Question

How can I determine whether a digital/analog signal system is linear, time invariant/variant, memoryless, causal, invertible, and stable? I am still a little bit confused after reading lecture notes on how to figure out the attributes of a signal system.

Answer 1

Solution

Linearity:

If the signal satisfies superposition.i.e., both additive and homogeneity property ,then it is said to be linear.

Additive property- let y(t)=x(t)

y1(t)=x1(t)

y2(t)=x2(t)

Hence[ y1+y2](t) = x1(t) + x2(t) = y1(t)+ y2(t)

Homogeneity property- for any arbitrary constant A

Ay(t)= Ax(t)

Time invariance

If input delay is same as output delay.i.e, no dependence on time.

y(t-a) = x(t-a)

Memoryless

If the signal corresponds only to the present time instant ,it is said to be memoryless. Any delay or advance in time is a memory based signal

Example:

y(t)= x(t) is memoryless

y(t)=x(t-a) has memory.

Causality

If signal has no value in the time advance form or a right sided signal is causal.i.e,

y(t)={0, t<0}

Invertibility

A signal is said to be invertible only if it has distinct outputs for distinct inputs.

Example :

Differentiation is non invertible as it gives zero for every constant input. But integration is invertible.

Stability

The signal is stable only if it is bounded or bounded input gives finite and bounded output.

Example

Ramp function - y(t)= t is not bounded hence not stable