《信号与系统 Signals and Systems》课程教学资料（英文版）ps2

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Comput 6.003: Signals and Systems--Fall 2003 PROBLEM SET 2 Issued: September 11. 2003 Due: September 19, 2003 Reading Assignments: Lectures #3-4 PS#2: Chapter 2 of O&w Lectures #5-6& PS#3: Chapter 3 of o&w Exercise for home study (not to be turned in, although we will provide solutions) (E1)O&W1.3(a) (E2)O&W2.33(a-(i) (E3)O&W244(a) Problems to be turned Problem 1 Compute the convolution y[n]=r[n]* h[n] of each of the two following pair of signals a).[ n] and hn are depicted below 4-3-2 (b).xn=un+4]-un-1],bn]=22-m

�

     "! # $ %'&)(+*-,/.0%'12,3+45768%'9:,/*6?71@A;81B%'%'*;8*Fx 3+4™xF;Ê@A1)(+6Êx'y ± (2²:³ƙÁÃÂlÄ (+1BCȅÁÃÂlÄ (+*̙Á¹~ Ë ÂMÄ¥³ {

Problem 2 Compute the convolution y(t)=r(t)* h(t) for each of the following pairs of (a).(t)=e-u(t+1),h(t)=e2(-t) (b). r(t) and h(t)are depicted below (t) h(t) t t roblem 3 The following are impulse responses of either discrete-time or continuous-time LTI systems. Determine whether each system is causal and / or stable. Justify your answer (b).h(t)=(1-t)-e-u(t (c).hm]=[1-(0.99)]n] (d),.h(t)=c5[u(t-1)-u(t-100 Problem 4 Consider the LtI system initially at rest and described by the difference equa tion yn]-oyn Find the response of this system to the input depicted below 2

|Ž¤r©™Wˆ`b h D3A.0&EB,/%|,/›%Z9G3A12¾A3A6ÊEB,/;Ê3A1¿ ±
� ²Å Æ ±
� ²7ÇÈ ±� ²4T3A*%=(+9 › 3+47,F›%¯4T3A6Ê683aÉ;81B@½&)(>;8*Fx3+4 xF;Ê@A1)(+6Êx'y ± (2²:³Æ ±� ²Å·Ì ±
� Í{Š²e}MÈ ±� ²7Å  TÌ ± Ë � ² ±zr²:³Æ ±� ²(>1CÈ ±� ² (+*F%"C%G&;89:,/%'CÀzM%'683aÉ"y  { Ë { { Æ ±
� ² �  { ~ € ¸ Ë { { È ±
� ² � |Ž¤r©™Wˆ`b N ›B%4T3A6Ê683aÉ;81@"(>*F%;8.0&E68x9 ›ºx x-,/%'.g;Êx9=(+ExF(+6…(+1C>3A*x,*F%Gx<,D(+1C0C%'x<9'*<;8zM%'C‘z Z,F›%CB;l%'*F%G19'%%pE)(Š¶ ,/;Ê3A1 ¿^ÁÃÂlÄ Ë { ~ ¿JÁÃÂ Ë {'ÄMÅ ~ŠÆ™ÁÏÂMÄ Ë Æ™ÁÏÂ Ë ~Ä  5;Ê1C‡,/›%"*F%GxF&M3A1x<%‚3+4 ,/›B;8xx x-,/%'.g,F3 ,F›%";81&BEB,C%'&B;89G,F%'CÀzl%G683aÉ Ë € Ë ~ Ë {  { ~ € ¸ ƙÁÃÂlÄ Â ~ { { { ~

Problem 5 Consider the cascade of LTI systems with unit sample responses h,In and h2n depicted below hin h2n Suppose we are gi ven the following information h2{m]=6m]-6{7-1 If the input is m]=n]-{7-2 then the output is as depicted below Find h1[n Problem 6 O&W 2.66, with the following corrections In part(a), replace(i)with (ii)1(t)=0 for t4 In part(a), the correct statement of (iv (iv)yn(t)=aI(t)*hi(t) is as large as possible at t= 4 After working this problem, make sure to read the brief paragraph on the top of p. 169 Problem 7 O&W 2.44(d

|Ž¤r©™Wˆ`b Ÿ D3+1xF;ÊC%'*Z,/›B%À9'(+xF9'(+C%¬3+4|x x.½&6Ê%‡*F%GxF&M3A1xF%Gx È  ÁÃÂlÄ(>1C È ÁÏÂMąC%'&B;89G,F%'CÀzl%G683aÉ"y ƙÁÃÂlÄ È  ÁÃÂlÄ È ÁÃÂlÄ ¿^ÁÃÂlÄ E&B&l3Ax,F;83A1^y � È ÁÃÂlÄ^Å
ÁÃÂlÄ Ë
ÁÏÂ Ë {GÄ � 4 ,/›%";Ê1&EB,;8x ƙÁÃÂlÄ^Å Ì5ÁÃÂlÄ Ë Ì5ÁÃÂ Ë ~Ä ,F›%'1¬,/›%"3AE,/&EB,;Êx(+xC%'&;Ê9G,F%'CÀzl%G683aÉ  { ~ € Ë { Ë ~ Ë € ¿JÁÃÂlÄ Â ~ { Ë { Ë ~ 5;Ê1CÈ  ÁÏÂMÄ¥³ |Ž¤r©™Wˆ`b H " ž ~³B}BÉ;Ê,F›‡,/›%4T3A6Ê683aÉ;81@¯9'3A*,/%G.½%G12,3>4 ± ;ʾB²D;8x ± ;ʾB²D¿  ±� ²Å Æ  ±
� ² ÇÈ  ±
� ²D;Êx(>x6 (+*, � Ÿ
4·,F%'*É3A* p;81B@ ,F›;8x&B*F3Az6Ê%'.À}.½( +%"xFEB*F%,/3 *F%'(+C¬,/›%"z*45&J³L{Aƒ³ |Ž¤r©™Wˆ`b " ž ~³¹¸+¸ ±Cr² €

The following two problems are from Comp Cter Erploratio Fs iF Sitfals aFa Srstems UsiFD Mn of nl Seco Fa eaitioF by Buck, Daniel, and einger(BDe). Please submit your code as a Problem 8 BDe eection 2.1 Problem g BDe eection 2.2 Reminder: The first 20 problems in each chapter of O&w have answers included at the end of the tedt. t onsider using these for additional practice, either now or as you study for

›%|4T3A686Ê3ŠÉ;Ê1@Q,É3 &*F3+z68%G.½x(+*1C;81@+%'* ±6Ê%=(+x&B,/%G*‚3+4ž ›)(=¾A%½(+1x-ÉD%G*FxQ;Ê19'6ÊEC%'C§(>,",/›B% %'1BC3+4…,F›% ,/%DC,'³D3A1xF;ÊC%'*Ex9G,/;Ê9'%>}%';L,/›%G*13aÉÐ3A*(>x A3AE¬x<,FEC 4T3+* ,/%Gx<,Fx'³ ¸