Properties of systems and interconnections of operations
In mathematical terms, a system can be viewed as an interconnection of operations that transforms an input signal into an output signal with properties different from those of the input signal. The signal may be continuous-time or discrete-time in nature.
The application of continuous-time signal be x(t) to the input of the system yields the output signal:
y(t)=H{x(t)}
Correspondingly, for the discrete-time case,
y[n]=H{x[n]},
Where the discrete-time signal x[n] and y[n] denote the input and output signals respectively.
Properties of systems
Some of the basic properties of a system are:
1. Stability
A system is bounded-input, bounded-output (BIBO) stable if and only if every bounded input results in a bounded output. The output in a BIBO system does not diverge if the input does not diverge.
To put the condition for BIBO stability formally, consider a continuous-time system whose input-output relation can be represented as:
|y(t)|<=My < infinity, for all t.
Whenever the input signal x(t) satisfies the condition:
|x(t)|<=Mx < infinity, for all t.
My and Mx represent finite positive numbers.
2. Memory
A system is said to possess memory if its output signal depends on past or future values of the input signal. The temporal extent of past or future values on which the output depends defines how far the memory of the system extends into the past or the future.
On the other hand, a system is said to be memoryless if its output signal depends only on the present value of the input signal.
For example, a resistor is memoryless as the current i(t) flowing through it in response to the applied voltage v(t) is defined by i(t)=1\Rv(t).
Here, R is the resistance of the resistor. In contrast, an inductor has memory, since the current i(t) flowing through it is related to the applied voltage v(t).
3. Invertibility
A system is said to be invertible if the input of the system can be recovered from the output. We may view the set of operations needed to recover the input because a second system is connected in cascade with the given system. For example, the output signal of the second system is equal to the input signal applied to the given system. On a formal basis,
let H represent the continuous-time system, with the input signal x(t) producing the output signal y(t).
y(t) can be applied to a second continuous-time signal represented by the operator H.
The output signal is defined by:
Hinv{y(t)}=Hinv{H{x(t)}}
HinvH=I.
Where I denote the identity operator. The output of the system can be described by the identity operator as exactly equal to the input.
The equation, HinvH=I expresses the condition the new operator Hinv must satisfy in relation to the given operator H in order for the original input signal x(t) to be recovered from y(t).
Hinv is called an inverse operator, and the associated system is called an inverse system.
Generally, the problem of finding the inverse of a given system is difficult. In any event, a system is not invertible unless distinct inputs applied to the system produce a distinct output. Meaning, there must be one to one mapping between the input and the output signals for a system to be invertible.
Identical conditions must be there for a discrete-time system to be invertible.
4. Linearity
A system is said to be linear in terms of the system input (excitation) x(t) and the system output i.e. response y(t) if it satisfies the following two properties of superposition and homogeneity.
- Superposition
Let us assume a system that is initially at rest. And the system is subjected to an input x(t)=x1(t), producing an output y(t)=y1(t). Suppose next that the same system is subjected to a different input x(t)=X2(t)
- Homogeneity
Let us assume a system that is initially at rest, and suppose an input x(t) results in output y(t).
Then the system is said to exhibit the property of homogeneity.
Though, whenever the input x(t) is scaled by a constant factor a, the output y(t) is scaled by the same constant factor a.
When a system violates either the principle of superposition or the property of homogeneity, the system is said to be non-linear.
Let us assume an operator H, the system operator H must commute with the summation and amplitude scaling. The communication can be only justified if the operator H is linear.
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