Chapter 1 Continuous-time Signals and systems
Chapter 1 Continuous-time Signals and Systems
§11 Introduction Any problems about signal analyses and processing may be thought of letting signals trough systems. f(t) h(t) y(t) o From f(t) and h(t), find y(t), Signal processing o From f(t)and y(t, find h(t, System design o From y(t h(t), find f(t), Signal reconstruction
§1.1 Introduction Any problems about signal analyses and processing may be thought of letting signals trough systems. h(t) f(t) y(t) ❖From f(t) and h(t)，find y(t)， Signal processing ❖From f(t) and y(t) ，find h(t) ，System design ❖From y(t) and h(t)，find f(t) ， Signal reconstruction
§11 Introduction There are so many different signals and systems that it is impossible to describe them e one by one .The best approach is to represent the signal as a combination of some kind of most simplest signals which will pass though the system and produce a response. Combine the responses of alll simplest signals, which is the system response of the original signal .This is the basic method to study the signal analyses and processing
§1.1 Introduction ❖There are so many different signals and systems that it is impossible to describe them one by one ❖The best approach is to represent the signal as a combination of some kind of most simplest signals which will pass though the system and produce a response. Combine the responses of all simplest signals, which is the system response of the original signal. ❖This is the basic method to study the signal analyses and processing
s 1.2 Continue-time Signal . B All signals are thought of as a pattern of e variations in time and represented as a time function f(t) In the real-world. any signal has a start Let the start as t0 that means f(t)=0t<0 Call the signal causal
§1.2 Continue-time Signal ❖All signals are thought of as a pattern of variations in time and represented as a time function f(t). ❖In the real-world, any signal has a start. Let the start as t=0 that means f(t) = 0 t<0 Call the signal causal
Typical signals and their representation . Unit Step u(t(in our textbook u(t)) ult) t>0 0t<0 u(t) u(t) is basic causal signal, multiply which with any non-causal signal to get causal signal
Typical signals and their representation ❖Unit Step u(t) (in our textbook (t)) 1 0 0 0 ( ) = t t u t u(t) 1 0 t u(t- t0 ) 1 0 t t 0 ❖u(t) is basic causal signal, multiply which with any non-causal signal to get causal signal
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 a0 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)