University of California at Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences

Transcription

1 A Uiversity of Califoria at Berkeley College of Egieerig Departmet of Electrical Egieerig ad Computer Scieces U N I V E R S T H E I T Y O F LE T TH E R E B E LI G H T C A L I F O R N I A EECS : Sigals ad Systems Fall Semester 998 Examples of Simple LTI Systems Cotiuous-Time, Secod-Order Lowpass Filter xt () R L C yt () This circuit is govered by the differetial equatio: d y dt R -- dy L dt We ca trasform the differetial equatio ito a stadard form by defiig the atural frequecy ω LC ad the dampig costat ζ ( R ) C L: The system exhibits three regimes of behavior: Uderdamped < ζ < Critically damped ζ Overdamped ζ > yt () LC xt () LC d y dy dt ζω ω dt yt () ωxt () Impulse Respose Overdamped: if ζ >, characteristic equatio has two distict, real roots, ad: ht () ω e ω ( ζ ζ )t e ω ( ζ ζ )t ut () ζ Uderdamped: if ζ <, characteristic equatio has two distict, complex cojugate roots, ad: A error i the overall sig of this equatio was corrected ht () ω e ζω t e j ζ ω t e j ζ ω t ω ut () e ζω t si j ζ ζ Critically Damped: if ζ ζ ω t u() t, characteristic equatio has oe real root of multiplicity two, ad: ω t ht () ω te ut ()

2 Examples of Simple LTI Systems of 6 8 ζ 5 77 Impulse Respose h(t) ω t 8 ζ 5 6 Step Respose s(t) u(t) ω t

3 Examples of Simple LTI Systems 3 of 6 Frequecy Respose As log as ζ >, the frequecy respose Hjω ( ) exists ad is give by: Hjω ( ) ω ( jω) ζω ( jω) ω log log log SecodOrder Lowpass Filter ζ 5 ω / ω ζ 77 ω / ω ζ ω / ω ζ 5 ω / ω ζ 77 ω / ω ζ ω / ω

4 Examples of Simple LTI Systems 4 of 6 Cotiuous-Time, Secod-Order Badpass Filter xt () R L C yt () This circuit is govered by the differetial equatio: d y dt dy RC dt Defiig ω LC ad η ( R) L C: yt () LC dx RCdt The frequecy respose Hjω ( ) d y dy dx dt ηω ω dt yt () ηω dt is give by: Hjω ( ) ηω ( jω) ( jω) ηω ( jω) ω (I circuit aalysis, it is customary to defie the quality factor: Q ω RC ) η log log log SecodOrder Badpass Filter η (Q 5) 3 4 ω / ω η (Q 5) 3 4 ω / ω η (Q 5) 3 4 ω / ω η (Q 5) ω / ω η (Q 5) ω / ω η (Q 5) ω / ω

5 Examples of Simple LTI Systems 5 of 6 3 Causal, FIR Approximatio to Discrete-Time, Ideal Lowpass Filter Ideal Lowpass Filter Frequecy Respose: He j, W ( ), where W < π, ad He ( j( π) ), W < π He ( j ) W Impulse Respose (ocausal, IIR): h [ ] ----sic W π π Causal, FIR Approximatio W ----sic W , N Trucated Impulse Respose (ocausal, FIR): h truc [ ] π π, > N Shifted, Trucated Impulse Respose (causal, FIR): W ----sic W π ( N) π, N h truc,shift [ ] h truc [ N] N Frequecy Respose: H truc,shift ( e j W ) ---- sic W ( N) j e π π, <, > N W π/, N 8 W π/, N h ts [] h ts [] H ts (e j ) 5 H ts (e j ) arg[h ts (e j )] 4 arg[h ts (e j )]

6 Examples of Simple LTI Systems 6 of 6 4 Simple Discrete-Time, Secod-Order System Differece Equatio: y [ ] rcosθy[ ] r y [ ] x [ ], r >, θ π Impulse Respose (IIR): Critically damped: For θ, characteristic equatio has oe real root of multiplicity two, ad: h [ ] ( )r u [ ] Uderdamped: For < θ < π, there are two distict, complex cojugate roots, ad: h [ ] r )θ] siθ [ ] For θ π, there is oe real root of multiplicity two, ad: h [ ] ( ) ( r) u [ ] Frequecy Respose: For < r <, He ( j ) exists ad is give by: He j ( ) ( rcosθ)e r e j r 5, θ r 5, θ π/ r 5, θ π h[] h[] h[] H(e j ) H(e j ) H(e j ) arg[h(e j )] arg[h(e j )] arg[h(e j )]