Statement I: The scaling theorem relates scale changes in frequency domain to the consequent changes in scale in the time domain. Statement II: The scaling theorem is not applicable to an impulse signal.

87) Given below are two statements 

Statement I: The scaling theorem relates scale changes in frequency domain to the consequent changes in scale in the time domain. 

Statement II: The scaling theorem is not applicable to an impulse signal. 

In light of the above statements, choose the correct answer from the options given below 

1. Both Statement I and Statement II are true [Option ID = 3177] 

2. Both Statement I and Statement II are false [Option ID = 3178] 

3. Statement I is true but Statement II is false [Option ID = 3179] 

4. Statement I is false but Statement II is true [Option ID = 3180] 

Answer: 1

Solution

Statement I: CORRECT

The Scaling Theorem (or Time-Scaling Property) states that if a signal is compressed in time, its frequency spectrum expands, and vice-versa. This is known as the reciprocal relationship between time and frequency.

If x(t) ↔ X(ω),
Then x(at) ↔ (1/|a|) * X(ω/a)

The scaling theorem is valid for impulse responses, but it is not applicable to an impulse signal. The theorem relates scale changes in the frequency domain to changes in the time domain, but since an impulse signal is present at the origin, any scaling would only affect its area, not its position. Therefore, the scaling theorem does not apply to impulse signals.

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