Homework 4 SOLUTION EE235, Summer 202. Causal and Sable. These are impulse responses for LTI sysems. Which of hese LTI sysem impulse responses represen BIBO sable sysems? Which sysems are causal? (a) h() δ( + 2) h() d h( 2) > 0 Bounded No causal (b) h() u( ) u( ) d 0 d h() > 0 for < 0 No Bounded No causal (c) h() cos() cos() d h() 0 for < 0 No Bounded No causal (d) h() e 2+2 e 2+2 d No Bounded h() 0 for < 0 No causal (e) h() e 2+2 u() e 2+2 u() d 0 e 2+2 d 2 e2 h() 0 for < 0 Bounded Causal (f) h() d h() for < 0 No Bounded No Causal 2. Calculae y() f() g() for he signal pairs below, and skech your resuls.
(a) f() u() u( ) g() u() u( 3) Soluion using graphical convoluion: 3 f() g( ) Then se up he inegrals wih proper limis 0 < 0 0 d 0 < < f() g() d > and 3 < 0 0 3 d 0 < 3 < 0 3 > 0 < 0 0 < < < < 3 4 3 < < 4 0 > 4 [ u() u( ) ] + [ u( ) u( 3) ] + (4 ) [ u( 3) u( 4) ] Soluion using properies of convoluion: Disribuing he convoluion, we ge f() g() u() u() u() u( 3) u( ) u() + u( ) u( 3) Le w() be he convoluion of u() u(). Consider he graphical convoluion. When < 0, he signals do no overlap, so he convoluion will be 0 for < 0 u() u( ) When > 0, he signals do overlap. u() u( ) 2
The overlap occurs for 0 < <, so Now express f() g() in erms of w(): w() u() u() u()u( )d d, > 0 0 0, < 0 {, > 0 0, < 0 u() 0.5 w() 2 f() g() w() w( 3) w( ) + w( 4) u() ( 3)u( 3) ( )u( ) + ( + 4)u( + 4) ***Noe ha he form is differen from he graphical mehod soluion, bu i is acually he same funcion. The resul mus look like he following plo: 2 f() g() 2 2 3 4 5 (b) f() e α u() g() e β u() for α > 0, β > 0, α β f() g() For < 0, here is no overlap f()g( )d e α u()e β( ) u( )d e α u() e β( ) u( ) For > 0, here is an overlapping region e α u() e β( ) u( ) 3
f() g() u() 0 e α e β( ) d u()e β e (β α) d 0 u()e β β α e(β α) 0 u()e β β α [e(β α) ] u() β α [e α e β ] 3. Consider he following inerconnecion of LTI sysems h () and h 2 (). (a) Express he overall sysem response of h T OT () in erms of he impulse responses h () and h 2 (). { [h h T OT () () h () + h () ] } h 2 () + h () h 2 () h () h 2 () h 2 () (b) Assume ha sysem is he inverse of sysem 2, so ha h () h 2 () δ(). Simplify your answer o (a). { [h h T OT () () h () + h () ] } h 2 () + h () h 2 () h () h 2 () h 2 () { [h () h () + h () ] } h 2 () + h () h 2 () h 2 () { [h () + δ() ] } + h () h 2 () h 2 () { [δ() + h2 () ] } + δ() h 2 () 2δ() (c) From your answer in (b), wha is he purpose of he overall sysem? This sysem amplifies he inpu by 2. 4. Define he area under a coninuous-ime signal f() as A f f()d. 4
Show ha if y() x() h(), hen A y A x A h. A y y()d [x() h()]d x()h( )dd ( ) x()h( )d d ( ) x() h( )d d ( ) x() h(z)dz d A h x()d A h A x Alernaively, you could noe ha convolving y() wih a signal which was jus a consan everywhere would give you a signal which was a consan A y everywhere. Le he signal which is a consan everywhere be w(). A y w() y() w() Then use properies of convoluion So hen A y A x A h A y w() [x() h()] w() x() [h() w()] x() A h w() [x() w()]a h A x A h w() 5. Deermine if each of he following saemens abou LTI sysems is rue or false. Wrie a senence jusifying your answer. (a) If an LTI sysem is causal, hen i is sable. False Consider a counerexample: h() u() (causal bu no sable) or h() u( ) (sable bu no causal). Causaliy and sabiliy are independen properies. (b) The cascade of a noncausal LTI sysem wih a causal LTI sysem mus be noncausal. False Consider a counerexample: he causal sysem h () δ( 2) and he non-causal sysem h 2 () δ( + 2). Their cascade yields he sysem h() h () h 2 () δ() which is causal. (c) Time-reversing he inpu o a LTI sysem resuls in a ime-reversed oupu. False Consider a counerexamle: le he sysem be h() δ( ). When he inpu is x(), he oupu is y() x( ). If he saemen were rue, we expec ha reversing he inpu should oupu x( ). Bu, when he inpu is x( ), he oupu is x( ( )) x( ), disproving he saemen. (Any LTI sysem whose impulse response is no an even funcion should give a couner-example.) 5
BONUS LEARNING: Nonlinear disorion in music (no poins) A good audio sysem is linear: you ask i for wice he volume, and wice he volume comes ou. Check ou some audio equipmen, such as a speaker or ipod (be careful wih your ears!) do you hink is linear over is volume range? Wha kind of audio signal would be bes o es his? Usually consumer audio sysems are percepually linear in he middle of he volume range, wih noiceable nonlinear disorion a op volumes. Some musicians use nonlinear disorion in heir music, for example clipping he ampliude of a signal. Then he scaling propery doesn hold, ha is, if you pu in wice a paricular frequency of sinusoidal signal he signal is clipped o be he same volume, no wice as loud. One band ha specializes in disorions is Congoronics. You can check ou heir hand-buil mikes and oher acousic equipmen a: www.crammed.be/konono, or lisen o heir music a emusic.com (search for Congoronics). Here is wha Richard Gehr of emusic has o say abou Congoronics (review edied for relevance o his class): The sree has is own uses for echnology, Neuromancer auhor William Gibson famously declared. The groups sound is amplified well pas he poin of disorion wih microphones fashioned from magnes found in abandoned cars. The likemb are accompanied by percussion consising of cooking pos, in cans and more car pars. The resuling music resembles a cross beween a New Orleans marching band, a village wedding combo and he Jimi Hendrix Experience. Their homemade PA sysem, designed o elevae he band above heir neighborhood sree sounds, is loud and disored enough o wake he ancesors. 6