Question

2. (10 pts.) Consider a system with input c() and output y(t) = x(2t) - 2(t-2). Determine whether this system is time-invariant and/or linear. Justify your answers.

Answer 1

Time in variant System → If the system response remain same for comsindered input isrespective of time of the application of the input. It as time invariant System. والله يه as (2 t) - (-2) - Yt) the given System perform time scaling by and time delay by '? input. to the -> ct) then Now, if the input iů yet)= x(2+) – X(t-2) delay by 26(2t-to) - Oct-to-2-0 by to" S time delay Sot - to) to-ta time scaled to xft-to) yt) output ylt) it 2 = 7+)- x/t-2) Replace "t" with toto" in Y-to) 3124-to)) ellt-to)-2) yt-to) X (2+-2to) x(t-to-2) - » from equation 0 & ① yet-to) + x[t-to); delay by "to" which mean in the input does not result in same dolay in output. Herre System is not Time invariant

yet) X (4) c/t -) • is Said to linear, if it follow both lancia Principle of Homogenity System Super position Homogeneity principle S 2 output Y(t) a xect's } input yet) yet) ca → Subor position principle S. (t) yolt) S Xt yat) gifts & Yalts Hi(t) + X(t) in the given example or Question, System perform scaling time scali & time delay.

s a X(t) x) y(t) 3062t) 24-2) ý input is multiplied with some constant a 2 (2 t) a Y't-2) = a [x24)– 24-] a yet) Sapto principle follow Hence Homogenicty s 4,4) Gift) = 5,2t) – 584-2) s X2€) yalt) Sc(2+) X₂ (+-2) 3, @+x, 6) $x, 2+ + 2,H) - (x-1)+Y,-)) [ x,(24) – 2,4-2) + **) = x2 4-2] Yot) + y2lt) Hence Superposition principle follow - System follow both principle of Homogeniety and Superposition. Hence the give system is a linear System