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On Apr 21, 7:24 am, Gene Fuller wrote:
K7ITM wrote: On Apr 20, 10:10 pm, Roy Lewallen wrote: Correction: Roy Lewallen wrote: Superposition means the following: If f(x) is the result of excitation x and f(y) is the result of excitation y, then the result of excitation (x + y) is f(x + y). . . That should read: Superposition means the following: If f(x) is the result of excitation x and f(y) is the result of excitation y, then the result of excitation (x + y) is f(x) + f(y). . . ^^^^^^^^^^^ I apologize for the error. Thanks very much to David Ryeburn for spotting it. Roy Lewallen, W7EL I guess that's the definition of linearity. I'm not sure I've heard it called superposition before, but rather that the superposition theorem is a direct result of the linearity of a system. I trust that's a small definitional issue that doesn't really change what you're saying. Cheers, Tom Tom, For most purposes the terms superposition and linearity are interchangeable. However, for the purists there is a difference. A system that is deemed linear requires that it has the properties of both superposition and scalability. These properties are essentially the same for simple systems, but they are not necessarily the same when considering complex values. I found some clear examples in a book, "The Science of Radio", by Paul Nahin. One example, y(t)=Re{x(t)} describes a system which obeys superposition, but not scaling. Hint: try a scaling factor of "j". That system is not linear. Another example is y(t)=[1/x(t)]*[dx/dt]^2. That system obeys scaling, but not superposition. Again, it is not linear. The bottom line is that superposition is necessary, but not sufficient to ensure linearity. You are correct that the definitional issue is not relevant to the current RRAA discussion. 73, Gene W4SZ Thanks, Gene--those examples are helpful. I'll retract what I posted last night. Based on the "necessary but not sufficient" statement above, we can say that superposition does hold in a linear system, so if we specify a linear system, we do have that guarantee. Cheers, Tom |
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