ISRO EC 2023 Q.21: solution and answer

ISRO 2023 · ECQ.21 · 1 mark · MCQ

Which of the following is NOT TRUE, given that "→" represents fourier transform, "*" represents convolution, $x (n) \to X (e^{j ω})$ and $y (n) \to Y (e^{j ω})$ ?

(A)
$a x (n) + b y (n) \to a X (e^{j ω}) + bY (e^{j ω})$
(B)
$x (n - n_{d}) \to e^{j ω n_{d}} (e^{j ω})$ (the printed option has no X before the bracket)
(C)
$e^{j ω_{0} n} x (n) \to X (e^{j (ω - ω_{0})})$ (printed with the letter w for the shift frequency)
(D)
$x (n) * y (n) \to X (e^{j ω}) . Y (e^{j ω})$

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Answer and solution

ANSWER

Option (B) is not true: a delay of $n_{d}$ samples multiplies the spectrum by $e^{- j ω n_{d}}$ , not $e^{+ j ω n_{d}}$ .

Official answer: B or CSource: ISRO's revised and final answer key of 16 February 2024

Write the DTFT as $X (e^{j ω}) = \sum_{n = - \infty}^{\infty} x (n) e^{- j ω n}$ and test each statement against it.
(A) Linearity: the sum is linear in $x (n)$ , so $a x (n) + b y (n) \to a X (e^{j ω}) + bY (e^{j ω})$ . This is true.
(B) Time shift: $\sum_{n} x (n - n_{d}) e^{- j ω n}$ with $m = n - n_{d}$ gives $e^{- j ω n_{d}} \sum_{m} x (m) e^{- j ω m} = e^{- j ω n_{d}} X (e^{j ω})$ . The exponent must be $- j ω n_{d}$ , but the option has $+ j ω n_{d}$ , and as printed it does not even contain the factor $X$ . So this statement is false.
(C) Frequency shift: $\sum_{n} e^{j ω_{0} n} x (n) e^{- j ω n} = \sum_{n} x (n) e^{- j (ω - ω_{0}) n} = X (e^{j (ω - ω_{0})})$ . This is true.
(D) Convolution in time becomes multiplication in frequency: $x (n) * y (n) \to X (e^{j ω}) Y (e^{j ω})$ . This is true.
The statement that is not true is (B).

KEY IDEA

A delay by $n_{d}$ in time multiplies the DTFT by $e^{- j ω n_{d}}$ (a linear phase), while multiplying by $e^{j ω_{0} n}$ in time shifts the spectrum to $X (e^{j (ω - ω_{0})})$ .

WATCH OUT FOR

Mixing up the signs of the time-shift and frequency-shift properties: the delay gives $e^{- j ω n_{d}}$ , and a positive modulation $e^{j ω_{0} n}$ moves the spectrum to $ω - ω_{0}$ .

From the ISRO EC 2023 paper. Every question of this paper · The full paper (PDF, ISRO) · All 3 ISRO DTFT & DFT questions, with notes