EE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
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1 EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of a car ime volage across a circui componen ime price of a cerain sock day In general, a signal can be a funcion of some variable oher han ime, for example, locaion. Moreover, a signal can be a funcion of more han wo variables, for example, brighness of a compuer screen as a funcion of he x-y coordinaes. However, we will focus on signals ha are funcions of only one variable and ha is ime.
2 2 Coninuous-ime signal: Defined a every poin in ime such as he velociy of a car or volage across a circui componen in he previous figure. We use he noaion x(). Discree-ime signal: Defined only a se of discree poins in ime such as he price of a cerain sock in he previous figure. We use he noaion x[n]. Commens: In applicaions, some signals may naurally arise as coninuous ime signals such as he volage across a circui componen measured by an analog device. In oher applicaions, some signals may naurally arise as discree ime signals such as he daily change in he price of a cerain sock. An imporan class of discree-ime signals arise from sampling a coninuous-ime signal, ha is, by recording he signal value a cerain discree ime insances and hrowing away he res of he signal. This is imporan because in many modern applicaions signals are processed by digial compuers ha require discree-ime daa. For example, in many modern auomaic conrol sysems, measuremens are coninuous-ime signals, bu hey are firs convered o discree-ime and hen fed o a digial microprocessor ha compues he necessary conrol signal which is also discree-ime.
3 3 Signal energy and power: Energy and average power of a coninuous-ime signal x() over a finie inerval 2 : Energy : E [, 2 ] = 2 x() 2 d, Average Power : P [, 2 ] = 2 2 x() 2 d. Energy and average power of a coninuous-ime signal x() over he infinie inerval < < : T Energy : E (, ) = lim x() 2 d, Average Power : P (, ) = T T T 2T lim x() 2 d. T T Energy and average power of a discree-ime signal x[n] over a finie inerval n n n 2 : Energy : E [n,n 2 ] = n 2 n=n x[n] 2, Average Power : P [n,n 2 ] = n 2 x[n] 2. n 2 n n=n Energy and average power of a discree-ime signal x[n] over he infinie inerval < n < : Energy : E (, ) = lim N N n= N x[n] 2, Average Power : P (, ) = 2N + lim N N n= N x[n] 2. Energy or power of a signal may no always be relaed o physical energy. Neverheless, his erminology is sill useful. Three imporan class of signals defined over (, ): ) Those wih finie energy, i.e., E (, ) <. These signals mus have zero average power, i.e., P (, ) =. For example, x() = e or x[n] = e n. 2) Those wih finie power, i.e., P (, ) <. If P (, ) >, hen hese signals mus have infinie energy, i.e., E (, ) =. For example, x() = sin() or x[n] = sin(n). 3) Those wih infinie power and infinie energy. For example, x() = or x[n] = n.
4 4 Transformaions of ime: Consider a signal x(), for example, he one shown below. x( ) x(/2) x(2) x( ) x() Time reversal: x( ) plays backward in ime Time scaling: x(2) plays wice as fas Time scaling: x(/2) plays a half speed Time shif: x( ) plays afer uni delay Transformaion of ime for discree-ime signals are analogous.
5 5 Example: Obain x( 2/3 + ). One soluion: Time reversal : y() := x( ) Time scaling by 2/3 : z() := y(2/3) = x( 2/3) Time delay by 3/2 : w() := z( 3/2) = y(2/3 ) = x( 2/3 + ) x() x( ) x( 2/3) x( 2/3+)
6 6 Anoher soluion: Time advancemen by : y() := x( + ) Time reversal : z() := y( ) = x( + ) Time scaling by 2/3 : w() := z(2/3) = y( 2/3) = x( 2/3 + ) x() x(+) x( +) x( 2/3+)
7 7 Periodic signals: x() is periodic wih period T > if x( + T ) = x() for all. The smalles T such ha x() has his propery is called he fundamenal period of x(). x[n] is periodic wih period N > if x[n + N] = x[n] for all n. The smalles N such ha x[n] has his propery is called he fundamenal period of x[n]. Examples: sin(2π )+sin(4π ) cos(nπ/2)+cos(nπ) n Coninuous-ime signal above is periodic wih fundamenal period T =. Discree-ime signal above is periodic wih fundamenal period N = 4.
8 8 Even and odd signals: x() is even if x() = x( ) for all. x() is odd if x() = x( ) for all. Examples: An even signal. cos(2π ).exp( /2 ) An odd signal. sin(π /2).( 4<=<=4) For an odd signal x(), we always have x() = x() =. Every real valued signal x() can be wrien as he sum of an even signal and odd signal: x() = x e () + x o () where x e () = x() + x( ) 2 and x o () = x() x( ). 2 x e () and x o () are referred o as even and odd pars of x(), respecively. Evenness and oddness and he relaed conceps are analogously inroduced for discree-ime signals.
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