Understanding Band-Pass Filter Parameters

A band-pass filter allows signals within a specific frequency range to pass through while attenuating signals outside that range. Key parameters include:

• Center Frequency ($f_c$): The frequency at the middle of the passband.
• Bandwidth (BW): The range of frequencies between the lower and upper half-power points (also known as -3dB points) where the filter's gain is at least 50% of the maximum gain. It is calculated as $BW = f_H - f_L$, where $f_H$ is the upper half-power frequency and $f_L$ is the lower half-power frequency.
• Quality Factor (Q): This value indicates the selectivity or 'sharpness' of the filter. A higher Q factor means a narrower bandwidth relative to the center frequency, resulting in better selectivity. It is defined by the ratio of the center frequency to the bandwidth:

  \( Q = \frac{f_c}{BW} \)

Analyzing the Effect of Increased Bandwidth

The question asks what happens if the bandwidth (BW) of a band-pass filter increases. Let's examine the relationship defined by the Quality Factor formula:

$$ Q = \frac{f_c}{BW} $$

From this formula, we can see the relationship between Q and BW:

• Inverse Relationship: Q is inversely proportional to BW, assuming the center frequency ($f_c$) remains constant or does not increase proportionally.
• Impact of Increase: If BW increases, the denominator in the formula gets larger. Consequently, the value of Q must decrease.

Therefore, if the bandwidth of a band-pass filter increases, the Q factor decreases.

Evaluating Other Options

• The roll-off rate increases: The roll-off rate describes how quickly the filter's attenuation increases outside the passband. This is primarily determined by the filter's order (e.g., a 2nd-order filter has a steeper roll-off than a 1st-order filter), not directly by an increase in bandwidth itself.
• The half-power frequency decreases: An increase in bandwidth means the range between the lower ($f_L$) and upper ($f_H$) half-power frequencies widens ($BW = f_H - f_L$). If the center frequency ($f_c$) remains constant, $f_L$ would typically decrease while $f_H$ increases, or vice-versa depending on how the bandwidth is increased symmetrically or asymmetrically. Simply stating that 'the' half-power frequency decreases is not universally true for all half-power frequencies, and the primary effect is the widening of the band.
• The center frequency decreases: The center frequency ($f_c$) is a separate characteristic from the bandwidth. While the definition of bandwidth relates to frequencies around the center frequency, an increase in bandwidth does not inherently cause the center frequency itself to decrease.

Conclusion

Based on the inverse relationship between the Quality Factor (Q) and Bandwidth (BW) as shown in the formula $Q = \frac{f_c}{BW}$, an increase in bandwidth leads to a decrease in the Q factor. This indicates that the filter becomes less selective.