OpenStax-CNX module: m Elemental Signals. Don Johnson. Perhaps the most common real-valued signal is the sinusoid.

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1 OpenSax-CNX module: m0004 Elemenal Signals Don Johnson This work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License.0 Absrac Complex signals can be buil from elemenal signals, including he complex exponenial, uni sep, pulse, ec. This module presens he elemenal signals in brief. Elemenal signals are he building blocks wih which we build complicaed signals. By deniion, elemenal signals have a simple srucure. Exacly wha we mean by he "srucure of a signal" will unfold in his secion of he course. Signals are nohing more han funcions dened wih respec o some independen variable, which we ake o be ime for he mos par. Very ineresing signals are no funcions solely of ime; one grea example of which is an image. For i, he independen variables are x and y (wo-dimensional space). Video signals are funcions of hree variables: wo spaial dimensions and ime. Forunaely, mos of he ideas underlying modern signal heory can be exemplied wih one-dimensional signals. Sinusoids Perhaps he mos common real-valued signal is he sinusoid. For his signal, A is is ampliude, f 0 is frequency, and φ is phase. 2 Complex Exponenials The mos imporan signal is complex-valued, he complex exponenial. s () = Acos (2πf 0 + φ) () s () = Ae i(2πf0+φ) = Ae iφ e i2πf0 (2) Here, i denoes. Ae iφ is known as he signal's complex ampliude. Considering he complex ampliude as a complex number in polar form, is magniude is he ampliude A and is angle he signal phase. The complex ampliude is also known as a phasor. The complex exponenial canno be furher decomposed ino more elemenal signals, and is he mos imporan signal in elecrical engineering! Mahemaical manipulaions a rs appear o be more dicul because complex-valued numbers are inroduced. In fac, early in he wenieh cenury, mahemaicians hough engineers would no be sucienly sophisicaed o handle complex exponenials even hough hey grealy simplied solving circui problems. Seinmez Version 2.29: Jul 6, :56 pm hp://creaivecommons.org/licenses/by/.0 hp://

2 OpenSax-CNX module: m inroduced complex exponenials o elecrical engineering, and demonsraed ha "mere" engineers could use hem o good eec and even obain righ answers! See Complex Numbers 2 for a review of complex numbers and complex arihmeic. The complex exponenial denes he noion of frequency: i is he only signal ha conains only one frequency componen. The sinusoid consiss of wo frequency componens: one a he frequency f 0 and he oher a f 0. Euler relaion: This decomposiion of he sinusoid can be raced o Euler's relaion. cos (2πf) = ei2πf + e (i2πf) 2 sin (2πf) = ei2πf e (i2πf) 2i (3) (4) e i2πf = cos (2πf) + isin (2πf) (5) Decomposiion: The complex exponenial signal can hus be wrien in erms of is real and imaginary pars using Euler's relaion. Thus, sinusoidal signals can be expressed as eiher he real or he imaginary par of a complex exponenial signal, he choice depending on wheher cosine or sine phase is needed, or as he sum of wo complex exponenials. These wo decomposiions are mahemaically equivalen o each oher. Acos (2πf + φ) = R ( Ae iφ e i2πf) (6) Asin (2πf + φ) = I ( Ae iφ e i2πf) (7) 2 "Complex Numbers" <hp://cnx.org/conen/m008/laes/>

3 OpenSax-CNX module: m Figure : Graphically, he complex exponenial scribes a circle in he complex plane as ime evolves. Is real and imaginary pars are sinusoids. The rae a which he signal goes around he circle is he frequency f and he ime aken o go around is he periodt. A fundamenal relaionship is T = f. Using he complex plane, we can envision he complex exponenial's emporal variaions as seen in he above gure (Figure ). The magniude of he complex exponenial is A, and he iniial value of he complex exponenial a = 0 has an angle of φ. As ime increases, he locus of poins raced by he complex exponenial is a circle (i has consan magniude of A). The number of imes per second we go around he circle equals he frequency f. The ime aken for he complex exponenial o go around he circle once is known as is periodt, and equals f. The projecions ono he real and imaginary axes of he roaing vecor represening he complex exponenial signal are he cosine and sine signal of Euler's relaion ((3)).

4 OpenSax-CNX module: m Real Exponenials As opposed o complex exponenials which oscillae, real exponenials (Figure 2) decay. s () = e τ (8) e Exponenial τ Figure 2: The real exponenial. The quaniy τ is known as he exponenial's ime consan, and corresponds o he ime required for he exponenial o decrease by a facor of e, which approximaely equals A decaying complex exponenial is he produc of a real and a complex exponenial. s () = Ae iφ e τ e i2πf = Ae iφ e ( τ +i2πf) (9) In he complex plane, his signal corresponds o an exponenial spiral. For such signals, we can dene complex frequency as he quaniy muliplying. 4 Uni Sep The uni sep funcion (Figure 3) is denoed by u (), and is dened o be 0 if < 0 u () = if > 0 (0) u() Figure 3: The uni sep.

5 OpenSax-CNX module: m Origin warning: This signal is disconinuous a he origin. Is value a he origin need no be dened, and doesn' maer in signal heory. This kind of signal is used o describe signals ha "urn on" suddenly. For example, o mahemaically represen urning on an oscillaor, we can wrie i as he produc of a sinusoid and a sep: s () = Asin (2πf) u (). 5 Pulse The uni pulse (Figure 4) describes urning a uni-ampliude signal on for a duraion of seconds, hen urning i o. 0 if < 0 p () = if 0 < < () 0 if > p () Figure 4: The pulse. We will nd ha his is he second mos imporan signal in communicaions. 6 Square Wave The square wave (Figure 5)sq () is a periodic signal like he sinusoid. I oo has an ampliude and a period, which mus be specied o characerize he signal. We nd subsequenly ha he sine wave is a simpler signal han he square wave. A Square Wave T Figure 5: The square wave.