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

Let's analyze each assertion: Assertion A: Inverse relationship exists between the time and frequency domain representation of signal This assertion refers to the time-scaling property of the Fourier Transform.
If a signal $x(t)$ has a Fourier Transform $X(j\omega)$, i.e., $x(t) \leftrightarrow X(j\omega)$, then time-scaling the signal by a factor $a$ results in: $$x(at) \leftrightarrow \frac{1}{|a|} X\left(j\frac{\omega}{a}\right)$$ This property shows that if a signal is compressed in the time domain (i.
e., $a>1$), its frequency spectrum expands (i.e., $\omega/a$ means the frequencies are spread out).Conversely, if a signal is expanded in the time domain (i.e., $0<a<1$), its frequency spectrum compresses.This demonstrates an inverse relationship between the duration of a signal in the time domain and its bandwidth in the frequency domain.
This is also related to the uncertainty principle for signals, which states that a signal cannot be arbitrarily localized in both time and frequency simultaneously.
Thus, Assertion A is true.Assertion B: A signal must be necessarily limited in time as well as frequency domains This assertion is false.
According to the Paley-Wiener theorem and the uncertainty principle, a signal cannot be strictly band-limited (finite extent in frequency) and strictly time-limited (finite extent in time) simultaneously, unless the signal is identically zero.
For example, an ideal low-pass filter has a rectangular shape in the frequency domain (finite bandwidth), but its inverse Fourier Transform (sinc function) extends infinitely in the time domain.
Similarly, a rectangular pulse in the time domain (finite duration) has a sinc function in the frequency domain, which extends infinitely.
Therefore, a signal cannot be "necessarily limited" in both domains simultaneously, unless it's a trivial zero signal.
Based on the analysis, Assertion A is true and Assertion B is false.