Electronic Signal and System problem: Ratio of signal energy

[nidhalc]

Homework Statement

x(t) is input into a perfect lowpass filter with frequency response H(ω), having a bandwidth of BHz and a passband gain of 1. For B = (2πT)^-1 Hz, calculate the ratio of the output signal energy to the input signal energy.

Homework Equations

x(t) = Ae^-|t|/T

The Attempt at a Solution

I have found the input energy to be (A^2)/2 but I'm unable to find any equation which links the frequency response to the bandwidth in order to get the output energy. I have that |H(ω)| = gain. I've a feeling I'm missing something very basic here.

 

[KataKoniK]

Hi there,

You are correct that you need to use the frequency response of the lowpass filter to calculate the output energy. The key here is to understand how the frequency response relates to the bandwidth of the filter.

The frequency response of a lowpass filter is typically given by the equation |H(ω)| = 1/√(1+(ω/B)^2), where B is the bandwidth of the filter. This means that as the frequency ω increases, the magnitude of the frequency response decreases. In other words, the filter attenuates higher frequencies.

To calculate the output energy, you will need to use the input energy and the frequency response. The output energy is equal to the input energy multiplied by the squared magnitude of the frequency response. So the ratio of the output energy to the input energy is given by:

Output energy/Input energy = |H(ω)|^2 = 1/(1+(ω/B)^2)

To calculate the value of this expression, you will need to substitute in the value of B given in the problem (B = (2πT)^-1 Hz) and the frequency ω of the input signal. Once you have done this, you should be able to calculate the ratio of the output energy to the input energy.

I hope this helps. Let me know if you have any further questions.