GATE 2024 EC Q.56: solution and answer

GATE 2024 · ECQ.56 · 2 marks · MSQ

The radian frequency value(s) for which the discrete time sinusoidal signal $x [n] = A cos (Ω n + π /3)$ has a period of 40 is/are ______.

(A)
$0.15 π$
(B)
$0.225 π$
(C)
$0.3 π$
(D)
$0.45 π$

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Answer and solution

ANSWER

The frequencies with fundamental period 40 are (A) $0.15 π$ and (D) $0.45 π$ . The official key accepts A, D and also A, C, D, because $0.3 π$ has fundamental period 20, so 40 is also a period of that signal.

Official answer: A, D or A, C, DSource: Official answer key, GATE 2024 EC (IISc Bengaluru)

Periodicity condition (DT sinusoid): $cos (Ω n + ϕ)$ repeats after $N$ samples only if $Ω N = 2 π k$ for some integer $k$ . The phase $π /3$ has no effect on the period.
Fundamental period: write $\frac{Ω}{2 π} = \frac{k}{N}$ in lowest terms, so that $g cd (k, N) = 1$ . The denominator $N$ is then the fundamental (smallest) period.
(A) $\frac{0.15 π}{2 π} = 0.075 = \frac{3}{40}$ . This is already in lowest terms, so $N = 40$ . Correct.
(B) $\frac{0.225 π}{2 π} = 0.1125 = \frac{9}{80}$ , so $N = 80$ . Not 40.
(C) $\frac{0.3 π}{2 π} = 0.15 = \frac{6}{40} = \frac{3}{20}$ , so $N = 20$ . The fundamental period is 20, not 40. The signal does also repeat after 40 samples, because 40 is a multiple of 20. That is why the official key also accepts A, C, D, for candidates who read "a period of 40" as any period rather than the fundamental one.
(D) $\frac{0.45 π}{2 π} = 0.225 = \frac{9}{40}$ . This is already in lowest terms, so $N = 40$ . Correct.
Check: for (A), $0.15 π \times 40 = 6 π = 2 π \cdot 3$ . For (D), $0.45 π \times 40 = 18 π = 2 π \cdot 9$ . In both cases no smaller $N$ works, because $3$ and $9$ share no common factor with $40$ . Taking "period" to mean the fundamental period, the answer is (A) and (D).

KEY IDEA

A discrete-time sinusoid $cos (Ω n + ϕ)$ is periodic only when $Ω/2 π$ is rational. Its fundamental period is the denominator $N$ of $Ω/2 π = k / N$ once the fraction is reduced to lowest terms. The phase does not affect the period.

WATCH OUT FOR

One mistake is using the continuous-time formula $N = 2 π /Ω$ . For (A) that gives $13.33$ , and for (D) it gives $4.44$ , so both correct options get rejected as \"not 40\" or even as aperiodic. The other mistake is leaving $Ω/2 π$ unreduced: $0.3 π$ gives $6/40$ , which looks like period 40, but it reduces to $3/20$ , so the fundamental period is 20.

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GATE 2024 EC Q.56: solution and answer

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