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 8 6 8 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 ()
Examples of Simple LTI Systems of 6 8 ζ 5 77 Impulse Respose h(t) 6 4 - -4-6 -8-4 8 6 4 8 ω t 8 ζ 5 6 Step Respose s(t) 4 8 6 4 u(t) 77 4 8 6 4 8 ω t
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
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
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 3 4 4 h ts [] h ts [] 5 5 4 6 H ts (e j ) 5 H ts (e j ) 5 5 5 5 5 arg[h ts (e j )] 4 arg[h ts (e j )] 5 5 6 5 5 5 5
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 5 5 5 h[] h[] h[] 5 5 5 4 5 4 5 4 5 3 3 3 H(e j ) H(e j ) H(e j ) 5 5 5 5 5 5 arg[h(e j )] arg[h(e j )] arg[h(e j )] 5 5 5 5 5 5