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In some textbooks the channel defined to be not only the propagating medium, but it is considered to include some of the components of the transmitter and receiver, usually the analog parts. For example, you might read somewhere that the "channel introduces noise, power loss, etc." and this is true when the channel includes both the propagating medium as ...

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So, it's generally right that a receiver adds noise due to physics (Johnson-Nyquist noise). So, yes, in any system, at least part of the noise in the received signal is caused by the receiver. There's also noise that is background noise from e.g. cosmic sources that happen neither in the receiver nor the transmitter (and also aren't other transmitters ...

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...my question is, is there an elegant notation to express such operation? If you know that these operations are to act on orthogonal dimensions, you can define their 1D version and then use indexing to express the order by which the function is applied to the multidimensional signal. But if you are working with multidimensional signals over an orthogonal ...

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Okay, I am not quite understanding your question. Let's start with the definition of the Gaussian, aka the Bell Curve, in its general form. $$f(t) = \frac{1}{ \sigma \sqrt{2\pi}} e^{ -\frac{(t-\mu)^2}{2\sigma^2} }$$ $\mu$ is the mean, and represents where the peak occurs. $\sigma$ is the standard deviation, and identifies where the inflection points ...

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