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- The following CT systems are described using their input-output relationships between the input signal x ( t ) and output y ( t ). Determine if the CT systems are: (i) linear; (ii) time-invariant; (iii) stable; and (iv) causal. i) y t ( ) x t ( 2)
ii) (^) y t ( ) tx t ( 10)
iii) (^) ( ) 0 t< ( ) ( 5) t 0
y t x t x t
° ® °¯ ^ ^ t
iv) y ( t ) x ( ) d 2 x ( t )
o
o
t
t
³ O O^
v) 3 5 3 () 2 2 () 1
2 2
2 3
3 4
4 xt dt
d x yt dt
dy dt
d y dt
d y dt
d y
- The following DT systems are described using their input-output relationships between the input signal x [ k ] and output y [ k ]. Determine if the DT systems are: (i) linear; (ii) time-invariant; (iii) stable; and (iv) causal. i) y [ k ] 0. 5 x [ 2 6 k ] 0. 5 x [ 2 6 k ]
ii) [ ] [ ] 2 | []|
2
2
y k xm x k
k
m k
¦^
iii) y [ k ] 2 x [ k ] iv) y [ k ] 5 y [ k 1 ] 9 y [ k 2 ] 5 y [ k 3 ] y [ k 4 ] 2 x [ k ] 4 x [ k 1 ] 2 x [ k 2 ]
- For a CT linear, time invariant (LTI) system, an input x ( t ) produces output y ( t ) shown in Fig. P2.6. Sketch the outputs for the following set of inputs. i) 5 x ( t ) ii) 0. 5 x ( t 1 ) 0. 5 x ( t 1 ) iii) x ( t 1 ) x ( t 1 )
iv) 3 ()
xt dt
dx t
y ( t )
t
y ( t )
t
Figure P3: CT output y ( t ) for problem 3.
- For a DT linear, time invariant (LTI) system, an input x [ k ] produces output y [ k ] shown in Fig. P2.7. Sketch the outputs for the following set of inputs.
i) 4 x [ k − 1 ] ii) 0. 5 x [ k − 2 ]+ 0. 5 x [ k + 2 ] iii) x [ k + 1 ]− 2 x [ k ]+ x [ k − 1 ] iv) x [ − k ] y [ k ]
k
y [ k ]
k
Figure P4: DT output y [ k ] for problem 4.
- Determine if the following CT systems are invertible. If yes, find the inverse systems.
i) y ( t )= 3 x ( t + 2 )
ii) =∫ τ−
t yt x tdt 0
iii) y ( t )= x ( t ) iv) y ( t )=cos( 2 π t )
- Determine if the following DT systems are invertible. If yes, find the inverse systems.
i) y [ k ]= ( k + 1 ) x [ k + 2 ]
ii) ∑
=
k
m
yk xm 0
[ ] [ 2 ]
iii) y [ k ]= x [ k + 2 ]+ 2 x [ k + 1 ]− 6 x [ k ]+ 2 x [ k − 1 ]+ x [ k − 2 ] iv) y [ k ]+ 2 y [ k − 1 ]+ y [ k − 1 ]= x [ k ]
- For each of the following differential equations modeling an LTIC system, determine the zero-input response, zero-output response and overall response of the systems for the specified input x ( t ) and initial conditions. What is the steady state response of the LTIC system?
i) y t &&( ) + 4 y t & ( ) + 8 y t ( ) = x t & ( ) + x t ( ) with x t ( ) = e −^4 tu t ( ), y (0) = 0, and y &(0) =0. ii) (^) y t &&( ) (^) + 6 y t & (^) ( ) + 4 y t ( ) = x t & (^) ( ) + x t ( ) with x t ( ) = cos(6 ) ( ), t u t y (0) = 2, and y &(0) =0. iii) y t &&( ) + 4 y t ( ) = 5 ( ) x t with x t ( ) = 4 te − tu t ( ), y (0) = − 2, and y &(0) =0. iv) &&&& y t ( )^ + 2 && y t ( )^ + y t ( ) = x t ( ) with x t ( ) = 2 ( ), u t y (0) = 0, and y &(0) =1.
- Determine the output^ y ( t ) for the following pairs of input signals^ x ( t ) and impulse responses^ h ( t ). i) x ( t )= u ( t ) h ( t )= u (− t ) ii) (^) x ( t )= u ( t )− 2 u ( t − 1 )+ u ( t − 2 ) h ( t )= u ( t + 1 )− u ( t − 1 ) iii) x ( t )= e^2 t^ u (− t ) h ( t )= e −^3 tu ( t )
iv) x ( t )= sin( 2 π t )( u ( t − 2 )− u ( t − 5 )) h ( t )= u ( t )− u ( t − 2 )
- Determine whether the LTIC systems characterized by the following impulse responses are memoryless, causal and stable. Justify your answer.
