Properties of Systems - Signals and Systems - Electrical Engineering (EE)

Memory

Memory is a property relevant only to systems whose input and output signals share the same independent variable (for example, both are functions of time).

A system is said to be memoryless if its output at any value of the independent variable depends only on the input at that same value of the independent variable.

• Example: the system described by y(t) = 5 x(t) is memoryless because the output at time t depends only on the input at the same time t.
• Physical example: an ideal resistor is a memoryless system if we regard voltage as input and current as output (Ohm's law relates them instantaneously).
• By contrast, a system that does not satisfy this property is said to have memory.

How to identify memory

• For a memoryless system, changing the input at an instant can change the output only at that instant. If a change in the input at some instant produces a change in the output at a different instant, the system has memory.

Important remark. A system description may appear to indicate memory but, after algebraic simplification, be memoryless. For example, consider the system

Y(t) = X(t − 5) + { X(t) − X(t − 5) }

Although the expression contains terms with X(t − 5), direct simplification gives

Y(t) = X(t)

so the system is actually memoryless. This shows that the same system can have multiple algebraic descriptions; only the simplified, equivalent description should be used to judge memory.

Examples

• y(t) = x(t) is memoryless.
• y[n] = x[n − 5] has memory: the output at index n depends on the input five indices earlier.

Linearity

Linearity is a fundamental property often expressed by the principle of superposition. A system is linear if a linear combination of inputs produces the same linear combination of corresponding outputs.

Formally, if a system operator is denoted by T{·}, linearity means that for arbitrary (allowed) signals x1(t) and x2(t), and arbitrary scalars a and b,

T{a x1(t) + b x2(t)} = a T{x1(t)} + b T{x2(t)}

Linearity is meaningful even when input and output have different independent variables (for example, discrete input and continuous output), and the definition is analogous for discrete-time systems.

Passive linear circuit elements-capacitor, inductor, resistor-or linear combinations of them are linear systems when voltage is the input and current is the output, as they obey superposition within their linear operating ranges.

Additivity and Homogeneity

Linearity can be decomposed into two separate properties:

1. Additivity: For any two input signals X1(t) and X2(t), the system satisfies
   
   i.e. the response to the sum of two inputs equals the sum of the responses to the inputs.
2. Homogeneity (Scaling): For any input X(t) and scalar α, the system satisfies
   
   i.e. scaling the input scales the output by the same factor.

Both properties together are equivalent to linearity.

Proof outline that additivity and homogeneity imply linearity

Assume the system is additive and homogeneous.

For inputs x1(t) and x2(t) and scalars a, b:

By homogeneity, the system response to a x1(t) is a y1(t), and to b x2(t) is b y2(t).

By additivity, the response to their sum is a y1(t) + b y2(t).

Hence T{a x1 + b x2} = a T{x1} + b T{x2}, which is linearity.

Conversely, linearity implies additivity and homogeneity by choosing scalars appropriately in the linearity relation: setting both scalars = 1 yields additivity; setting one scalar = 0 yields homogeneity.

Additivity and Homogeneity are independent

• There exist systems that are additive but not homogeneous, and systems that are homogeneous but not additive.
• Examples illustrating these facts are given below.

This example is additive but not homogeneous for complex scaling (it might be homogeneous for real scalars but fails for complex scalars); see the specific expression in the figure for details.

The figure above shows a system that is homogeneous but not additive. From such examples one can generalise classes of systems that satisfy only one of the properties.

Worked examples

1. The system y(t) = t·x(t)is linear.

   Reason: For inputs x1(t) and x2(t) with outputs y1(t) = t x1(t) and y2(t) = t x2(t), the response to a x1 + b x2 is

   t (a x1(t) + b x2(t)) = a t x1(t) + b t x2(t) = a y1(t) + b y2(t).

2. The system y(t) = (x(t))2is not linear.

   Reason: squaring is nonlinear; the response to a sum does not equal the sum of responses, and scaling fails homogeneity.

Shift Invariance

Shift invariance (also called time-invariance) applies to systems whose input and output signals have the same independent variable. It formalises the idea that the system's characteristics do not change with shifts along the independent variable axis.

Definition: If an input x(t) produces output y(t), and for every shift t0 the shifted input x(t − t0) produces the correspondingly shifted output y(t − t0), the system is shift-invariant.

Formally, for every permissible x(t) and every t0,

If T{x(t)} = y(t) then T{x(t − t0)} = y(t − t0)

In other words, shifting the input by t0 shifts the output by the same amount for shift-invariant systems. This property need not hold for all systems.

The two inputs x(t) and x(t − t0) are different signals; a general system could map them to outputs that are not shifts of each other. If the mapping preserves shifts, the system is shift-invariant; otherwise it is shift-variant.

Examples

• Discrete-time and continuous-time versions of the property are analogous: if y[n] = T{x[n]}, then shift invariance means T{x[n − n0]} = y[n − n0] for all integer shifts n0.

Stability

Stability describes whether the output of a system remains bounded when the input is bounded. Several mathematical notions of stability exist; the most commonly used in signals and systems is BIBO stability (Bounded-Input, Bounded-Output).

Informally, a stable system does not produce unbounded outputs in response to bounded inputs. A small bounded input leads to a predictable, bounded response.

Physical intuition

• Example: an ideal mechanical spring (with elongation proportional to applied tension). If tension as a function of time is the input and elongation as a function of time is the output, the system is intuitively stable: a bounded tension produces a bounded elongation.
• Different notions of stability (Lyapunov stability, asymptotic stability, BIBO stability) are not always equivalent; the choice depends on the system description and context.

BIBO Stability (formal statement)

Definition (BIBO): A system is BIBO stable if every bounded input produces a bounded output. That is, if there exists a finite constant M_x such that |x(t)| ≤ M_x for all t (or |x[n]| ≤ M_x for discrete time), then there exists a finite constant M_y (possibly dependent on M_x) such that |y(t)| ≤ M_y for all t.

This definition applies to continuous-time, discrete-time, and hybrid systems. BIBO stability is a practical criterion used widely in signal processing and control.

Summary

• Memory: A memoryless system's output at any instant depends only on the input at that instant; systems that depend on past or future input values have memory.
• Linearity: A system is linear if it satisfies superposition. Linearity is equivalent to additivity and homogeneity together; these two properties can be independent.
• Shift invariance: A system is shift-invariant if input shifts produce identical output shifts; otherwise it is shift-variant.
• Stability: BIBO stability requires bounded inputs to produce bounded outputs; other stability notions exist for specific contexts.