The roll-off factor (β:beta) in raised cosine filter represents the ratio of:

Concept:

The roll-off factor (β) in a raised cosine filter represents the ratio of:

Correct Option Explanation:

The correct answer is option 4: excess bandwidth to the Nyquist bandwidth.

The raised cosine filter is a type of filter used in digital communication systems to shape the transmitted signal's spectrum, ensuring that intersymbol interference (ISI) is minimized. The roll-off factor, β, is a critical parameter in the design of the raised cosine filter. It determines the excess bandwidth used by the filter beyond the minimum required Nyquist bandwidth.

The Nyquist bandwidth is the minimum bandwidth required to transmit data without ISI, given by the symbol rate (Rs). However, practical filters cannot achieve a perfect rectangular frequency response, so additional bandwidth is introduced. This additional bandwidth is characterized by the roll-off factor (β).

Mathematically, the total bandwidth (B) of a raised cosine filter is given by:

B = Rs × (1 + β)

Where:

• Rs is the symbol rate (also known as the Nyquist rate).
• β is the roll-off factor, which ranges from 0 to 1.

The roll-off factor (β) represents the ratio of the excess bandwidth to the Nyquist bandwidth. In other words, it indicates how much additional bandwidth is introduced by the filter relative to the minimum required Nyquist bandwidth.

Additional Information 

Option 1: Excess power of system to the minimum Nyquist bandwidth

This option is incorrect because the roll-off factor (β) does not represent a ratio involving power. The roll-off factor is strictly related to bandwidth and not power. It describes the additional bandwidth used by the filter beyond the Nyquist bandwidth, not the excess power of the system.

Option 2: Additional bandwidth to total system bandwidth

This option is incorrect because the roll-off factor (β) specifically represents the ratio of excess bandwidth to the Nyquist bandwidth, not the total system bandwidth. The total system bandwidth includes the Nyquist bandwidth and the excess bandwidth. Therefore, this option does not accurately describe the roll-off factor.

Option 3: Total available bandwidth to the Nyquist bandwidth

This option is incorrect because the roll-off factor (β) represents the ratio of excess bandwidth to the Nyquist bandwidth, not the total available bandwidth. The total available bandwidth is the sum of the Nyquist bandwidth and the excess bandwidth, so this option does not accurately describe the roll-off factor.