If a discrete signal represented by x[n] = αn u(n), then what is the value of the signal g(n) = x(n) - αx(n - 1)?

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UPSC IES Electrical 2022 Prelims Official Paper
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  1. g(n) = δ(n)
  2. g(n) = δ(n - 1)
  3. g(n) = αδ(n - 1)
  4. g(n) = δ(n - α)

Answer (Detailed Solution Below)

Option 1 : g(n) = δ(n)
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Detailed Solution

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Calculation: 

The graph of x[n] = αn u(n),

 F3 Madhuri Engineering 31.10.2022 D13 

(α > 0)

⇒ it starts from n=0.

 The graph of αx(n - 1),

F3 Madhuri Engineering 31.10.2022 D13

⇒ it starts from n = 1.

 g(n) is obtained by subtraction of both graphs.

F3 Madhuri Engineering 31.10.2022 D14

Hence, 

g(n) = δ(n).

Alternate Method

g(n) = x(n) - αx(n - 1) where (α > 0)

⇒ g(n) = αn u(n) - α*αn-1 u(n-1)

⇒ g(n) = αn u(n) - αn u(n-1)

 g(n) = δ(n).

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