6 dB corresponds to a frequency response with magnitude equal to:

Explanation:

Decibel (dB) and Frequency Response

Definition: The decibel (dB) is a logarithmic unit used to express the ratio of two values, commonly used in acoustics and electronics to describe the gain or loss in a system. The frequency response of a system shows how the amplitude of the output signal varies with frequency.

Understanding 6 dB: A 6 dB increase corresponds to a doubling of the amplitude of the signal. This is because the decibel scale is logarithmic. Specifically, a gain of 6 dB means that the output signal's amplitude is twice that of the input signal. The formula for converting a magnitude ratio to decibels is:

dB = 20 * log₁₀(magnitude ratio)

To find the magnitude ratio that corresponds to 6 dB, we rearrange the formula:

6 dB = 20 * log₁₀(magnitude ratio)

Dividing both sides by 20 gives:

0.3 = log₁₀(magnitude ratio)

To solve for the magnitude ratio, we take the antilogarithm (base 10) of both sides:

magnitude ratio = 10^0.3

Using a calculator, we find:

magnitude ratio ≈ 2

Therefore, a 6 dB increase corresponds to a magnitude ratio of approximately 2.

Correct Option Analysis:

The correct option is:

Option 1: 2

This option correctly describes the magnitude ratio corresponding to a 6 dB increase in the frequency response. As calculated, a 6 dB gain means that the output signal is twice the amplitude of the input signal, which matches the magnitude ratio of 2.

Additional Information

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

Option 2: 3

This option is incorrect. A 6 dB increase corresponds to a magnitude ratio of 2, not 3. Using the formula mentioned earlier, a 3 dB increase corresponds to a doubling of power, but not of magnitude. For magnitude, a 6 dB increase is required to double it.

Option 3: 1

This option is incorrect. A magnitude ratio of 1 means no change in amplitude, corresponding to 0 dB. Therefore, it does not represent a 6 dB increase.

Option 4: 0

This option is also incorrect. A magnitude ratio of 0 would imply no output, which is not possible in the context of a 6 dB increase. The magnitude ratio corresponding to a 6 dB increase is 2, as calculated.

Conclusion:

Understanding the relationship between decibels and magnitude ratios is crucial in fields such as acoustics and electronics. A 6 dB increase corresponds to a doubling of the magnitude of the signal, leading to a magnitude ratio of 2. This logarithmic scale allows for a more manageable representation of large changes in signal strength. Thus, the correct answer to the question is option 1, which accurately reflects the magnitude ratio corresponding to a 6 dB increase in frequency response.