Question

Question : 15 -Signals:

\(y(t) = Real[x(t)]\)

Check this system for Linearity and for Time-Invariance...

Answer 1

a system is saying to be time invariant if/
for input
x(t) mapped to y(t)
then
x(t-k) is mapped to y(t-k) such a response means it is time invariant

Answer 2

Linearating
L(c1x1+c2x2)=c1L(x1)+c2L(x2)
it is called a linear transformation when
the response of 2 inputs together is the sum of the respond of each input applied separetely

Answer 3

L(c1x)=L(x+x+...c1 times).... so the definition above also covers constant multiple case

Answer 4

2 or more* inputs*

Answer 5

You are simply telling the definition or theoretical thing, but I have difficulty in solving it.. :)

Answer 6

Linearating is not a word.
maybe Linearizing

Answer 7

where i come from we call that linearting

Answer 8

@Miracrown

Answer 9

Suppose \(\large x(t) = e^{j \omega t}\) a complex signal:

Then :

\[x(t) \rightarrow y(t) \\x(t) \rightarrow \Re[x(t)] = \cos( \omega t)\]

Here, if I check homogeneity, then, let us say I multiply with \(j\),

\[j \cdot x(t) \rightarrow \Re[j \cdot x(t)] = -\sin( \omega t) \ne j \cdot y(t)\]

Answer 10

If I multiply by j in this case, the real part will become Imaginary and Imaginary can become Real, and that changes the output.. May be, this is the way to prove it Non-Linear.. :(

Answer 11

@miracrown ..