SYNTHESIS OF SIGNAL USING THE EXPONENTIAL FOURIER SERIES

Transcription

1 SYNTHESIS OF SIGNAL USING THE EXPONENTIAL FOURIER SERIES Sadro Adriao Fasolo ad Luiao Leoel Medes Abstrat I 748, i Itrodutio i Aalysi Ifiitorum, Leohard Euler ( ) stated the formula exp( jω = os( ω jsi( ω. This formula is essetial i the teleommuiatio theory. The ore of Fourier Trasform (FT) is a futio as exp( jω. The FT theory states that ay periodi sigals a be deomposed i a sum of a os( ω ad b si( ω, where the FT theory estimates the oeffiiets a ad b. I all text about FT theory the graphis are made usig the osie ad sie urves i time domai. This paper provides the light iside o proess of sigal geeratio the sum of omplex expoetials. A dediated omputatioal tool was developed o CBuilder platform to allow the studets to produe ad aalyze the sigal i the time ad phasor domais. The fasor are omplex expoetial rotatig i a omplex plae. I the simulatio it is possible to sum up to seve ompoets, with differet values of amplitude ad phase Idex Terms euler, fourier series, omplex plae. INTRODUCTION This paper presets a eduatioal approah for the study about sigals geeratio usig the expoetial Fourier series. This approah is realized usig a simulatio program developed with CBuilder platform, whih allows the user to defie the amplitude, frequey of the ompoets whih will produe the sigal. The simulator allows oe to aalyze graphially the rotatio ad the ombiatio of the phasors. The paper outlie is as follows. First, i the ext setio, a brief review of Euler s idetities ad the fasor represetatio is show. The, the expoetial Fourier Series is overviewed. Next, the simulatio program is desribed. After that, two examples of simulatio are show. Fially, i the last setio, the olusio is preseted. EULER S IDENTITIES I the aalysis of liear systems we are partiularly iterested i sigals that a be represeted as futio omplex expoetial. Physially, this futio a be obtaied desribig the rotatio of a vetor with uit legth o the plae omplex, as show i Figure []. The ed poit of the vetor revolves outerlokwise at a agular rate of ω radias per seod. θ ( = ω t ω = FIGURE. COMPLEX PLANE AND PHASOR. πf rad/s To foud the projetios o the axis is eessary add a ouple of vetors, oe with agle θ = ω t ad other oe with θ = ω t, as preseted i Figure. θ ( = ω t θ ( = ω t ω = πf rad/s FIGURE. A COUPLE OF PHASORS ROTATING IN OPPOSITE DIRECTION. Thus, the projetio o the real axis is the resultat of the additio of two vetors with same agular frequey but rotatig i oppose diretio. Thus, exp( jω exp( jω os( ω = () Its is possible observe i Figure that this sum results is i a value that orrespods twie of the projetio o the real axis value, while the positive imagiary part is aeled by the egative imagiary part. Beause of this, there is a Sadro Aadriao Fasolo, INATEL, Av. João de Camargo, 50, , Sata Rita do Sapuaí, MG, Brazil, sadro.fasolo@iatel.br Luiao Leoel Medes, INATEL, Av. João de Camargo, 50, , Sata Rita do Sapuaí, MG, Brazil, luiaol@iatel.br

2 ormalizig fator equals i (). This value is alled osie of agle θ = t. ω si( ω θ ( t(s) os( θ ( ) FIGURE 3. THE COS(X) FUNCTION. Similarly, the projetio o the imagiary axis is obtaied by exp( jω exp( jω si( ω = () j I Figure 4, it is possible to observe that this sum results i a value that orrespods to the twie the value of projetio o the imagiary axis multiplied by the imagiary elemet j =. I the aalog form, the real parts have same value but opposite sigal ad aeled it other. Beause of this, there is a ormalizig fator equals j i (). This value, is alled sie of agle θ = ω t. si( θ ( ) FIGURE 4. THE SIN(X) FUNCTION. Addig the terms o the right-had side ad the terms o the left-had of () e () we a represet de term exp( ω by j exp( jω = os( ω j si( ω (3) The projetio o real axis is the osie futio ad de projetio o imagiary axis is the sie futio, as a be see i Figure 5. FIGURE 5. THE SIN(X) AND COS(X) BY COMPLEX EXPONENTIAL FUNCTION. COMPLEX FOURIER SERIES The futio f ( a be expressed as a summatio of omplex expoetial terms, eah with its ow magitude, phase, ad agular rate (frequey). Eah omplex expoetial term a be represeted by a phasor. The summatio of phasor (the real part, by ovetio) desribe the istataeous values of omplex-valued sigal usig the expoetial Fourier series represetatio. It a be easily show that a set of expoetial futios of form exp( jω must be subjet of the ostraits T 0 T m = exp( jmω exp( jω = (4) 0 m Thus, the set of futios φ ( = exp( jω = 0,,,... (5) forms a orthogoal set over the iterval ( 0, T ) if ω = π /T. A arbitrary sigal f ( a be expressed i terms of a fiite summatio of omplex expoetials by [] = F exp( jω t < t < t (6) = where the oeffiiets F is the projetio of f ( o eah omplex expoetial of the orthogoal futios set. Thus

3 F t exp( j dt t < t < t t t t = ω (7) THE SIMULATION PLATAFORM Figure 6 shows the developed omputer tool. The graphial iterfae shows the phasors (blak lie), the phasor resultat (red lie) ad the projetio o real axis (blue lie). The amplitude of sigal i time domai is the projetio o real axis. The harateristi of this program are: Number of tos: 9 DC level. Frequeies of tos: DC,,,3,4,5,6,7,8 ad 9 Hz. Rage of amplitude:.5v A. 5V. Rage of phase: π θ π. Number of pre-programmable sigals: 4 Pre-programmable sigals: square, half sie, modulus of sie ad triagle. Plot i time domai. Plot i phasor domai. FIGURE 6. THE HZ COSINE SIGNAL WITH 45º DELAY IN SIMULATION PLATAFORM. EXAMPLE Figure 7 shows the simulatio for a square sigal. I this example, the symmetri square waveform a be expressed i terms of a sum de osie futios with frequeies odd multiple of the fudametal frequey [3]. 4 si[( ) t] = π = 0 (8) The omplex otatio i this simulatio is give by 4 = exp( jω exp( j3ω π 3 exp( j5ω exp( j7ω exp( j9ω where ω = π rad s. / (9)

4 FIGURE 7. THE SQUARE SIGNAL IN SIMULATION PLATAFORM. EXAMPLE Figure 8 shows the simulatio for the half sie. I this example, half sie waveform a be expressed i terms of a sum de osie futios with frequeies eve multiple of the fudametal frequey [3]. = π si( os( π = ( )( ) (0) = [ exp( jω ] π () exp( jω exp( j4ω exp( j6ω π where ω = π rad / s The omplex otatio of this simulatio is give by

5 FIGURE 8. THE HALF SINE SIGNAL IN SIMULATION PLATAFORM. ACKNOWLEDGMENT [3] Spiegel, R. Murray, Mathematial Hadbook of Formulas ad Tables, Shaum s Outlie Series, MGraw Hill, 973. The authors would like to thak INATEL for the fiaial support. CONCLUSIONS This paper has preseted a didati simulator to aalyze the sigal geeratio usig the expoetial Fourier series. The simulatio platform has preseted a effiiet solutio for studets to self-lear the oepts about orthogoal sigals, sigals sum, Fourier Series ad phasor maipulatio. REFERENCES [] Stremeler, F., G., Itrodutio to Commuiatio Systems, Third editio, Addiso-Wesley,99. [] Lathi, B., P., Commuiatio Systems, Joh Wiley & Sos, I., 968.