Time Convolution with Impulse Response

Abstract

The main idea here is simple. We know how to decompose a signal in terms of convolution with the delta (impulse) function. Assume now we have the impulse response of a system/circuit h(t). To find the response due to an arbitrary signal x(t), all that needs to be done is take the impulse response h(t) and time convolve it with the new input signal x(t)! All these follow from linearity and superposition. This is an extremely powerful concept. The main premise is as follows: characterize the system/circuit only once and obtain the impulse response; next apply as many variants of input stimuli and figure corresponding response without even having to re-characterize the system again! We demonstrate the convolution process on the parallel RC circuit. First we figure the impulse response. Then we figure the unit step response via convolution. We show all the intermediate steps and we observe the solution being built one convolution step at a time. Next we repeat the process on a different stimulus—this time a periodic pulse. Then again on a causal sinusoid. We observe in all cases better accuracy for smaller convolution time steps. We wrap the chapter with an applied example on a parallel RLC network. It is critical that the reader comprehends the idea of taking the impulse response, scaling it, and then adding it to the next offset impulse response till all the input stimuli have been sampled. This is best explained graphically as has been done here.