GATE 2024 IN Q.53: solution and answer

GATE 2024 · INQ.53 · 2 marks · NAT

The 4-point DFTs of two sequences $x [n]$ and $y [n]$ are $X [k] = [1, - j, 1, j]$ and $Y [k] = [1, 3 j, 1, - 3 j]$ , respectively. Assuming $z [n]$ represents the 4-point circular convolution of $x [n]$ and $y [n]$ , the value of $z [0]$ is ________ (rounded off to nearest integer).

Note: The DFT of a $N$ -point sequence $x [n]$ is defined as
$X [k] = n = 0 \sum N - 1 x [n] e^{\frac{- j 2 π nk}{N}}$

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Answer and solution

ANSWER

$z [0] = 2$

Official answer: 2Source: Official answer key, GATE 2024 IN (IISc Bengaluru)

Circular convolution property of the DFT: a 4-point circular convolution in $n$ becomes a point-by-point product in $k$ , so $Z [k] = X [k] Y [k]$ for $k = 0, 1, 2, 3$ .
Multiply term by term: $Z [0] Z [1] Z [2] Z [3] = (1) (1) = 1 = (- j) (3 j) = - 3 j^{2} = 3 = (1) (1) = 1 = (j) (- 3 j) = - 3 j^{2} = 3$ so $Z [k] = [1, 3, 1, 3]$ .
Inverse DFT: the forward DFT given in the question has no scale factor, so the $\frac{1}{N}$ goes on the inverse: $z [n] = \frac{1}{4} \sum_{k = 0}^{3} Z [k] e^{\frac{j 2 π nk}{4}}$ .
At $n = 0$ every exponential equals 1, so $z [0]$ is the average of the DFT samples: $z [0] = \frac{1}{4} (1 + 3 + 1 + 3) = \frac{8}{4} = 2$
Check: $Z [k]$ is real and $Z [1] = Z [3]$ , which is what a real, circularly even $z [n]$ produces, so the result is consistent. $z [0] = 2$ .

KEY IDEA

A circular convolution of length $N$ becomes a product of the $N$ -point DFTs, $Z [k] = X [k] Y [k]$ . Setting $n = 0$ in the inverse DFT turns every exponential into 1, so $z [0] = \frac{1}{N} \sum_{k} Z [k]$ . You never have to recover $x [n]$ or $y [n]$ .

WATCH OUT FOR

Leaving out the $\frac{1}{N}$ in the inverse DFT gives $z [0] = 8$ . Because the forward DFT given here has no scale factor, the whole $\frac{1}{4}$ belongs to the inverse. The other slip is a sign error in $j \cdot j$ : $(- j) (3 j) = - 3 j^{2} = + 3$ , not $- 3$ . With that error $Z [1]$ and $Z [3]$ come out as $- 3$ , and the sum gives $z [0] = - 1$ .

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GATE 2024 IN Q.53: solution and answer

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