GATE 2024 EE Q.53: solution and answer

GATE 2024 · EEQ.53 · 2 marks · NAT

If the energy of a continuous-time signal $x (t)$ is $E$ and the energy of the signal $2 x (2 t - 1)$ is $c E$ , then $c$ is ______ (rounded off to 1 decimal place).

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Answer and solution

ANSWER

$c = 2.0$ . The energy of $2 x (2 t - 1)$ is $2 E$ .

Official answer: 2.0Source: Official answer key, GATE 2024 EE (IISc Bengaluru)

Energy definition: $E = \int_{- \infty}^{\infty} ∣ x (t) ∣^{2} d t$ . Let $y (t) = 2 x (2 t - 1)$ .
Amplitude scaling: $∣ y (t) ∣^{2} = 4 ∣ x (2 t - 1) ∣^{2}$ , so the energy picks up a factor $2^{2} = 4$ .
$E_{y} = 4 \int_{- \infty}^{\infty} ∣ x (2 t - 1) ∣^{2} d t$
Substitute $τ = 2 t - 1$ , so $d t = \frac{d τ}{2}$ and the limits stay $- \infty$ to $\infty$ (the scale factor $2$ is positive): $E_{y} = 4 \cdot \frac{1}{2} \int_{- \infty}^{\infty} ∣ x (τ) ∣^{2} d τ = 2 E$
Check with the general rule: for $y (t) = A x (a t - b)$ , the shift $b$ leaves energy unchanged and the time scaling by $a$ divides it by $∣ a ∣$ , so $E_{y} = \frac{A ^{2}}{∣ a ∣} E$ . Here $A = 2$ , $a = 2$ : $\frac{4}{2} = 2$ .
So $c = 2.0$ .

KEY IDEA

For $y (t) = A x (a t - b)$ , energy scales as $E_{y} = \frac{A ^{2}}{∣ a ∣} E_{x}$ . Amplitude enters squared. Compressing time by $∣ a ∣$ divides the energy by $∣ a ∣$ , and a time shift changes nothing.

WATCH OUT FOR

Students often multiply by $∣ a ∣$ instead of dividing: compressing the signal makes it shorter, so it holds less energy. That mistake gives $4 \times 2 = 8$ . Others forget to square the amplitude and get $2/2 = 1$ , or think the shift of $- 1$ changes the energy.

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GATE 2024 EE Q.53: solution and answer

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