Typical signals and their representation Sinusoidal A sin(ot+o) f(t)=A sin(ot+p)=A sin(2nft+o) A-Amplitude f-frequency (Hz,cycle per second) o=2af angular frequency (radians/sec) -start phase(radians)
Typical signals and their representation Sinusoidal A sin(ωt+φ) f(t) = A sin(ωt+φ)= A sin(2πft+φ) A - Amplitude f – frequency (Hz, cycle per second) ω= 2πf angular frequency (radians/sec) φ – start phase(radians)
Typical signals and their representation sin/cos signals may be represented by complex exponential )) A Asin(at+p)= cos(t+p)=号 (eo+o）+ejo+p)） Euler's relation e)-cos(@t+p)+jsin(ot+p)
Typical signals and their representation sin/cos signals may be represented by complex exponential ( ) 2 cos( ) ( ) 2 sin( ) ( ) ( ) ( ) ( ) ω ϕ ω ϕ ω ϕ ω ϕ ω ϕ ω ϕ + − + + − + + = + + = − j t j t j t j t e e A A t e e j A A t Euler’s relation cos( ) sin( ) ( ) ω ϕ ω ϕ ω ϕ = + + + + e t j t j t
Typical signals and their representation Sinusoidal is basic periodic signal which is important both in theory and engineering. Sinusoidal is non-causal signal.All of periodic signals are non-causal because they have no start and no end. f (t)=f(t+mT) m=0,士1，士2，…，士o0
Typical signals and their representation Sinusoidal is basic periodic signal which is important both in theory and engineering. Sinusoidal is non-causal signal. All of periodic signals are non-causal because they have no start and no end. f (t) = f (t + mT) m=0, ±1, ±2, ···, ±∞
Typical signals and their representation Exponential f(t)=eat a is real a <0 decaying Q=0 constant a>0 growing
Typical signals and their representation Exponential f(t) = eαt •α is real α < 0 decaying α = 0 constant α > 0 growing
Typical signals and their representation Exponential f(t)=eat a is complex a=0+jo f(t)=Aeat =Ae(o+jo)t =Aeot cosωt+jAeot sinωt o 0,sinusoidal o >0,growing sinusoidal o<0,decaying sinusoidal (damped)
Typical signals and their representation Exponential f(t) = eαt •α is complex α = σ + jω f(t) = Aeαt = Ae(σ + jω)t = Aeσ t cosωt + j Aeσ t sinωt σ = 0, sinusoidal σ > 0 , growing sinusoidal σ < 0 , decaying sinusoidal (damped)