Let us take a system where the input is $V_{i}(t)$ and output is $V_{o}(t)$ and the impulse response of the system be $I(t)$ where $t$ represents time domain and $w$ be frequency. we get $\tag{1}V_{o}(s)= I(s) \cdot V_{i}(s)$
I am using the bilateral laplace transformation and CTFT (Continuous time fourier transform) by the given formulas: $ V_{o}(s)=\int_{-\infty}^{\infty} V_{o}(t) e^{-st} dt $ and $ F(V_{o})= V_{o}(w)=\int_{-\infty}^{\infty} V_{o}(t) e^{-jwt} dt $
since $s= a+jw$ we can write: $ \tag{2} V_{o}(s)=\int_{-\infty}^{\infty} (V_{o}(t) \cdot e^{-at}) e^{-jwt} dt $
By frequency convolution property:$\tag{3} V_{o}(s)=\frac{1}{2\pi}( V_{o}(w) * F(e^{-at}))$ $\tag{4}I(s)= \frac{1}{2\pi}(I(w)* F(e^{-at})) $ $\tag{5}V_{i}(s)= \frac{1}{2\pi}(V_{i}(w)* F(e^{-at})) $
since $V_{o}(t)= V_{i}(t) * I(t)$ and $V_{o}(w)=V_{i}(w)\cdot I(w)$, substituting this in the equation 3 gives $\tag{6} V_{o}(s)=\frac{1}{2\pi}([V_{i}(w)\cdot I(w)] * F(e^{-at})) $ Now combining equation 4 , 5 and 1 we get: $\tag{7}V_{o}(s)= \left[\frac{1}{2\pi}(I(w)* F(e^{-at}))\right] \cdot \left[\frac{1}{2\pi}(V_{i}(w)* F(e^{-at}))\right] $
Though equation $6$ and $7$ represent the s-domain of $V_{o}(t)$ they seem to be not equal.
So my questions are:
Where have I went wrong in the above steps?
How can you write $(a\cdot b)*c$ in terms of $a*c$ and $b*c$. where ($\cdot$) is multiplication and $(*)$ is convolution.
Kindly solve this issue. Thank you in advance.