Solution manual for an introduction to signals and systems 1st edition by stuller

Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION CHAPTER 1.2 - WAVEFORMS AND DATA CT1.2.1 (a) (b) (c) (d) CT1.2.2 (a) (b) (c) CT1.2.3 (a) (b) (c) (d) CT1.2.4 (a) (b) (c) (d) Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller INTRODUCTION DT1.2.1 (a) (b) (c) (d) DT1.2.2 (a) (b) (c) DT1.2.3 (a) (b) (c) DT1.2.4 (a) (b) (c) Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION 1.3 - SLTI SYSTEMS CT1.3.1 a) B" > Ò C" > œ E- "  7B" > cos #1J- >  B# > Ò C# > œ E- "  7B# > cos #1J- >  +B" >  ,B# > Ò E- "  7e+B" >  ,B# > f cos #1J- >  The output shown on the right immediately above does not equal +C" >  , C# > Therefore, the transformation from B > to C > is not linear b) B > Ò C > œ E- "  7B > cos #1J- >  B >  Ò E- "  7B >  cos #1J- >  The output shown on the right immediately above does not equal C >  Therefore, the transformation from B > to C > is not time invariant c) The AM transmitted waveform, C > œ E- "  7B > cos #1J- >  , at time > depends only on the value of the message B † at time >Þ Because it does not depend on values of the message that are future to time >, the transformation from B > to C > is causal d) The transformation from B > to C > is stable because every bounded B > produces a bounded output C > We can show this by considering any bounded input B > lB > l Ÿ F Then lC > l œ lE- "  7B > cos #1J- >  l œ lE- cos #1J- >   7B > E- cos #1J- >  l Ÿ lE- llcos #1J- >  l  l7llB > llE- llcos #1J- >  l Ÿ lE- l  l7lFlE- l  ∞ CT1.3.2 a) B" > Ò C" > œ EB" > cos #1Jo >  B# > Ò C# > œ EB# > cos #1Jo >  Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION +B" >  ,B# > Ò Ee+B" >  ,B# > fcos #1Jo >  The output shown on the right immediately above equals +C" >  ,C# > Therefore, the transformation from B > to C > is linear b) B > Ò C > œ EB > cos #1Jo >  B >  Ò EB >  cos #1Jo >  The output shown on the right immediately above does not equal C >  Therefore, the transformation from B > to C > is not time invariant c) The DSB-SC transmitted waveform, C > œ EB > cos #1Jo >  , at time > depends only on the value of the message B † at time >Þ Because it does not depend on values of the message that are future to time >, the transformation from B > to C > is causal d) The transformation from B > to C > is stable because every bounded B > produces a bounded output C > We can show this by considering any bounded input B > lB > l Ÿ F Then lC > l œ lEB > cos #1Jo >  l œ lEllB > llcos #1Jo >  l Ÿ lElF  ∞ CT1.3.3 a) B" > Ò C " > œ EcosŒ#1Jo >  7( B" - -  > >9 B# > Ò C # > œ EcosŒ#1Jo >  7( B# - -  > >9 +B" >  ,B# > Ò EcosŒ#1Jo >  7( e+B" -  ,B# - f -  > >9 The output shown on the right immediately above does not equal +C" >  , C# > Therefore, the transformation from B > to C > is linear Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION b) B > Ò C > œ EcosŒ#1Jo >  7( B - -  > >9 B >  Ò EcosŒ#1Jo >  7( >7 B - -  >9 The output shown on the right immediately above does not equal C >  Therefore, the transformation from B > to C > is not time invariant > c) The FM transmitted waveform, C > œ EcosŠ#1Jo >  7'>9 B - -  9‹, at time > depends only on the value of the message B † at, or before, time >Þ Because it does not depend on values of the message that are future to time >, the transformation from B > to C > is causal d) The transformation from B > to C > is stable because every bounded B > produces a bounded output C > We can show this by considering an input B > Then lC > l œ »EcosŒ#1Jo >  7( B - -  » œ lEl»cosŒ#1Jo >  7( B - -  »l > > >9 >9 Ÿ lEl CT1.3.4 a) B" > Ò C" > œ ( ∞ B# > Ò C# > œ ( ∞ >ß - B" - - ∞ >ß - B# - - ∞ +B" >  ,B# > Ò ( ∞ ∞ >ß - e+B" >  ,B# > f - The output shown on the right immediately above equals +C" >  , C# > Therefore, the transformation from B > to C > is linear b) B > ÒC > œ( ∞ >ß - B - ∞ B >7 Ị ( ∞ >ß - B -7 ∞ Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION The output shown on the right immediately above does not equal C >  Therefore, the transformation from B > to C > is not time invariant c) The waveform, C > œ C > , at time > depends on B - for all -, including those values that are greater than >Þ Therefore, the transformation from B > to C > is noncausal d) From the hint we have C > œ ¸( ∞ ∞ >ß - B - -¸ Ÿ ( á2 >ò - ááB - á.- Now if B > is bounded, so that lB > l Ÿ F, for all >, then C > œ Ÿ F( á2 >ò - - We can see from the above that the system is generally unstable It is stable if and only if ( ∞ ∞ ¸2 >ß - ¸ -  ∞ for all > CT1.3.5 The modified system is linear and time varying, in general However, because >ß - œ ! for -  >, we have B > ÒC > œ( > >ß - B - ∞ so that C > does not depend of values of the input that are future to time >Þ Therefore the revised system is causal The system is still, in general, unstable following a derivation similar to that in problem CT3.4, we can show that a necessary and sufficient condition for the system to be stable is ( > á2 >ò - -  ∞ CT1.3.6 The modified is still linear However, it is time invariant We can see this by writing B > ÒC > œ( ∞ ∞ >  - B - - œ ( ∞ " B >  " " ∞ where the second integral was obtained by setting >  - œ " and integrating with respect to "Þ It follows that B >7 Ị( ∞ " B >   " " œ C >  ∞ which proves time-invariance The system is still, in general, unstable following a Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION derivation similar to that in problem CT3.4, we can show that a necessary and sufficient condition for the system to be stable is ( ∞ ∞ ¸ >  - ¸ -  ∞ The above can be written equivalently as ( ∞ ∞ ¸2 > ¸.>  ∞ CT1.3.7 The condition stated in this problem is necessary and sufficient for the system to be BIBO stable We established this result in the solution to problem CT3.4 DT1.3.1 a) We can see from the figure that and B" c8d Ò Cc#8d B# c8d Ò Cc#8  "d The transformations are both linear Consider the transformation B" c8d Ò Cc#8d and assume that B"" c8d Ò C" c#8d, and B"# c8d Ò C# c#8d Then +B"" c8d  ,B"# c8d Ò +C" c#8d  ,C# c#8d Similarly for B# c8d Ò Cc#8  "d b)Both transformations are time-invariant for if B" c8d Ò Cc#8d then Similarly if then B" c8  7d Ò Cc#  d B# c8d Ò Cc#8  "d B# c8  7d Ò Cc#   "d c) Both transformations are not causal For example, Cc#d œ B" c"d and Cc$d œ B# c#dÞ Both of these values for C depend on input values that are in the future d) Both transformations are BIBO stable The output values equal shifted versions of the input values Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION DT1.3.2 In the DT version of an analog suppressed carrier double-sideband communication system, a message sequence, Bc8d, is transformed to Cc8d œ EBc8dcos #10o  where E, 0o and are constants Is the I-O relation between Bc8d and Cc8d a) linear? Justify your answer b) time-invariant? Justify your answer c) causal? Justify your answer d) stable? Justify your answer a) (1) B" c8d Ò C" c8d œ EB" c8dcos #10o  B# c8d Ò C# c8d œ EB# c8dcos #10o  +B" c8d  ,B# c8d Ò E +B" c8d  ,B# c8d cos #10o  The response on the right hand side of the I-O relation immediately above equals +C" c8d  ,C# c8dÞ Therefore, the I-O relation is linear b) B c8d Ò Cc8d œ EBc8dcos #10o  B c8  7d Ò EBc8  7dcos #10o  The response on the right hand side of the I-O relation immediately above is not equal to Cc8  7dÞ Therefore, the I-O relation is time invariant c) We can see from the I-O relation B c8d Ò Cc8d œ EBc8dcos #10o  that Cc8d does not depend on future values of the input, i.e on Bc8  d for "ò #ò ỏ ị Therefore, the I-O relation is causal d) lCc8dl œ ¸EBc8dcos #10o  ¸ Ÿ ¸E¸¸Bc8d¸ It follows that if Bc8d is bounded, i.e lBc8dl Ÿ F , then Cc8d Ÿ EF Therefore, the system is stable Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION DT1.3.3 a) B" c8d Ò C" c8d œ ∞ B# c8d Ò C# c8d œ +B" c8d  ,B# c8d Ị 2c8ß 7dB" c7d 7œ∞ ∞ 2c8ß 7dB# c7d 7œ∞ c8ß 7de+B" c7d  ,B# c7df ∞ 7œ∞ The response on the right hand side of the I-O relation immediately above equals +C" c8d  ,C# c8dÞ Therefore, the I-O relation is linear b) B c8d Ò Cc8d œ B c8  7d Ò ∞ c8ß dB c5 d 5œ∞ c8ß dB c5  7d ∞ 5œ∞ The response on the right hand side of the I-O relation immediately above is not equal to Cc8  7dÞ Therefore, the I-O relation is time invariant c) We can see from the I-O relation B c8d Ị Cc8d œ ∞ c8ß dB c5 d 5œ∞ that Cc8d depends on values of the input B c5 d that occur for all , including those that are future to (i.e including  8ĐÞ Therefore, the I-O relation is not causal d) From the hint we have lCc8dl œ ¸ ∞ c8ò 7dBc7dá It follows that if Bc8d is bounded, i.e lBc8dl Ÿ F , then Cc8d F á2 c8ò 7dááBc7dá á2 c8ò 7d¸ In general, the system is unstable It is stable if and only if Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION á2 c8ò 7dá  ∞ DT1.3.4 The condition c8ß 7d œ ! for  makes the system causal We can see this by using this condition to write the output as Cc8d œ 5œ∞ c8ß dB c5 d Thus, Cc8d depends only on the values of Bc5 d for which Ÿ DT1.3.5 The condition c8ß 7d œ c8  7d makes the system time invariant (shift invariant) To see this, write the I-O relation as Bc8d Ò Cc8d œ ∞ 5œ∞ c  d B c5 d œ ∞ 2c3dB c8  3d 3œ∞ where we changed the index of summation from to  œ It follows from the above that Bc8  7d Ò ∞ c3dB c8   3d œ Cc8  7d 3œ∞ which proves time invariance DT1.3.6 The condition is necessary and sufficient for the I-O relation to be stable This result was established in Problem 3.3 1.4 - ENGINEERING MODELS CT1.4.1 a) B" > Ò C" > œ EB" > ß B# > Ị C# > œ EB# > +B" >  ,B# > Ò Ec+B" >  ,B# > d œ +C" >  ,C# > B > Ị C > œ EB > ß b) B >7 Ò EB >7 œ C >7 c) Because B > Ò C > œ EB > , the output, C > , does not depend on future values of the input 10 Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://TestbankDirect.eu/ INTRODUCTION d) C > œ EB > is bounded if B > is bounded because |C > | œ |E||B > | CT1.4.2 a) B" > Ò C" > œ ( > ∞ +B" >  ,B# > Ị ( B" - - ß B # > Ò C # > œ ( > ∞ > B# - ∞ c+B" -  ,B# - d - œ +C" >  ,C# > b) B > ÒC > œ( > B - -ß ∞ B >7 Ị ( >7 B - - œ C >7 ∞ > c) Because B > Ò C > œ '∞ B - -, the output, C > , depend= only on the present and past values of the input d) C > œ( > ∞ B - - œ œ !à '0> - œ >à >! > ! A plot is given in the solution to problem CT1.2.1a (Set > œ B > in the figurĐ CT1.4.3 a) B" > Ị C" > œ +B" >  ,B# > Ò B" > ß B # > Ị C # > œ B # > > > e+B" >  ,B# > f œ +C" >  ,C# > > b) B > ỊC > œ B>ß > B >7 Ò B >7 œ C >7 > c) Because B > Ò C > œ > B > , the output, C > , does not depend on future values of the input d) C > is a pulse having amplitude "% and duration % !à C > œœ" %à >  ! and >   % !Ÿ>% As % decreases, C > , increases without bound The bounded output, B > , produces an 11 Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://TestbankDirect.eu/ INTRODUCTION unbounded output in the limit % Ä ! CT1.4.4 a) B" > Ò C" > œ B" >7o ß B# > Ị C# > œ B# >7o +B" >  ,B# > Ò +B" >7o  ,B# >7o œ +C" >  ,C# > B > Ò C > œ B >7o ß b) B >7 Ò B >7 7o œ C >7 c) The word “delay” implies that 7o  ! in C > œ B >7o This means that C > , does not depend on future values of the input d) C > œ B >7o is bounded if B > is bounded because |C > | œ |B >7o | CT1.4.5 a) b) Change the variable of integration to α by setting - œ >  α CT1.4.6 a) B" > Ò C" > œ > > ( B" - - ß B # > Ò C # > œ ( B # - >7 >7 +B" >  ,B# > Ò > ( c+B" -  ,B# - d - œ +C" >  ,C# > >7 b) > B > Ò C > œ ( B - -ß >7 >X B >X Ị ( B - - œ C >X >X 7 > c) Because C > œ 71 '>7 B - - C > , depends only on values of the input that occur from time >  to time > Therefore, C > does not depend on future values of the input 12 Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://TestbankDirect.eu/ INTRODUCTION d) > > |C > | œ » ( B - -» Ÿ ( lB - l >7 >7 If B > is bounded, then lB - l Ÿ F and C > is bounded: > ( F - Ÿ F >7 |C > | Ÿ DT1.4.1 a) B" c8d Ò C" c8d œ EB" c8dß B# c8d Ị C# c8d œ EB# c8d +B" c8d  ,B# c8d Ò Ec+B" c8d  ,B# c8dd œ +C" c8d  ,C# c8d b) B c8d Ò Cc8d œ EB c8dß Bc8  7d Ò EB c8  7d œ Cc8  7d c) Because B c8d Ò Cc8d œ EB c8d, the output, Cc8d, does not depend on future values of the input d) Cc8d œ EB c8d is bounded if Bc8d is bounded because |Cc8d| œ |E||B c8d| DT1.4.2 a) B" c8d Ò C" c8d œ +B" c8d  ,B# c8d Ị B" c5 dß B# c8d Ị C# c8d œ 5œ∞ 5œ∞ B# c d 5œ∞ e+B" c5 d  ,B# c5 df œ +C" c8d  ,C# c8d b) B c8d Ò Cc8d œ Bc5 dß 5œ∞ c) Because B c8d Ị Cc8d œ past values of the input B c8  7d Ò Bc5 d œ Cc8  7d 5œ∞ Bc5 d, the output, Cc8d, depend= only on the present and 5œ∞ 13 87 Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://TestbankDirect.eu/ INTRODUCTION d) Cc8d œ DT1.4.3 a) Ú !à Bc5 d œ Û Bc5 d œ  "à Ü 5œ! 8! 5œ∞ 8 ! B" c8d Ò C" c8d œ B" c8  7o dß B# c8d Ị C# c8d œ B# c8  7o d +B" c8d  ,B# c8d Ò +B" c8  7o d  ,B# c8  7o d œ +C" c8d  ,C# c8d b) B c8d Ò Cc8d œ Bc8  7o dß B c8  7d Ò B c8   7o d œ C c8  7d c) The word “delay” implies that 7o  ! in Cc8d œ Bc8  7o d This means that Cc8d, does not depend on future values of the input d) Cc8d œ Bc8  7o d is bounded if Bc8d is bounded because |Cc8d| œ |Bc8  7o d| DT1.4.4 By definition, a) ?Bc8d œ Bc8d  Bc8  "d B" c8d Ò C" c8d œ ?B" c8d œ B" c8d  B" c8  "d B# c8d Ò C# c8d œ ?B" c8d œ B# c8d  B# c8  "d +B" c8d  ,B# c8d Ò ?e+B" c8d  ,B# c8df œ +C" c8d  ,C# c8d b) B c8d Ò ?Bc8d œ Bc8d  Bc8  "d 14 Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION Bc8  7d Ò ?Bc8  7d œ Bc8  7d  Bc8   "d œ Cc8  7d c) Cc8d depends only on Bc8d and Bc8  "d It does not depend on future values of the input d) We have ¸Cc8d¸ œ ¸?Bc8d¸ œ ¸Bc8d  Bc8  "d¸ Ÿ ¸Bc8d¸  ¸Bc8  "d¸ If Bc8d is bounded, ¸Bc8d¸ Ÿ F, then Cc8d is bounded because ¸Cc8d¸ œ Ÿ F  F DT1.4.5 a) Change the index of summation: œ  b) DT1.4.6 a) B" c8d Ò C" c8d œ 8 " " B" c7dß B# c8d Ị C# c8d œ B# c7d R 7œ8R " R 7œ8R " +B" c8d  ,B# c8d Ò " c+B" c7d  ,B# c7dd œ +C" c8d  ,C# c8d R 7œ8R " b) " B c8d Ị Cc8d œ Bc7dß R 7œ8R " 9Þ c) Because Cc8d œ " R 8Q " B c8  Q d Ò Bc7d œ Cc8  Q d R 7œ8Q R " Bc7d Cc8d, depends only on values of the input that occur from  R  " to Therefore, Cc8d does not depend on future values of the input 7œ8R " 15 Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION d) 8 " " |Cc8d| œ » Bc7d» Ÿ lBc7dl R 7œ8R " R 7œ8R " If Bc8d is bounded, then lBc7dl Ÿ F and Cc8d is bounded: |Cc8d| Ÿ " FŸF R 7œ8R " MISCELLANEOUS PROBLEMS a) The plots are shown below The input, Bc8d, is depicted by the “ ‚ ” marks The output, Cc8d, is depicted by the solid circles with stems b) The transformation is not linear Consider for example, B" c8d !ị" ề C" c8d "!ò B# c8d œ !Þ" Ị C# c8d œ "! for which B" c8d  B# c8d œ !Þ# Ị "! Á C" c8d  C# c8d c) The transformation is shift invariant because Cc8d œ J Bc8d where J + is a fixed function of a real number + d) The transformation is causal because Cc8d depends only on Bc8d for each e) The transformation is stable because lCc8dl Ÿ $& for every inputÞ a) The inequality l/c8dl Ÿ "# ? follows from an inspection of the figure b) Consider first the quantizer of Figure 16 There we see that the step size is 10 and ‚ 10 œ )! œ # ‚ %! This result has the form #, ? œ #E where E œ %!, ? œ 10, and ? , œ $ Also for Figure 17, #, ? œ #E which rearranges to #E œ #, 16 Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller INTRODUCTION The figures for parts (c) and (d) are shown below (c) (d) ?‰ We see from figure (d) that ˆ #E decreases by ' dB when , is increased by dB 17 Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller INTRODUCTION 18 ... the form #, ? œ #E where E œ %!, ? œ 10, and ? , œ $ Also for Figure 17, #, ? œ #E which rearranges to #E œ #, 16 Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller. .. Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION DT1.3.2 In the DT version of an analog suppressed carrier double-sideband... >  , C# > Therefore, the transformation from B > to C > is linear Solution Manual for An Introduction to Signals and Systems 1st Edition by Stuller Full file at https://./ INTRODUCTION b) B