Introduction to systems - Continuous and Discrete Time Signals and Systems - page 97

2.10 The following DT systems are described using their input-output relation-
ships between input x [ k ] and output y [ k ]. Determine if the DT systems are
(a) linear, (b) time-invariant, (c) stable, and (d) causal. For the non-linear
systems, determine if they are incrementally linear systems.

y ( t )

1

(i) y [ k ] = ax [ k ] + b ;
(ii) y [ k ] = 5 x [3 k − 2];
(iii) y [ k ] = 2 x [ k ] ;
(iv) y [ k ] = k

t

−1 1

Fig. P2.11. CT output y ( t ) for
Problem 2.11.

x [ m ];

m =−∞

k + 2

(v) y [ k ] = x [ m ] − 2 x [ k ] ;

m = k − 2

(vi) 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] .
(vii) y [ k ] = 0 . 5 x [6 k − 2] + 0 . 5 x [6 k + 2] .

2.11 For an LTIC system, an input x ( t ) produces an output y ( t ) as shown in
Fig. P2.11. 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)

d x ( t )
d t

+ 3 x ( t ) .

y [ k ]

2.12 For a DT linear, time-invariant system, an input x [ k ] produces an output
y [ k ] as shown in Fig. P2.12. 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];

4

2

(iii) x [ k + 1] − 2 x [ k ] + x [ k − 1];
(iv) x [ − k ] .

−1

k

−2

1 2

−2

2.13 Determine if the following CT systems are invertible. If yes, find the
inverse systems.
(i) y ( t ) = 3 x ( t + 2);

Fig. P2.12. DT output y [ k ] for
Problem 2.12.

d y ( t )
d t

(iv) + y ( t ) = x ( t );

t

(v) y ( t ) = cos(2 π x ( t )) .

(ii) y ( t ) = x ( τ − 10)d τ ;

−∞

(iii) y ( t ) = x ( t ) ;

2.14 Determine if the following DT systems are invertible. If yes, find the
inverse systems.
(i) y [ k ] = ( k + 1) x [ k + 2];

k

(ii) y [ k ] = x [ m + 2];

m = 0

∞

(iii) y [ k ] = x [ k ] δ [ k − 2 m ];

m =−∞

(iv) y [ k ] = x [ k + 2] + 2 x [ k + 1] − 6 x [ k ] + 2 x [ k − 1] + x [ k − 2];
(v) y [ k ] + 2 y [ k − 1] + y [ k − 2] = x [ k ] .

d x ( t )
d t

d y ( t )
d t

2.15 For an LTIC system, if x ( t ) → y ( t ), show that → . Assume

that both x ( t ) and y ( t ) are differentiable functions.