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<∞