Convolution Question: Are We Talking About The Same Thing?

[David A. Heiser]

First Gautam Sethi used the term "convolution" for the product to two (uniform) densities. Aniko responded with a definition of convolution as the sum of two random variables. Then Jan de Leeuw stated that "convolution is the distribution of the sum". Herman Rubin stated that "convolution is the distribution of the sum". The idea that "convolution" represents the distribution of the sum of two distributions continues in the additional E mails.   Everyone here is talking about "convolution" in an entirely different sense than what it is used in engineering. That is why I am confused.   The convolution or Faltung integral is defined as follows: (Re: any advanced engineering mathematics text. Microsoft Outlook Express only transmits plain text or rich text, and therefore I cannot give the expression symbolically.)   y(t) = {the integral from zero to t of} f(t-lambda) times g(lambda) d(lambda).     Where t is a positive, finite time value.                 f is a function of time, describing an input (re: an x characteristic)                 g is a function of time, describing an fixed, operator on f (re: a parameter characteristic)                 y is the result, or output as a function of time.                 lambda is a variable of integration, with units of time.   Taking the Laplace transforms of the above equation we have L(y) = L(f) times L(g)   We can also take Fourier transforms.   The convolution integral is the fundamental mathematical description of all control systems from aircraft flight to industrial processes. With current numerical computer methods, the above equations can be solved for any process.   I tried this back in 1962. I tried defining the time statistical variations of the flow rate of solid oxidizer into a continuous mixing process by means of power spectral density. Given the transfer function of the mixing machine (obtained by pulse measurement methods), I would be able to arrive at estimates of the statistical variations in mixed propellant composition. I didn't get it finished, because I didn't have the computer tools and the knowledge at the time. This was also beyond the understanding of management and of the customer, and there was no interest to pursue it.   Now when you are referring to "convolution" as a sum of distributions, how does it fit in?   DAHeiser