Solution 8 :
By definition ;
![](Solution_Template3_clip_image001.jpg)
Replacing x (t) by cx(t) where c is a real
number ,
![](Solution_Template3_clip_image002.jpg)
Similarly putting ,
, we get
![](Solution_Template3_clip_image004.jpg)
![](Solution_Template3_clip_image005.jpg)
![](Solution_Template3_clip_image006.jpg)
Thus ,
obeys homogeneity and additivity .
Implies
is LINEAR .
NEXT , substituting x (t) by x (t-t0)
![](Solution_Template3_clip_image007.jpg)
Now , Substituting ,
,we get
![](Solution_Template3_clip_image009.jpg)
Hence
need not be equal to
for all x (t) and t0.
Hence
is shift-variant .
Let
and let x (t) = u (t)
Now ,
![](Solution_Template3_clip_image015.jpg)
Even though x (t)=0 for t < 0 ,
is non-zero for t < 0 .
This counterexample shows
that the auto correlating
system is non-causal.
(b) The system remains linear
(homogeneous and additive) and non-causal. The system
becomes shift invariant.
![](Solution_Template3_clip_image001.jpg)
Replacing x (t) by x (t-t0) , we get
and
![](Solution_Template3_clip_image017.gif)
Hence
for all signals x (t) and all values of t0
.
Thus
is a shift-invariant system .
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