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

The output shown on the rightimmediately above equals+C" > ,C# >... ∞ > so that C > does not depend of values of the input that are future to time Therefore the>Þ revised system is cau

(a) (b) (c) (d)

CT1.2.2

(a) (b) (c)

CT1.2.3

(a) (b) (c) (d)

CT1.2.4

(a) (b) (c) (d)

CHAPTER 1

1.2 - WAVEFORMS AND DATA

CT1.2.1

(a) (b) (c) (d)

DT1.2.2

(a) (b) (c)

DT1.2.3

(a) (b) (c)

DT1.2.4

(a) (b) (c)

1.3 - SLTI SYSTEMS

CT1.3.1

a)

B >" ÒC > œ E "  7B >" - " cos # J > 1 - 9

B ># ÒC > œ E "  7B ># - # cos # J > 1 - 9 +B >  ,B >" # Ò E "  7- e+B >  ,B >" # f cos 1# J > - 9 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 > ÒC > œ E "  7B >- cos # J > 1 - 9

B > 7 ÒE "  7B > - 7 cos # J > 1 - 9 The output shown on the right immediately above does not equal C >  7 Therefore, the transformation from B > to C > is not time invariant

c) The AM transmitted waveform, C > œ E "  7B >- cos # J > 1 - 9 , at time > depends only on the value of the message B † at time Because it does not depend on>Þ 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 # J > 1 - 9 l

œ lE-cos # J > 1 - 9  7B > E-cos # J > 1 - 9 l

Ÿ lE ll- cos # J > 1 - 9 l  l7llB > llE ll- cos # J > 1 - 9 l

Ÿ lE l  l7lFlE l  ∞-

-CT1.3.2

a)

B >" ÒC > œ EB >" " cos # J > 1 o 9

B ># ÒC > œ EB ># # cos # J > 1 o 9

a)

C > œ E # J >  7 B 

>

>

9

# J >  7 B 

C > œ E

>

>

9

+B >  ,B >" # Ò EcosŒ# J >  71 o ( e+B" -  ,B# - f - 9

>

>

9

The output shown on the right immediately above does not equal +C" >  ,C# > Therefore, the transformation from B > to C > is linear

The output shown on the rightimmediately above equals+C" > ,C# > Therefore, the transformation from B > toC > is linear

b)

B > ÒC > œEB > cos #1Jo>9

B >7

The output shown on the right immediately above does not equal C >7 Therefore, the transformation from B > toC > is not time invariant

c) The DSB-SC transmitted waveform, C > œEB > cos #1Jo>9 , 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 > toC > is causal

d) The transformation from B > toC > is stable because every bounded B > produces a bounded output C > We can show this by considering any bounded inputB >

lB > lŸF Then

lC > lœlEB > cos #1Jo>9 lœlEllB > llcos #1Jo>9 l

ŸlElF ∞

B > Ò C > œ EcosŒ# J >  71 o ( B - - 9

>

>

9

B > 7 ÒEcosŒ# J >  71 o ( B - - 9

>

>

9

7

The output shown on the right immediately above does not equal C >  7 Therefore, the transformation from B > to C > is not time invariant

c) The FM transmitted waveform, C > œ EcosŠ# J >  71 o '>B - - 9‹, at time >

>

9

depends only on the value of the message B † at, or before, time Because it does not>Þ 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 E» cosŒ# J >  71 o ( B - - 9 »œ lEl»cosŒ# J >  71 o ( B - - 9 »l

>

>

>

>

Ÿ lEl

CT1.3.4

a)

2 >ß B

B >" C > œ" "

∞

∞

-Ò

2 >ß B

B ># C > œ# #

∞

∞

-Ò

+B >  ,B >" # Ò( 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 > Ò C > œ( 2 >ß- B - -

∞

∞

B >7 Ò( 2 >ß- B  - 7

-∞

∞

C > œ Ÿ F( ¸2 >ß ¸.

∞

∞

- -

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

2 >ß- œ ! for -  >, we have

B > Ò C > œ( 2 >ß- B - -

∞

>

so that C > does not depend of values of the input that are future to time Therefore the>Þ 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

∞

>

2 >ß-  ∞

-CT1.3.6

The modified is still linear However, it is time invariant We can see this by writing

B > Ò C > œ( 2 > - B - - œ( 2 " B > " "

where the second integral was obtained by setting > - œ" and integrating with respect

to It follows that"Þ

B > 7 Ò( 2 " B >  7 " " œ C > 7

∞

∞

which proves time-invariance The system is still, in general, unstable following a

The output shown on the right immediately abovedoes not equal C >7 Therefore, the transformation from B > toC > is not time invariant

c) The waveform, C > œC > , at time > dependson B - for all -, including those values that are greater than >Þ Therefore, the transformation from B > toC > is noncausal d) From the hint we have

C > œ¸(



∞

∞

2 >ß- B - -¸Ÿ(



∞

∞

¸2 >ß- ¸¸B - ¸.-Now if B > is bounded, so that lB > lŸF, for all >, then

( ¸ ¸

∞

∞

2 > -  ∞ -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

DT 3.1 1.

a) We can see from the figure that

and

B 8#c dÒC #8  "c d The transformations are both linear Consider the transformation B 8"c dÒC #8c d and assume that B""c d8 ÒC #8"c d, and B"#c d8 Ò C #8#c d Then

Similarly for B 8#c dÒ C #8  "c d b)Both transformations are time-invariant for if

then

B 8  7"c dÒ C # 8  7c d

Similarly if

B 8#c dÒC #8  "c d then

B 8  7#c dÒC # 8  7  "c d c) Both transformations are not causal For example, C # œ B "c d "c d and C $ œc d

B # Þ#c d Both of these values for depend on input values that are in the future.C d) Both transformations are BIBO stable The output values equal shifted versions of the input values

derivation similar to that in problem CT3.4, wecan show that a necessary and sufficient condition for the system to be stable is

DT 3.2 1.

In the DT version of an analog suppressed carrier double-sideband communication system, a message sequence, B 8c d, is transformed to

C 8 œ EB 8c d c dcos 1# 0 8 o 9 (1) where , and are constants Is the I-O relation between E 0o 9 B 8c d and C 8c d

a) linear? Justify your answer

b) time-invariant? Justify your answer

c) causal? Justify your answer

d) stable? Justify your answer

a)

B 8#c dÒC 8 œ EB 8#c d #c dcos # 0 8 1 o 9 +B 8  ,B 8"c d #c dÒE +B 8  ,B 8"c d #c d cos # 0 8 1 o 9 The response on the right hand side of the I-O relation immediately above equals +C 8  ,C 8 Þ"c d #c d Therefore, the I-O relation is linear

b)

B 8c dÒ C 8 œ EB 8c d c dcos # 0 8 1 o 9

B 8  7c d ÒEB 8  7c dcos # 0 8 1 o 9 The response on the right hand side of the I-O relation immediately above is not equal to

C 8  7 Þc d Therefore, the I-O relation is time invariant

c) We can see from the I-O relation

B 8c dÒ C 8 œ EB 8c d c dcos # 0 8 1 o 9 thatC 8c d does not depend on future values of the input, i.e on B 8  5c d for 5 œ "ß #ß á Þ Therefore, the I-O relation is causal

d)

lCc d8 lœ EB 8¸ c dcos # 0 8 1 o 9 ¸ ¸ ¸¸Ÿ E B 8c d¸

It follows that if B 8c d is bounded, i.e lB 8 l Ÿ Fc d , then

Cc d8 Ÿ EF Therefore, the system is stable

DT 3.3 1.

a)

7œ∞

∞

B 8# C 8 œ# 2 8ß 7 B#

7œ∞

∞

2 8ß 7 +B 7  ,B 7

7œ∞

∞

The response on the right hand side of the I-O relation immediately above equals +C 8  ,C 8 Þ"c d #c d Therefore, the I-O relation is linear

b)

B 8c dÒ C 8 œc d 2 8ß 5 Bc d

5œ∞

∞

c d5

B 8  7c d Ò 2 8ß 5 Bc d  7

5œ∞

∞

c5 d

The response on the right hand side of the I-O relation immediately above is not equal to

C 8  7 Þc d Therefore, the I-O relation is time invariant

c) We can see from the I-O relation

B 8c dÒ C 8 œc d 2 8ß 5 Bc d

5œ∞

∞

c d5

that C 8c d depends on values of the input Bc d5 that occur for all , including those that are5 future to (i.e including 8 5  8ÑÞ Therefore, the I-O relation is not causal

d) From the hint we have

lCc d8 lœ¸ 2 8ß 7 Bc d c d7¸Ÿ ¸2 8ß 7 Bc d¸¸ c d7¸

It follows that if B 8c d is bounded, i.e lB 8 l Ÿ Fc d , then

Cc d8 Ÿ F ¸2 8ß 7c d¸

7œ∞

∞

In general, the system is unstable It is stable if and only if

¸ c d¸

7œ∞

∞

2 8ß 7  ∞

DT 3.4 1.

The condition 2 8ß 7 œ !c d for 7  8 makes the system causal We can see this by using this condition to write the output as

C 8c dœ 2 8ß 5 Bc d 5

5œ∞

8

c d

Thus, C 8c d depends only on the values of B 5c d for which 5 Ÿ 8

DT 3.5 1.

The condition 2 8ß 7 œ 2 8  7c d c d makes the system time invariant (shift invariant) To see this, write the I-O relation as

B 8c dÒC 8 œc d 2 8  5 Bc d œ 2 3 Bc d

5œ∞

3œ∞

c d5 c8  3d

where we changed the index of summation from to 5 8  5 œ 3 It follows from the above that

B 8  7c dÒ 2 3 B 8  7c d

3œ∞

∞

c  3 œ C 8  7d c d

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)

B > ÒC > œEB > ß B >7

c) Because B > Ò C > œ EB > , the output, C > , doesnot depend on future values of the input 9

d) C > œ EB > is bounded if B > is bounded because |C > œ E B > 9| | || |

CT1.4.2

a)

B >" C > œ" B" ß B ># C > œ# B#

>

>

(

-+B >  ,B >" # +B"  ,B# œ +C >  ,C >" #

∞

>

(

b)

B

C >

9

c) Because B > ÒC > œ'∞> B - - , the output, C > , depend only on the present= and past values of the input 9

d)

œ >à >   ! (∞> - - œ '>

-0

A plot is given in the solution to problem CT1.2.1a (Set 2 > œ B > in the figureÞÑ

CT1.4.3

a)

B > C > œ . B > ß B > C > œ . B >

.>

+B >  ,B > . +B >  ,B > œ +C >  ,C >

.>

b)

B > C > œ . B > ß B > . B > œ C >

c) Because B > Ò C > œ .>.B > , the output, C > , does not depend on future values of the input 9

d) C > is a pulse having amplitude and duration "% %

C > œ !à >  ! >  

à ! Ÿ > 

%

As decreases, % C > , increases without bound The bounded output, B > , produces an

unbounded output in the limit % Ä !.

CT1.4.4

a)

B >" Ò C > œ B >" " 7o ß B ># Ò C > œ B ># # 7o

+B >  ,B >" # Ò+B >" 7o  ,B ># 7o œ +C >  ,C >" # 9 b)

B > ÒC > œ B >7o ß B >7 ÒB > 7 7o œ C >7 9 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 9

d) C > œ B >7o is bounded if B > is bounded because |C > œ B >| | 7o | 9

CT1.4.5

a)

b) Change the variable of integration to by setting α - œ > α

CT1.4.6

a)

B >" C > œ" B" ß B ># C > œ# B#

>

>

(

1 1

+B  ,B œ +C >  ,C >

>

>

d c

(

-7

1

7

9 b)

B œ C >X

B > Ò C > œ ( B - - ß B >X Ò (

1 1

>X

>

9

c) Because C > œ 1' B C > , depends only on values of the input that occur

>

-from time > 7 to time Therefore, > C > does not depend on future values of the input 9

|C > œ| »17( B - - » Ÿ 17( lB - l

If B > is bounded, then lB - l Ÿ F and C > is bounded:

|C > Ÿ| 17( F Ÿ F

->

>

7

9

DT1.4.1

a)

b)

Bc d8 Ò Cc d8 œ EBc d8ß Bc8  7dÒ EBc8  7 œd Cc8  7d 9 c) Because Bc d8 ÒCc d8 œ EBc d8, the output, Cc d8, does not depend on future values of the input .9

d) Cc d8 œ EBc d8 is bounded if Bc d8 is bounded because |Cc d8|œ E B| || c d8 9|

DT1.4.2

a)

5œ∞

5œ∞

8 8

5œ∞

8

c d

c d

c d

b)

Bc d8 ÒCc d8 œ Bc d5 ß Bc8  7dÒ Bc d5 œ Cc8  7d

5œ∞

5œ∞

9

c) Because Bc d8 Ò Cc d8 œ Bc d5 , the output, Cc d8, depend only on the present and=

5œ∞

8

past values of the input 9

œ B œ

C

8  !

!à

8   !

B œ 8  "à

c d

5

Ú Û Ü

5œ∞

8

5œ!

8

DT1.4.3

a)

b)

Bc d8 ÒCc d8 œ Bc8  7odß Bc8  7dÒBc8  7  7odœ Cc8  7d 9 c) The word “delay” implies that 7o  !in Cc d8 œ Bc8  7od This means that Cc d8, does not depend on future values of the input 9

d) Cc d8 œ Bc8  7od is bounded if Bc d8 is bounded because |Cc d8|œ B| c8  7od| 9

DT1.4.4

By definition,

?B 8 œ B 8  B 8  "c d c d c d a)

b)

Bc d8 Ò ?B 8 œ B 8  B 8  "c d c d c d

a)

B

R

7œ8R "

7œ8R "

c d

7œ8R "

8

b)

8

8Q 8Q

9Þ c) Because Cc d8 œ R" Bc d c d7 C8, depends only on values of the input that occur

7œ8R "

8

from 8  R  " to Therefore, 8 Cc d8 does not depend on future values of the input 9

Bc87dÒ?Bc87dœBc87dBc87"dœCc87d 9 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¸ŸF, then Cc8d is bounded because

¸Cc8d¸œ ŸFF 9

DT1.4.5

a) Change the index of summation: 7œ85

b)

7œ8R "

7œ8R "

8 8

is bounded:

and

is bounded, then

IfBc d8 lBc d7l Ÿ F Cc d8

|C |Ÿ " F Ÿ F

R

c d8

7œ8R "

8

9

MISCELLANEOUS PROBLEMS

1

a) The plots are shown below The input, B 8c d, is depicted by the “‚” marks The output, C 8c d, is depicted by the solid circles with stems

b) The transformation is not linear Consider for example, B 8 œ !Þ""c d ÒC 8 œ "!ß"c d

B 8 œ !Þ"#c d ÒC 8 œ "!#c d for which B 8  B 8 œ !Þ#"c d #c d Ò"! Á C 8  C 8"c d #c d c) The transformation is shift invariant because C 8 œ J B 8c d c d where J + is a fixed function of a real number +

d) The transformation is causal because C 8c d depends only on B 8c d for each 8 e) The transformation is stable because lC 8 l Ÿ $&c d for every inputÞ

2

a) The inequality l/ 8 l Ÿc d " 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

8‚10œ )! œ # ‚ %! This result has the form #,?œ #E where E œ %!, ?œ10, and , œ $ Also for Figure 17, #,?œ #E which rearranges to #E? œ #,