Introduction to systems - Continuous and Discrete Time Signals and Systems - page 88

(iii) Increasing ramped output:

y [ k ] = kx [ k ] .

The input-output relationship is expressed as follows:
x [ k ] = 1

k y [ k ] .
The input signal can be uniquely determined for all time instant k , except at
k

= 0. Therefore, the system is not invertible.
(iv) Summer:

y [ k ] = x [ k ] + x [ k − 1] .

Following the procedure used in Example 2.8(iv), the input signal is expressed
as an infinite sum of the output y [ k ] as follows:

x [ k ] = y [ k ] − y [ k − 1] + y [ k − 2] − y [ k − 3] +−
= ∞

( − 1) m y [ k − m ] .

m = 0

The input signal x [ k ] can be reconstructed if y [ m ] is known for all m ≤ k .
Therefore, the system is invertible.
(v) Accumulator:

k

y [ k ] = x [ m ] .

m =−∞

We express the accumulator as follows:

k − 1

y [ k ] = x [ k ] + x [ m ] = x [ k ] + y [ k − 1]

m =−∞

or

x [ k ] = y [ k ] − y [ k − 1] .

Therefore, the system is invertible.

2.2.6 Stable and unstable systems

Before defining the stability criteria for a system, we define the bounded prop-
erty for a signal. A CT signal x ( t ) or a DT signal x [ k ] is said to be bounded in
magnitude if

x ( t ) ≤ B x < ∞

for t ∈ ( −∞, ∞ );

CT signal (2.48)

DT signal x [ k ] ≤ B x < ∞ for k ∈ ( −∞, ∞ ) , (2.49)

where B x is a finite number. Next, we define the stability criteria for CT and
DT systems.
A system is referred to as bounded-input, bounded-output (BIBO) stable if
an arbitrary bounded-input signal always produces a bounded-output signal. In
other words, if an input signal x ( t ) for CT systems, or x [ k ] for DT systems,