e 2t u(t) e 2t u(t) =?

Transcription

1 EE : Signals, Sysems, and Transforms Fall 7. Skech he convoluion of he following wo signals. Tes No noes, closed book. f() Show your work. Simplify your answers. g(). Using he convoluion inegral, find he convoluion of he signal f() = e u() wih iself. e u() e u() =?. A coninuous-ime LTI sysem is described by he equaion, y() = Z x(τ) dτ where x is he inpu signal, y he oupu signal. (a) Accuraely skech he impulse response of he sysem. (b) Accuraely skech he sep response of he sysem. (c) Wha is he dc gain of he sysem?. I is observed of some coninuous-ime LTI sysem ha he inpu signal x() = u() produces he oupu signal y() = u() + e sin(π) u(). (a) Wha are he poles of he LTI sysem? (b) Wha is he dc gain of he sysem? (c) Find he impulse response h() of he sysem.

2 . A causal coninuous-ime LTI sysem is described by he equaion y () + y () + ( + π ) y() = π x() 7. The signal x(): x() where x is he inpu signal, and y is he oupu signal. (a) Find he impulse response of he sysem. (b) Accuraely skech he pole-zero diagram. (c) Find he form of he sep-response as far as you can wihou compleing parial fracion expansion or inegraion. 6. Consider a coninuous-ime LTI sysem wih he impulse response has he Fourier ransform X f (): h() X f () - (a) Find he frequency response H f (). (b) Skech he frequency response magniude H f (). Indicae he frequencies where H f () equals zero. (c) Skech he frequency response phase H f (). (d) Find he oupu signal y() produced by he inpu signal π π π π π π π π Accuraely skech he signal g() ha has he specrum G f (): x() = + cos(π). (e) If x() is a periodic signal wih a fundamenal period of seconds and Fourier series coefficiens c k, G f () x() = X k c k e jko, hen wha is he oupu signal y() when x() is he inpu signal? π π π π π π π π Noe ha he specrum G f () is an expanded version of X f (). Specifically, G f () = X f (. ).

3 8. The signal x() has he specrum X f () shown. X f () π π π π The signal x() is used as he inpu o a coninuous-ime LTI sysem having he frequency response H f () shown. H f (). A coninuous-ime LTI sysem has he impulse response h() = δ() sinc() The inpu signal x() has he specrum X f () shown, X f () π (.π) (.π) (π) (π) π π π π π π π π π π π π Accuraely skech he specrum Y f () of he oupu signal. 9. The impulse response of a coninuous-ime LTI sysem is obained by muliplying wo signals, f() and g(). where h() = f() g() f() = sinc(), g() = sinc(). (a) Accuraely skech he frequency response H f () of he sysem. (b) Wha kind of sysem is H f () (LPF, HPF, BPF, BSF, or none of hese)? (c) Can his sysem be implemened wih a finie order differenial equaion? Explain. Find he oupu signal y().. The firs six seconds of he impulse responses of eigh causal coninuousime sysems are illusraed below, along wih he pole/zero diagram of each sysem. Bu hey are ou of order. Mach he figures wih each oher by compleing he able (copy he able ino your answer book). Impulse Response Pole/Zero Diagram

4 IMPULSE RESPONSE IMPULSE RESPONSE POLE/ZERO DIAGRAM POLE/ZERO DIAGRAM IMPULSE RESPONSE IMPULSE RESPONSE POLE/ZERO DIAGRAM POLE/ZERO DIAGRAM IMPULSE RESPONSE IMPULSE RESPONSE 6 POLE/ZERO DIAGRAM POLE/ZERO DIAGRAM IMPULSE RESPONSE 7 IMPULSE RESPONSE 8 POLE/ZERO DIAGRAM 7 POLE/ZERO DIAGRAM 8. 6 TIME (SEC) 6 TIME (SEC)

5 . The frequency responses of eigh causal coninuous-ime sysems are illusraed below, along wih he pole/zero diagram of each sysem. Bu hey are ou of order. Mach he figures wih each oher by compleing a able. POLE ZERO DIAGRAM POLE ZERO DIAGRAM FREQUENCY RESPONSE FREQUENCY RESPONSE POLE ZERO DIAGRAM POLE ZERO DIAGRAM FREQUENCY RESPONSE FREQUENCY RESPONSE POLE ZERO DIAGRAM POLE ZERO DIAGRAM FREQUENCY RESPONSE FREQUENCY RESPONSE 6 POLE ZERO DIAGRAM 7 POLE ZERO DIAGRAM FREQUENCY RESPONSE 7 FREQUENCY RESPONSE