Linearity, Time Invariance, Causality, ETC.

[dashkin111]

Homework Statement

Is the following input/output (x is input, y is output) system linear, time invariant, causal, and memoryless? Answer yes or no for each one.

Homework Equations

$$y(t)=2x(t)+3$$

The Attempt at a Solution

My instinct tells me it's linear, but for some reason I have trouble showing it mathematically.It is linear if when you add scaled values of the input x(t), it equals the sum of the same scaled outputs. But it doesn't work out to be linear if I go by that definition

 

[wildman]

LOL. Of course you can't prove it. It’s a trick question. The word linear is used differently in polynomial algebra and in signal processing.

In algebra a linear equation in one that is of the form y(t) = ax(t)+b.
In signal processing it is one that satisfies homogeneity and additivity. You are obviously talking signal processing (I recognize the language) so forget what you have learned about in algebra and apply the two tests up above. Clue: does cy(t) c(ax(t) +b) = acx(t) + b?

 

[wildman]

opps, I mean cy(t) = c(ax(t) +b) = acx(t) + b? Are they equal (of course not!)