11/22/11 - Laplace Transforms and the Convolution Integral

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11/22/11 - Laplace Transforms and the Convolution Integral

			Convolution

			Measurements (or observations -- which are really the same thing) can never be exact.

			No system has an instantaneous time response. That is, if a system is driven with a Delta-function input, the system will respond for some finite amount of time. This is called the impulse-response of the system, and if the system is a linear system, then the impulse response totally characterizes how the system will respond to any input.

			Likewise, your visual system does not have infinite resolution. There is a limit to how fine the detail is that you can see with your naked eye. This is how Apple markets the iPhone 4 with the "retina" display: http://www.apple.com/iphone/features/retina-display.html Your visual acuity (the amount of detail you can see) depends on the "impulse-response" or the "point-spread function" of your eye. This is never a delta function because of the limitations of the optics (cornea/lens) in your eye and the finite size of the light detectors (rods and cones) in your retina. The sequence of graphics below show what your eye sees when viewing a sinusoidally-varying intensity pattern (a sinusoidal "bar" chart). Note that the yellow rectangle (whose width represents the sharpness of your vision) has an area of 1, just like the delta function we saw last time. That means as the rectangle gets wider (indicating that you see less detail), the amplitude has to decrease to maintain unit area.

			The yellow rectangle on the left depicts your eye's impulse response, which you could think of as the image of an infinitesimal point source of light (such as a star) on your retina. The blue sine wave is the intensity profile of the bar chart (which is depicted below as a light intensity band of blue light). Note that the color here is not significant; it's only to easily distinguish input from output. The red band below is what your eye "sees" with the given impulse response, and the red curve is the profile of that red intensity plot. In the first case, because the impulse response is much narrower than the variation in intensity in the bar chart, your eye sees something very close to the actual input.

			As the impulse response of your eye broadens (meaning your vision is less sharp because your eye integrates over a larger area), what you see changes significantly until, finally, you detect no variation in intensity at all, as shown in the graphics below.

			The technical name for what you just witnessed is convolution. What your eye (or any of your senses or any detector) actually sees is the convolution of the actual input with the impulse response of your eye. The mathematical definition of the convolution is given by the following convolution integral:

			The meaning of this integral can be understood graphically by taking the convolution of two rectangle functions, as follows:

			If f(t) is a rectangle, then f (-t) is the reflection of that rectangle about the y-axis, and f(u-t) is the sliding of that reflected rectangle to the right by the amount u. So the convolution of the two rectangles involves reflecting one of them about the y-axis and then sliding it to the right, across the other rectangle, multiplying the two rectangles together and finding the area of their product (the yellow portion in the graph below). That sliding, multiplying and integrating process continues until the sliding rectangle has moved all the way across the fixed rectangle. The convolution is shown as the blue triangle.

			Similarly, here is the convolution of a triangle and a rectangle:

			The convolution of the triangle and rectangle is shown as the blue curve at the bottom of the figures.

			Impulse response of an RC circuit:

			To find out how the RC circuit responds to a rectangular pulse input (driving function) with width w, we take the convolution of the impulse response q(t) with the rectangular function f(t) = H(t)-H(t-w):

			Hard way -- do the convolution integral

			In the graph below, w = 2:

			Easy way -- use the Convolution Theorem

			In the graph below, w = 2:

			Graphic to Summarize the Idea of Convolution

			Homework