《信号与系统 Signals and Systems》课程教学资料（英文版）lecture 4 Representation of ct signals

Signals and Systems Fall 2003 Lecture #4 16 September 2003 Representation of CT Signals in terms of shifted unit impulses 2. Convolution integral representation of CT Lti systems 3. Properties and examples 4. The unit impulse as an idealized pulse that is short enough": The operational definition of d(t

Signals and Systems Fall 2003 Lecture #4 16 September 2003 1. Representation of CT Signals in terms of shifted unit impulses 2. Convolution integral representation of CT LTI systems 3. Properties and Examples 4. The unit impulse as an idealized pulse that is “short enough”: The operational definition of δ(t)

Representation of CT Signals Approximate any input x(t) as a sum of shifted, scaled pulses X( 0△ ()=x(k△),k△<t<(k+1)△

Representation of CT Signals • Approximate any input x ( t) as a sum of shifted, scaled pulses

6△(t) (t) has unit area x(k△) (k△)06△(t-k△)△ k△个 (k+1)△ ()=∑(k△A△(t-kA)△ k=- l limit as△→0 x(T)6(t-T) The Sifting Property of the Unit Impulse

has unit area The Sifting Property of the Unit Impulse

Response of a ctlti system CT LTI 6△(t)一h△(t) (1)=∑m(k△△(t-kA△ ∑m(k△△(-k△△ k k Impulse response 6(t)—h(t) Taking limits△0 x(7)6(t-r)dr-)y(t) (rh(t-r)d Convolution Integra

Response of a CT LTI System LTI ⇒

Operation of cT Convolution g(t)=.(t)* h(t Fli Slide →h(t-T) Multiply (Th(t-T) Integrate x(T)h(t-7) Example: CT convolution x(1) h1) Xt h(t-τ) ,t+1 t+2

Example: CT convolution Operation of CT Convolution

Time Interval x(t)·h(t-0) Output t2 yt)=0

-1 -1 0 0 1 1 2 2

PROPERTIES AND EXAMPLES 1)Commutativity (t)*hb(t)=b()*(t) 2) (t)*8(t-to)=c(t-to) Sifting property: a(t)8(t=at 3)An integrator y(t)=/(r)dr So if input i t)=6(+) It g(t)=h(t) h(t)=/6(7)dr=u( That is ()=x(t)米h()=|m(t)*(t) ITdT 4)Step response s(t)=0(t)米h()=h()*(t)

PROPERTIES AND EXAMPLES 1) Commutativity: 2) 4) Step response: 3) An integrator:

DISTRIBUTIVITY h1(t)+h2( y(D)=x()*[h1()+h2(1) y()=x(t)*h/(t)+x(1)*h2(t)

DISTRIBUTIVITY

ASSOCIATIVITY h1(t) h2(2) y()=[x(t)*h1(D)*h4() h1()*h2(t) y()=x(2)*[h1()*h2() Commutativity h2(t)*h1(t) y()=x()*[h2(D)*h1( x() h2(t) h1(t) y()=[x()*h2()*h(

ASSOCIATIVITY

Causality: CT LTI system is causal h(t)=0,t<o Stability: T LTI system is stable台∫sb(r)dr<∞