Chapter 1 Continuous-time Signals and Systems
Chapter 1 Continuous-time Signals and Systems
§1 Introduction Any problems about signal analyses and processing may be thought of letting signals trough systems ft) 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 and 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
§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 l simplest signals which is the system response of the original signal This is the basic method to study the signal nalyses 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
81.2 Continue-time Signal ☆ All sig nals 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)=0t0 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 o Unit Step u(t(in our textbook u(t) t>0 t<0 u(t-to) ●☆u( 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 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 growing
Typical signals and their representation Exponential f(t) = eαt •α is real 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)