Typical signals and their representation Sinusoidal asin(ot+o) f(t=Asin(ot+()=Asin (2rit+o) A-Amplitude f-frequency (Hz) o=2If angular frequency(radians/sec) cp-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 sin/cos signals may be represented by complex exponential Asin( at+)=.(e/(ot+p)-e /(0+) Acos(at+)==(e/(or+o)+e (or+9)) Euler's relation e/f(ot+p)=cos(at+)+jsin( ot+)
Typical signals and their representation ❖sin/cos signals may be represented by complex exponential ( ) 2 cos( ) ( ) 2 sin( ) ( ) ( ) ( ) ( ) + − + + − + + = + + = − j t j t j t j t e e A A t e e j A A t ❖Euler’s relation cos( ) sin( ) ( ) = + + + + e t j t j t
Typical signals and their representation o 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 ❖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)=et a is real 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 ☆ Exponential f(t) ea is complex a=0+jo f(t)=Ae at=Aero+jo) = Aegt cos at +jAe sin t 0=0. 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)