Vision_Notes_4 - IV Linear systems analysis Linear systems...

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57 IV. Linear systems analysis Linear systems analysis refers to a set of mathematical techniques that can be used to analyze and describe input/output systems that satisfy certain linearity assumptions. The inputs and outputs of linear systems are represented by functions. These functions can be of any number of variables, but for present purposes there will only be one variable, time (t) or space (x) , or two variables, space-space (x,y) , or space-time (x,t) . For example the input functions might represent the variation in light level on a single receptor over time, the spatial distribution of light entering the eye, or the spatial distribution of neural activity transmitted from the eye to the brain along the optic nerve. The output functions might represent the electrical response of the receptor over time, the spatial distribution of light falling on the retina, or the spatial distribution of neural activity of a group of cells receiving inputs from the optic nerve. The letter L will be used to represent a linear system; i(t) , i(x,y) , etc. will represent input functions; and o(t) , o(x,y) , etc. will represent output functions. Thus, we can write in the one-dimensional case, o ( t ) = Li ( t ) {} (4.1) or in the two-dimensional case, o ( x , y ) = ( x , y ) (4.2) The curly brackets are used as a reminder that a linear system takes a whole function as input and gives a whole function as output. For example, the optical system of the eye can be regarded as a linear system that transforms the two-dimensional light distribution in an object plane, i(x,y) , into a two-dimensional light distribution in the image plane, o(x, y) . Figure 4.1 illustrates graphically a one-dimensional input function, a linear system, and a one dimensional output function. To make the figure concrete the reader might imagine that the left side represents a temporal signal (rectangular pulse) turned on shortly after time 0, while the right side represents the response of the linear system to this signal. A. Constraints defining linear systems In order for a one-dimensional system to be linear, it must satisfy the following constraint, La i 1 ( t ) + bi 2 ( t ) = aL i 1 ( t ) + bL i 2 ( t ) (4.3)
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58 where i 1 (t) and i 2 (t) are two arbitrary input functions and a and b are arbitrary constants (scalars). In words, the constraint is that the response of a linear system to the sum of two input function equals the sum of the responses to each input taken alone, and also scaling an input function by some amount scales the output function by exactly the same amount. In Figure 4.1B, another pulse has been added to the one in Figure 4.1A. If the system is linear then the response must be the sum of the individual responses (as shown). In Figure 4.1C, the input pulse in Figure 4.1A has been doubled in amplitude. If the system is linear the response should double in amplitude (as shown).
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This note was uploaded on 08/12/2008 for the course PSY 380E taught by Professor Geisler during the Fall '07 term at University of Texas.

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Vision_Notes_4 - IV Linear systems analysis Linear systems...

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