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• Discrete Time Signal Processing 3rd Edition Solution Manual Free Download

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• Step 1 of 38

  Criteria for stable system:

  A system is said to be stable when a finite input sequence isapplied to the system, a finite or bounded output sequence has tobe produced.

  Criteria for causal system:

  A system is said to be causal when the present values of theoutput sequence depends on the present and the past values of theinput sequence. If the present value of the output sequence dependson the future values of the input sequence then it is said to benon-causal system.

  • Criteria for stability needs to be modified as input sequence which has finite, bounded value for all time index n, output of system should have also bounded values for all n, because the finite input sequence can be interpreted as infinite time index cannot be selected to test stability.
• Step 2 of 38

  Criteria for linear system:

  Linearity is the property of an element describing a linearrelationship between cause and effect. The property is acombination of both the homogeneity (scaling) property and theadditivity property.

  The homogeneity property requires that if the input ismultiplied by a constant, then the output gets multiplied by thesame constant. The homogeneity property is satisfied only if thefollowing equation holds true.

  The additivity property requires that the response to a sum ofinputs is the sum of the responses to each input appliedseparately. The additivity property is satisfied only if thefollowing equation holds true.

• Step 3 of 38

  Criteria for time-invariance system:

  A time –invariant system is a system, that transforms the timeshift or delay in the input sequence to the time-shift or delay inthe output sequence.

  Criteria for memoryless system:

  A system is said to be memoryless when the value of the outputsequence depends only on the present values of the inputsequence.

• Step 4 of 38

  (a)

  Consider the following system, defined at the number in the sequence:

  The output sequence is and the input sequence is.

  Check for the stability of the function as follows:

  The output sequence,is the result of the multiplication of two sequences and,which is the input sequence itself. So apply a finiteinput,then the stability of the function depends upon the function because is finite in nature. Therefore, the functionis stable ifis stable.

• Step 5 of 38

  Check for the causality of the function as follows:

  The output sequence,is the result of the multiplication of two sequences and,which is the input sequence itself. For a given value in the input sequence, the output sequence value depends on thatpresent value. Therefore, the system is causal.

• Step 6 of 38

  Check for the linearity of the function as follows:

  The property is a combination of both the homogeneity (scaling)property and the additivity property.

  For two separate inputs the system produces the following response

  And

  Check for the additivity property of the function as follows:

  Hence, additivity property is satisfied.

  Check for the homogeneity property of the function as follows:

  Hence, homogeneity property is satisfied.

  As the additivity and homogeneity property is satisfied, thesystemis a linear system.

• Step 7 of 38
  Check for the time-invariance property of thefunction as follows:

  Determine the outputof the system for the delayed input,

  Now delay the output by samples given by the equation,

  Now compare and,observe that

  Therefore, the system is not time-invariant.

• Step 8 of 38

  Check whether the system is memoryless or not as follows:

  The output sequence,is the result of the multiplication of two sequences and.The output sequenceat every value of depends only on the input at the same value. Therefore, thesystem is memoryless system.

  Thus, the system is causal, linear and non time-invariant,memoryless and stable ifis finite in nature.

• Step 9 of 38

  (b)

  Consider the following system, defined at the number in the sequence:

  The output sequence is and the input sequence is.

  Check for the stability of the function as follows:

  The value of is as follows:

  The function may not settle to a finite value. Hence, it is unstable.

  Therefore, the functionis unstable.

• Step 10 of 38

  Check for the causality of the function as follows:

  The output sequence is

  Here, the information about is incomplete that is whether is greater or less than is not known. Therefore, the system is non-causal.

• Step 11 of 38

  Check for the linearity of the function as follows:

  The property is a combination of both the homogeneity (scaling)property and the additivity property.

  For two separate inputs the system produces the following response

  And

  Check for the additivity property of the function as follows:

  Hence, additivity property is satisfied.

  Check for the homogeneity property of the function as follows:

  Hence, homogeneity property is satisfied.

  As the additivity and homogeneity property is satisfied, thesystemis a linear system.

• Step 12 of 38
  Check for the time-invariance property of thefunction as follows:

  Determine the outputof the system for the delayed input ,

  Now delay the output by samples given by the equation,

  Now compare and,observe that,

  Therefore, the system is time-invariant.

  • this is time variant. you have solved it wrong ! -_-
  • It is time invariant, anon. If you have a different answer, back it up.
• Step 13 of 38

  Check whether the system is memoryless or not as follows:

  The output sequence is

  Here, the information about is incomplete that is whether is greater or less than is not known. Therefore, the system is not memoryless.

  Thus, the system is noncausal, linear and time-invariant, notmemoryless and unstable.

• Step 14 of 38

  (c)

  Consider the following system, defined at the number in the sequence:

  The output sequence is and the input sequence is.

  Check for the stability of the function as follows:

  The expression for is,

  The function settles to a finite value because a small interval enclosing is taken into account. Hence, it is stable.

  Therefore, the systemis stable.

• Step 15 of 38

  Check for the causality of the function as follows:

  The output sequence is

  For a given value in the input sequence, the output sequence value depends on thepresent and future values. Therefore, the system is noncausal.

  Therefore, the system is non-causal.

• Step 16 of 38

  Check for the linearity of the function as follows:

  The property is a combination of both the homogeneity (scaling)property and the additivity property.

  For two separate inputs the system produces the following response

  And

  Check for the additivity property of the function as follows:

  Hence, additivity property is satisfied.

  Check for the homogeneity property of the function as follows:

  As the additivity and homogeneity property is satisfied, thesystemis a linear system.

• Step 17 of 38
  Check for the time-invariance property of thefunction as follows:

  Determine the outputof the system for the delayed input,

  Now delay the output by samples given by the equation,

  Now compare and,observe that

  Therefore, the system is time-invariant.

• Step 18 of 38

  Check whether the system is memoryless or not as follows:

  The output sequence is

  Therefore, the system acts as a memoryless system only ifbecause then only its output depends on the present valuesof the input.

  Thus, the system is non-causal, linear and time-invariant,memoryless ifand stable.

• Step 19 of 38

  (d)

  Consider the following system, defined at the number in the sequence:

  The output sequence is and the input sequence is.

  Check for the stability of the function as follows:

  Apply a finite input,then the output is the shifted version of the input by samples which is finite. Therefore, the systemis stable.

• Step 20 of 38

  Check for the causality of the function as follows:

  For a given value ofin the input sequence, the output sequence value depends on thatpresent value except for the values, because for those values the output depends on the future values.Therefore, the system is causal only for.

  • this is wrong! it is only causal if n is less than or equal to zero.
  • this is right, you can check for the solution at the back of the book and it says when n0 is greater or equal to zero.
• Step 21 of 38

  Check for the linearity of the function as follows:

  The property is a combination of both the homogeneity (scaling)property and the additivity property.

  For two separate inputs the system produces the following response

  And

  Check for the additivity property of the function as follows:

  Hence, additivity property is satisfied.

  Check for the homogeneity property of the function as follows:

  As the additivity and homogeneity property is satisfied, thesystemis a linear system.

• Step 22 of 38
  Check for the time-invariance property of thefunction as follows:

  Determine the outputof the system for the delayed input,

  Now delay the output by samples given by the equation,

  Now compare and,observe that

  Therefore, the system is time-invariant.

• Step 23 of 38

  Check whether the system is memoryless or not as follows:

  For a given value ofin the input sequence, the output sequence value depends on thatpresent value only if , for other values the output depends on the past values.Therefore, the system is memoryless only for.

  Thus, the system is causal only for, linear, time-invariant, memoryless forand stable.

• Step 24 of 38

  (e)

  Consider the following system, defined at the number in the sequence:

  The output sequence is and the input sequence is.

  Check for the stability of the function as follows:

  Apply a finite input,then the output is also finite in nature. Therefore, thefunctionis stable.

• Step 25 of 38

  Check for the causality of the function as follows:

  For a given value of in the input sequence, the output sequence value depends on thatpresent value. Therefore, the system is causal.

• Step 26 of 38

  Check for the linearity of the function as follows:

  The property is a combination of both the homogeneity (scaling)property and the additivity property.

  For two separate inputs the system produces the following response

  And

  Check for the additivity property of the function as follows:

  Hence, additivity property is not satisfied.

  Check for the homogeneity property of the function as follows:

  Hence, homogeneity property is not satisfied.

  As the additivity and homogeneity property is not satisfied,the systemis a nonlinear system.

• Step 27 of 38

  Check for the time-invariance property of the function as follows:

  Determine the outputof the system for the delayed input,

  Now delay the output by samples given by the equation,

  Now compare and,observe that

  Therefore, the system is time-invariant.

  Check whether the system is memoryless or not as follows:

  The output sequenceat every value of depends only on the input at the same value. Therefore, thesystem is memoryless system.

  Thus, the system is causal, nonlinear, time-invariant,memoryless and stable.

  (f)

  Consider the following system, defined at the number in the sequence:

  The output sequence is and the input sequence is.

  Check for the stability of the function as follows:

  The output sequence,is the result of the multiplication of a constant with theinputadded to a constant. So apply a finite input,then the output will also be finite since the multiplication andaddition of a constant to the input does not make the functioninfinite. Therefore, the systemis stable.

• Step 28 of 38

  Check for the causality of the function as follows:

  For a given value of of the input sequence, the output sequence depends on that presentvalue. Therefore, the system is causal.

• Step 29 of 38

  Check for the linearity of the function as follows:

  The property is a combination of both the homogeneity (scaling)property and the additivity property.

  For two separate inputs the system produces the following response

  And

  Check for the additivity property of the function as follows:

  Check for the homogeneity property of the function as follows:

  As the additivity and homogeneity property are not satisfied,the system is a nonlinear system.

  • it is linear when b=0
• Step 30 of 38

  Check for the time-invariance property of the function as follows:

  Determine the outputof the system for the delayed input,

  Now delay the output by samples given by the equation,

  Now compare and,observe that

  Therefore, the system is time-invariant.

  Check whether the system is memoryless or not as follows:

  The output sequenceat every value of depends only on the input at the same value. Therefore, thesystem is memoryless system.

  Thus, the system is stable, causal, non-linear if andtime-invariant and memoryless.

• Step 31 of 38

  (g)

  Consider the following system, defined at the number in the sequence:

  The output sequence is and the input sequence is.

  Check for the stability of the function as follows:

  The output sequence,is the result of the time reversal form of the input sequenceitself. So apply a finite input,then the output is also finite but it is time –reversal form.Therefore, the functionis stable.

• Step 32 of 38

  Check for the causality of the function as follows:

  For a given value of the input sequence, the output sequencevalue may depend on the future value as follows:

  For example, consider .Then evaluate the output for

  For all , the output depends on the future values. Therefore, the systemis non-causal.

  Check for the linearity of the function as follows:

  The property is a combination of both the homogeneity (scaling)property and the additivity property.

  For two separate inputs the system produces the following response

  And

  Check for the additivity property of the function as follows:

  Hence, additivity property is satisfied.

  Check for the homogeneity property of the function as follows:

  Hence, homogeneity property is satisfied.

  As the additivity and homogeneity property is satisfied, thesystemis a linear system.

• Step 33 of 38

  Check for the time-invariance property of the function as follows:

  Determine the outputof the system for the delayed input.

  Now delay the output by samples given by the equation,

  Now compare and,observe that

  Hence, the system is time-variant.

  Check whether the system is memoryless or not as follows:

  The output sequence depends on the future values for all . Therefore, the system is not a memoryless system.

  Thus, the system is stable, noncausal, linear andtime-variant and not memoryless.

  • So this is time invariant because y1 = y[n-n0]
  • *y1[n]=y[n-n0]
  • time variant, dude.
• Step 34 of 38

  (h)

  Consider the following system, defined at the number in the sequence:

  The output sequence is and the input sequence is.

  Check for the stability of the function as follows:

  The output sequence , is the result of the addition,which is the input sequence itself and a unit step function definedfrom , multiplied by a constant. So apply a bound input,then the system produces a bounded output because the signal is bounded.

  Therefore, the system is stable.

• Step 35 of 38

  Check for the causality of the function as follows:

  The output sequence value depends on the present value of inputsequence , though it depends on future value of step function . Since the output depends on present value of input sequence,therefore, the system is Causal.

• Step 36 of 38

  Check for the linearity of the function as follows:

  The property is a combination of both the homogeneity (scaling)property and the additivity property.

  For two separate inputs the system produces the following response

  And

  Check for the additivity property of the function as follows:

  Hence, additivity property is not satisfied.

• Step 37 of 38
  Check for the homogeneity property of thefunction as follows:

  Hence, homogeneity property is satisfied.

  As the additivity property is not satisfied and homogeneityproperty is satisfied, the systemis a nonlinear system.

• Step 38 of 38

  Check for the time-invariance property of the function as follows:

  Determine the outputof the system for the delayed input,

  Now delay the output by samples given by the equation,

  Now compare and,observe that

  Therefore, the system is time-variant.

  Check whether the system is memoryless or not as follows:

  The output sequenceat every value of depends only on the input at the same value. Therefore, thesystem is memoryless.

  Thus, the system is causal, nonlinear and time-variant,memoryless andstable.

  • Can you explain why the system is time variant because y1[n] does not equal y[n - n0]?
  • for y1 you're replacing just x[n] with x[n-n0] for y[n-n0] you're replacing all n's with n-n0 therefore y1 and y are not equal

Discrete Time Signal Processing 3rd Edition Download Software

Discrete Time Signal Processing 3rd Edition Solution Manual Free Download

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