8/17/2019 dsic ch.1 part 2

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For the following systems check 1-linerarity 2- causal or non casual

3-Time variance 4-Memoryless or with memory 5-stability

غز – الجمع سع  

Islamic University - Gaza

Faculty of EngineeringDepartment of Electrical Engineering

Signals and Linear System Discussion (EELE 3110)Fall-2014

Eng. Yousef Shaban

8/17/2019 dsic ch.1 part 2

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 1. Indicate whether the following systems are causal, linear, memor yless, and/ or   time 

invar iant  by circling the corr ect  o ptions.  (A system  may have mor e than one of

these  pr o per ties.)  Justif y your answer . 

(a) y(t) = x(t− 2)+x(2− t)

Causality:  The system  is NOT causal because for any t < 1, the out put

depends on a f utur e input, e.g. y(0) = x(−

2) + x(2). 

Linear ity:  The system  is linear because if

y1(t)  =  x1(t−  2) + x1 (2 −  t), 

y2(t)  =  x2(t−  2) + x2 (2 −  t),  and 

x(t)  =  αx1(t) + βx2 (t), 

then the output y(t) corresponding to the input x(t) is

y(t) =  x(t −  2) + x(2 −  t) 

=  (αx1+ βx2 )(t−  

2) + (αx1+ βx2 )(2−  

t) =  αx1(t −  2) + βx2 (t−  2) + αx1 (2 −  t) + βx2(2 −  t) 

=  α (x1 (t−  2) + x1(2 −  t)) + β (x2 (t−  2) + x2 (2 −  t)) 

=  αy1(t) + βy2(t). 

Memorylessness: The system is NOT memoryless because the output at

time t depends on input values at times other  than t. 

Time Invariance: The system  is time  invar iant.  To see this,  we let  y(t) bethe output corresponding to the input x(t) and let xa (t) = x(t −  a)

1 . Then

the output ya(t) corresponding to the input signal xa (t) is

ya(t) =  xa (t−  2) + xa (2 −  t) 

=  x((t −  a) −  2) + x(2 −  (t−  a)) 

=  y(t −  a). 

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 (b)

Causality: The system is NOT causal because for any t < 2, the out putdepends on a f utur e  input. 

Linear ity:  The system is linear  

Memorylessness: The system  is NOT memoryless because the output at

time t depends on input values at times other   than t. 

Time Invariance: The system is NOT time invar iant.  To see this, we let 

x(t) = u(t)−  u(t −  1) so that

Constant for all t, so in particular, y(t-3)=1 for any value of t. However, if we let

So , system is time varying.

2

)()(   dt t  xt v  

1)()(

)1()()(

1

0

  dt dt t  xt  y

t ut ut  x

 

2

10)(3

)4()3()3()(3

t  y

t ut ut  xt  y

 

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(c) y(t) = (dx/ dt)(t) Causality:  The system  is causal. 

Linear ity:  The system  is linear because if

y1(t) = (dx1/ dt)(t),  and

y2(t) = (dx2/ dt)(t), 

and x(t) = αx1 (t) + βx2(t), then the output y(t) corresponding to the

input x(t) is

y(t) =  (d(αx1+ βx2)/ dt)(t) 

=  α(dx1 / dt)(t) + β(dx2/ dt)(t) 

=  αy1(t) + βy2(t). 

Memorylessness: The system  is memoylessTime Invariance: The system  is time  invar iant.  To see this,  we let  y(t) be

the output corresponding to the input x(t) and let xa (t) = x(t−  a). Thenthe output ya(t) corresponding to the input signal xa (t) is

ya(t) = (dxa/ dt)(t) = (d(x)/ dt)(t −  a) = y(t−  a).

(d) y(t) = x(t/ 3) 

Causality:  The system  is NOT causal because if t < 0, then the out put depends on f utur e  values of the input, e.g. for t = 3 we have y(− 3) = 

x(−

1). Linear ity:  The system  is linear because if

y1(t) = x1(t/ 3),  and

y2(t) = x2 (t/ 3) 

and x(t) = αx1 (t) + βx2(t), then the output y(t) corresponding to the

input x(t) is

y(t) =  (αx1+ βx2)(t/ 3) 

=  αx1 (t/ 3) + βx2(t/ 3) 

=  αy1(t) + βy2(t) 

Memorylessness: The system  is NOT memoryless because the output at

time t depends on input values at times other  than t. 

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Time Invariance: The system  is time  invar iant.  To see this,  we let  y(t) bethe output corresponding to the input x(t) and let xa (t) = x(t−  a). Then

the output ya(t) corresponding to the input signal xa (t) is

ya (t) = xa (t/ 3) = x((t−  a)/ 3) = y(t−  a)

(e) y(t) = cos(x(t)) Causality:  The system  is memoryless, hence causal. 

Linear ity:  The system  is NOT linear because if

y1(t)  = cos(x1(t)), and 

y2(t)  = cos(x2(t)), 

and x(t) = αx1 (t) + βx2(t), then the output y(t) corresponding to the

input x(t) is

y(t) =  cos(αx1 (t) + βx2(t)) 

=  cos(αx1 (t)) cos(βx2(t))−  sin(αx1(t)) sin(βx2(t)) 

=  α cos(x1 (t)) + β cos(x2 (t)) 

Memorylessness: The system  is memoryless because the output at time  t 

depends on input values at only time t. 

Time Invariance: The system  is time  invar iant.  To see this,  we let  y(t) bethe output corresponding to the input x(t) and let xa (t) = x(t−  a). Thenthe output ya(t) corresponding to the input signal xa (t) is

ya(t) = cos(xa (t)) = cos(x(t−  a)) = y(t−  a).