At a particular frequency \omegaω, the amplitude of the input signal frequency spectrum is 20\text{dB}20dB. The bode plot of your filter at the same frequency \omegaω is -10\text{dB}−10dB. If you were to measure the amplitude of the output signal frequency spectrum at \omegaω, what would the value be? Give your answer in \text{dB}dB, and omit units from your answer.
To determine the amplitude of the output signal frequency spectrum at a specific frequency, we can use the concept of gain in the frequency domain.
In this scenario, we are given that the amplitude of the input signal frequency spectrum at frequency ω is 20 dB. We also know that the bode plot of the filter at the same frequency ω is -10 dB.
To find the output signal's amplitude at ω, we can calculate the gain of the filter. The gain is the difference between the input and output signal amplitudes in decibels.
The gain in decibels can be calculated using the formula: Gain (dB) = 20log10(Ao/Ai), where Ao is the output amplitude and Ai is the input amplitude.
Re-arranging the formula, we can find Ao: Ao = Ai * 10^(Gain(dB)/20).
Substituting the values into the formula, we have: Ao = 10^(20 dB/20) = 10^1 = 10.
Therefore, the amplitude of the output signal frequency spectrum at ω is 10 dB.