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![]() Richard Harrison wrote: K7ITM wrote: "If you say that in the system (y = mx + b) that b is the response to zero input (x=0) then you will conclude the system is nonlinear." Why? the factor (b) is a constant, a value to be added to (mx) to total a value for (y). Because, from 3.3, 'Network Theorems,' in "The Electrical Engineering Handbook," Ed. Richard C. Dorf, Boca Raton: CRC Press LLC, 2000 (you can find it online), "The superposition condition: If the input to the system, e1, causes a response, r1, and if an input to the system, e2, causes a response, r2, then a response, r1 + r2, will occur when the input is e1 + e2." If you take the response to x0=0 to be b, then the response to, say, x1=1 must be m+b, and to x2=2 must be 2*m+b. Then for superposition, and therefore for linearity, the response to (x1+x2)=3 must be m+b+2*m+b = 3*m+2*b, which it is not: it is 3*m+b. I do not suggest a change in the definition of superposition or of homogeneity (which seems to be simply a subset of superpostion anyway) or of linearity. I only suggest that "response" be interpreted as the change in output which occurs when you go from input a to input b. I've generally not seen "response" to be very well defined in those texts which define linearity, though it's quite possible I've missed it. In the PDF file for all of Chapter 3 of the book quoted above, it is certainly not. I guess I'd prefer, basically, to say that the following relationship defines a linear system (MIMO even) -- but it is NOT sufficient to describe all linear systems: dx/dt = Ax + Bu y = Cx + Du where u is a vector of independent input variables, y is a vector of dependent output variables, x represents state variables for the system, and A, B, C and C are matrices of coefficients which are constant in a time-invariant system, but which may be variable with time in a system which is time variant. Cheers, Tom (mx) is a straight line. Every value of (b) produces a straight lline parallel with lhe line y=mx when b=0. Factor (b) is merely the offset value of the sloped line in the x direction. y=mx+b is listed in math books as a defining example of a linear equation. (When plotted, a linear equation produces a straight line.) y=mx+b has a special name: "The point slope formula". Perfectly descriptive, too. To clarify everything, graph a few values for yourself. Best regards, Richard Harrison, KB5WZI |
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