GATE 2018 IN Q.34: solution and answer

GATE 2018 · INQ.34 · 2 marks · MCQ

Let $y [n] = x [n] * h [n]$ , where $*$ denotes convolution and $x [n]$ and $h [n]$ are two discrete time sequences. Given that the z-transform of $y [n]$ is $Y (z) = 2 + 3 z^{- 1} + z^{- 2}$ , the z-transform of $p [n] = x [n] * h [n - 2]$ is

(A)
$2 + 3 z + z^{- 2}$
(B)
$3 z + z^{- 2}$
(C)
$2 z^{2} + 3 z + 1$
(D)
$2 z^{- 2} + 3 z^{- 3} + z^{- 4}$

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Answer and solution

ANSWER

$P (z) = z^{- 2} Y (z) = 2 z^{- 2} + 3 z^{- 3} + z^{- 4}$ , which is option (D).

Official answer: DSource: Official answer key, GATE 2018 IN (IIT Guwahati)

Convolution property of the z-transform: $y [n] = x [n] * h [n] \Rightarrow Y (z) = X (z) H (z) = 2 + 3 z^{- 1} + z^{- 2}$ .
Time-shifting property: $h [n - 2] ⟷ z^{- 2} H (z)$ .
Apply the convolution property to $p [n] = x [n] * h [n - 2]$ : $P (z) = X (z) [z^{- 2} H (z)] = z^{- 2} X (z) H (z) = z^{- 2} Y (z)$
Substitute $Y (z)$ : $P (z) = z^{- 2} (2 + 3 z^{- 1} + z^{- 2}) = 2 z^{- 2} + 3 z^{- 3} + z^{- 4}$
Time-domain check: $Y (z)$ is a finite polynomial in $z^{- 1}$ , so $y [n] = 2 δ [n] + 3 δ [n - 1] + δ [n - 2]$ , i.e. ${2, 3, 1}$ for $n = 0, 1, 2$ . Delaying one operand of a convolution delays the result by the same amount, so $p [n] = y [n - 2] = {2, 3, 1}$ for $n = 2, 3, 4$ , whose z-transform is $2 z^{- 2} + 3 z^{- 3} + z^{- 4}$ . This is option (D).

KEY IDEA

Convolution in time becomes multiplication in the z-domain, and a delay of $k$ samples multiplies by $z^{- k}$ . Delaying either operand of a convolution by 2 samples delays the output by 2 samples: $x [n] * h [n - 2] = y [n - 2]$ , so $P (z) = z^{- 2} Y (z)$ .

WATCH OUT FOR

Using $z^{+ 2}$ for the delay $h [n - 2]$ , which gives $z^{2} Y (z) = 2 z^{2} + 3 z + 1$ (option C). A delay (shift right) multiplies by $z^{- 2}$ , not $z^{2}$ . Another slip is to shift only part of the expression, which produces mixed-power options such as (A) or (B).

From the GATE 2018 IN paper. Every question of this paper · Download the full paper · All 12 z-transform questions, with notes

GATE 2018 IN Q.34: solution and answer

Answer and solution

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