According to the book: signals and systems by HWEI P. HSU, chapter 1, A system is linear if it is:
1) additive, $T[x_1 + x_2] =y_1+y_2$
2) homogeneous, $T[cx] =cy$ , c = scalar
Question 1:
I will use the methodology described in the book: signals and systems edition 2 by Oppenheim, chapter 1, to prove why y(t)=x(t)x(t−1) is not linear. I would like to request a feedback , is my proof correct or wrong? If wrong, can you show me?If it is correct(!), is there a better way to prove it?
$x_1(t)->y_1(t) = x_1(t)\cdot x_1(t-1)\\ x_2(t)->y_2(t) = x_2(t)\cdot x_2(t-1)\\ Let \,\,\, x_3(t) = x_1(t)+ x_2(t)\\ x_3(t)->y_3(t) = x_3(t)\cdot x_3(t-1),below\,we\,rewriting\, the\,same...\\ y_3(t) = T[x_3(t)] = x_3(t)\cdot x_3(t-1),\,below\, we\, begin\,replacing...\\ y_3(t) =\overbrace{\big[x_1(t)\cdot x_1(t-1)+x_2(t)\cdot x_2(t-1)\big]}^{x_3(t)}\cdot \overbrace{\big[x_1(t-1)\cdot x_1(t-2)+x_2(t-1)\cdot x_2(t-2)\big]}^{x_3(t-1)} $
This result does not obey the additive property because of t-2, it will never be the same as: $y_3(t) = y_1(t)+ y_2(t) = x_1(t)\cdot x_1(t-1) + x_2(t)\cdot x_2(t-1)$
Question 2: After going through the theory content included in the above two books(chapters 1) i conclude that the following proof of homogeneous property is wrong: $$y(t)=T[c\cdot x(t)]=c\cdot x(t)\cdot c \cdot x(t-1)=c^2\cdot x(t)\cdot x(t-1)=c^2\cdot T[x(t)]$$
Reason: I was not able to discover any single piece of theory from the books which would clearly say that we can multiply constant c as shown above. The definition of a linear system does not even imply such a thing. I suspect that i can test the homogeneous property using the same proof as in question 1, i did not use a,b coefficients above there :
$$x_3(t) = ax_1(t)+b x_2(t)\\ y_3(t) =\overbrace{\big[ax_1(t)\cdot x_1(t-1)+bx_2(t)\cdot x_2(t-1)\big]}^{x_3(t)}\cdot \overbrace{\big[ax_1(t-1)\cdot x_1(t-2)+bx_2(t-1)\cdot x_2(t-2)\big]}^{x_3(t-1)}$$
i do not completely solve this, i suspect that if i do the distribution law and rearrange the terms it is not scalable. Am i right? If not can you show the theory and a solid proof i can (hopefully) understand?
It is not clear what your domain is.
If $x$ is a constant function, or is (a non zero) constant on some interval $[t_0,t_0+1]$, then we see that $(Tx)(t_0+1) = x(t_0)^2 = T (-x)(t_0+1)$, hence this violates the $T (\lambda x) = \lambda Tx$ requirement for linearity.