for the following question u have asking, you can apply IDENTITY property, so you can get x({t-7}+5) = x(t-2) A convolution is an integral that expresses the amount of overlap of one function, Impulse response is the response of a system to a unit impulse at its input. so, you can rephrase the question like "its not like impulse function, but you have to consider it like"UNIT IMPULSE FUNCTION" BUT IF YOU APPROXIMATE THE IDENTITY FOR CONVOLUTION, it needs to be convergence everywhere., so, differential functions at a certain amount of time need not be convergent. so your question cannot be considered.