Question

Which among the below assertions is precise in accordance to the effect of time scaling? A: Inverse relationship exists between the time and frequency domain representation of signal B: A signal must be necessarily limited in time as well as frequency domains

• A. \( \text{A is true \& B is false} \)
• B. \( \text{A is false \& B is true} \)
• C. \( \text{Both A \& B are true} \)
• D. \( \text{Both A \& B are false} \)

Hint:

The time-scaling property of the Fourier Transform is a fundamental concept in signal processing. Remember that compression in one domain leads to expansion in the other, and vice versa. This is a direct consequence of the mathematical relationship between the time and frequency representations. Also, be aware of the implications of the uncertainty principle and the Paley-Wiener theorem, which state that a non-zero signal cannot be simultaneously time-limited and band-limited. This understanding is crucial for analyzing signal properties and limitations in various communication and processing systems.

Answer 1

The Correct Option is A

To solve this problem, we need to analyze the assertions related to the effect of time scaling on a signal and determine their correctness based on signal processing concepts.

1. Understanding Time Scaling:

- Time Scaling: Time scaling refers to stretching or compressing a signal in time. For a signal \( x(t) \), a time-scaled signal \( x(at) \) is obtained by multiplying the time variable by a constant \( a \). If \( |a| > 1 \), the signal is compressed, and if \( |a| < 1 \), the signal is stretched.

- Effect of Time Scaling on the Frequency Domain: The Fourier transform of \( x(at) \), where \( a \) is a scaling factor, results in a frequency-domain representation that is inversely proportional to \( a \). That is, the time-domain scaling factor \( a \) results in a frequency-domain scaling of \( \frac{1}{|a|} \).

2. Analyzing the Assertions:

- Assertion A: "Inverse relationship exists between the time and frequency domain representation of a signal." This is true. The scaling of a signal in the time domain by \( a \) leads to a corresponding inverse scaling in the frequency domain by \( \frac{1}{|a|} \). If the time domain is compressed (i.e., \( |a| > 1 \)), the frequency domain is expanded (i.e., the frequency components are spread out more), and vice versa.

- Assertion B: "A signal must be necessarily limited in time as well as frequency domains." This is false. A signal can be infinite in time and still have a finite bandwidth (frequency domain), or it can be finite in time and still have a wide spectrum. Therefore, a signal is not required to be limited in both domains simultaneously.

Final Answer:

The correct answer is \( \text{A is true \& B is false} \).