The discrete-time Fourier transform is always periodic in ω with period: 

Explanation:

Discrete-Time Fourier Transform (DTFT)

Definition: The discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values, typically a function of time. The DTFT is used to analyze the frequency content of discrete signals. It transforms a sequence of discrete-time signals into a continuous function of frequency.

Mathematical Representation: The DTFT of a sequence \( x[n] \) is defined as:

\( X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} \)

where:

• \( x[n] \) is the discrete-time signal.
• \( \omega \) is the normalized angular frequency (in radians per sample).
• \( j \) is the imaginary unit.

Periodicity of the DTFT: One of the key properties of the DTFT is its periodicity. The DTFT is always periodic in \( \omega \) with a period of \( 2\pi \). This means that:

\( X(\omega + 2\pi) = X(\omega) \)

This periodicity arises due to the nature of discrete-time signals, which are inherently sampled. The Nyquist-Shannon sampling theorem indicates that the highest frequency that can be uniquely represented by a sampled signal is half the sampling rate, leading to a periodicity in the frequency domain.

Correct Option Analysis:

The correct option is:

Option 2: \( 2\pi \)

This option correctly identifies that the DTFT is periodic in \( \omega \) with a period of \( 2\pi \). This periodicity is a fundamental characteristic of the DTFT and is crucial for understanding the frequency representation of discrete-time signals.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \( 4\pi \)

This option is incorrect because the DTFT is not periodic with a period of \( 4\pi \). The correct periodicity is \( 2\pi \), as discussed above.

Option 3: \( \pi \)

This option is incorrect because \( \pi \) is not the period of the DTFT. While \( \pi \) is half the period of \( 2\pi \), it does not represent the full periodicity of the DTFT.

Option 4: \( \pi/2 \)

This option is incorrect as well. \( \pi/2 \) is a quarter of the period \( 2\pi \) and does not represent the periodicity of the DTFT.

Conclusion:

Understanding the periodicity of the DTFT is essential for analyzing the frequency content of discrete-time signals. The DTFT is always periodic in \( \omega \) with a period of \( 2\pi \), which is a fundamental property derived from the nature of discrete-time signals. This periodicity allows for the analysis and representation of signals in the frequency domain, providing insights into their spectral characteristics.