real consan (.7) 6. Frequency shif xe πf X(f + f ), where f is a real consan (.8) 7. Time and frequency differeniaion d p x d p (jπf) p X(f), ( jπ) p x dp X(f) df p, p is a real number 8. Convoluion (.9) x x X (f)x (f); x x = G (f) G (f) (.) Parseval s heorem x x d = X (f)x (f)df (.) Table.: Properies of he Fourier ransform for complex analog signals.

6 5 CHAPTER. COMPLEX SIGNALS x(n) X(f); x (n) X (f); x (n) X (f). Lineariy ax (n) + bx (n) ax (f) + bx (f), a and b consans (.). The symmery propery is no relevan 3. The scaling propery is no relevan 4. Time reversal x( n) X( f) (.3) 5. Time shif x(n + n ) X(f)e jπfn T, n is an ineger number (.4) 6. Frequency shif propery x(n)e jπfn T X(f + f ) (.5) 7. Differeniaion (jπn T ) p x(n) dp X(f) df p (.6) 8. Convoluion x (n) x (n) X (f)x (f); x (n)x (n) X (f) X (f) (.7) Parseval x (n)x (n) = f s f s/ f s/ X (f)x (f)df (.8) Table.3: Properies of he Fourier ransform for complex digial signals.

7 .3. ANALYTIC SIGNALS 5 Figure.3: Filraion of complex signals... Linear processing of complex signals A complex signal consiss of wo real signals - one for he real and one for he imaginary par. The linear processing of a complex signal, such as filraion wih a ime-invarian linear filer, corresponds o applying he processing boh o he real and he imaginary par of he signal. The filraion wih a filer, which impulse response is real, corresponds o wo filraion operaions - one for he real and one for he imaginary par of he signal. Filering a complex signal using a filer wih a real-valued impulse response can be reaed as wo separae processes - one for he filraion of he real and one for he filraion of he imaginary componen of he inpu signal: h (a + jb) = h a + jh b. (.9) If he filer has a complex impulse response, hen he operaion corresponds o 4 real filering operaions as shown in Fig..3 An example of an ofen-used filer wih complex impulse response is he filer given by: { h m (n) = N ejm π N n n N (.3) oherwise The ransfer funcion of he filer H m (f) = N sin π(fn T m) sin π(f T m/n) e jπ(n )(f T m/n), (.3) is a funcion of he parameer m. Figure.4 illusraes boh he impulse response and he ransfer funcion of he filer..3 Analyic signals An analyic signal is a signal, which specrum is one-sided. For analog signals his means ha heir specrum is for f > or f <. Analyic discree-ime signals have a specrum which is for fs < f < and in he corresponding pars of he periodic specrum, or < f < fs and he corresponding pars of he periodic specrum. The so-inroduced condiion for an analyic signal gives he connecion beween he real and he imaginary par of he complex signal.

8 5 CHAPTER. COMPLEX SIGNALS Real par Imaginary par h R (n). h I (n) n 5 5 n N = 6.8 H (f), N= f/f s 3 arg H (f) f/f s Figure.4: Impulse response and ransfer funcion of a complex filer used o carry ou he Discree-ime Fourier ransform.

9 .3. ANALYTIC SIGNALS Analyic analog signals If a real signal x wih frequency specrum X(f) is aken as a saring poin, hen he following relaions will be valid for he respecive analyic signal z x and is specrum: X(f) for f > z x Z x (f) = X(f) for f = (.3) for f <. This relaion can be expressed in a more compac form as : Z x (f) = [ + sgn(f)]x(f). (.33) Since he sgn(f) is he fourier specrum of he funcion j π (j π sgn(f)), hen he above equaion is equivalen o: ( z x = δ + j ) x (.34) π Here we inroduce he signal x H, known as he Hilber ransform of x and given by: I can be seen ha x H = x π. (.35) z x = x + jx H. (.36) Noice ha he complex conjugae zx is also analyic wih specrum given by for f > Zx(f) = X() for f = (.37) X(f) for fz < and ha consequenly: hen If z x is wrien in he form x = (z x + z x). (.38) z x = a z exp(jθ) (.39) x = a z cos(θ z ) and x H = a z sin(θ) (.4) If he analyic signal z x is filered wih a filer wih a real impulse response, hen he oupu signal y will be: y = h z x = h x + jh x H. (.4) If he Hilber ransform of h is denoed by h H, hen one ges: y = x (h + jh H ) = x z h. (.4) This operaion is some imes useful when one wans o work wih analyic signals, bu has only a real signal o sar wih. Noice ha h H is usually noncausal. Using he symmery propery of he Fourier ransform i can be shown ha he real and imaginary pars of he specrum of a real-signal form a Hilber pair, ha is each can be obained from he oher using a Hilber ransform. This, and a number of oher properies of he Hilber ransform can be found in Table??. The symbol H is used in he able o denoe he Hilber ransform, and he resul is x = H{x}. sgn(f) reurns he sign of he argumen. I reurns + if f >, if f <, and if f =.

10 54 CHAPTER. COMPLEX SIGNALS G(f). G (f)..5 f g f.5 f.5 f Figure.5: Consrucion of an analyic signal.3. Analyic discree-ime signals If a real discree-ime signal x(n) X(f) is used as a basis, hen he corresponding analyic signal 3 z x (n) will be given as X(f) for pf s < f < (p + ) fs z x (n) Z x (f) = X(f) for f = pf s (.43) oherwise, where p is an ineger number. Le s consider he following specrum: for pf s < f < (p + ) fs Z (f) = for f = pf s for (p )f s < f < pf s. (.44) The discree ime signal ha corresponds o his specrum is I follows direcly ha z x = j nπ sin (n π ). (.45) ( δ(n) + j nπ sin (n π ) ) g(n). (.46) By analogy wih he relaions in Secion.3. we denoe he signal x H (n) = x(n) nπ sin (n π ) (.47) as he Hilber ransform of x(n). The relaion beween z x (n), x(n) and x H (n) is given by: z x (n) = x(n) + jx H (n). (.48) Similarly o Secion.3. i can be shown ha h(n) z h (n) = x(n) z h (n), (.49) where z h (n) = h(n) + jh H (n) = h(n) + jh(n) nπ sin (n π )..4 Insananeous ampliude and frequency Le s consider he band-limied real signal x wih a band limi f g. The ampliude specrum of such a signal is shown in he lef sub-plo of Fig The use of he erm analyic in relaion o discree-ime signals leads o mahemaical difficulies. Many of hem can, however, be circumvened if one applies he fac, ha a given digial signal corresponds o an equivalen analog signal.

11 .4. INSTANTANEOUS AMPLITUDE AND FREQUENCY 55. Lineariy H{ax + bx } = a x + b x, a and b consans (.5). Time shif H{x( + )} = x( + ) (.5) 3. Applying wo imes he Hilber ransform gives he original signal H{H{x}} = x (.5) 4. The inverse Hilber ransform x = H {x} = x du = x π( u) π (.53) 5. Even/odd propery x even x odd x odd x even (.54) 6. Conservaion of energy x d = x d (.55) 7. Orhogonaliy x x = (.56) 8. Modulaion if H{x cos(πf )} = x sin(πf ) (.57) X(f) = { X(f) f F, f > F oherwise 9. Convoluion H{x h} = x h = x h (.58). Specrum of a real signal H{X R (f)} = X I (f), H{X I (f)} = X R (f) (.59) where x X R (f) + jx I (f), and x is real and causal. Table.4: Properies of he Hilber ransform.

12 56 CHAPTER. COMPLEX SIGNALS.5.8 g a z g H ψ z Figure.6: Insananeous ampliude and phase for an analyic signal. For his paricular case an analyic signal can be creaed by frequency-shifing he specrum of he signal wih an offse f, where f > f g. The frequencyshifing operaion is illusraed in he righ sub-plo of Fig..5. In oher words, we creae a signal z = xe jπf, which is analyic if f > f g. I can be represened wih a real and imaginary par as: z = x + jx H, (.6) where x H is he Hilber ransform of he real signal x. Using he Euler s relaions, we find ha: x = x cos(πf ), and x H = x sin(πf ). (.6) If we assume ha x ges only non-negaive values (x ), hen i follows immediaely ha: a z = x and θ z = πf. (.6) The insananeous ampliude of z is hen x, and he value θ z/π is said o be he insananeous frequency of he signal. An example is given in Fig..6. In he case ha x can boh be posiive and negaive, he ampliude a z will be equal o he absolue value of x, a z = x. The phase θ z of he analyic signal will jump from wih ±π in hose ime insances, when x changes sign. The firs derivaive wih respec o ime θ will sill be proporional o he insananeous frequency, excep for he poins of disconinuiy as shown in Fig..7. The conceps of insananeous ampliude and frequency can be ransferred formally o he ypical case, in which he analyic signal z is given in he form a z exp(jθ). Figure.8 shows an example in which θ varies non-linearly in ime. These conceps are also applied o non-analyic signals, and he same values can be defined for discree-ime signals. The phase of he signal is given by: θ = arcan x H x. (.63) The insananeous frequency is proporional o he derivaive of he phase wih respec o ime, and can be found from: θ π = π xx H x Hx x + x H (.64)

13 .4. INSTANTANEOUS AMPLITUDE AND FREQUENCY g a z g H ψ z Figure.7: Insananeous ampliude and phase for an analyic signal wih a poin of disconinuiy..5.8 g R a z g I ψ z Figure.8: Insananeous ampliude and phase of a signal wih non-linear variaion of θ z.

14 58 CHAPTER. COMPLEX SIGNALS.5 ( r).5 g R r.5 g I g I.5.5 g R Figure.9: Complex FM signal x F M = exp(jπr ).4. Linear FM signal The complex signal exp(jπr ), where r is a consan has some ineresing properies. From he expression i can be seen ha he ampliude of he signal is, and ha he phase of he signal is given by: θ = πr. (.65) The signal is illusraed in Fig..9. The specrum of he signal is formally given by: X F M (f) e jπr e jπf d = e jπf /r j e jπr( f/r) d = r e jπf /r. (.66) The ampliude specrum of he signal is consan (frequency independen), and is phase specrum varies quadraically wih frequency. Noice he similariy beween he signal and is specrum.

15 .4. INSTANTANEOUS AMPLITUDE AND FREQUENCY 59 The insananeous frequency f i is found o be: f i = θ = r. (.67) π The insananeous frequency f i increases linearly wih ime a rae defined by he consan r (measured for example in Hz/sec.). This is he reason why his signal is called linear frequency modulaed signal/pulse. Figure.9 shows he signal as a funcion of ime. I can be seen, ha his signal does no belong o he se of signals wih finie energy. However, he signal x = aexp(jπr ), (.68) where a is appropriaely chosen real signal, does. The specrum X(f) of his signal can be found from: X(f) = e jπf /r ae jπr(f/r ) d. (.69) In oher words, he shape of he ampliude specrum X(f) is deermined by he convoluion of wo funcions A(f) and exp(jπrf ). While he resul of his convoluion canno always be calculaed direcly, one can for very large values of he parameer r use an approximaion, which is described in he following paragraphs. Consider he following signal: p = p j exp(jπp ). (.7) I can be shown ha p δ, when p. Correspondingly one can show, ha for large values of he produc rt, where T is he duraion of a, we have: ( ) f j X(f) a r r e jπf /r (.7) if a(f) is coninuous. Under hese condiions, X(f) has almos he same shape os a (appropriaely scaled hough). Consider he Fourier ransform pair: a exp(jπr ) X a (f) j r exp( jπf /r), (.7) where a X a (f). When he produc rt is small, one can use similar consideraions as above and show ha he signal a exp(jπr ) has an envelope, whose shape is given by X a (f). X a (f) mus be appropriaely delayed and scaled. Figure. shows an example of he resul from he convoluion operaion for a recangular signal, whose duraion is varied. Noice ha if a is ime-limied, i.e. sars a = and ends a =, hen he a FM-signal wih a sar and end frequencies r and r is creaed. This can also be achieved by applying he frequency-shif propery of he Fourier ransform.

16 6 CHAPTER. COMPLEX SIGNALS g R cos(π r ) ( r).5 g I sin(π r ) r g g * exp(jπ r ) x 3 g * exp(jπ r ) g 3 * exp(jπ r ) g g x Figure.: Averaged complex FM signals

17 .5. EXAMPLE PROBLEMS 6.5 Example problems Example Plo he magniude of he ransfer funcion of a high-pass filer, which is described by: Soluion Using he rules of lineariy and ime shif we obain: The specrum is illusraed in Fig.. y(n) = x(n) x(n ) (.73) Y (f) = X(f)( e jπf ) {z } H(f) H(f) = e jπf = e jπf sin(πf) {z } = H(f) (V/Hz) f (oscillaions/second) Figure.: The ransfer funcion of he sysem from Example.