Which Method of Convolution (If Any) Is Most Appropriate Here?

Question

I need to convolve (or otherwise get the impulse response h(t) of) the input signal $x(t) = 2u(t)$ and $y(t) = cos(4t) + 2e^{(t-1)}$. I have tried the Fourier Transform and the Laplace Transform, but they are both very complex to evaluate.

Is there a simpler way, or is taking the convolution the best approach? I'm occasionally confused on how to recognize which method is best when finding the impulse response, so any general advice on that would be appreciated.

Thanks for any and all help provided!

Answer 1

$$x(t)*y(t)=\int_{-\infty}^\infty{x}(\tau)y(t-\tau)\,d\tau$$

For all $t<0$, $x(t)=0$ and for all $t>0$, $x(t)=2$.

$$x(t)*y(t)=\int_0^\infty(2\cos(4t-4\tau)+4e^{(t-\tau-1)})\,d\tau$$

From here treat $t$ as constant, and use u-substitution and the linearity of the integral to solve the integral.

Answer 2

I have tried the Fourier Transform and the Laplace Transform, but they are both very complex to evaluate.

I don't know what you have tried for Laplace transform, but the Laplace transform of $x*y$ is: $$ \mathcal{L}\{(x*y)(t)\} = \mathcal{L}\{(x(t)\}\cdot \mathcal{L}\{y(t)\} = \frac{2}{s}\left( \frac{s}{s^2+4^2} + \frac{2}{e(s-1)}\right), $$ which if using partial fraction, can be represented as: $$ \mathcal{L}\{(x*y)(t)\} = \frac12 \frac{4}{s^2+4^2} + \frac{4}{e} \frac{1}{s-1} - \frac{4}{e}\frac{1}{s}. $$ Now every term on the right side can be found in any Laplace transform table, so you can take the inverse Laplace transform to get back to $(x*y)(t)$.