How Does a Linear System Respond to a Sine Wave Input Using Fourier Transform?

[jaguar515]

Find the response of the of the linear system at rest
y' + y = x , where x = sin(t)
using the Fourier transform and formulas :

Y(jw)=H(jw)X(jw)
y(t)=h(t)*x(t) convolution

 

[SGT]

Find the response of the of the linear system at rest
y' + y = x , where x = sin(t)
using the Fourier transform and formulas :

Y(jw)=H(jw)X(jw)
y(t)=h(t)*x(t) convolution

If the FT of y(t) is $$Y(j\omega)$$ what is the FT of y'(t)?

 

[ravenprp]

The response of a linear system is the output of the system when a specific input is applied. In this case, we are given a linear system with a differential equation y' + y = x, where x is an input function defined as sin(t). To find the response of this system at rest, we can use the Fourier transform and the convolution formula.

First, we can apply the Fourier transform to both sides of the equation to get:

jωY(jω) + Y(jω) = X(jω)

Next, we can solve for Y(jω) by dividing both sides by jω + 1:

Y(jω) = X(jω) / (jω + 1)

Now, we can substitute the given input function x(t) = sin(t) into the equation to get:

Y(jω) = sin(ω) / (jω + 1)

To find the response at rest, we need to evaluate this expression at t = 0, which means ω = 0. Plugging this into the equation, we get:

Y(j0) = sin(0) / (j0 + 1) = 0 / 1 = 0

Therefore, the response of the linear system at rest is 0. This means that when the system is not being driven by an external input, the output will be 0. This makes sense intuitively, as a system at rest should not have any output.

To further confirm this result, we can also use the convolution formula to find the response. The convolution formula states that the output y(t) can be found by convolving the input x(t) with the impulse response h(t):

y(t) = h(t) * x(t)

In this case, the impulse response h(t) can be found by taking the inverse Fourier transform of Y(jω):

h(t) = F^-1{Y(jω)} = F^-1{sin(ω) / (jω + 1)}

To evaluate this inverse Fourier transform, we can use the formula for the inverse Fourier transform of sin(ω) / ω, which is a shifted delta function:

F^-1{sin(ω) / ω} = πδ(t)

Applying this to our equation, we get:

h(t) = πδ(t + 1)

Finally, we can convolve this impulse response with the input x(t) = sin(t)