Typical signals and their representation Sinusoidal asin(ot+o) f(t=asin(ot+)= Asin(nit+o) A- amplitude f- frequency (hz o=2f angular frequency(radians/sec) a-start phase(radians)
Typical signals and their representation Sinusoidal Asin(ωt+φ) f(t) = Asin(ωt+φ)= Asin(2πft+φ) A - Amplitude f - frequency(Hz) ω= 2πf angular frequency (radians/sec) φ – start phase(radians)
Typical signals and their representation oo sin/cos signals may be represented by complex exponential A sin( at+o)=c(e/tonto j(at+o) A cos( at+o)=(e/tp)+ e (at+o) ☆ Euler' s relation s ej(ax+p)=cOS(@t+o)+ jsin(at+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 ☆Si inusoidal is basic periodic signal which is important both in theory and engineering &sinusoidal is non-causal signal. All of eriodic signals are non-causal because they have no start and no end f(t)=f(tmT)m=0,±1,±2,…,如
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 .a is rea a <0 decaying a=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 冷 Exponential f(t)=e a is complex a=0+io f(t=Ae at= Allot jo)t Aeot cos ot +i aeot sin ot sInusoidal 0>0, growing sinusoidal 0<0, decaying sinusoidal(damped)
Typical signals and their representation Exponential f(t) = eαt •α is complex α = σ + jω f(t) = Ae αt = Ae(σ + jω)t = Aeσ t cos ωt + j Aeσ t sin ωt σ = 0, sinusoidal σ > 0 , growing sinusoidal σ < 0 , decaying sinusoidal (damped)