Linearity, time invariance, causality

[Quincy]

Homework Statement

For each of the following systems, determine whether or not the system is linear, time-invariant, and causal.

a) y[n] = x[n]cos(0.2*PI*n)
b) y[n] = x[n] - x[n-1]
c) y[n] = |x[n]|
d) y[n] = Ax[n] + B, where A & B are constants.

Homework Equations

The Attempt at a Solution

I know that they're all causal, because they all depend on present or past values of n, I don't know how to determine if they're linear and time-invariant, the book is terrible at explaining it.

 

[swraman]

L/TI are really important concepts for signals and systems - and they're pretty easy to get too.

To test for linearity: does scaling the input x[n] by a constant A scale the output by the same constant. in equation form:

so for the first one:

y[n] = x[n]cos(0.2*PI*n)

Scaling the input by A:

= Ax[n]cos(0.2*PI*n)
= Ay[n]

so this system is linear.

To satisfy time invariance, the output of the system should be shifted by a time T if the input is shifted by the same time T.

y[n] = x[n]cos(0.2*PI*n)

Sift the input signal by a time T

= x[n + T]cos(0.2*PI*n)
!= y[n+T]

because of the cosine's dependence on n this system is not time invariant.

Hope this helps, applying these rules to the systems are fairly simple and will tell you if a system is Linear/TI. Keep in mind that the linear scaling test must work for all (real and complex) values of A and the time sifting must work for all T in order for the system to L/TI.