Chapter 1 Continuous-time Signals and Systems
Chapter 1 Continuous-time Signals and Systems
Introduction Any problems about signal analyses and processing may be thought of letting signals trough systems. f(t) h( 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
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
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 the simplest signals which will pass through the system and produce a response.Combine the responses of all simplest signals,which is the system response for the original input signal. This is the basic approach to study the signal analyses and processing
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 the simplest signals which will pass through the system and produce a response. Combine the responses of all simplest signals, which is the system response for the original input signal. This is the basic approach to study the signal analyses and processing
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()=0 t<0 Call such signals causal
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 such signals causal
Typical signals and their representation Unit Step u(t)(in our textbook u(t)) t>0 t<0 tu(t-to) 0 to 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 µ(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 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 growing
Typical signals and their representation Exponential f(t) = eαt •α is real 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)