Convolution Is Fancy Multiplication
Convolution is a correlation with a reversed signal.<p>Correlation is a generalized dot product: multiplying corresponding pairs of values from two signals, and then adding the factors together. The result is zero if the signals are orthogonal (like the dot product of two vectors in 2D or 3D that are at 90 degrees).<p>The intuition behind the reversed signal comes from processing in the time domain. There are application in which we convolve a new signal with a segment based on a captured historic signal. That function is reversed so that the newest sample from the new signal correlates with the least recently captured sample in the historic signal.<p>For instance, we can use an impulse signal (like a clap) to capture the echoes from an acoustic space. That signal can then be convolved with audio data in order to give it a reverberation similar to what would be produced by that acoustic space. The captured impulse response has to be reversed, because echoes come in reverse. In the impulse capture, the longest echo is found on the far right. When we apply the impulse to simulate the echo, it has to be on he far left, because it has to correlate with an old sample of he input.<p>E.g. if we are at t = 0, and want to hear the 500 ms echo, that echo corresponds to a past sound sample that is now at t = -500.    When are capturing the impulse response, then we issue the gunshot or clap at t = 0, and the 500 ms echo comes at t = 500.<p>The earliest reflections have to apply to more recent signal, but they are based on the least recently captured data.