GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 06 Summer 07 Problem Set #5 Assiged: Jue 3, 07 Due Date: Jue 30, 07 Readig: Chapter 5 o FIR Filters. PROBLEM 5..* (The LTI property.) For each system below, with iput x[ ] ad output, do both of the followig: Determie whether the system is LTI (liear ad time-ivariat). (Aswer YES/NO). If YES, specify its impulse respose h[ ]. If NO, explai why it is ot LTI. = x[ ]. = x[ ]. (c) = x[ ] + x[ ]. = 3x[ ]. (e) = max{x[ ], 0}. (I other words, egative iputs are replaced by zeros.) (f) The three-poit ruig-media filter = media{x[ ], x[ ], x[ ]}. (The media of three umbers is the middle umber after sortig.) PROBLEM 5..* The followig differece equatio defies a LTI discrete-time system: = 3x[ ] x[ ] x[ ]. Determie the impulse respose h[ ]. Sketch a stem plot of the impulse respose h[ ]. Label each ozero value. P (c) Fid the filter coefficiets {b k } i the FIR represetatio = M b k =0 kx[ k ]. Fid the order M of the filter. (e) Fid the legth L of the filter. (f) Sketch a stem plot of the output whe the iput is the uit step, x[ ] = u[ ]. (Hit: There are oly four ozero outputs.) (g) Sketch a stem plot of the output whe the iput is x[ ] = u[ ] u[ 6], as sketched below. Use covolutio. x[ ] 0 3 5
PROBLEM 5.3.* The figure below depicts a cascade coectio of a 3-poit ruig average filter with a -poit ruig average, i which the output y [ ] of the first filter is the iput x [ ] to the secod filter, ad the output y [ ] of the secod filter is the overall output : Sketch a stem plot of the impulse respose h [ ] of the 3-poit ruig average filter. Fid the impulse respose h[ ] of the overall cascaded system (from x[ ] to ). Specify your aswer i the form of a carefully labeled stem plot of h[ ] versus. PROBLEM 5..* Suppose the sequece x[ ] = u[ ] u[ 6] is fed as a iput to a FIR filter whose impulse respose is h[ ] = u[ ] u[ ], leadig to a output. Sketch a stem plot of the iput x[ ]. Sketch a stem plot of the impulse respose h[ ]. (c) Compute the output usig covolutio. Sketch a stem plot of the result. x[ ] 3-POINT y [ ] = x [ ] -POINT RUNNING AVERAGE Fill i the blaks i the followig MATLAB code so that it cofirms your (c) aswer: x = oes(, ); h = oes(, ); y = cov(x, h); stem(y); RUNNING AVERAGE h [ ] h [ ] PROBLEM 5.5.* If the output of a causal -poit ruig average filter is the legth-9 sequece = {... 0 3... 6 8 0...} illustrated i the figure below, what is the iput sequece x[ ]? (Hit: x[ ] is ozero oly for {0,... 5}.) x[ ] -POINT RUNNING AVERAGE 6 0 8 6 3 5 6 7 8 9 0 0 5 0 5 0 3 5 6 8
PROBLEM 5.6.* (This problem looks at what happes whe a siusoid is fed ito a LTI filter, which is the topic of Chapter 6. However, for this problem the siusoid ad the filter are both simple eough that the solutio ca be foud usig the tools of Chapter 5 aloe.) Suppose that the siusoidal sigal x[ ] = cos( ) ˆ is applied to a two-poit averager with impulse respose h[ ] = 0.5 [ ] + 0.5 [ ]: x[ ] = cos( ) ˆ -POINT AVERAGER (c) Assumig that ˆ = Sketch a stem plot of the iput sigal x[ ] = cos( ), ˆ ad sketch a stem plot of the output sigal. Assumig that ˆ = 0.5 Sketch a stem plot of the iput sigal x[ ] = cos( ), ˆ ad sketch a stem plot of the output sigal. Assumig that ˆ = 0.5 The output sigal ca be writte i the form: = Acos( ˆ 0 + ). Fid umeric values for the parameters ˆ 0, A, ad. Assumig that ˆ = /3 Fid the umerical value of the output y[06] that takes at time = 06.
PROBLEM 5.7. The figure below depicts a cascade coectio of two liear time-ivariat (LTI) systems, i which the output y [ ] of the first system is the iput x [ ] to the secod system, ad the output y [ ] of the secod system is the overall output. x[ ] y [ ] = x [ ] LTI# LTI# Suppose that the first LTI system has a impulse respose h [ ] = ( ) for {0,,... 7}, where the parameter is a uspecified positive umber. (The impulse respose is zero for other values of.) For example, if = 0.8 the the impulse respose for the first system is as sketched below: h [ ] 6 0 6 3 5 7 This first system has the effect of blurrig the iput sigal x[ ]. I a attempt to deblur ad recover the iput x[ ], a egieer proposes to pass the output of the first system ito a secod LTI system that is defied by the followig differece equatio: (c) (e) (f) = x [ ] + x [ ]. Fid the differece equatio that relates the iput x [ ] ad output y [ ] of the first system. Fid the impulse respose h [ ] of the secod system. Determie the impulse respose h[ ] = h [ ] h [ ] of the overall cascade system. Fid a sigle differece equatio that relates to x[ ] i the overall cascade system. Commet o the degree to which the egieer s goal of x[ ] is achieved. For example, are there values of the parameter for which x[ ]? Are there values of the parameter for which the ultimate goal of exact equality ( = x[ ]) is achieved? Suppose that = 0.5. Specify umeric values for the vector bb i the first lie below so that the overall cascade system ca be implemeted i MATLAB via the followig: bb = [??? ]; yy = firfilt(bb, xx); where xx is a vector cotaiig the ozero values of the iput x[ ], ad where yy is a vector cotaiig the ozero values of the output.
PROBLEM 5.8. Cosider a discrete-time sequece x[ ] that is periodic with period N 0 =, satisfyig x[0] = 0, x[] =, x[] =, ad x[3] = 3. As show below, this sequece is fed as a iput to a LTI system, producig the output : x[ ] LTI If the LTI system is a L-poit ruig average filter, what value or values of the parameter L will result i a output that is costat for all time, say = c? Fid the output y[06] at time = 06 if the impulse respose h[ ] of the LTI system is as show below: h[ ] 0 3 7 PROBLEM 5.9. So far we have see that a FIR filter ca be described i oe of three equivalet ways: By its differece equatio, by its impulse respose h[ ], ad by its filter coefficiets {b k }. I part below you are give a differet FIR filter described by oe of the three; specify the remaiig two. h[ ] = 0.5 (u[ ] u[ 5]), where u[ ] is the uit step. b 0 = b = b = b 3 = /. (All other b k values are zero.) (c) = x[ ] + x[ ] + x[ ] + x[ 3]. PROBLEM 5.0. A system is said to be causal whe its curret output at time depeds oly o the curret iput x[ ] ad the past iputs {x[ ], x[ ], x[ 3],...}. I particular, the curret output of a causal system does ot deped o future iputs. Determie whether or ot each of the systems through (g) i Prob.. are causal. PROBLEM 5.. A quick-ad-easy test for liearity is to see if doublig the iput doubles the output. If the system fails this test, it is defiitely ot liear. If the system passes this test, it is usually liear. But there are exceptios. Cosider the followig system: = max(x[ ], 0). I words you might say that this system replaces all of the egative values i the iput sequece by zero. Does the system satisfy the doublig the iput doubles the output property? Is the system liear? If ot, why ot?
PROBLEM 5.. Systems that are both liear ad time-ivariat are special they are completely characterized by their impulse respose. However, ot all systems are both liear ad time-ivariat. For each of the systems described below, determie whether or ot they are liear, ad whether or ot they are time-ivariat. Summarize your aswers by writig YES or NO i each etry of this table: LINEAR TIME-INVARIANT Justify your aswers i each case: If you aswer YES, provide a proof; if you aswer NO, provide a specific couterexample. = [ ]x[ ]. = x[ ] +. (c) = x []. = cos(0. x[ ]). (e) = 0.5x[ ] + 0.5x[ ]. (f) = P k = x[ k ]h[ k ], where h[ ] = 0.5. (g) The three-poit ruig-media filter = media{x[ ], x[ ], x[ ]}. (The media of three umbers is the middle umber after sortig.) (c) (e) (f) (g)