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The outputs of four systems (S1, S2, S3 and S4 ) corresponding to the input signal sin(t) , for all time t , are shown in the figure. Based on the given information, which of the four system is/are definitely NOT LTI (linear and time-invariant)?
  • a)
    S4
  • b)
    S1
  • c)
    S3
  • d)
    S2
Correct answer is option 'A,C'. Can you explain this answer?
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Most Upvoted Answer
The outputs of four systems (S1, S2, S3 and S4 ) corresponding to the...
Assume a system (S) having input x(t) and output y(t) as shown below
According to question, four systems (S1 ,S2 ,S3 ,S4 ) are given with input signal sint .
Method 1
For system (S1 ) :
Here, y(t) = -sint = -x(t)
Thus, y(t) ∝ x(t)
S1 directly follow proportional relationship between input x(t) and output y(t) it means S1 follow both homogeneity and additivity property. Hence S1 is linear.
For S1 , y(t) = -x(t), it shows no change in time instant, it means system S1 is time invariant.
Hence, system S1 is linear and time invariant.
For system (S2 ) :
Here, y(t) = sin(t + 1) = x(t + 1)
Thus, y(t) ∝ x(t + 1)
S2 directly follow proportional relationship between input x(t) and output y(t) it means S2 follow both homogeneity and additivity property. Hence S2 is linear.
For S2 , y(t) = x(t + 1)
Delayed response for S2 , y(t - t0 ) = x(t - t0 + 1)
Response to delayed input x(t - t0) in system S2 , y(t,t0 ) = x(t - t0 + 1)
Thus, delayed response y(t - t0 ) and response to delayed input y(t,t0 ) are same for system S2 , that is
why system S2 is time invariant.
Hence, system S2 is linear and time invariant.
For system (S3 ) :
Here, y(t) = sin2t = x(2t)
Thus, y(t) α x(2t)
S3 directly follow proportional relationship between input x(t) and output y(t) it means S3 follow both homogeneity and additivity property. Hence S3 is linear.
For S3 , y(t) = x(2t)
Delayed response for S3 , y(t - t0 ) = x(2t - t0 )
Response to delayed input x(t - t0 ) in system S3 ,y(t, t0 ) = x(2t - 2t0 )
Thus, delayed response y(t - t0 ) and response to delayed input y(t,t0 ) are same for system S3 , that is why system S3 is time variant.
Hence, system S3 is linear but time variant.
For system (S4 ) :
Thus, y(t) ∝ square of x(t)
S4 directly follow square relationship between input x(t) and output y(t) it shows S4 gives non-linear relationship between input x(t) and output y(t) . Hence S4 is non-linear.
For S4 , y(t) = x2(t), it shows no change in time instant t, it means system S4 is time invariant.
Hence, system S4 is non-linear and time invariant.
Hence, the correct option is (A) and (C).
Method 2
For LTI system, the response to sinusoidal or complex exponential input is a sinusoidal or complex exponential output at same frequency as the Input. It means LTI system cannot change output frequency when sinusoidal or complex exponential input is applied.
According to question, all four system (S1, S2, S3, S4) have sinusoidal input sint ,
For system (S1 ) :
Thus, x(t) = sint , frequency of input is 1 rad/sec.
y(t) = sint , frequency of output is 1 rad/sec.
So, frequency of input x(t) and output y(t) is not changed.
Hence S1 is definitely LTI.
For system (S2 ) :
Thus, x(t) = sint , frequency of input is 1 rad/sec.
y(t) = sin(t + 1) , frequency of output is 1 rad/sec.
So, frequency of input x(t) and output y(t) is not changed.
Hence S2 is definitely LTI.
For system (S3 ) :
Thus, x(t) = sint , frequency of input is 1 rad/sec.
y(t) = sin(2t) , frequency of output is 2 rad/sec.
So, frequency of input x(t) and output y(t) is not same.
Hence S3 is not definitely LTI.
For system (S4 ) :
Thus, x(t) = sint , frequency of input is 1 rad/sec.
, frequency of output is 2 rad/sec.
So, frequency of input x(t) and output y(t) is not same.
Hence S4 is not definitely LTI.
Hence, the correct option is (A) and (C).
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Community Answer
The outputs of four systems (S1, S2, S3 and S4 ) corresponding to the...
Assume a system (S) having input x(t) and output y(t) as shown below
According to question, four systems (S1 ,S2 ,S3 ,S4 ) are given with input signal sint .
Method 1
For system (S1 ) :
Here, y(t) = -sint = -x(t)
Thus, y(t) ∝ x(t)
S1 directly follow proportional relationship between input x(t) and output y(t) it means S1 follow both homogeneity and additivity property. Hence S1 is linear.
For S1 , y(t) = -x(t), it shows no change in time instant, it means system S1 is time invariant.
Hence, system S1 is linear and time invariant.
For system (S2 ) :
Here, y(t) = sin(t + 1) = x(t + 1)
Thus, y(t) ∝ x(t + 1)
S2 directly follow proportional relationship between input x(t) and output y(t) it means S2 follow both homogeneity and additivity property. Hence S2 is linear.
For S2 , y(t) = x(t + 1)
Delayed response for S2 , y(t - t0 ) = x(t - t0 + 1)
Response to delayed input x(t - t0) in system S2 , y(t,t0 ) = x(t - t0 + 1)
Thus, delayed response y(t - t0 ) and response to delayed input y(t,t0 ) are same for system S2 , that is
why system S2 is time invariant.
Hence, system S2 is linear and time invariant.
For system (S3 ) :
Here, y(t) = sin2t = x(2t)
Thus, y(t) α x(2t)
S3 directly follow proportional relationship between input x(t) and output y(t) it means S3 follow both homogeneity and additivity property. Hence S3 is linear.
For S3 , y(t) = x(2t)
Delayed response for S3 , y(t - t0 ) = x(2t - t0 )
Response to delayed input x(t - t0 ) in system S3 ,y(t, t0 ) = x(2t - 2t0 )
Thus, delayed response y(t - t0 ) and response to delayed input y(t,t0 ) are same for system S3 , that is why system S3 is time variant.
Hence, system S3 is linear but time variant.
For system (S4 ) :
Thus, y(t) ∝ square of x(t)
S4 directly follow square relationship between input x(t) and output y(t) it shows S4 gives non-linear relationship between input x(t) and output y(t) . Hence S4 is non-linear.
For S4 , y(t) = x2(t), it shows no change in time instant t, it means system S4 is time invariant.
Hence, system S4 is non-linear and time invariant.
Hence, the correct option is (A) and (C).
Method 2
For LTI system, the response to sinusoidal or complex exponential input is a sinusoidal or complex exponential output at same frequency as the Input. It means LTI system cannot change output frequency when sinusoidal or complex exponential input is applied.
According to question, all four system (S1, S2, S3, S4) have sinusoidal input sint ,
For system (S1 ) :
Thus, x(t) = sint , frequency of input is 1 rad/sec.
y(t) = sint , frequency of output is 1 rad/sec.
So, frequency of input x(t) and output y(t) is not changed.
Hence S1 is definitely LTI.
For system (S2 ) :
Thus, x(t) = sint , frequency of input is 1 rad/sec.
y(t) = sin(t + 1) , frequency of output is 1 rad/sec.
So, frequency of input x(t) and output y(t) is not changed.
Hence S2 is definitely LTI.
For system (S3 ) :
Thus, x(t) = sint , frequency of input is 1 rad/sec.
y(t) = sin(2t) , frequency of output is 2 rad/sec.
So, frequency of input x(t) and output y(t) is not same.
Hence S3 is not definitely LTI.
For system (S4 ) :
Thus, x(t) = sint , frequency of input is 1 rad/sec.
, frequency of output is 2 rad/sec.
So, frequency of input x(t) and output y(t) is not same.
Hence S4 is not definitely LTI.
Hence, the correct option is (A) and (C).
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The outputs of four systems (S1, S2, S3 and S4 ) corresponding to the input signal sin(t) , for all time t , are shown in the figure. Based on the given information, which of the four system is/are definitely NOT LTI (linear and time-invariant)?a)S4b)S1c)S3d)S2Correct answer is option 'A,C'. Can you explain this answer?
Question Description
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