i'm not sure what you are asking but if you calculate the convolution between a pulse and a train of impulses, what you get is a train of pulses, alas, a periodic signal based on the pulse... is that what you wanted?
I think what SM is asking is to have a single pulse as input and sustained oscillation as output. I think one way can be like this:
you have an oscillator or an astable multistable multivibrator. Use a D latch (asynchronous) followed by a buffer (current ammlification to the bias supply of the oscillator).
i'm not sure what you are asking but if you calculate the convolution between a pulse and a train of impulses, what you get is a train of pulses, alas, a periodic signal based on the pulse... is that what you wanted?