Combined impulse responses I'm looking for a simple way to show that you can form a single impulse response that is the equivalent $M$ other impulse responses, ie:

$$
h\left(t\right)=h_0\left(t\right)\star h_1\left(t\right)...\star h_{M-1}\left(t\right)
$$

I know convolution is multiplication in the frequency domain, so I cooked up this "equation":

$$
h\left(t\right)=\mathcal F^{-1}\left\{\prod_{i=0}^{M-1} \mathcal F\left\{h_{i}\right\}\right\}(t)
$$

With $\mathcal F^{-1}$ being the inverse Fourier transform operator, and $\mathcal F$ being the Fourier transform operator. Expressing myself in math notation isn't really my strong side, what I intended to show was that the combined impulse response $h(t)$ equals the sequences multiplied together in the frequency domain. 


Does it make any sense? 
Or is there perhaps a simple way to show this? 
What about circular convolution in the frequency domain, and infinite impulse responses in continuous time?
You can readily use the associativity of the convolution to explain this fact. Assume you send a signal $x(t)$ through a chain of convolutions $h_i(t)$:

$$ y(t) = (((x(t)*h_1(t))*h_2(t))*h_3(t)) $$

By associativity, this becomes

$$ y(t) = x(t) * ((h_1(t)*h_2(t))*h_3(t)) $$

I.e. you can see this operation as if sending your signal $x(t)$ through a single filter with impulse response $h(t)=h_1(t)*h_2(t)*h_3(t)$.