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From the impulse response of a system can we decide whether it is time variant or not? also can we say anything about its linearity?

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Yes we can decide. you have to perform two test. ifY=H(x); and if x=x1+x2 then y= y1+y2 and x= a*x1 then y= a*y1, that is you have to check additivity and , homogenity

Yes